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s g f = (ImeN gf+i/m- B y Theorem 8.7 for martingales with reversed time, we have (1) (■, w) is bounded on every finite interval in R+for almost every u> G £2. Remark 15.10. Every quasimartingale on a standard filtered space (Q, 5, {5t},P) is in C(R+ x £2). Note also that C(R + x £2) c B(R+ x Q) c L ^ ( R + x Q, ^ , P) c L'1<^(R+ X £2, (J,A, P) for every yl G A' 0C (Q,5, {5 ( },P). The last set inclusion is from the fact that for every t G R+ and w G £2, P-A(-, <*>) is a finite measure on ([0, t], 93[o,<]) so that for every measurable process O on (£2,5, -P) we have ®2(s,uj)dA(s,u>) < oo =► / • X, then M^tX = <3> • Mx and V^tX = ) < Ai^(^i) + A 2 y(6)Then the left derivative (D~(p)(0 and the right derivative (D+ ) = m, 5) V( we have by the substitution z = C ( r — m)
E[/ 0 (X t0 )-■•/„(*<„)|gf +0 ] =
lim E[/o(X (o ) • • • / „ ( X ( J | g f + 1 / J
a.e. on (£2,gfo, P),
CHAPTER 3. STOCHASTIC
278
INTEGRALS
for arbitrary versions of the conditional expectations. Now according to Lemma 13.19 (2)
E[/ 0 (X to ) • • • / . ( J C J l S f l = MX*) ■ ■ ■ fk-dXtk_,MXs)
a.e. on (Q,fff, P )
and similarly for m £ N so large that s + 1/m < tk we have (3)
E[/0(Jt4o)---/n(Xt„)|^f+1/J = M%%) ■ ■ ■ fk-iiXtk_,MX3+l/m)
a.e. on ( f l , ^ , P ) .
^ Note that since the filtered space (£2,5, {5 t }, P ) is augmented, such conditions as 'a.e. on ( Q , j f , P ) ' , 'a.e. on (£2,#f,. 0 ,P)\ and 'a.e. on ( A ^ + i / m , P ) ' are all equivalent to the condition 'a.e. on (£2,5, P)'- From (1) and (3), we have (4)
E[/o(X, 0 ) ■ • • /„(X ( J|5f + o] = Jim, MXto) ■ ■ ■ /*-i(X { t _,)^(X s + 1 / m ) = /oW-/w(I
l M
MX,)
a.e. on (£2, s f , P ) ,
where the second equality is by the continuity of X and the continuity of ip on Rd. Thus we have E[/ 0 (X % ) ■ ■ ■ fn{XtJ\^] = E[/ 0 (X ( „) • ■ ■ fn{XtJ\$f] by (4) and (2). ■ Observation 13.21. Let G be an open set in a metric space (S, p). Then there exists a sequence {/, : n £ N} of continuous functions on 5 such that 0 < /„ < 1 on S for n € N and lim fn = 1Q on 5. n~-*-oo
Proof. If G = S, then /„ = 1 on S for n g N will do. Suppose G 4 S. Then Gc is a nonempty closed set. Let p{x, Gc) = infj,6G<: p(x, y) for x € S. Then p(a;, G c ) is a continuous function of x € 5 and p(i, Gc) = 0 if and only if x G G c . For n G N, let / n be defined by p(:r, Gc) + n ' Then / n is continuous on 5 and satisfies the condition 0 < /„ < 1 on S. If x G G, then i £ G c , /»(x, G c ) > 0, and jim^ /„(z) = 1. If x $ G, then x £ G c , p(x, G c ) = 0, /„(x) = 0, and lim fn(x) = 0. This shows that lim fn = \Q on S. ■ n—too
n—foo
Proposition 13.22. Let X = {Xt : t € K+} be a d-dimensional Brownian motion on a complete probability space (£2,5, P). Let'Jibe the collection of all the null sets in (£2, #, P)
§/3. ADAPTED
279
BROWNIANMOTIONS
and let g f = CT{XS : s G [0,i]} and g f =
E[l / 1 |gf + 0 ] = E [ l / 1 | g f ]
for every A G g f
Now since g^,, C g< for r. > s, (1) implies (2)
E[ly4|gf+0]=E[lyl|gf]
for every A G gf+o-
Since 1A G EpUlgf+o] for A in gf+0, (2) implies 1A G E t l ^ g f ], that is, 1A is gfmeasurable, in other words, A G g f . Thus g ^ , C g s and therefore gfo = gf. Therefore it remains to prove (1). Now g f = c ( g f U 9t) and according to Lemma 13.7 we have g f = cr(UTS7;<7(XT)) where Tt is the collection of all finite strictly increasing sequences r in [0, t], X T = (Xtn... ,Xtn) for r = {<],...,<„}, and UTer,cr(XT) is a 7r-class of subsets of Q. Let us prove (1) first for the particular case where A G cr(XT) for some r € %. Now if A G
E[lx-,(e)|gf+0] = E[lx-,(E)|gf]
for every £ G <8 r *.
d
Let G i , . . . , Gn be open sets in R By Observation 13.21, there exist sequences {/,-,* : fc G N}, j = 1 , . . . , n, of continuous functions on Rd, bounded between 0 and 1 such that lim fjj, = 1 G for j = 1 , . . . , n. According to Lemma 13.20 we have k
(4)
E [ / U ( X ( | ) ■ ■ • / n ,*(X ( J|gf + 0 ] = E [ / M ( X t l ) • • ■ / M ( J £ i J | 3 f 1.
By the Conditional Bounded Convergence Theorem, we have (5)
lim E[/,, fc (X ( ,) • • ■ / ^ ( X „ ) | g f + 0 ]
K—►OO
= E[lGl(X(|)---lGn(XtJ|gf+0]
a.e. on(O,gf + 0 ,P),
for arbitrary versions of the conditional expectations, and similarly (6)
limE[/1,*(X,)---/„,*(XtJ|gf] k—t-oo
= E[lG|(Xt,)---lG„(X(J|gf]
a.e. o n ( Q , g f , P ) .
280
CHAPTER 3. STOCHASTIC
INTEGRALS
By (4), (5) and (6), we have (7)
E[l G l (X ( | ) • ■ • low(*f„)l&>] = E[1 G ,(X ( | )- • • l G „ ( X ( J | S f ]■
Since XT = (Xtl,...,
Xt„), we have
1 G , ( ^ « I ) • ■ • 1G„(-X'*„) = Ijf-'(Gi)' ' ' ^"'CGW =
=
*-X^HOt)n-rtX-nHon)
l(X,,,...,Artn)-'(G,x-xG„) = lxr'(G,x-xG„)
so that by (7) we have (8)
E[lx-i{GiX...xGn)\3f+0]
= E [ l x - i ( G l X . . . x G n ) | 5 s ].
Let £> be the collection of all members of QSrd such that
(9)
E[lx-iCD)lCo] = E[l^-lcD)|fff ].
It is easily verified that D is a d-class. For instance, if £>fc G ID,fcG N, and Dk | , then lim Z)A G ID by the Conditional Monotone Convergence Theorem. Let D(d} and D(nd) be the collections of all open sets in Rd and Rnd respectively. Now since the collection of members of 9 3 ^ of the type Gi x • • ■ x Gn where G j , . . . ,Gn G Did) is a 7r-class, and since 3? contains this 7r-class according to (8), we have a{d
x • • • x Gn : Gu ..., Gn € Did)} C D
by Theorem 1.5. Now every member of Oind) is a countable union of sets of the type G\ x • ■ • x Gn where G\,..., Gn G O and thus we have cr(Q
Dw}.
Therefore <8ind = a(0{nd)) C D. Thus (9) holds for every E G 23E„d. This proves (3) and therefore (1) holds for every A G u{XT). From arbitrariness of r G 7^, (1) holds for every A G U T6Tt o-(^ T )It is easily verified that the collection of all members A of 5 for which (1) holds is a d-class of subsets of Q.. Now since this
E[1A\^+0}=E[1A\^]
foreveryAGgf.
113. ADAPTED
281
BROWNIANMOTIONS
Also (11)
E[l w |5f + 0 ] = E [ l N | 5 f ]
for every Ne
%
since the left side consists of all extended real valued functions on £2 which are equal to 1 except on a null set in (Q,#s+o, P), the right side consists of all extended real valued functions on £2 which are equal to 1 except on a null set in (£2, gf, P), and the collections of all the null sets in these two measure spaces are the same 9t. By (10) and (11). the equality (1) holds for every A G g f U 9L Now by the completeness of the probability space (£2, 5, P), an arbitrary subset of a member of 9t is again a member of 91. This implies that g f U 91 is a 7r-class. Then since (1) holds for every A in the ir-class g f U 91, it holds for every A in cr(^x U 9t) = g f b y the same argument as above. ■ By Remark 13.18 and Proposition 13.22 we have the following theorem. Theorem 13.23. Let X = {Xt : t G R+} be a d-dimensional Brownian motion on a com plete probability space (£2, g, P)- Let 91 be the collection of all the null sets in (£2, g, P) and let $? = a{Xs : s G [Q,t]} md$f = a($? U 9t)/ort G R+. Then (Q, 5, {lg},P) is a standard filtered space and X is an {5f } -adapted d-dimensional Brownian motion on it. We show next that conditions 2° and 3° in Definition 13.17 for an {JJ-adapted ddimensional Brownian motion is equivalent to a condition on the conditional expectation of the characteristic function of Xt — Xs with respect to g 3 . For this we need the following lemma. Lemma 13.24. Let X be a d-dimensional random vector on a probability space (Q, 5, P) and let
E[e^x>l
a.e.on(n,<5,P).
2) If there exists a complex valued function i/> on Rd such that for every y eRd we have (2)
E[ei
then {<&, X} is independent and thus ip = ipx-
on(n,<3,P),
282
CHAPTER 3. STOCHASTIC
INTEGRALS
Proof. 1) If { 0 , X} is independent, then {0, e^v<x>} is independent for every y G K"1 since e*'^) is a!BEd-measurablefunction onK . Thus Etc^*-*' 10] = Ete'^*)] = ^x(y)
a.e. on (Q, 0 , P).
2) Conversely suppose (2) holds. To show the independence of { 0 , X}, it suffices to show the independence of {1G, X} for every G G 0 . According to Kac's Theorem, the ddimensional random vector X and the random variable 1 G constitute an independent system if and only if for every y G Rrf and z £ l , w e have E i y ^ ' ^ M * ' 1 0 ^ ] - E[e'^"^]E[e'( 2,lG> ]. To verify this last equality note that E[(*(WW»4o»] = E[E[e i{+<2 ' lG>} |0]] = E[ei{*'lG)E[ei{y-x)\®]]
since 1 G is 0-measurable
= E l e ^ ^ ^ C v ) ] = ^(y)E[e'
I
Proposition 13.25. Let X = {Xt : t G R+} be a d-dimensional stochastic process on a filtered space ( 0 , 5 , {$t\,P)- Then conditions 2° and 3° in Definition 13.17 are equivalent to the condition that for any s, t G K+ such that s < t and y g R we have E[e'<**-*«>[S,] = e - ^ ( i - s )
a.e. on (Q,&„P).
Proof. 1) Assume 2° and 3° in Definition 13.17. The independence of the system {$s,Xt — X,} implies that of {&, e'te'*'-*^}. Thus E[e«v.x,-x.)
|
5 J =
£[„*<**,-*.>] = e - ^ « - * >
a.e.
on (Q, & , P),
where the last equality is from the fact that the probability distribution Px,-x, of Xt — Xs is the d-dimensional normal distribution Nd(0, (t — s) ■ I) so that by the Image Probability Law Ery<».*.-*.>] =
I
JM.d
jMpXt_xAdx)
and this last integral is the characteristic function of Nd(0, (t — s) ■ I) which is equal to e -W-
e Rd b y T h e o r e m D . 5 .
§/3. ADAPTED BROWNIAN
283
MOTIONS
2) Conversely assume the condition in the Proposition. Then by Lemma 13.24, the system {$s, Xt - Xs} is independent, and the characteristic function
= e~^
for y € Rd
so that the probability distribution of Xt - Xs is the d-dimensional normal distribution jVd(0,(f - 8) ■ I). ■ As an immediate consequence of Definition 13.17 and Proposition 13.24 we have the following theorem. Theorem 13.26. Let X = {Xt : t e K + } be a d-dimensional stochastic process on a stan dard filtered space (Q,5, {&}, P). Then X is an {$t}-adapted d-dimensional Brownian motion on the filtered space if and only if 1°. Xt is ^-measurable for every t £ M+> 2°. for every s, t 6 R+ such that s < t and y &Rd we have
3°. every sample function ofX is continuous on R+.
Theorem 13.27. An {&} -adapted d-dimensional process X = {Xt : t € R+} on a standard filtered space (Q, 5, {5t}, P) is an {$t}-adapted d-dimensional Brownian motion on the filtered space if and only if it satisfies the following two conditions. jo {ei(!/,jrt)+J4-« : t e R+} is a martingale on the filtered space for every y g Rd, 2°. every sample function ofX is continuous on R+. Proof. It suffices to note that condition 2° in Theorem 13.26 is equivalent to the condition that for every s, t £ R+ such that s < t and y € Rd we have E[e '
= 6%.*.)+^
a .e.
on (Q,ff.,P). ■
284
CHAPTER 3. STOCHASTIC
INTEGRALS
Let X = {Xt : t G R+} be an {3t}-adapted d-dimensional Brownian motion on a stan dard filtered space (£2, 5, {&}, P). For a fixed t0 G R + , let 0 , = &0+1 for t e R+. Then (£2,3, {©(}, P ) is a standard filtered space, and if we let F< = Xto+t for t 6 R+, then Y = {Yt : t 6 R+} is a {©(}-adapted d-dimensional process on (£2,5, {<SJ, F ) with initial probability distribution PYo = PXt . For s,t G R+ such that a < *, the system {&t, Yt Ys} = {5s+t0,Xt+t0 - Xs+t0} is independent. Also Py.-y. = PxH,0-x„,0 = Nd(0, (t - s) ■ I). Thus Y is a {0 ( }-adapted d-dimensional Brownian motion on (£2,3, {<St}, P). This shows that if X is an adapted d-dimensional Brownian motion, then at any time t0 G R+ it starts anew as an adapted d-dimensional Brownian motion Y with Px,0 as its initial probability distribution but otherwise the probability distribution of Y does not depend on the proba bility distribution of X in the time interval [0, to)- This property of an adapted Brownian motion is a particular case of the following theorem. Theorem 13.28. (Strong Markov Property) Let X = {Xt : t G R+} be an {$t}-adapted d-dimensional Brownian motion on a standard filtered space (£2, 3 , {fit}, P)- For a finite stopping time T on the filtered space, let &t = $T+t and Yt = Xr+t for t G R+. Then (£2, 3 , {&t},P) is a standard filtered space and Y = {Yt : t G R+} is a {<3>t]-adapted ddimensional Brownian motion on the standard filtered space with initial distribution Py0 = PxTProof. Clearly {<St : t G R+} is a filtration on the probability space (£2,5, P), that is, an increasing system of sub-a-algebras of 3- To show that this filtration is augmented, we show that ©o contains all the null sets in (£2,3, P)- Now <S0 = 3 r and 3 r consists of all A e doo such that A n {T < t] G 3t for every t € R+. Let N be an arbitrary null set in (£2,3, P). Then N G 3o C 3oo since 3o is augmented. The completeness of the measure space (£2,3, P) implies that N D {T < t] is a null set in it and thus iV n {T < t} G 3 t since fo is augmented for every t G R+. Therefore TV G 3 T = @0. This shows that the filtration {&t '• t £ R+} is augmented. Let us show the right-continuity of the filtration {<St : t G R+}, that is, for every io G R+ we have n„>4o<Su = <Sto, in other words, nu>t03r+u = $T+t0- To show this, we show that if A G 3T+U for all u > tQ, then A G 3T+( 0 - NOW since the filtration {& : t G R+} is right-continuous, according to Theorem 3.4, A G 3r+u if and only if A G 3oo and A n {T + u < t) 6 5* for every i G R+, and similarly A G 3r+t0 if and only if A G 3oo and A H {T + <0 < <} 6 3* for every i G R + . Note that with fixed t G R+, we have {T + u < i} \ as u j £0, and {T + u < i} c {T +10 < i) for u > t0. If io e {T + t0 < t}, then T(w) + 1 0 < i so that T(w) + u < t and thus w G {T + u < i}
§13. ADAPTED
285
BROWNIANMOTIONS
for some u > f0. Therefore {T +1 0 < i} = U u>(o {T + u < t} = lim{T + u < t} and thus A n lim{T + u < t] = A n {T +1 0 < t) for any A C £2. Now let A € fo^ for all u > t0. Then since ,4 D {T + u < t} G fo for every u > t 0 we have A n lim{T + u < i} G £ ( , that uito
is, A n {T + 1 0 < t} e 5(, for every t G R+, and therefore A G 5r+t0. This proves the right-continuity of {<Sf : t 6 R+}. Let us show that F is a {<5(}-adapted d-dimensional Brownian motion on the stan dard filtered space ( 0 , 5 , {(5 ( },P). Now since Yt = XT+U <S, = 5 T + ( , and X T + ( is i m measurable,!^ is (St-measurable for every t G R+, that is, Y is {0(}-adapted. Clearly every sample function of Y is continuous. According to Theorem 13.27, it remains to show that {ei
|<s s] = e>(y.n>+i#»
a e
on
(Q^^P),
in other words,
(1)
E [ e %^«)+¥«
|5T+J] = e«WW**£«
a e o n (Q5 5 T + S I P )
The Jx+s-measurability of Xj+S implies that of the right side of (1). Thus it remains to verify that for every A G ST+S we have
(2)
I e*<****>*¥« dP = /" e****"*^* dP.
Now according to Remark 13.28, {e^"*'> +lj ^' : t G R+} is a martingale on the filtered space (Q, g, {&}, P ) for every y G Rd. For n G N, consider the stopping time Tn = TAn. By Theorem 8.10 (Optional Sampling for Bounded Stopping Times) we have E[e
.
= e.
a e
o n
(
Qi5Tn+s)P).
By the gTn+s-measurability of e i T" , the last equality reduces to (3)
E [ e * ' x ^ ^ ; ? r » + J = e''(^M>+1^s
a.e. on (Q, 5 r „ + „ P ) .
If A G 5 T + S , then ,4 n {T + s < Tn + s) G 5r B « by Theorem 3.9. But {T + s
f JAn{T
ti
dP = / JAn{T
e'^XT"^'dP.
286
CHAPTER 3. STOCHASTIC
INTEGRALS
Since (4) holds for every n e N and since the finiteness of T implies that U ne N{T < n) = £2 and consequently lim lAu{T
[Ill] 1-Dimensional Brownian Motions A 1-dimensional Brownian motion is a particular case of d-dimensional Brownian motion we considered thus far and will be referred to simply as a Brownian motion. We shall show that if X = {Xt : t 6 R+} is an {5(}~aclapted Brownian motion on a standard filtered space (£2,5, {&}, P ) and if X 0 = 0 a.e. on (£2,5, P ) then X £ M£(£2, 5, {&}, P ) and its quadratic variation process is given by [X]t = t for < 6 R t . Lemma 13.30. Le? X = {X t : i 6 R+} foe a Brownian motion on a probability space (£2,5, P ) wifn an arbitrary initial distribution PXg. Then for every t G R+, we have (1)
E(X4) = E(X 0 ) = /
x0PXo(dx0),
J*.
provided the integral exists, and (2)
E(X 2 ) = E(X02) + t= f x20PXo(dx0) +1. m
Thus X is an L\-process if and only if / m xoPxoC^o) 6 R and X is an L2-process if and only if f& xlPXo(dx0) < oo. In particular, if PXo is a unit mass at some a € R, then E(X ( ) = a and E(X(2) = a2 + tfor t G R+ and X is an Lj-process.
287
§13. ADAPTED BROWNIAN MOTIONS
Proof. For t > 0, the probability distribution Px, of the random variable Xt is given by (3) of Proposition 13.16 as PX,(E) = (27ri)-'/ 2 / PXo{dx0) I e x p l - l ^ " ^ 1 2 } Jm JE { 2 t — to J for E e 93K. NOW from (3)
mddx),
(2^)- , / 2 ^exp|-iyjm L (di)=l,
(4) we have for any xo € R,
(2nt)-V2J^xexpLl-]-^f£\rnL(dx)
(5) =
(2^)-1/2^(x-xo)exp|-^|x~Xo|2|mL(rix)
+ (27rt)" 1/2 / xo expI _l\x~xo\ JR [ 2 t
J
\mL(dx)
= x0 Then by the Image Probability Law and by (3) and (5), we have E(Xt)
=
( XtdP= ( xPXl{dx) Jn Jm = J^PXo(dx0)h2nt)-^^x
=
exp|-^J~Io|2|mL(^)|
/ x 0 PXo(dx0),
proving (1). Similarly from (2irt)-'/2 J x2 exp I ~ ~
(6)
\ mL(dx) = t,
we have (7)
(27rr)-,/2|ffiX2exp|-^^^}mL(rfx) f = (27ri)- 1/2 J {{x - x 0 ) 2 + 2x0(x - x 0 ) + x 2 } exp I = t + x\.
2 1 |x — xol l — l - \ mL{dx)
288
CHAPTER 3. STOCHASTIC
INTEGRALS
By the Image Probability Law, (3), and (7), we have
E(Xf)
= f XfdP= f x2PxMx) = [ PXo(dxt,)|(27rt)- " 2 / x1 expl-
1 \x -x0\ - \ mL{dx) > t 2
= / xlPXo(dx0) + t, Jm
proving (2). ■ Proposition 13.31. Let X = {Xt :teR+}bean {ff(}-adapted Brownian motion on a standard filtered space (Q,ff, {ffj, P)- If XQ is integrable then X is a martingale and if Xo is square-integrable then X is an Lx-martingale on the filtered space. If XQ = 0 a.e. on (Q, ff, P) then X G M|(Q, ff, {$t}, P) and a quadratic variation process [X] of X is given by [X]t = tfort G Rt. Proof. If X 0 is integrable, then by Lemma 13.30, E(Xt) = E(X 0 ) G R for all t G R+ and thus X is an Lj-process. To show that X is a martingale, let s, t G R+ be such that s < t. Then E[Xt - Xs |ff.] = E[X ( - X.] = 0 a.e. on (Q, ffs, P ) , where the first equality is by the independence of {ffs, Xt—Xs} according to 2° of Definition 13.17 and the second equality is by the fact that the probability distribution of Xt — Xs is given by iV(0, t — s) according to 3° of Definition 13.17. If Xo is square-integrable then X is an L2-process by Lemma 13.30. IfX 0 = 0a.e. on (Q,ff, P) thenX G M^(£2,ff, {ff ( },P) by Definition 11.3. To find a quadratic variation process [X], let a stochastic process A = {At : t G R+} be defined on the filtered space by setting A(t,ui) = t for (t,ui) G R+ x Cl. Then A is trivially a continuous increasing process on the filtered space. To show that A is a quadratic variation process of X, it remains to verify that X 2 — A is a null at 0 rightcontinuous martingale. Clearly X 2 — A is null at 0 and continuous. To show that it is a martingale, we show that for s, t G K+ such that s < t we have E[X 2 - At |ff,] = X 2 - As or, equivalently,
E[X 2 - X 2 |ff,] = t - s
a.e. on (Q,ffs, P ) , a.e. on (£2, g „ P).
But E[X 2 - X 2 | f f J = E[{X t - X,} 2 |ff,] = E[{Xt-Xs}2] =t - s a.e. on(Q,ff„P),
§13. ADAPTED
289
BROWNIANMOTIONS
where the first equality is by the martingale property of X, the second equality is by the independence of {# s , {Xt - X s } 2 } implied by that of {&, Xt -Xs}, and the third equality is by the fact that the probability distribution of X, - Xs is given by 7V(0, t - s). This shows that A is a quadratic variation process of X. ■ Proposition 13.32. Let X = {Xt : t g R+} be an {&}-adapted d-dimensional Brownian motion on a standard filtered space (Q, 5, {J ( }, P). Then its components X ( i ) ,i = l,...,d, ore {5 ( } -adapted 1 -dimensional Brownian motions on the standard filtered space with ini tial distributions PXM = Px0 o -K~X where 7T; is the projection ofRd = Ri x • ■ ■ x Rd onto R for i = \,...,d. Proof. By Proposition 13.4, X w is an {JJ-adapted process on the filtered space. Also X ( , ) is a continuous process. Thus according to Theorem 13.26, to show that X ( , ) is an {5t }-adapted Brownian motion it remains to verify that for every s, t g R+ such that s < t and any y; € R we have (1)
E [ e t M ' , - ^ ' ' > | 5 J = e-^<<-*>
a.e.on(A&,P).
But since X is an {5t}-adapted d-dimensional Brownian motion, Theorem 13.26 implies that for every y € Rd we have E[e , '< ! '^-^>|5J = e - 1 ^ ( t - s )
(2)
a.e. on ( Q , & , P ) .
d
With the choice y = ( 0 , . . . , 0, yu 0 , . . . , 0) g M. , (2) reduces to (1). Regarding the initial distribution PXM of X ( , ) , we have PJM{E} =
= P o (X< !) )-'(£) = P o fo o X 0 ) - ' ( £ )
P o X 0 -' o TT-'CE) = P x „ o Tfl(J5)
for E 6 » a . I
Proposition 13.33. Lef X = {X t : t g R+} be an {$t}-adapted d-dimensional Brownian motion on a standard filtered space {Q.,$,{$t},P). 7fE(|X 0 | 2 ) < oo, then the components X ( l ) , . . . , X w of X are continuous Li-martingales on the filtered space. For every s,t g R+ such that s < t we have (1)
E[X ( ( i ) -X<'>|S s ] = 0
a.e.
OTI(Q,S.,P),
and (2)
E[{X<° - * « } { X t 0 ) - X « } | & ] = M * - 5> °-e- o n ( Q - 5 S , F ) .
290
CHAPTER 3. STOCHASTIC
INTEGRALS
In particular when X 0 = 0 then X ( 1 ) , . . . , X(d) £ M£(Q,ff,{fft}, P ) and their quadratic variation processes are given by (3)
[X«\Xlj)]t = 6iJ
forteR+.
Proof. By Proposition 13.32, X ( 1 ) ,... ,X(d) are {ffj-adapted Brownian motions on the filtered space. Since |X 0 | 2 = E t i \X®\2. if E(|X 0 | 2 ) < oo then E(|X^| 2 ) < oo for i = 1 , . . . , d and thus by Proposition 13.31, X ( 1 ) , . . . , X W ) are L2-martingales. The equation (1) is the martingale property of X ( , ) . To prove (2), recall that since X is an {ffj-adapted d-dimensional Brownian motion, {3s, Xt - Xs} is an independent system by Definition 13.17. Then since the mapping 7Tij(xi, ...,xd) = XiXj of Rd into K is a ©j^/OSi-measurable mapping, the system {&, {X t w - Xf }{Xf - X ^ } } is an independent one. Thus E [ { X f - Xf}{X? - X « } | g s ] = E[{X« - X « } { X f - ! « } ] = 8ij(i - s) a.e. on (£2,ffs, P), since the probability distribution of Xt — Xs is the normal distribution iV(0, (t — s) ■ I) whose covariance matrix is given by (t — s) ■ I according to Theorem D.5. This proves (2). IfX 0 = 0, then X^ 0 = 0 so that X ( , ) G M§(Q,ff, {ffJ,P). Letz,; = 1,... ,d be fixed. Consider V e V(Q,ff, {ffj, P ) defined by Vt = S{)jt for t e R+. Then
E[{xM> - vt} - {xf xp - vs} m = E[X ( (,) X ( 0) - X«X<-»|ff s ] - fiy(i - s) = E[{X((l> - X<»}{X« - Xjp) |ff.] - *„-(* - s) = 0 a.e. on(Q,ff s ,P), where the third equality is by the martingale property of X w and X 0 ' and the last equality is by (2). Thus X W X W — V is a right-continuous null at 0 martingale. This proves (3). ■
[IV] Stochastic Integrals with Respect to a Brownian Motion If B = {Bt : t e R+} is an {ffj-adapted null at 0 Brownian motion on a standard filtered space (ii,ff,{ff«},P), then B e M|(£i,ff, {ff(},P) and its quadratic variation process is given by [B]t = t for t € M+ according to Proposition 13.31. Thus for our
113. ADAPTED BROWNIAN
291
MOTIONS
[B] G A(£2,§, {&}, P), the family of Lebesgue-Stieltjes measures {/<(B](*, w) : w G £2} on (R+, fBiJ determined by [P] is simply Wsj(*i«) - n»£
for every w € £2,
where mi, is the Lebesgue measure. Since a null at 0 Brownian motion B is a particular case of martingales in M§(£2, 3 , {&}, P), the results in § 12 concerning stochastic integrals with respect to M G M | ( Q , 5 , {5,}, P) apply. We show below some implications of the special property of the quadratic variation process [B] of B. Observation 13.34. Letp G D,oo). Consider the collection of all measurable processes X = {Xt : t G R+} on a probability space (£2,5, P) satisfying the following integrability condition (1)
\X{s,w)\p{mL
/
x P)(d(s,u>)) < oo
for every t € K+.
This condition is equivalent to the condition (2)
/
i[0,m]x(!
\X(s,u)\"(mL
x P)(d(s,ij)) < oo
for every m € N.
By the Tonelli Theorem, we have (3)
/
|Ja^)|"(mLxP)(d(5,uO) = E [ /
J[0,m]xn
U[0,m]
\X(s)\pmL(ds)
.
The condition E[/ [ 0 m , |X(s)|pmi,(d.s)] < oo implies that / [ 0 m ] \X(s)\pmL(ds) < oo a.e. on (£2,5, P). Thus for every m G N there exists a null set Am in (£2,5, P) such that /[0,m] \X(s,uj)\pmL(ds) < oo forw G AJj,, Then with the null set A = Um6NAm we have (4)
/
\X(s,u)\pmL(ds)
< oo
for every i G R+ when w G Ac.
The condition (5)
/
|X - F | p d(m L x P) = 0
for every i G R+
is an equivalence relation in the collection of all measurable processes on (£2, 5, P). Let L p coCR^ x Q, mi x P ) be the linear space of the equivalence classes of all measurable processes on (£2, #, P ) satisfying (1) with respect to this equivalence relation. The element
292
CHAPTER 3. STOCHASTIC
INTEGRALS
0 g Lp]0O(R+ x fi,m[ x P) is the equivalence class of all measurable processes X on (Q, 5, P) satisfying the condition / \X(s,uj)\p(mLxP)(d(s,Lo)) V[o,(]xn
(6)
=0
for every t £ R+.
A measurable process X on (Q, 5, P) satisfies condition (6) if and only if there exists a null set A in (Q., 5, P) such that for every u> e Ac we have / \X(s,u>)\pmL(ds) = 0 i[0,t]
(7)
for every t G R+.
This follows by the same argument as in Observation 11.16. Note that condition (7) is equivalent to the condition that there exists a null set A in (Q, $, P) such that for every u £ Ac we have (8)
X(-,to) = 0
a.e. onCK+^m^mL).
Definition 13.35. Let p £ [l,oo). In the linear space LPi00(R+ x Q,,mL x P ) of the equivalence classes of measurable processes X = {Xt : t 6 R+} on a probability space (Q, 5, P ) satisfying the condition / \X(s,ui)\v(mL ./[o,i]xn
(1)
x P)(d(s,u))<
oo for every t 6 R+,
we define (2)
||X||^*F=[/
JX|»d(mLxP)]1/P = E [ /
L/[0,t]xi2
J
|X(5)|»m£(<&)
-ll/p
U[o,t]
for every t £ R+, and define
(3)
ii^ii P m i, , < p =i:2- m {ii^ii P m 4 x P A i }meN
Remark 13.36. The functions || • \\™$ x P for t e R+ and || ■ ||££, xP on L Pi00 (R + x Si, m t x P ) defined above have the following properties. 1) || • \\™tXP is a seminorm on LPj00(R+ x Q, mL x P ) for every t e R+. 2) For X, XM <E L ^ C R * x Q, m L x P), n G N, we have ton | | X « - X | | - ^ = 0 « , ton ||X<"> - X | | - x P = 0 for every m 6 N.
§13. ADAPTED
293
BROWNIANMOTIONS
3) || • ||p,oo is a quasinorm on L P|00 (R + x Q, mL x P). Proof. These statements are proved in the same way as Remark 11.4 for the seminorm | | and the quasinorm | • | m on the space M 2 (Q, 3 , {& }, P). ■ Let the Banach space L p ([0, m] x £2, £r(95[0,m] x 3), m L x P) for m G N be abbreviated as Lp([0, m] x Q). The function || • ||™4*p is only a seminorm on the space LPi0O(R+ x Q,m L x P), but it is a norm on L p ([0,m] x £2) and l p ([0,m] x £2) is complete with respect to the metric associated with this norm. We use this fact to show that the space LPi00(K+ x Q , m t x P ) is complete with respect to the metric associated with the quasinorm
II • \\%>xP-
Theorem 13.37. Let p G [1, oo). The space LPi00(R+ x £2,mi x P) is a complete metric space with respect to the metric associated with the quasinorm || ■ ||p"£,xP on LP|O0(R+ x £2, mL x P). Proof. Let {XM : n G N} be a Cauchy sequence in LPi00(R+ x Q,mL x P) with respect to the metric associated with the quasinorm || • \\™^p Let m G N be fixed and let X^ be the restriction of Xin) to [0, m] x Q for every n G N. Now for every e > 0 there exists N G N such that \\XM - Xm\\™gp < 2-me for nJ>N, and thus 2- m {||X ( t ) ) - XZWW
A 1} < 2 -
£
for
nJ>N.
We may assume without loss of generality that e < 1. Then we have
pq2)-X$A%?P<*
for
nJ>N.
Thus {X^ : n G N} is a Cauchy sequence in the Banach space Lp([0,m] x fil) and therefore there exists Y{m) e l p ([0,m] x Q) such that Jinn \\X$} - Y(m)\\pnJkxP = 0. Let us take an arbitrary real valued representative function of Y<m) and fix it for m G N. Now for m = 1, X$ converges to Yw in the Lp-norm on £ p ([0,1] x £2) and therefore there exists a subsequence {ni,*} of {n} such that X,",'*' converges to Ym on [0,1] x il- A{ where Ai is a null set in [0,1] x £2. Then since X^,k converges to Ym in the Lp-norm on L p ([0,2] x Q), there exists a subsequence {n2,k} of {n^/t} such that X^'* converges to y(2) on [0,2] x £2 — A2 where A2 is a null set in [0,2] x Q. containing Ai. Thus proceeding inductively we obtain a subsequence {nkxk } of {n} such that for every m G N the sequence
294
CHAPTER 3. STOCHASTIC
INTEGRALS
X^jk converges both in the Lp-norm of Lp([0, m] x £2) and pointwise on [0, m] x Q. — Am where A m is a null set in [0, m] x Q. and contains A ] , . . . , A m _i. Thus y (m) = Y(m-\-, on [0, m - 1] x Q. - A(m_D for m > 2. Let A = Um£NAm- Let us define a function Y on R + x Q b y setting vu
F(f
v
/ * W * . w>
'w)-l0
for
(*•w) e [°> m] x Q - A m and m 6 N
fbr(*fW)6A.
The function F is well defined and is a measurable process on (£2,5, P ) satisfying the condition | | y | | £ £ x P = | | y ( m ) | | ^ x P < oo so that Y G L ^ O ^ t x Q , m L x P ) . Also lira H X ^ ^ - Y\\™£p = Urn | | X ( " ^ ) - y ( m ) | | ^ x P = 0
fc—>-00
"
for m € N.
K—t'OO
This implies lim IIX*"'*' - Y\\™£p = 0 according to Remark 11.18. ■ fc—^oo
In Proposition 2.13 we showed that every left- or right-continuous adapted process on a filtered space is a progressively measurable process. More generally we showed in Propo sition 2.23 that every well-measurable process, and in particular every predictable process, on a filtered space is a progressively measurable process. Theorem 13.38. Let X = {Xt : t 6 R + } be a progressively measurable process on an augmented filtered space (Q, J , {§t},P). If there exists a null set A in (Q, 5 , P ) such that (1)
\X(s, ui)\m.L(ds) < oo for every t € K+ when u> € A c ,
/ ■/[<>,«]
then there exists a predictable process Y on the filtered space such that (2)
X{-,UJ) = Y{-,UJ)
a.e. o/i (R+.QSn^mi) w/ien w g Ac.
/w particular if X is in LP|00(R+ x Q, m L x P)for some p € [0, oo), then (I) is satisfied and X and Y are equivalent in this space. Proof. Since X is a progressively measurable process, it is an adapted measurable process according to Observation 2.12. Since the filtered space is augmented, A e J , for every t € R+. For every n e N, define a real valued function XM on K+ x Q. by setting ri\
YMH , A _ j
n
U
/
,
X(s,u)mL(ds)
for (i, w) 6 K+ x Ac for(<,oj)GR + x A.
§75. ADAPTED BROWNUN
295
MOTIONS
Let us show first that Xin) is an adapted process. Now since X is progressively measurable, foreveryi € R+ the restriction ofX to [0,t]xQ is
=
n
<
n/ n/
Jf(3,w)mt(
|l [ t _i i ] (5)-l [ t o _i ( o J (5)||X(s,u;)|m L (£fs)
[0,t 0 +l]
1
[0,to+l]
[(-J- 1 t]A[(o-- ! -,(o] ( ' s) ll X(s ' a;) l mL(d ' s) -
Since lim l, t _± (] ^ [( _ j . t ,(3) = 0 and since X(-,u>) is integrable on [0, t0 + 1], we have by the Dominated Convergence Theorem lim \Xin)(t,u>) - XM(t0,u)\ t—*tf)
(n)
= 0. This proves the
continuity of X (-, u>) at
F ( i w ) = liminfX ( " ) (i,u;) 71—+OO
for (t,w) G R+ x n .
Since X ( n ) is an 6/QJis-measurable mapping of R+ x Q. into R for every n G N, Y is an 6/9%-measurable mapping of R+ x Q into R. Then {Y = ±co} G 6 so that if we define a real valued function Y on R+ x Q by setting (5)
.y
Y
&u>-U
{I
wheny(t,o;)GR when7(t, W ) = ±oo
then Y is an 6/93a-measurable mapping of R+ x £1 into R, that is, Y is a predictable process on the filtered space. By (1), for every u> G Ac the sample function X(-, u) is Lebesgue integrable on every finite interval in R+. Then by Lebesgue's Theorem the indefinite integral of X(-,u>) is
296
CHAPTER 3. STOCHASTIC
INTEGRALS
differentiable with derivative equal to X(t,ui) for a.e. t G R+. Thus recalling the definitions of X ( n ) and Y by (3) and (4) respectively, we have for every u e A c (6)
X(-,U)
= Y(;UJ)
= Y(-,UJ)
a.e. on(R + ,Q5m„m L ),
where the second equality is from the fact that X is real valued. This proves (2). If X is in LP?00(R+ x a, mL x P), then X satisfies (1) and thus by (6) for every u 6 A c we have \Y(s,uj)\1'mL(ds)=
/ J[0,t]
f
\X(s,u)\"mL(ds)
for every t G K+,
\X(s,u>)\"mL(ds)
< oo
J[0,t]
and then \Y(s,u)\pmL(ds)}
E[/ J[0,t]
=E\ [ J
for every t G E + .
U[0,t]
This shows that F is in L P)00 (E + x Q, mL x P). From (5) we also have E [/
U[0,t]
\Y{s,u>)-X{s,w)\>mL{ds) = 0 for every t G
This shows according to (5) of Observation 13.34 that X and Y are equivalent processes in L ^ C R * x Q, mL x P). ■ For quadratic variation processes of stochastic integrals with respect to Brownian mo tions we have the following. Proposition 13.39. Let B = {Bt : t G R+} be an {$t}-adapted d-dimensional Brownian motion on a standard filtered space (Q, 5, {& }, P) with BQ = 0 and let BM, i = 1 , . . . , d, be its components. Let X w be a predictable process in L2>00(R+ x Q, mi x P)for i = 1 , . . . , d. Then for i,j = 1 , . . . , d, there exists a null set A in (Q, $,P) such that on Ac we have for every ( 6 1 + (1)
[ X w . B ( , ) , X w . B^]t = 6tJ I
X^(s)X^(s)mL(ds),
J[0,t)
and thus for any s,t G R+, s < t, we have (2)
E[{(X W . B(% - (X (i) . B (, ' ) )J{(X (J ' ) . B«)« - (X°'> . BU))S} | & ] = E[(X (i) • B%(X^ • S 0 ' ^ - (X (i) . B(i))s(XU) • S ° ' ) S | 5 S ] = SijElf
A- w (u)X 0 ) («)m 1 (du)|ff J ]
a.e. o n ( n , 5 „ P ) ,
297
114. EXTENSIONS OF THE STOCHASTIC INTEGRAL and in particular E[{(X (,) . P<")t - ( I d . B®) S }{(X W . Bu\
(3)
- ( X w . B°>),}]
X < , , ( u )X ( j ) («)m L (du)].
= 6ijE,[[ J(s,(]
Proof. According to Proposition 13.33, [.BW,.BW]4 = 6i}jtiort G R+. Thus the Proposition is a particular case of Theorem 12.16. ■ Corollary 13.40. Let B and Bli\ i = 1,... ,d, be as in Proposition 13.39. Let X{i) and YM be predictable processes in L2|00(M+ x Q., TTIL X P) for i = l,...,d. Then for X = E t , X ( i ) • P ( i ) and Y = £? =1 *"W • # W « M|(Q,g, {&}, P) f/iere e x t o a n«« sef A in (£2, 5, P ) SMC/I that on Ac we have for every t G K+
(1)
£X(,,(5)ym(5)mL(ck)
[*,n(=/
anrf thus for any s,t G R+, s < £, we nave
(2)
E[{x( - xs}{Yt - r j | 5 s ] = E[x(y - x,y,|3f,] VX('V)y(,)(«)^L(d«)|5s]
= E[/
a.e. on (£},&, P).
Proof. Since X w • P ( " is in M§(Q,#, {&}, P), so is X = E t i ^ ( 0 • £ W - Similarly for y . Thus [X, Y] is defined and according to Proposition 11.26, we have [X, Y]t = [ £ X ( , ) • S w , £ y ( i ) • P ( , ) ] 4 = £ [X c0 • B®, X w »-P 0 ) ] ( i=l
=
Y'Sijf
.'=1
X{i)(s)Yu)(s)mdds)=
i,i=\
I
tl(i)(S)y(')(S)mt(dS)
where the third equality is by Proposition 13.39. ■
§14
Extensions of the Stochastic Integral
[I] Local L2 -Martingales and Their Quadratic Variation Processes For M G M 2 (fi, 5, {&}, P ) and a predictable process X on the filtered space satisfying the integrability condition X G L2,oo(R+ x il,filMhP), that is, E[/ [ 0 A X 2 (s) d[M](s)] < oo
298
CHAPTER 3. STOCHASTIC
INTEGRALS
for every t £ R + , the stochastic integral X ; M of X with respect to M was defined in Definition 12.9 as an element in M 2 (Q, 3 , {& }, P). We show below that if X satisfies the weaker integrability condition that / [0 (] X\s) d[M](s) < oo for every t £ R+ for almost every u £ Q, then there exists a sequence {X ( n ) : n £ N} in L2,oo(R+ x Q, ^ [ M ] , P ) such that XM converges pointwise on R+ x Ac to X and X ( n ) • M converges pointwise on R+ x Ac where A is a null set in (Q, 3 , P). We then extend the definition of stochastic integral with respect to M by defining the limit of the pointwise convergence of the sequence {XM • M : n £ N} in M 2 (Q,3, {3<}, P) as the stochastic integral of X with respect to M. This leads to the definition of local martingales. Definition 14.1. Let X = {Xt : t £ R+} be an adapted process on a filtered space (£2, 3 , { 3 J , P) and let {Tn : n £ N} be an increasing sequence of stopping times on the filtered space such that Tn | co a.e. on (Q, 5, P). We say that X is a local martingale with respect to the sequence {T„ : n £ N} ifXTnA is a martingale on the filtered space for every n £ N. We say that X is a local L-i-martingale with respect to {Tn : n £ N} ifXT"n is an L2-martingale on the filtered space for every n £ N. Thus if X is a local martingale with respect to {T„ : n £ N}, then since Tn(u>) | oo for every u £ A D where A is a null set in (Q, 5, P) we have lim XTnA(t, w) = X(t, UJ) for n—►oo
(t,u>) £ R+ x Ac. Thus X is the limit of pointwise convergence on R+ x Ac of a sequence of martingales. Note that a martingale is always a local martingale with respect to the sequence of stopping times {Tn : n £ N} where Tn = oo for every n £N. Note also that the fact that X is a local martingale with respect to an increasing sequence of stopping times {T„ : n £ N} such that Tn | oo a.e. on (£2,3, P) does not imply that it is a local martingale with respect to every other increasing sequence of stopping times tending to oo almost surely. Observation 14.2. Let X = {Xt : t £ R + } be a local martingale on an augmented fil tered space ( Q , 3 , {3(}j-P)- If there exists a nonnegative integrable random variable Y on ( Q , 5 , P ) such that \X(t,u)\ < Y{w) for (t,w) € R + x Ac where A is a null set in (Q, 3 , P), then X is in fact a martingale. If the random variable Y2 is integrable, then X is an L2-martingale. Proof. Since X is a local martingale, there exists an increasing sequence of stopping times {Tn : n e N} on the filtered space such that Tn f oo on A£ where Ao is a null set in (Q, 3 , P) and XTnA is a martingale on the filtered space for every n £ N. Then for any pair
§ 14. EXTENSIONS OF THE STOCHASTIC
INTEGRAL
299
s,t £ R+ such that s < t, we have according to Theorem 8.12 (1)
E[X T n A t |5 s ] = A-r„As
a.e. on(£2,5 s ,P).
Since T„ | ° ° on A£, we have lim X TnA( = Xt on A£. Also |Xr„A1l < Y on Ac. Thus by the Conditional Dominated Convergence Theorem, we have (2)
Jun E[XTnM |5.] = E[Xt |ff,]
a.e. on (O,&, P ) .
We have also lim XT„ A S = Xs on Aj;. Since the filtered space is augmented, the null set Ao is in 5 S - Thus (3)
lim XTnA, = X,
a.e. on (£2, 5 „ P ) .
Letting n -> 00 in (1) and using (2) and (3), we have E [ X , | 5 J = X, a.e. on (£2,5S, P). This shows that X is a martingale. If Y2 is integrable, then since X2 < Y2 on Ac we have E(X(2) < E(Y2) < 00 for every t e K + s o that X is an L2-process. ■ Note that for a local martingale X, the right-continuity and the continuity of X(-, u) for an arbitrary ui £ £2 is equivalent to the right-continuity and continuity of X T " A (-, u>) for all n £ N. Lemma 143. Let {Sn : n £ N} an^ {Tn : n £ N ) t e two increasing sequences of stopping times on a right-continuous filtered space (£2,5, {St}, P ) such that S n T °° and Tn f 00 a.e. orc(Q,5,P)- | f X is a local martingale with respect to the sequence {Sn : n £ N}, f/ien if is a /oca/ martingale with respect to the sequence of stopping times {Sn A T„ : n £ N}. 7f X is a right-continuous local Lr-martingale with respect to the {Sn : n £ N}, r/ien if is a local L2-martingale with respect to {Sn A T , : n £ N}. Proof. Clearly 5„ A T„ j 00 a.e. on (£2,5, P)- If X is a local martingale with respect to {Sn : n £ N}, then for every n £ N, X 5 " A = {X SnAt : t £ K+} is a martingale. Thus ^S„AT„A _ {jfSjiATnAi : i e R+} is a martingale by Theorem 8.12. This shows that X is a local martingale with respect to {5„ A T„ : n £ N}. Suppose X is a local Z/2-martingale with respect to {S„ : n £ N}. Then for every n £ N, X S " A = {XS„A4 : t £ R + } is an L2-martingale so that (X S " A ) 2 = {X| n A t : t £ R + } is a submartingale on the filtered space. Since Sn A t and 5„ A Tn A t are stopping times on the filtered space and Sn A t > Sn A T„ A t for i £ R+, Theorem 8.10 (Optional Sampling with Bounded Stopping Times) implies that E[XfnA4|5s„AT„A<] > X| n A T n A t
300
CHAPTER 3. STOCHASTIC
INTEGRALS
a.e. on (Q,5s„AT„At,-P) and consequently we have E[Xf nATnAi ] < E[X| n A f ] < oo. This shows that x s " A T n A is an Ir2-process and is thus an Z/2-martingale. Since this holds for every n G N, X is a local L2-martingale with respect to {Sn A T„ : n G N}. ■ Definition 14.4. Let M20C(Q, 5, {&}, P), abbreviated as M20C,feetfzecollection of equiva lence classes of all right-continuous local Li-martingales X = {Xt : t £ R+} with Xo = 0 almost surely on a standard filtered space (Q,5, {5,}, P). Le* M 2 '' oc (Q, 5, {&}, P), afcbreviated as M2''°c,feer/ze subcollection consisting of almost surely continuous members of M'2°c. Observation 14.5. The fact that M 2 0C (fi,5, {&}, P) is a linear space, that is, aX + bY G M2oc for X, F £ M^ c and a, b G R, can be shown as follows. Now if X € M2DC the clearly aX G M20C for a £ R. Thus it suffices to show that if X, F G M ^ then X + F G M2DC. Suppose X, F G M2DC. Then there exist two increasing sequences of stopping times {Sn : n G N} and {Tn : n G N} such that 5„ | °° and r n T °° a.e. on ( a , 5 , P ) and X S " A and YTNA are L2-martingales for every n G N. By Lemma 14.3, X SwAl- " A and F S w A T " A are L2-martingales for every n G N. But (X + F) s " A r » A = x s " A T " A + F S " A T " A Thus (X + y) s « A r n A i s an L2-martingales for every n G N. This shows that X + F is a local I/2-martingale. Since X and F are right-continuous and null at 0, so is X + F . Therefore X + F G M2oc. Definition 14.6. Let A'°C(X2,5, {5t},P) be #«e collection of equivalence classes of all stochastic processes A on a standard filtered space (Q, 5, {5t}, P ) satisfying conditions 1° and 4°, but not necessarily condition 2° that A be an L\-process, of Definition 10.1. Let V'°C(Q, 5, {5 t },P) be the linear space of equivalence classes of all stochastic pro cesses V on a standard filtered space (£2,5, {5«}, P) satisfying conditions 1° and 4°, but not necessarily condition 5° that | V | be an L\-process, of Definition 11.10. We write A C ' /0C (Q,5, {5,}, P) amf V C ' ,0C (Q,5, {&},F)/br fte subcollections o/A' 0 < : (0,5, {&}, P) and V'0C(Q, 5, {5t},P) respectively consisting of almost surely continuous members. In what follows we write A G A'0C(Q, g, {J,}, P) for both an equivalence class and an arbitrary representative of the equivalence class. Similarly for V G V'°C(Q, 5, {5;}, P). Lemma 14.7. If A', A" G A'~(£l,ff, {&},P), fnen A' - A" £ V' o c (ft,3, {3,},P). / / V G V' o c (Q,5, {St},P), then V = A' - A" where A', A" G A ,0C (Q,g, {&},P).
§ 14. EXTENSIONS OF THE STOCHASTIC
INTEGRAL
301
Proof. The proof parallels that of Theorem 11.12. ■ Theorem 14.8. For every X in M20C(Q, 3 , {&}, P), there exists an equivalence class A in A'°C(Q, 3 , {3 t }, P ) such that X2-Aisa right-continuous null at 0 local martingale. IfY is also in M20C(Q, 3 , {&}, P), then there exists an equivalence class V in V'0C(Q, 3 , {St}, P) such that XY — V is a right-continuous null at 0 local martingale. Proof. Suppose X and Y are in M2oc(£2,3, {&},P) and are local L2-martingales with respect to two increasing sequences of stopping times {Sn : n G N} and {T„ : n G N} such that 5„ T oo and T„ f oo a.e. on (Q., 3, P ) respectively. Let Rn = Sn A Tn for n G N. Then both X and Y are local L2-niartingales with respect to the sequence of stopping times {R^; n G N} according to Lemma 14.3. Let Ai be a null set in (Q, 3 , P ) such that Rn(u;) f oo for a; G Aj. Now for every n G N, since X fi " A and F * " " are in M 2 (£2,3, {3 ( }, P), according to Proposition 11.22, there exists a unique natural quadratic variation process V(n) of I ' * " " and y H " A in V(Q,3, {gt},P) such that X*" A Y R " A - VM is a right-continuous null at 0 martingale. Similarly there exists a unique natural y ( n + " G V(Q, 3 , {3t}, P) such that jf/^^Ay^+iA _ y(n+i) i s a right-continuous null at 0 martingale. Then (XR"*'AYR"*'A y ( " + I ) ) f l " A is a right-continuous null at 0 martingale by Theorem 8.12. But R„. < P„ + i implies that
(X*"
+lAyfl„+|A _ y(n+lh.R„A _ T£-K„Ay.R„A _
/y(n+l)\RnA
Thus by the uniqueness of natural quadratic variation process of XR"A and YR"A we have yM
_ /y(n+lKfi„A
In what follows we write V(n) for an arbitrarily fixed representative of the equivalence class. Then by the last equality there exists a null set A2 in (D, 3 , P) such that (1)
Vr(",(-,u;) = (V("+1,)finA(-,u>)
for all n G N when w G A|.
Iterating (1) and using the fact that Rn A P„ + i = P„, we have (2)
V
forn,p G N when a; G AJ.
Since V (n) G V(£2,3, {&}, P ) for every n G N, there exists a null set A3 in (Q, 3 , P ) such that if u G A3 then y ( n ) (-,^) are functions of bounded variation on every finite interval of R+ for all n G N. Let A = U L A.% Let t G R+ be fixed. Since P„(u>) f 00 for w G Ac, there
CHAPTER 3. STOCHASTIC
302
INTEGRALS
exists Nw G N such that s < Rn(uj) for 5 G [0, t] when n > Nu. Then by (2) we have for W6A 1 , Vin\s, w) = V (n+P, (s, w)
(3)
for s G [0, i], p G N, n > JVW.
(n,
Thus lim V (t, w) exists in K for every t € R+ when u> G Ac. Let us define a real valued n—»-oo
function V on R+ x Q by setting lim VM(t, u) (4) V w ) «' 10
for (t, w) G R+ x Ac for«,u,)eR+xA.
Since F ( n ) is an adapted process, ^ is fo-measurable on O for every < G R+. Since the filtration is augmented the null set A is in g ( for every t G R+. Then Vt defined by (4) is 5t-measurable. Thus V is an adapted process on the filtered space. Let us show that for u> € Ac, V(-, ui) is a function of bounded variation on every finite interval in R+. Let t e R+ be fixed. Let N G N be so large that RN(u) > t. Then by (3) and (4) we have V{S,LJ) = V(N\S,L>) for s G [0,t]. Since V(Ar>(-,o;) is a function of bounded variation on every finite interval of R+ and in particular on [0,f], V(-,io) is a function of bounded variation on [0,t]. We have thus shown that if u> 6 Ac then V(-, u>) is a function of bounded variation on every finite interval in R+. This proves that the equivalence class represented byVisinV'~(A$,{&},P). Finally to show that for every n € N, (XY — V)RnA is a right-continuous null at 0 martingale, note that for (t, w) G R+ x Ac we have (XY -V)RnA{t,u)
= (XY)R"A(t,u>)-VRnA(i,uj) R A = (XY) " (t, u) - (lim V(*>)n"A(i, w)
by (4)
A:—t-oo
= (XY) R " A (i,a;)- lim(V w ) H " A (t,u>) = (iy)fl»A{{,w)-(VC))(t,u)
by (2).
(n)
Since (JfY)*"* — (V ) is a right-continuous null at 0 martingale and since A e 5 , for every t G R+, the last equality implies that ( 1 7 - V)fi»A is a right-continuous null at 0 martingale. This shows that XY - V is a right-continuous null at 0 local martingale. The existence of A in A,0C(Q, 5, {5/}, P ) such that X2 - A is a right-continuous null at 0 martingale is proved likewise. ■ Corollary 14.9. Let X,Y G M£ i o c (0,& {&},P). 77je« r/iere exisf arc equivalence class A G A c - ' 0 C ( O , 5 , { 3 t } , P ) a ^ a n e ^ v a Z e n c e d a w y 6 VC''DC(Q, 5, { & } , P ) sucfc that X2 — A and XY — V are almost surely continuous null at 0 local martingales.
§ 14. EXTENSIONS
OF THE STOCHASTIC
INTEGRAL
303
Proof. Since X,Y g M^ 0 C (Q,5, {3<},P), the processes X*" A and Y fl " A in the proof of Theorem 14.8 are in Mf(Q,3, {&},P). Then by Proposition 11.27, there exists V (n) in V c (fi,5, {&}, P ) such that X f i " A y*" A - V<"> is an almost surely continuous null at 0 martingale. The almost sure continuity of VCn) implies that the process V defined by (4) in the proof of Theorem 14.8 is almost surely continuous. The existence of A in AC''0C(Q, 5, {& }, P) is proved likewise. ■ Definition 14.10. Let X, Y g M ^ Q , 5 , {&}, P). An equivalence class A in A'oc(Q,5J> {&}> P ) weft fftaf X2 — A ijr a right-continuous null at 0 /oca/ martingale is called a quadratic variation process ofX and we write [X]for it. An equivalence class V in V' oc (£2,5, {5i}, P ) weft fftaf X Y — V is a right-continuous null at 0 local martingale is called a quadratic variation process ofX and Y and we write [X, Y]for it. In what follows we write [X] for both an equivalence class of processes and an arbi trary representative of an equivalence class. Similarly for [X, Y]. The existence of [X] in A'° C (Q,5, {$,}, P ) and [X, Y] in V'~(Q,5, {&}, P) for X and Y in M^(fl,5, {&}, P) was proved in Theorem 14.8. Theorem 14.11. LetX,Y g M£ , o c (n,& {&},P)andfef {S n : n g N} and {Tn : n g N} oe increasing sequences of stopping times such that Sn f oo, T„ j oo almost surely as n -> co, andX S " A , Y T " A g Mf/or n € K Let R„ = Sn A Tnfor n e KThen there exists a null set A i« (Q, #, P ) JMC/Z that for every u> g Ac we have (1) lim (E,u) = fiix,Y)(E,uj)for E e <8m, t g R+, 71—►OO
"
(2) T lim n{XRn^]{E,Lj) = fHx](E,ij)forE g 05^, (3) T lim n[YRnA](E, w) = jUmCB. w ) / " ' ^ g » » , . In particular fort g R+ OIK/W g Ac, we have (4) Jirn [X fl " A , Y fl " A ](i, u) = [X, Y](t, w), (5) T lim[X*" A ](t,a/) = [XKi,a;), n—*oo
(6)
T Hm[YR"*](t,u>) = [Y](t,Lj). n—t-oo
Proof. As we showed in the proof of Theorem 14.8, there exists a null set Ai in (Q, 5, P) such that on R+ x A, we have (7)
Jim [X"»A, Y*" A ] = [X, Y] Jirn [X fl " A ] = [X], and Jim [Yfl"A] = [Y].
Since X f i ~' A g M | ( Q , 5 , {&},P) and (X H -' A ) f l " A = X*" A , Lemma 12.23 implies that //[JT^AJCW) < ^xH^iAjC-jW) for w g A£ where A2 is a null set in (fi,5,P). Thus
304
CHAPTER 3. STOCHASTIC
INTEGRALS
^(xfi„«](-,u) f as n —♦ oo and in particular for any a, 6 G R+ such that a < b we have T lim|iyfW(a,6],w)=T 71—►OO
lim{[X*" A ](&,^)-LY*" A ](a,a>)} Tl—►OO
= [X](6, u) - [X](a, w) = / i m ( ( o , &],«). Thus ^[x«»A]((a,6],w) < /j[Jf]((a,&],u>). From this and by the same argument for Y it follows that (8)
^i[XRn^(E,iv)
and/j,[YRn^](E,ij)<
filX)(E,Lj)
for .E G QSJL, •
c
Let A = A! U A2. Let u;A and t G R+ be fixed and let 0 be the collection of all members E G 23(o,(] such that lim /itjfj«n*)yju,A}(13,w) = / i ^ y ^ E , w). Since iJ n (w) f ooasrc —> oo, for every s £ [0, t] we have by (7) lim [XR"n,YR"A](s,uJ)
= lim [X,Y]R"A(S,CJ)
=
[X,Y](s,u).
Thus for a, 6 G (0, t] such that a < 6 we have lim mx^Ay*.*j((a, Tl—»-00
L
J
k],w)«
lim{[Xfl"A,rR"A](6,u.)-[XR"A,rR"A](a,a;)} Tl—t-OO
= [X, F](&, w) - [X, K](o, w) = Mtx,n((«, &].«>)■ Let 3 be the semialgebra of subsets of (0, t] consisting of subinterval of (0, t] of the type (a, 6] and 0. We have just shown that 3 C 85. Since 3 is also a 7r-class and since <J(3) = 25(0,<], if we show that 0 is a d-class then we have 93(o,i) C © by Theorem 1.7. To show that 0 is a
l e N with -Eo = 0- Then {Fk : fc G N} is a disjoint sequence in 0 . Let us define step functions f„,n G N, / and g on (0, oo) by setting / n (x) == ^/Li[xflnA,yH„A](Ft,a;) a; gG (fc (fc --- l,fc]and fc €g N Jn(X) XKnA,yJInAI(i')t,W) tfor or X l,fc]andfc N < /(x) = H[X,Yi(Fk,Lj) forx g (fc -- l,fc]andfc g N (9) i g(x) = {/j.[X](Fk,u)ij,[Y]{Fk,Lo)}1'1 forx g (fc -- l,fc]andfc g N. /(aO = Mtjr,yj(iPfc,w) forx g (fc - l,fc] and fc g N Since i ^ g w) n—*oo ,/2 ,/2 < {fJ. < {/^[Jf < {A*m(J*. {^ m (E t ,u/) w )) )^ / i [[y y f«„A i . »](**,«)}' *n*](-Ejfc, ) (F fc ,u ] (F fc ,a;)} 1/2 < y u [ y ) (.F, : ,uj)} «)}' /2 , [X«„A w)wri(JPfci
I
114. EXTENSIONS OF THE STOCHASTIC
INTEGRAL
305
by Theorem 12.7 and (8). Thus |/„(x)| < g(x) for x G (0, oo). To show that g is Lebesgue integrable on (0, oo), note that / '
/(0
'°
o)
g(,x)mL{dx)=Y,{»m{Fk,Lo)ii[Y]{Fk,Lj)yi2 k&
< { £ »m
<
Wn(P^)},/2 = {W](£^)}l/2{f[n(£,W},/2
km
W]((0,a^}
1/2
W ] ( ( 0 , < ] , w ) } , / 2 < oo.
Thus by the Dominated Convergence Theorem we have lim / *-"*>
J(0,oo)
fn(x)mL{dx)
= \
f(x)mL(dx),
7(0,00)
in other words, n'il^ 12 /i[Jrfi"\Y*""]CF*:,w) = E l*lX,Y](Fk,Ui), AeN
jteN
that is, „'i m D ^[^ R " A .i" ! " A )(- B ' u; ) =
VIX,YI(.E,U).
This shows that E G 0 and completes the proof that 0 is a d-class. Therefore 93(o,t] C © and thus (1) holds for every E G 93(o,tj- By the equalities //[^H„A yR„A]({0},a;) = 0 and /■i[*,y]({0}> OJ) = 0, (1) holds for every E g ®(o,t]- The equalities (2) and (3) are proved in the same way as (1). Finally (4), (5), and (6) follow from (1), (2), and (3) respectively by letting E = [0,t]. I Proposition 14.12. Let X e M ^ c ( a , 5 , {ft},P). IfE([X]a) < oo for some a € M+, then X is an Li-martingale on [0, a]. Thus if X is in M2°c(£2,5, {ft}, P ) tfzen -X" is m M 2 (fi, 3, {& }, P ) if and only if [X] is in A(Q, » , {ft}, P). Proof. Since X G M2°C(Q, 5, {ft},P), there exists an increasing sequence of stopping times {Sn : n € N} such that 5„ f oo almost surely as n —► oo and X 5 n A is in M 2 (Q, 5, {ft}, P ) for n G N. Now [X] G A' M (a, 3 , {ft},P) and X 2 - [X] is a rightcontinuous null at 0 local martingale. Thus there exists an increasing sequence of stopping times {Tn : n € N}suchthatT„ | oo almost surely and ( X 2 - [ X ] ) T n A is aright-continuous null at 0 martingale. Let P„ = 5„ A Tn for n G N. Then there exists a null set A in (£2,5, P ) such that RJp) T °o for a; G Ac, X H » A G M 2 (Q,ft {ft},P), and (X 2 - [X])*" A is a right-continuous null at 0 martingale. Thus for any s,t G [0,a] such that s < t we have E C X 2 ^ - [X]ft.Atlft) = -X/LA. - [X]R^S a.e. on ( Q , f t , P ) . In particular for
CHAPTER 3. STOCHASTIC
306
INTEGRALS
s = 0, integrating the last equality and recalling that X0 = 0 and [X]0 = 0, we have E ( X | n A i - [ X ] f i n A ( ) = 0sothat E(X£„ At ) = E([X]«„ At ) < E([X] ( ) < E([X]a) < oo. Thus sup n e N ||X flnA (|| 2 < oo and therefore {XRnM : n G N} is uniformly integrable by Theorem 4.12. Now since X f l n A is a martingale we have (1)
E(X f i „ A i |&) = X f l n A s
a.e. o n ( n , 5 s , P ) .
Now Rn(ui) T oo for a; G Ac implies that lim XRnM = Xt on Ac. This convergence and the n—»-oo
uniform integrability of {XRnM : n G N} imply that Inn ||XK„At - X ( ||i = 0 by Theorem 4.16. Thus (2)
limE(X R „ A ( |3 s ) = E ( X t | 3 s )
a.e. o n ( Q , 5 s , P ) .
n—*oo
Since the filtration is augmented we have A 6 5 S . Thus letting n —> oo in (1), we have E(X ( |5 S ) = Xs a.e. on ( f i , 3 s , P ) by (2). This shows that X is a martingale on [0,o]. Finally since lim X£ nA( = X\ on Ac, we have by Fatou's Lemma E(X?) < liminf E(XR
Ai )
< E ( [ X ] J < oo.
n—*oo
This shows that X is an L2-martingale on [0, a]. ■
[II] Extensions of the Stochastic Integral to Local Martingales Observation 14.13. For A G A'0C(Q, 5, {&}, P), consider the collection of all measurable processes X = {Xt : t G R+} on the (Q, 5, P ) such that for every u e A ' where A is a null set in (£2,5, P ) depending on X we have /
X2(s,w)(iA(ds,u)
< oo forevery i G R+.
-/[0,t]
For such processes X and Y the condition that there exists a null set in (Q., 5, P ) depending on X and F such that for ui G Ac we have /
|X(s,uO-l%s,u;)|V4(ds,w) = 0 for every t G R +
Jl0,t]
is an equivalence relation.
§ 14. EXTENSIONS OF THE STOCHASTIC
INTEGRAL
307
Definition 14.14. For A G A' oc (n, 3 , {&}, P), toL^(R+ x £2, ^ , P ) oe rne linear space of equivalence classes of measurable processes X = {X t : t G R + } o« (£2, 5, P ) rac^ f/iar for OJ G Ac w/iere A u a nn// ser in (£2, 5, P) we have /
X 2 (5,cj)^(
Note that L ^ C R * x £2, ,/„, P ) c L|%(R + x £2, p.A, P ) for A 6 A(Q, 5, {&}, P ) by (3) of Observation 11.16. Lemma 14.15. Let M e Mj' oc (£2,3, {&}, P)TOtfiar[M] G A ^ Q , 5, {5,}, P ) and let X = {X ( : t 6 R+} oe a predictable process on the filtered space such that X G L^CR* x £2, ^[M], P)- Let {an : n € N} be a strictly increasing sequence in R+ such that an | 00 as n —> 00. For every n G N, /er Tn fee a nonnegative valued function on £2 defined by Tn(w) = inf{i E l + : /
X 2 (5,w)/i (M] ((is,w) > a„} A a„ / o r u £ a .
J[0,(]
Ler {Sn : n G N} be an increasing sequence of stopping times such that Sn f 00 a.e. on (£2,3,P)asn -> 00 andMs»A 6 M^(£2,3, {&},P).ro that [M S " A ] G A c (£2,3, {&},P). Then 1) {Tn : n G N} is an increasing sequence offinite stopping time on the filtered space and Tn 1 00 a.e. on (£2, 5, P ) as n —» 00, 2^X [TnI is a predictable process and XlT"^ G L2l00(R+ x Q,fi[Msk^,P) for every k G N. Proof. In Proposition 3.5 we showed that the first passage time of an open set in R+ by an adapted right-continuous process on a right-continuous filtered space is a stopping time. Now if Z = {Zt : t G R+} is an adapted continuous process on a right-continuous filtered space, then the function on the sample space defined by inf{i G R+ : Zt > an) is equal to inf{< G R+ : Zt > an}, that is, the first passage time of the open set (an, 00), and is therefore a stopping time for every n G N. The continuity of Z also implies that at the stopping time defined above, Z is equal to an. Let us construct an adapted continuous process Z on our standard filtered space (£2,5, {&}, P ) which is equivalent to the process Ul0:t]X\s)plM](ds):teR+}. Since X G L ^ C R * x £2, p.{M], P), there exists a null set Ai in (£2,5, P) such that for every u> G A, we have (1)
/
X2{s,uj)ij.[M]{ds,uj) < 00
for every < G R+.
308
CHAPTER 3. STOCHASTIC
INTEGRALS
Since [M] is an almost surely continuous process there exists a null set A2 in (Q, 5 , P ) such that [M](-,u>) is a continuous monotone increasing function on R+ for u> G A£. Let A = Ai U A2 and define a real valued function Z on R+ x Q by setting / / [0 „ X2(s, u)fi[M)(ds, u) *(t,u)-y
m (>
7
for (i, u) G R+ x Ac for(t,u/)GR+xA.
Now since X is a predictable process, it is a measurable process and then so is X2 Thus {/[0 j] X2(s)fi[M](ds) : t e R + } is an adapted process by Theorem 10.11. Since the filtered space (Q,$, {&}, P ) is augmented, A g & for every t € ! + . Thus Z(t, ■) defined by (2) is 5(-measurable. This shows that Z is an adapted process. For u> € Ac, [M](-,u>) is a continuous function on R + and therefore fj,lM]({s},uj) = 0 for every s € R+. This implies that / [0j] X 2 (s,o;)//[M](d5,ai) is a continuous function oft G R+. Thus we have shown that 2 is an adapted continuous process. For every n £ N, let P„ be a nonnegative valued function on O defined by (3)
Rn(u) = inf{t eR+:Z(t,u)>
an} A an
forw G i2.
By the continuity of Z(-,UJ) for every u> G £1, inf{t G K+ : Z(t,w) > a„} = inf{t G R+ : Z(f,w) > a n } , which is the first passage time of the open set (a n , co) in E and hence a stopping time by Proposition 3.5. Thus Rn is a stopping time. Clearly Pn(tj) | as n —> 00 for every w G £2. Let us show that actually P„(a;) f 00 for every CJ G Ac. Let u G A c . If Z(-,u>) is bounded on R+, say Z(t,ui) < am for some m G N for all t G R+, then inf{t G R+ : Z{t,w) > an} = inf{0} = 00 for n > m and thus Rn{u>) = 00 A an = an for n > m and then P„(u;) f 00 as n —> 00. On the other hand if Z(-,u) is not bounded on R + , then lim Z(t,u>) = 00. If R„(ui) T 00 does not hold, then there exists m G N such that /—too
R„(u>) < am for all n G N. Then for n > m wehaveinfjt G R+ : Z(t,u>) > an}Aan < am and therefore inf{t G R+ : Z(t,u>) > an} < a m f o r n > m. Thus lim Z(t,u>) = 00. But the *Tam
continuity of Z(-, w) implies lim Z(t, u>) = 2(a m , u) and then Z(a m , w) = 00 contradicting Z(am:tj) = f[0am]X2(s,u)iJ,iM](ds,u)) < CXD. This shows that Rn(u>) f co for this case also. Now by (2), Z(-,u) = J[QAX2(s,u))fMlM](ds,uj) forw G Ac. Thus for il„ defined by (3) we have Rn{ui) = Tn(uj) for w G Ac. Since Rn is a stopping time on a standard filtered space, this implies that Tn is a stopping time on the filtered space by Lemma 3.17. Clearly Tn(u>) t for every w G Q. Since il„(w) t 00 for u» G Ac, we have Tn(u>) T co for w G Ac.
§ 14. EXTENSIONS OF THE STOCHASTIC
INTEGRAL
309
Consider the truncated process X[Tn]. Since X is a predictable process, X[Tn] is a pre dictable process by Observation 3.35. To show that XlT"] £ L2i00(R+ x Q, /i[MsfcA,, P ) for anyfcG N, note first that since XlT"] is a predictable process, it is a measurable process by Proposition 2.23. Next, by Observation 3.34, for every w e £1 we have v-py,,
m
( s
lA_f
X ( * , w ) for 5 e [0,T„(w)] '^-i0 for«e(T«(W),oo).
According to 2) of Remark 14.11 we have ^[MsfcA](-,w) < JU[M](-,^) on (R+,*BaJ for u 6 A c t where Afc is a null set in (£2,5, P). Thus for every w e (A U A*)0 and f G R + , we have by (4) and the fact that Tn(uj) = R\.{D) (5)
(X [ r " 1 ) 2 ( 5 ,a;) / i [ M 5 t . ] (d S ,^))< /
/ -'[0,t]
=
(X^"..^
}
(d
)}
J[0,t]
^ 2 (5,^V[M](rf5,aj)) = Z(t A Pi„(u;),t<;) < a„,
/ ./[0,tAR„M]
where the last inequality holds since by the definition of Rn by (3) if t < R„(LO) then Z(tARn(u),u>) = Z{t,u) < a„andif< > i?„(w)then Z(MP n (cj),a;) = ^(P^(w),w) = a„ by the continuity of Z(-,UJ). Since (5) holds for w G (A U A,t)c, we have (XlT"])2(s)d[MSkA](d.s) < E(a n ) = an < oo,
E I J[0,t]
for every < G M+. This shows that XlTn] G L2,oo(K+ x Q, M[Ms*Ai. -f)- ■ Theorem 14.16. Ler M G M^' o c (n,5, { 5 J , P ) and let X = {Xt : t G K + }fcea pre dictable process on the filtered space such that X G L ^ C R t x Q, /j.[M], P). Let {Sn : n G N} be an increasing sequence of stopping times such that Sn ] oo a.e. on (Q, 5, P) as n —> oo awd M S " A G M|(Q, 5, {&}, P ) / o r every " G N. Le/ {a„ : n G N} be a strictly increasing sequence in R+ such thatan | oo as n —> oo. Let {Tn : n G N} be an increasing sequence of stopping times defined by Tn{u) = inf{t G R+ : /
X2(s,L})plM](ds,u)
> a„} A a„
/oruefi.
V[0,(]
Then there exists Y e M^iQdAZt}^) such that YS»*T"* = X™ • Ms"'-for n G N and therefore there exists a null set A in (Q, 5, -P) sucfe /nat Y(t,u)
= lim (X (TnJ • M S " A )(i,a;) /or (i,u;) G R+ X Ac
every
310
CHAPTER 3. STOCHASTIC
INTEGRALS
Furthermore the process Y is not only unique in M^' °c(£2, J , {&}, P ) but is in fact inde pendent of the choices of the sequences {an : n £ N} and {Sn : n £ N}. In particular 1) if M £ Mc2(£2,$, {&},P) and X £ L ^ ( R + x £1, fi[M], P), thenthereexists Y £ M2'' 0C (Q,5, {5;}, P)sue/! rta?F TnA = X[T"> • M/orevery n e N a n r f t h e r e exists a null set A in (Q, 5, P ) such that Y(t,u) = lim (X [ r " ] • M)(t, w)/or (t, UJ) £ R+x Ac, n—*oo
and!) if M £ Mc2'loc(n,$, {$t},P) and X £ L2,<x,(IR+ x Cl,filM],P), then there exists Y £ M2',oc(fi, 5, {&}, P) .sudi f/iaf F S " A = X • Ms"*for every neNand there exists a null set A in (Q, £, P ) such that Y(t, u>) = lim (X ; MSnA)(t, uS)for (t, w) £ R+ x Ac. n—>-oo
Proof. Since M £ M^'^CQ, 5, {5«},P), it has an almost surely continuous quadratic variation process [M] by Corollary 14.9. By Lemma 14.15, T„ t oo a.e. on (Q, 5 , P ) and the truncated process X[Tn] is a predictable process and X [ 7 "' 6 L2i0O(R+ x £2, /i[M], P)According to Lemma 12.23, p.[Ms„^(-,u!) < H[M)(-,U) on (R+, 18i+) for a.e. u> € Q. This implies that Xt7""1 6 L2,00(K+ x Q, ^[Msn^, P) for every n £ N. Thus X [ T " ] • M s " n is defined and is in M|(Q, j , {5 t }, P). Consider the subset {(•) < SnATn} = {(t,ui) £ R+ x Q. : t < Sn(uj) A T„(w)} of R+ x £2. Since Sn A T„ t on £2, {(•) < S n A T„} | as n -> oo. Let A be a null set in (fi, 5, P ) such that Sn(uj) f oo and T„(u>) | oo as n —> oo for w S Aj;. If a; £ AQ then for any i € R+ we have i < S„(UJ) A Tn(w) so that (t,u>) £ {(•) < 5„ A Tn} for sufficiently large n £ N. Thus R+ x AJ C UneKi{(0 < 5„ A T n }. For each n £ N, define a real valued function Y (,l) on {(•) < Sn A T n } l~l {R+ x AJ} by setting (1)
YM(t,u)
= {X^.Ms^){t,Lj)
for (t,u/) £ { ( ■ ) < £ > A T n } n { R + x A < } .
For any pair k, n £ N such that k < n, there exists according to Theorem 12.22 and Theorem 12.24 a null set Ak
foru, £ AcKn.
Let us show that (3)
YM(t,Lv) = Yik\t,uj)
for«,o;)6{(-)<5fcATfc}n{R+x(AoUAM)c}.
Now for (t,u>) £ {(■) < Sk A Tt} n {R+ x (Ag U AfcT„)c}, we have Y{k\t,Lo)
= (X [ T t l .M s * A )(t,o;)
by(l)
= ((X[T"1)[Tfc] • (M s " A ) SkA )(t, LJ) by Observation 3.36 = (I|r"'iMs'A)(SiHAWAf,u) by (2) = (X [Tnl • Ms"A)(i, w) since i < S^w) A Tfc(u;) = y ( n ) (t,w) by(l).
§ 14. EXTENSIONS OF THE STOCHASTIC
INTEGRAL
311
This proves (3). Now let A = A0 U (Uk,n6rj,fc<„AA,n). Define a real valued function Y on M+ x Q. by setting (4)
Y(t
W
)=(r(n)(<'a') 10
f
or(i^)e{(-)<S„ATn}n{R+xAc}andneN f o r ( i , u ; ) G R + x A.
Note that Y is well-defined on R+ x Ac by (3) and hence it is well-defined on R+ x Q. To show that Y is in M^oc{Q., 5, {& }, P), let us show first that Y is adapted, that is, for every 5 G R+, Y(s, ■) is & -measurable. Now since 5„ A Tn is a stopping time on a rightcontinuous filtered space, we have {s < Sn/\Tn} = {Sn ATn < s}c G fo by Theorem 3.4. Also since the filtered space is augmented, the null set A is in 5 S . Thus {s < S„ A T„} fl Ac is in &,. Since on this 5S-measurable set Y(s, ■) = Y{n\s, •) = (XlT»] • M S " A )(5, ■) which is 5s-measurable, Y(s, •) is 5S-measurable on this set. Since this holds for every n G N and since U„ £ N{S < SnATn}r\Ac = Ac, Y(s, •) is fo-measurable on Ac. Then since Y(s, ■) = 0 on A, F ( s , ■) is 5s-measurable on Q.. This shows that Y is adapted. Next we show that with our increasing sequence of stopping times {Sn A Tn : n G N} such that 5„ A T„ | oo on A£ we have F S " A T " A G M£(ft, 5, {&}, P) by showing (5)
yS„AT„/x _ Z [ T „ ] ,
MS„A
Now an arbitrary (t, u>) G R+ x Ac is in {(•) < Sm A T m } n {R+ x Ac} for some m G N and we can assume that m > n without loss of generality. Then by (4) and (1) we have Y(t,u) = (X [Tml • MSmA)(.t,v) and therefore yS„AT„A(i; w )
=
(X[Tm]
.
M
SmA)(5B(W)
A TB(W) A tj
w )
5
= ( ( i W ) W , ( M - r t ( f , u ) by (2) = (X [Tnl • M s " A )(t,u;) since m > n. This shows that Fs"AT"A(-,u>) = (X [T "'.M s " A )(-,w)foro; G Ac. Thus (5) holds. Therefore Y G M2''°c(£2,5, {&}, P). For w G Ac we have 5„(w) A T n (u) j o o a s n - t o o s o that for any < G R+ we have Sn{u>) A T„(u>) A t = t for sufficiently large n G N and thus F(t,w) = lim F 5 " AT " A (i,u>). n—»-oo
To show that Y is independent of the choices of the sequences {an : n e N} and {5„ : n G N}, let {a'n : n G N} be another strictly increasing sequence in R + such that a'n t oo and let T„ be defined by 2^(w) = inf{< G R+ : /
X2(s,u;)WM](
for w G fl.
CHAPTER 3. STOCHASTIC
312
INTEGRALS
Let {S'n : n g N} be the increasing sequence of stopping times such that S'n f °° a.e. on ( Q , 5 , P ) a s n -> ooandM s « A g M£(Q,3, {&},P) for every n £ N. Then by what we showed above there exits Z G M ^ ' C n , 5, {& }, P ) such that £ S ; , A 7 > _ jjrp;]
# M
S;A
fof e v e r y n
€
N
Let us show that there exists a null set A in (Q, 5, P) such that K(-, u) = Z(-,bj)forui £ Ac. Let A0 be a null set in (Q, 5, P) such that Sn A Tn A S'n A T^ t °° as n -> oo on A|j. Now yS„AT„AS>TiA _ (Jf[Tn] . [T 1 ra
= (X " )
J^S„f\^.S'„AT^
S A S
. (M " ) > = X 1T " Ar "] • M S " A S - A ,
where the second equality is by Theorem 12.22 and Theorem 12.24. Similarly 7-S„AT„AS;AT^A _
-v-[T„AT^]
JW-S„AS;A
Thus r s " AT " AS " AT " A (-,w) = Z S " A T " A 5 " A T " A (-,U>) for (j G A^ where A„ is a null set in (Q, 5, P). Let A = U neZ+ A n . Then YS»AT"AS"Ar>(., w) = Z s " A T ^ A ^ A ^ W ) f o r e v e r y n G N when w e Ac. Since S„ A T„ A S'n A T^ | oo as n —> oo on Ac, by letting n —* oo in the last equality we have Y(-,u>) = Z(-,u>) for u g Ac. This proves the independence of Y from the sequences {an : n g N} and {S n : n g N}. Finally when M G M|(Q, 5, {&}, P) and X G I^,(R+ x Q, W M ] , P), we let S„(u) = oo for all w G Q and n G N. Then M S " A = M and V S " AT " A = YT"A for every n G N. Similarly when M G M ^ Q , 5, {&},P) and X g L2:00(R+xQ,fj,[MhP),we\etTn(Lj) = oo for all u G Q and n G N so that X[T"] = T and YS"AT»* = y s " A for every n g N. I Definition 14.17. Let M £ M%'°c(Cl,$, {5<},P) and to X = {X t : t g R + } fee a predictable process on the filtered space such that X g lS£l0(R+ x Q, /U[M], P)- i ^ ' {5 n : n g N} fee £«e increasing sequence of stopping times such that Sn | oo a.e. on (Q, 5, P ) as 7i -» oo and M S " A g M.%Q,$, {$t},P)for every n £ N. Let {an : n £ N} fee a strictly increasing sequence in E+ such that an f oo as n —» oo. For every n g N, /ef T„feedefined by Tn(u>) = inf{t G R+ : /
X 2 (s,w)^ M (ds,u;) > a„} A a n
/ o r u g O.
We de/me rfee stochastic integral X • M of X with respect to M to be the unique Y in MC/°C(Q,$, {&}, P ) sucfe fnaf F S " A T " A = X ™ . Ms^ for every n £ N and thus Y(t, w) = lim (X [T " 1 • Ms"A)(t, Lj)for (i, w) g R+ x Ac wfeere A iy a nu« sef in (Q. 5, P).
114. EXTENSIONS OF THE STOCHASTIC INTEGRAL
313
As an alternate notation, we write J[01] X(s) dM{s)for the random variable (X ; t € R+.
M)tfor
For truncation by stopping times for the extended stochastic integral we have the fol lowing. Theorem 14.18. Let M G M£'0C(Q, 5, {&}, P) and fer X fee a predictable process on the filtered space such that X G L ^ ( R + x £2, /i [ M ] , P). Then for any stopping time T on the filtered space there exists a null set A in (£2, j , P ) swcn that for every w G Ac we nave (X • M TA )(-,u;) = (X . M ) T A ( - , W ) .
Proof. Let {S„ : n € N} and {Tn : n G N} be as in Theorem 14.16 so that there exists a null set Ai in (£2,5, P ) such that (1)
X • M = lim X [ r " ] • M S " A
onR+xA^.
n—*oo
Now since M 5 " " G M|(£2,5, {&},P) we have M T A S " A G M|(£2,£, {&},P) for every n G N and this shows that M T A G M j ' ' 0 ^ ^ 5, {5«},P). Since ^ [ M A] < // [M] accord ing to Theorem 14.11, the fact that X is in L^CR* x £2,^[Af],P) implies that X is in L4°|0(R+ x Q,/i [M TA 3 ,P). Thus according to theorem 14.16, X • M T A is defined and is in M2'' oc (£2,3,{fo},P). By Lemma 14.15, X17""1 G L2,oo(R+ x Q , ^ [ M S „ A ] ; P ) . Since /X[MTAS„A] < ^[M^A] by Lemma 12.23, we have X [Tnl G L2,oo(R+ x £2, ^ [ M TAS„A],P). Thus X [T " ] • M T A S " A G M|(£2,5, {&}, P) and according to Theorem 14.16 there exists a null set A2 in (£2,5, P ) such that (2)
X • M T A = lim X l T n l • M T A 5 " A
on R+ x AS.
n—►oo
By Theorem 12.22 there exists a null set A3 in (£2,5, P) such that for every n G N (3)
X [T " ] . M T A 5 " A = (X ( r " ] . M S " A ) A T
onRtx4
Let A = UL Ai. Then X . M T A = lim (X [Tnl . M S " A ) T A = (X . M) T A on E + x Ac by (2), (3)and(l). ■ Theorem 14.19. Let M G Mf00 (£2,5, {&}, P) and Zef X fee a predictable process on the filtered space such that X G 14%(K+ x £2, /i [ M ], P). Let T be a stopping time defined by T(w) = inf{< G R+ : / •/[0,t]
I 2 (s,u)ftM]((is,w)>tt}Aa
/cruG£2
CHAPTER 3. STOCHASTIC
314
INTEGRALS
where a > 0 and let S be an arbitrary stopping time. Then there exists a null set A in (Q, 5, P ) such that for every o> g Ac we have ( X [ T A S 1 • M)(-, u) = (X • M ) T A S A ( - , W) a«d in particular (X [ T ] • M)(-,w) = (X • M) TA (-,u;). Proof. Since M g M2'/oc(£2, $, {&}, P ) there exists an increasing sequence of stopping times {Sn : n g N} such that S n f oo on Af where A t is a null set in (£l,$,P) and M S " A g M^(Q,5, {&},P) for n g N. Let {Tn : n g N} be as defined in Theorem 14.16. By Lemma 14.15, X [ T ] , X [ r " ] g Lil00(R+ x £1,H{Usk«},P) for n,k g N. Note that for sufficiently large n g N we have T < Tn so that T A 5 < Tn and thus X [ T A 5 ] = ( X [TAS])[T„I _ (jsf [T„])[rASj B y t h i s f a c t a n d b y Theorem 12.22 there exists a null set A2 in ( 0 , 5 , P ) such that for sufficiently large n g N we have (1)
X [ T A S I . M 5 " A = (X [ T " ] ) [ T A 5 1 . M 5 " A = (X [T " ] . Ms^fASA
o n R + x AJ.
Since X m g L2l0o(K+ x n,^ [ M s„A],P), we have X [ T ] g L2<^(R+ x £ 2 , ^ [ M S „ A J , P ) and this implies that X [ T A 5 ) g L2%(R+ x Q, ^ M ^ . * ] , P). Then by Theorem 14.18, there exists a null set A3 in (£2, #, P ) such that for all n g N we have (2)
X [ T A S 1 • M S " A = (X [TAS1 • M) S " A
on R+ x A|.
On the other hand by Theorem 14.16 there exists a null set A3 in (Q, g, P ) such that for all n g N we have (3)
lim (X [T " ] • M S " A ) T A S A = (X s M) T A S A
on R+ x A3.
Let A = u t , A - Then letting n -* 00 in (1) and using (2) and (3) we have X [ r A S 1 • M = (X • M) T A S A on E + x Ac. In particular with S = 00, we have X m • M = (X ; M ) T A . ■ For the quadratic variation process of the extended stochastic integral we have the fol lowing. Theorem 14.20. LetM,N g MJloc(S2, 5, {&}, P ) and fef X and F foe predictable pro cesses on the filtered space such that X g L2°^)(M+ x Q, /i [ M ] , P ) and Y g 14°^(R+ x Q, p[tr\, P). Then there exists a null set A in (£2,5, P) .wen f t a on Ac we have for every t g K+ (1)
[X.M,Y.iV]t= /
X(s)Y(s)d[M,JV](s),
§ 14. EXTENSIONS
OF THE STOCHASTIC
INTEGRAL
315
and in particular (2)
[X*M\t=f
X\s)d[M](s). J[0,t)
Proof. Since M,N £ MC2M(Q.,$, {&},P), there exists by Lemma 14.3 an increasing sequence of stopping times {Sn : n £ N} such that Sn t oo on A£ where A0 is a null set in ( Q , g , P ) and MS»'\NS»A G M§(n,g, {&},P) for n € N. Let {an : n G N} be a strictly increasing sequence in R+ such that an t oo as n —> oo. Let {Tn : n G N} and {Tn:n e N} be two increasing sequences of stopping times defined by Tn(u) = inf{t G R+ : /
X2(s,L})/j.[M](ds,w)
> an} A an
foru;G£2,
Y2(s,uj)^m(ds,OJ)
> an] A an
forwGQ.
-'[0,1]
and T^w) = inf{t G R + : /
J[0,t]
By Lemma 14.15, Tn t oo and T^ | oo on AJ where Ai is a null set in ( Q , 5 , P ) and furthermore XiTn] G L2,oo(R+ x S 2 , ^ [ M ^ « ] , P ) and Y[T«] G L2,oo(K+ x £2,^[Ars«A],-p) for n G N. If we let Rn = Tn A Tn then {P„ : n G N} is an increasing sequence of stopping times, Rn t oo on AJ, X[flnl G L i ^ C ^ x £2, fi(Ms„»], P), and F[flT>1 G L2,oo(K+ x a, /i[Ns„A], P ) . Thus X[R"] • M S " A , r [ R » ] ' • iVs"A G M ^ Q , ^ , {5*}, P ) and by Theorem 12.17 there exists a null set A2 in (Q, 5, P ) such that on Aj we have for every t G K+ and every n G N (3)
[X [flnl • Ms"A, y [ f l " ] • iV s " A ], = /
X [R " 1 ( 5 )y [fi " 1 (5) d[M s " A , ATS" A ](s).
Now (4)
LY1*"1 . M S " A , y [ f l " ] . JVS"A]( = [(XlT"])[T"] . M S " A , ( y ™ ) ™ . ATS»A]( = [(XlT"] • M S " A ) T > , (y'7-"1 • Ar 5 » A f" A ] t = [(X.M)T»ASnA:rnA,(yra»iV)7'"AS^:r"A]i
by Theorem 12.22 by Theorems 14.18 and 14.19
= [(X • M) f i " A S " A , ( y - A0*" AS " A ] ( on R+ x A3 for all n G N where A3 is a null set in (Q, 5, P). Now since X • M, Y • TV is in M£''0C(Q, 5, {5,}, P), according to Theorem 14.11 there exists a null set A4 in (Q, 5, P ) such that on ACA and for every i G K+ we have (5)
lira [(X . M)*" A 5 " A , ( y • AT)*"**1*], = [X»M,Y*
N]t
316
CHAPTER 3. STOCHASTIC
INTEGRALS
and (6)
lim HrMS-A N^^{E,LJ)
= (i,M,N](E,u)
for E £ 93[o,t]-
Let A = UtoAj. For w G Ac we have (iJ„ A Sn)(oj) > t for sufficiently large n since (P n A Sn)(w) t oo for any t £ R+. Thus such n we have X [fl " 1 ( 5 ,o;)F [fl " 1 ( 5 ,u.)d[M 5 " A ,Ar S " A ]( 5 ,u;)= /
/ J[0,t]
X{s,u)Y(s,uj)d[M,
N](s,u).
J[0,t]
Letting n -► oo in (3) and applying (5) and the last equality we have (1). We have (2) as a particularcaseof(l). ■ Theorem 14.21. Let M £ M^'0C(Q, 5, {&}, P ) and let X and Y be two predictable pro cesses on the filtered space such that X and YX are in L^CK-f x Q., p.[M],P)- Then X • M £ Mc',oc(a,5, {&}, P), Y £ L^%(R+ x Q, Mrx.OT, P ) «> * « * F • (X • M ) exirte in M c ' /oc (f2,5, {5t}, P ) and furthermore there exists a null set A in (£1, #, P) such that for every u G Ac we have (Y • (X • M))(;u>) = (YX • M)(-,w). Proof. Let us show that y G L ^ ( R + x Q, /irx.Mli p )- B v Theorem 14.20, there exists a null set A, in (Q,5, P ) such that [X ; Af](t, w) = J[0)t]X2(s,u>)d[M](s, w) for i G R+ when u e AJ so that ^[jjf.M](-,^) is absolutely continuous with respect to ^(M](-,^) on (R+, 58M,.) with Radon-Nikodym derivative given by (
/ J[0,t]
= [
Y2(s,Lj)X2(s,Lj)d[M]( S,U))
< OO,
J[0,t]
the finiteness of the last integral being from the fact that YX £ L ^ C R * x Q, fj.[M], P). This shows that Y £ L ^ ( R + x £2, (J.iX»Mh P)Let {an : n £ N} be a strictly increasing sequence in R+ such that an | oo as n —» oo. Let {Tn : n G N} and {T^ : n G N} be two increasing sequences of stopping times defined by Tn{u)
= inf{t G R+: /
X2{s,uj)fj,[M](ds,ui)
> an} A an
foru) G Q,
J[0,t]
T^OJ)
= inf{i G R+ : /
Y 2 (s,uO W x.M)(ek,w) > o , } A a ,
^ [0,i]
= inf{t G R+ : /
Y 2 ^ , ^ ) ^ 2 ^ , ^ ) ^ ) ^ , ^ ) > a„} A an
forw G a .
§ 14. EXTENSIONS OF THE STOCHASTIC
INTEGRAL
317
Then Tn f oo and T^ | oo almost surely. Since both M and X • M are in M j oc (Q, 5, {3(}, P ) there exists by Lemma 14.3 an increasing sequence of stopping times {S„ : n e N} such that Sn f oo almost surely and both M S " A and (X • M) 5 " A are in M | ( Q , 5 , {&},P) for every n € N. Thus (X • M ) S " A T " A T > e M|(Q,5, {&},P) for every n G N. Now (1)
y • (X • M) = lim F17-"1 • (X • M) 5 " AT " A =
[T ]
[T ]
S A
lim F " • (X " • M " )
by Theorem 14.16 by Theorem 14.19 and Theorem 14.18
o n R t X A j where A2 is a null set in (Q, 5, P). Since X is in L ^ ( R + x Q, /i[M], P ) and |i[MS.A] < yU[M], we have X G L ^ C I ^ x Q, ^ [ M S„A], P). This implies that X[Tn] is in L ^ ( E + x il, nlMsnA}, P). Similarly since YX is in V2%(R+ x Q, ^ [ M ] , P), we have YX in Li,%(R+ x fi, /i[Afs„A], P ) and thus y[T»lX[T»1 is in 14% (Rt x Q, /iIWA,Aj, P). Therefore by Theorem 12.19 there exists a null set A3 in (Q, g, P) such that on R+ x A3 we have for alln 6 N y [ r " ] • (XlT"] • M S " A ) = YlT"]X[T"] • M 5 " A = (FX) [T " AT " 1 • M 5 " A .
(2)
Therefore if we let A = U?=1 A, then on R+ x Ac we have by (1) and (2) Y • (X • M) = lim (yx) [ T - A T " ] • M 5 " A = lim ((YX) [T " ] • M s " A ) r " A =
lim (YX) [T » ] • M S " AT " A = (YX) • M, n—*ao
where the second equality is by Theorem 12.19, the third equality is by Theorem 12.24, and the last equality is by Theorem 14.16. ■ Corollary 14.22. Let M e M^ o c (fi,g, {&}, P) a/uf Zef X " \ . . . , X<"» fee predictable processes such that X' 1 ', X ( , ) X ( 2 ) , . . . , X<" • • • X<"> € L!%(R+ x Q, R M ] ) P). T/ien (X ( n ) • • • • • (X (2) • (X ( , ) . M)) ■ • ■) = (X (1) • • • X ( n ) ) • M.
Proof. By iterated application of Theorem 14.21, we have the Corollary. ■
CHAPTER 3. STOCHASTIC
318
§15
INTEGRALS
Ito's Formula
[I] Continuous Local Semimartingales and Ito's Formula Definition 15.1. By a continuous local semimartingale we mean a stochastic process X = {Xt : t G R+} on a standard filtered space (Q., 3 , {3,}, P) such that X = X0 + M + V where M G M ^ C Q , 3 , {&},P), V G V c ' , o c (n,3, {&},P) and X0 is a real valued domeasurable random variable on (£2,3, P), The processes M and V are called the martin gale part and the bounded variation part of the continuous local semimartingale X. In particular, when X0 is integrable, M is in Mc2(£l, 3 , {&}, P) and V is in Vc(£i, 3 , {3<}, P) then X is called a continuous semimartingale. A continuous local semimartingale is also called a quasimartingale. Proposition 15.2. The decomposition of a quasimartingale X in Definition 15.1 is unique, that is, if X = X0 + M + V and X = Y0 + N + W where X0 and Y0 are real valued d0-measurable random variables on (Q, 3 , P), M and N are in Mj °c(£l, 5, {3t}, P), and V and W are in VC,'°C(Q, 3 , {3t}, P), then there exists a null set A in (Q, 3 , P) such that XQ(UJ) = YQ(u), M ( - , W ) = 2V(-,w) andV(;w) = W{-,u>)forw G Ac. Proof. Since M, N, V, and W are almost surely continuous processes, we may assume that they are continuous without loss of generality in the proof. Now since M is in MC2hc(Cl,3, {3t},P) there exists an increasing sequence of stopping times {Tn : n G N} such that T„ f almost surely and AfT"A is a null at 0 continuous L2-martingale for every n G N. By the continuity of M, the function Sn on Q defined by Sn(ui) = mf{t G R+ : \M(t,u)\
> n) = M{t G R+ : \M(t,u>)\ > n } ,
for ui G £1 and n G N is the first passage time of the open set (n, oo) in R and is therefore a stopping time by Proposition 3.5. Clearly {5„ : n G N} is an increasing sequence. Since M(-, a>) is a real valued continuous function on R+, it is bounded on [0, t\ for every t G R+. From this it follows immediately that Sn{u>) | oo as n —* oo for every w G il. Since V is a continuous process, so is its total variation process | V | . Thus if we define .R„(uO = i n f { t G R + : | V\(.t,u)>
n},
forw G Q. and n G N, then {R„ : n G N} is an increasing sequence of stopping times such that Rn(u>) T oo as n -+ oo for every u> G £2 for the same reason as for {Sn : n G N}. Let {T^-.ne N}, {S'n : n G N},and {R'n : n G N} be defined in the same way for N and
§75. ITO'S FORMULA
319
W in the place of M and V. Let Q„ = T„ A S„ A P_„ A Tn A S'n A < for n 6 N. Then {Q n : n 6 N} is an increasing sequence of stopping times such that Qn(u) T oo as n -> oo for u e A\ where Ai is a null set in (Q, g, P). Since M T " A ,AT T > e M ^ ( n , £ , { g ( } , P ) , we have MO-",JV«-« G M ^ U , { & } , P ) by Lemma 14.3. By Definition 11.10, it is clear that | V G » A | = | V | « " A . Since | V | « " A is bounded by n, so is | VQnA \. Thus | VQnA | is an i,-process and therefore V g " A is in Vc(fi, 3 , {& }, P). Similarly W«- A is in VC(Q, 5, {5,}, P). Now since X 0 + M + V = y „ + N + W and M, N, V, and W are null at 0, X0 = Y0, that is, there exists a null set A2 such that X0(u) = Y0(.u) for u e A j . Next, from M + V = N + W we have M — N = WV and then M 5 " " - Ar°"A = WQnA - F 0 " "
(1)
Since V Q " A and WQ"A are in VC(Q, g, {&}, P), so is W"3'"'1 - VQ"A Then there exist A and P. in A c (I2,g, {&}, P ) such that W Q " A - VQ"A = B - A. Then M Q " A + . 4 = AfQ"A + B. Now MclnA and iVt3"A are continuous martingales bounded by n so that they are uniformly integrable and thus are in the class (D) by Theorem 8.22. On the other hand A and B are continuous nonnegative submartingales so that they are in the class {DL). Thus MQ"A + A and JVQ"A + B are continuous submartingales of class {DL). Note also that since A and B are continuous increasing processes, they are natural by Theorem 10.18. Therefore by the uniqueness of Doob-Meyer Decomposition in Theorem 10.23, we have MQnA = NQnA. From this and (1), we have VQ"A = WQ"A also. This implies that there exists a null set A3 such that f MQ"A{-,u) = NQ"A{-,u) \VQ"A{-,ui) = WQ"A{-,u)
( )
f o r n e Nando; G A^ f o r n e N a n d w G A^.
Let A = U?=1 A,. Since Qn{u) t 00 as n -> 00 for u> G Ac, we have lim MQnA{t,u) n—HX>
= lim M{Qn{oj) A t,u) = M(t,w)
for (i,w) G R+ x Ac.
n—t-oo
Similarly for NT"A, VT"A, and WT"A Therefore letting n -► 00 in (2), we have M(-,u;) = N{-,LJ) and V{-,u) = W{-,u})foTu> G Ac. We have also Xa{w) = Y0{u) for u> G Ac. ■ Let C 2 (R) be the collection of all real valued continuous functions F with continuous derivatives F' and F" on R. Let X = X0 + M + V be a quasimartingale on a standard
CHAPTER 3. STOCHASTIC
320
INTEGRALS
filtered space (Q,5, {5 ( },P). Consider the real valued function F o I o n M + x f i . We have ( F o X) ( = (F o X)(t, •) = F o X ( and therefore F o X = {(F o X)t : t € R + } = { F o I , : f e R+}. Similarly for F' o X and F " o X. Ito's Formula shows that F o X is again a quasimartingale and gives its martingale part and bounded variation part in terms ofM, [Af]andV. Theorem 15.3. (Ito's Formula) Let X = X 0 + M + V be a quasimartingale on a standard filtered space (Q.,$,{$t},P) and let F e C2(R). Then for the real valued function F o X onR+xCl there exists a null set A in (£2,5, P) such that on Ac we have for every t £ R+
(FoX)4-(FoX)0 =
/
J[o,t]
(F'oX)(s)dM(s)+
Recall that fm](F'
f
J[o,t]
(F'oX)(s)dV(s)
+- f
2 Jio,t)
(F"oX)(s)d[M](s).
o X)(s) dM(s) is an alternate notation for ( F ' o X • M)t for t G E+.
Since M € Mf'^fi,ff, {5,}, P) and V G V c ' /D<: (n,5, { & } . - ? ) . t h e r e e x i s t s a n u l 1 s e t A i n ( Q , 5 , P ) such that M(-,w) and y(-,u;) are continuous on R+for u> G Ac. If we redefine M and V by setting M(-,u>) = 0 and V(-, w) = 0 for w G A, then since the filtered space is augmented M and V are still in M£I<,0(Q, 5, {&}, P ) and V c ' /oc (Q, #, {&}, P ) respectively but now every sample function of M and of V is continuous. Thus we assume that our M and V are not only almost surely continuous but are in fact continuous processes. For [M] 6 Ac'!°c(£l,5, {5t}, P), we select a representative which is a continuous process. Before we prove Ito's Formula, we show in Proposition 15.5 below that the stochastic integral {/[0(t](F' oX){s) dM(s) : t G R + } is defined and is in M£ foo (Q, 5 , {&}, P ) and the twoprocesses{/ [ 0 ( ] (F'oX)(s)dV(s): t G R + } and{/ [ 0 ( ] (F"oX)(s)d[M](.s): t G R + } are defined and are in Vc,'°c(£2,5, {& }, P). Then since ( F o X)o is a real valued 50-measurable random variable, F o X , as given by Ito's Formula, is indeed a quasimartingale. Observation 15.4. 1) If M e M2' / o c (Q,5,{&},P) and V G V C '' 0C
§75.
TTO'S FORMULA
321
processes and in particular measurable processes. 3) Since X(-,OJ) is a real valued continuous function on R+ for every w G ft, and since F, F', and F" are real valued continuous functions on R, {F o X)(-,u), {F' o X)(-,u), and (F" o X)(-,u) are real valued continuous functions on R+. This implies that every sample function of ( F o X), ( F ' o X), and (F" o X) is bounded on every finite interval in R+. 4) Since (F' o X ) and (F" o X) are measurable processes and since every sample func tion of these processes is bounded on every finite interval in R+, / [0 t](F' o X)(s) dV(s) and / [0 (] (i ? "oX)(s) d[M](s) are real valued fo-measurable random variables by Theorem 10.11. Thus {/[0 t]{F'oX)(s) dV(s): t € R + } and {/[(M](F"oX)0s) d[M](s) : t G R+} are adapted processes. (Note that our V G V c ' ,oc (ft,5, {5,},P)and[M] e Ac''°c(£l,$, {&},-?) are not Li-processes. Nevertheless Theorem 10.11 is still applicable since the fact that the process A in Theorem 10.11 is an L\-process was not needed in the proof of Theorem 10.11.) Proposition 15.5. 1) The two processes {f[ot]{F' o X)(s)dV(s) : t G R+} and UM(F" o X)(s) d[M](s): t G R+} are in Vc''°c(ft, 3 , {&}, P). 2) The process F' o X is in L^tR-f x ft, fJ,[M), P) so tnat tne stochastic integral Ulo,t](F' ° X ) ( 5 ) dM^ ^ e * * } ' 1 defined and is in M2''°c(ft, 3 , {&}, P). Proof. 1) Let us consider the process <1> = {< : t € R+} where
(F'oI)(i,u)^(s,w)
for(i,a;)GR + x f i .
J[0,t]
As we saw in Observation 15.4, <J> is an adapted process. Also since every sample function of F' o X and V is continuous, <&(•, u>) as given above is a continuous function on R+ for every u € ft. It remains to show that (•, w) is of bounded variation on [0,i] for every t G R+. Let w G ft and t G R+ be fixed. Since ( F ' o X)(-, w) is continuous on R+) we have C = sup s6[0 „ \(F' o X)(5, w)| < oo. Let T be the collection of all finite strictly increasing sequences r = {i 0 , • -• ,U} with t0 = 0 and t„ = t. Then
Ars]£|*(fe,«)-«(*fc-i,«)|
£
/
(F'oX)(s,L>)dV(s,u>)-
J[0,tk]
< y f
I JlO,tk-t)
|(JF'ojr)(3,w)|
< C | V | ( i , w ) < oo.
(F'oX){s,uj)dV(s,uj)
322
CHAPTER 3. STOCHASTIC
INTEGRALS
Thus sup Tg7; AT < oo. This shows that <£>(-,o») is of bounded variation on every finite interval in R+ and completes the proof that 4> is in V c ' /oc (Q, 5, {& }, P). We show similarly that ¥ is in V c ' ,oc (Q, #, {&}, P) for the process T = { 4 ' i : t e R + } defined by 4/(<,u))= /
(f"oI)(s,u)(i[M)(s,u)
./[0,i)
for(i,w)£K+xfi.
2) For every u> 6 Q, (P"oX)(-, u>) continuous and hence bounded on every finite interval in R+ so that for every t 6 R+wehave/ [0<] (F'oX) 2 (5,a')/i[ A f](d5,w) < oo. T h u s P ' o X i s in Li^CRt x Q, ft[M], P ) and consequently the stochastic integral {/ [0j] (P'oX)(.s) dM{s) : t € R + } is defined and is in M ^ Q , 5, {&}, P) by Definition 14.17. ■ Lemma 15.6. Let M = {Mt : t £ R + } be a null at 0 martingale on a filtered space (Q, 5, {& }, P ) 5«c^ r/ia/1M | < K on [0, t] x Q/or wwe t e R+ a«d if > 0. For 0 = t0< ■■■
E( 7 j ) < E(a,) = E\E[{M(t3) - M ( V , ) } 2 | V . " = E[E[M(i,) 2 - M(tj^f\^J]
= E[M(t3)2 - M(i,_,) 2 ]-
Thus (2)
E(5) < £ EtMCt,) 2 - M ^ . , ) 2 ] = E[M(<)2] < tf2. 3=1
To estimate E(5 2 ), let us write
(3)
S2 = {±lj}2 = ±1) J=l
j=l
2±li{±1]].
+ i=l
j=i+l
By the computations in (1) and (2), we have
(4)
EE(7 2 ) = EE(a J )
3=1
§/5.
LTO'S FORMULA
323
By the Conditional Holder Inequality and by (3) in the proof of Lemma 12.26 we have n.
n
E f y l & J < £ E[a;|ff ( J < (2K)2
£ j=i+l
a.e. on (Q,%n,P)
j=i+l
and thus (5)
E[£
7i{
i=l
= £
E
£
7 J }]
i=«+i
= £ E[E[ 7 ,{ £
Ti} j=i+i
i=i
|&J]
t7iE[ £ 7,15tj] < (2K)2 £ E( 7 .) = (2K)2E(S) < 4K4
i=l
j=i+l
(=1 2
2
By (3),(4), and (5), we have E(5 ) < (1 + AK )K2
■
Proof of Theorem 15.3. Step 1. Since the process F o I - ( F o A")0 is continuous and since the three processes Ul0yt](F' o X)(s) dV(s): t e R+}> {{I[0,t](F" o X)(s) d[M](s): t G R+} and {/[o,t](-Fv ° X)(s) dM(s) : i G R+} are almost surely continuous according to Proposition 15.5, to show that F o X — (F o X)0 is equivalent to the sum of the three processes it suffice to show according to Theorem 2.3 that for every t G R+ there exists a null set A( in (£2,5, P) such that on A£ we have (6)
(F o X)t - (F o X)0 =
I
(F'oX){s)dM(s)+
J[0,t]
I (F'oX)(s)dV(s) J[0,t]
+ l- f
(F" o
X)(s)d[M](s).
2 J[Q,t]
Let us consider first the case where Xo, M, [M] and | V | are all bounded, that is there exists K > 0 such that (2)
\X0(u)\,\M(t,u>)\,[M)(t,L>),\Vl(t,L>)
fortf.uOeR+xQ.
In this case, Xo is an integrable random variable, M is a martingale by Observation 14.2 so that M e M![(Q, 5, {&}, P), and \V| is an L r process so that V e V C (Q,£, {&}, P). Note also that since |V| < | V | , we have |X| = \X0 + M + V\ <3.ff o n R + x n . SinceF, F ' and F" are continuous on R+ they are bounded on the finite interval [-3K, 3K]. Thus there exists C > 0 such that sup ie[-3K,3tf]
|J?(x)|,
sup \F'{x)\, x£[-3K,3K]
sup *6[-3A:,3/n
|F"(x)|
CHAPTER 3. STOCHASTIC
324
INTEGRALS
Then (3)
\FoX\,\F'oX\,\F"oX\
onR+xQ.
The boundedness of | F ' o X | in particular implies that \F' oX\ isinL2i0O(R+ x ( l , / i M , P ) by Observation 11.19 so that (F' o X) • M e M|(Q, 5, {5,}, P) by Definition 12.9. For n 6 N, let A„ be a partition of R+ into subintervals by a strictly increasing sequence Unk '■ k £ Z+j in R+ such that tn n = 0 and <„ ^ T oo as fc —> co and lim IAJ = 0 where |A„ | = sup^j^i,,^ — <„,/t_i). Let t e R+ be fixed. For each n e N, let in,Prl = t where tn,k < t for A; = 0 , . . . , p„ — 1. Let us write (4)
F(X(t)) - F(X(0)) = £{nX-(<„,*)) -
F{X{tn^x))}.
it=i
Now for a, b e R, a ^ 6, according to Taylor's Theorem we have F(b) - F(a) = F'(a)(b -a) + -F"(c)(b - a) 2
for some c e (a A 6, a V 6).
Thus we have (5) =
F(X(tn,k, w)) - F(X(*„,*-,, w)) F , (Jr(t M _i,w)){A-(* n ,»,u;)-A , (< B> _i J w)}
+ ^F"(^,,(a;)){X(t„,,,cu) - I ( V l , w ) } 2 , where (6)
Z(t»^_i,w) A X(t n , t ,w) < ^ ( w ) < I ( ^ n , w ) V X(tn,A:,aj).
Now £„.*. may not be a random variable, that is, it may not be 5-measurable. However on the set {X(tnik) — X(tn^-\) 7^0} G §tnk we have from (5) the equality F\Uk)
= 2{F(X(tn,k)) - J , (JC(t„,t-i))}{A-(t Bit ) - 2F'(X(t n ,,_,)){X(i n ,,) - *(*„,*_,)}-•,
X(tn,k_,)}-2
which is 5 t nfc-measurable. Let us define a function G„^ on Q by setting Gnk(u)=fF"(Uk("))
10
foruj e {X{tn)k)-X{tn,k_x)iQ} otherwise.
§75.
ITO'S FORMULA
325
Then G„tk is an g (n k -measurable random variable and furthermore (7)
F d W - F d ^ n ) )
= F\X(tn>k^)){X(.tn:k)
-
X(tn
^Gn,k{X(tnik)-X(ta>k^)}2.
+
Note also that by (6), the Intermediate Value Theorem applied to the continuous function X(-,u>), and (3) we have (8)
sup |G„,it(w)| < sup \F"((nJe(u)))\ < C.
We now have (9)
F(X(t))-F(X(.Q))
=
Y,F'(X(tn^)){X(tn,k)-X(tn,k^)}
+ ^£G„,,{X(i„,A)-X(t„,A_,)}2 Let us write the first member on the right side of (9) as (10)
£ F\X{tnik_,)){X{tn,k)
-
k=\ Pn
X{tn,k-X)} Vn
=
Y.F'(X(tn,k-i)){M(tn,k)
=
S^ + S^
- M(i n ,*_,)} +£F'(X(i„,t_ 1 )){y(t„,*) - V(t„, t _,)}
By Proposition 12.16, ljm ||5j n> - ((F' o X) • M)(t)\\2 = 0. This implies that there exist a subsequence {ne} of {n} and a null set A' in (Q, #, P) such that (11)
\imS\n')(u>) = ((F'oX)»M)(t,uj)
f o r c e A'c.
Regarding S j 0 , note that since ( F ' o X)(-,w) is a continuous function and V(-,u>) is a continuous function of bounded variation on every finite interval in R+ for every u € £2, we have (12)
HmSfV)=/ n-.oo
*
J[0,t]
(F , oX)(5,w)rfy(s,w)
forceQ.
326
CHAPTER 3. STOCHASTIC
INTEGRALS
Let us write the second member on the right side of (9) as \Y.Gntk{X(tn,k)-X(tn^)}2
(13) =
£ Gn,k{M(tn>k)
- M(*Blik_,)}{V(*nit) - V(*»,*_i)}
k=\
+
- V(i n , fc _,)} 2 + \ £ Gn,k{M(tn,k)
\ £ Gn,k{V(tnM) 1
z
>t=i
- M(t n , fc _!)} 2
fc=i
= Sf,) + S,f) + 5f). Now by (8) we have |S
|M(tnii)-M(tB,tM)|£|V(t».*)-V(t»,*-i)|
C max
|M(* n , t )-M(t B ,fc_,)||V| t .
Since every sample function of M is continuous and hence uniformly continuous on [0, t], lim max \M(tn k) — M(tn k_i)\ = 0 on fl. Thus we have n-too A=l,...,p„
'
lim Sln\u) = 0 for w £ Q.
(14) Similarly we have
| 5 f | < ^ C max | V ( t M ) - V ( * ^ _ i ) | | 7 | , L
fc=l,...,p„
so that by the uniform continuity of every sample function of V on [0, i] we have (15)
lim S f V ) = 0
forojeQ.
71—HX)
To show Jiirn E(\S^n) - \ fM(F" a X)(s) d[M](s)\) = 0, let 5en) = \ £ ( * " ' ° *)(t»,*-i){M(i„, t ) - M(<„,,_,)} 2 , and Sf
= \ £ ( ^ " o X)(tn,k^){[M](tn,k)
- [M](i B ,*-i)}.
§15.
327
ITd'SFORMULA
For brevity, let us write An,k = {M(tn:k)-M(tnik_i)}2,
an=
max
Bn,k = {[M](tn,k)-[M]{tnik_l)},
pn=
max"B„,fc.
An
Now l^
n )
-5n
\Y,\Gn,k-{F"°X)(tn^)\{M{tn,k)-M{tn^)}2
< L
<
k=i
C max
|M(t„i4)-M(tnit_i)|£|M(t„ii)-M(*„,t_i)|
so that by Holder's Inequality and Lemma 15.6 we have E(|S
|M(t Bii ) - iW(in,fc_,)|}2]^ <
C E ^ ^ T T ^ .
By the uniform continuity of the sample functions of M on [0, t], we have lim an = 0 on n—*oo
D. Since an is bounded by 4if 2 we have lim E[a n ] = 0 by the Bounded Convergence n—*oo
Theorem. Therefore (16)
jirnEClSf'-S^'l^O.
Next we have s£° - Sf> = | £ £ , ( F " ° X X ^ , ^ ) ^ - BnJt} so that E(|Sj n ) - S^n)\2) = i £ E [ ( F " O X)(tn,k-i)2{An,k 4
+ 7 ^
^
- B n ,,} 2 ]
A:=l
E[(F" o I ) ( i n i ) _ , ) ( F " o I K i . w ) ! ^ - ^ J j i ^ - B„ | t }].
j,k=\,...,V„\]ik
For j ^ k, say j < k, we have E[(F" o X)(t„, J _i)(F" o X)(i n , Jt - 1 ){^„, J - B nJ -}{A n , t - -B„,fc}|&nk_,] = ( F " o X){tn,3-X){F"
o X X ^ - i M A ^ - B«j}E[{^» -
BnM)!&„,»_,],
a.e. on (£2, & n t _,, P). By the definition of [M] we have E [ { A n , f c - £ „ , a i& n , k _,] = E[{Jlf (*„,*) - M(i„,fc_,)}2 - {[M](t„,*) - [M](t„, t _,)}!&._,_,] = 0,
328
CHAPTER 3. STOCHASTIC
INTEGRALS
a.e. on (£2,5 (n , P). Thus the summands in the second sum in the expression for E(|S£n) - S\n)\2) above are all equal to 0 and therefore we have E ( | S f - S<">|2) = - £ E [ ( P " o X ) ( i n , , _ , ) 2 { ^ ^ - 5 n , A } 2 ] 4
i
k=\
p«
<
- Y, E[(F" o A-)(*„,*_,)2{ A 2 ,, + P 2 ,,}]
<
\c2Y,K[{M(tn,k)
<
^C 2 E[a„ JT{M(tn,k)
<
2
2 ,/2
^C E[a ] Z
<
- M(i„,*_,)} 4 + {[M](f„.*) - [M](i„,,_,)} 2 ] - M(t n ,*_,)} 2 ] + lc2E[(3n £{[M](* n ,*) - [M](t Bik _,)}]
E [ { £ { M ( i n , t ) - M(i„, fc _,)} 2 } 2 ] 1/2 + ic 2 E[^„[M](i)] Z
4=1
^C 2 E[ a 2 ] 1 / 2 v^2A^ 2 + ic 2 E[/3JM](t)],
by Lemma 12.26. By the uniform continuity of every sample function of M and [M] on [0, t] we have lim a 2 = 0 and lim 5„ = 0 on Q. Since M and [M] are bounded we have n—*oo
n—*oo
lim E[a 2 ] = 0 and lim E[/3n[M](t)] = 0 by the Bounded Convergence Theorem. Theren—►oo
n—>-oo
fore lim E(|S£n) - S f >|2) = 0 and consequently lim E(|Si n) - Si B) |) = 0.
(17)
71—*0O
Since every sample function of F" o X is continuous we have (18)
limS5°(w) = i /
"-*00
(F"oJt)(5,w)d[M](s,w)
foruefl.
2 J[o,t]
Now (16) and (17) imply that there exists a subsequence {n m } of {n^} and a null set A" in (£2,5, P ) such that (19)
lim \Sf\u>) - Sf\w)\ 77i—>-oo
= lim |S£n)(w) - S$n)(u>)| = 0 forw e AT. m—»-oo
*
Let A, = A' U A?. Replace n in (9) by nm. If we let m -> oo in (9), then by (10), (11), (12), (13), (14), (15), (18), and (19) we have (1) holding on Af. This proves (1) for the case where X0, M, [M] and I V | are all bounded.
§ 15.
UO '5 FORMULA
329
Step 2. Let us remove the assumption that XQ is bounded maintaining the assumption that M, [M], and | V | are bounded. Let X
( F o Xin))t
- ( F o X(n,)0 = /
( F ' o X (n) )( 5 ) dM(s)
J[0,t]
/
J[o,t]
(F'oXM)(s)dV(s)
+ ]- I
2 7[o,t]
(F"oXin))(s)d[M)(s).
Since the stochastic integral does not depend on the values of the integrand process at t = 0 as we noted after Definition 12.3, we have (F'oXM)(s)dM{s)
/ J[0,t]
= f
(F'oX)(s)dM(s).
J[0,t]
For a fixed u G £2, let n be so large that |-Xo(o>)| < n s o that ^ " ' ( C J ) = Xa(ui) and hence X
/ J[0,t]
and /
f
(F'oX)(s)dV(s),
J[0,t]
(F"oXM)(s)d[M](s)=
Ho.t]
I
(F"oX)(s)d[M](s
J[o,t]
On the other hand since lim X tn) (w) = X(u>) and since F is continuous, we have lim ( F o XM)(s,
n—*oo
w) = ( F o X)(S,UJ) for every 5 G R+. Let A = U n€N A n . Then letting
n—*oo
n —> oo in (20) we have (1) holding on Ac. Step 3. Let us remove the assumption that X0, M, [M], and \V|
are bounded. Let
inf{tGR+ \M(t)\ > n} An, 5 2 , n = inf{t G R+ [M](<) > n } A n , 53,„ = inf{* 6 R+ | V | ( i ) > n } An, 54,n = i n f i t G l ^ | ( F ' o X ) ( t ) | > n } A n , S,,n
Tn
=
= inf{i G R+ /
( F ' o X)2(s) d[M](s) > n} A n.
V[o,(]
Then {S;,„ : n G N} for i = l , . . - , 4 , and {T„ : n G N} are increasing sequences of stopping times, and since every sample function of M, [M], | V | and F'oX is continuous
330
CHAPTER 3. STOCHASTIC
INTEGRALS
by our assumption, we have S;,„ f oo and T„ | °o on Q. as n —» oo. Let R„ = 5i,„ A • ■ • A S4i„ A T„. Then {#„ : n g N} is an increasing sequence of stopping times such that Rn T oo on i2. Since the processes MRnA, [M] R " A , | F | f l » A and(F'o X)RnA are bounded by n, we have M*" A g Mg(0,ft {&},P), [M]*" A g A c (Q,5, {&},P), | ^ | H " A € V c (Sl,5,{5 t },PX and ( F ' o JO*" A g L2l0O(K+ x fl,ttM^»i,P). Note that [MR-A] = [M]R"A by Proposition 12.25 and | V * " A | = | V| T " A For every n g N, let X f i " A = X 0 +M R n A +V R n A . By Step 2 there exists a null set A in (Q., 5, P ) such that on A c we have for every n g N and every t g E+ ( f o X f i n A ) ( - ( F o Xfi"A)o = /
(21)
( f o XR"A)(s)
dMR"A(s)
J[0,t]
+
f
J[o,t]
( F ' o XRnA)(s)
dVR"A(s) + I- f
2 J[o,t]
(F"oXR"A)(s)d[MR"A](s).
Let us observe that if Y is an adapted process and 5 is a stopping time on a filtered space, and G is a real valued function on K, then for the stopped process YSA and the truncated process YlS] we have G o YSA = (G o Y)SA
(22) and
(F S A ) t 5 ] = F [ 5 ]
(23)
as can be verified easily. For u £ Ac and i g R+, for sufficiently large n g N we have Pn(u>) > t so that P„(w) A s = 5 for all s g [0, i]. For such n we have by (22) (24)
(FoX*" A )(s,w) = (FoX) f i " A (s,u;) = (FoX)(s,u;)
forsg[0,t],
and similarly C251 V '
' (p' o X*" A )(S,UJ) = ( F o X)(s,u>)
for s g [0,t]
{(f"oI«"»)(j,u) = ( f o I ) ( s , u )
forsg[0,t].
Regarding the first term on the right side of (21) we have {F' o XRnA) • M R " A = ( F o X) fl " A • M R » A = ( F ' o X) R " A • M H " Afl " A = ((F' o X) R " A • MRnA)RnA = « F ' o X)*" A ) [finl • M R " A = ( F ' o X)lRn] • M H " A = ( ( F o X) [ R n l • M) fl " A
§75.
ITd'S FORMULA
331
where the first equality is by (22), the third equality is by Theorem 12.24, the fourth equality is by Theorem 12.22, the fifth equality is by (23), and the last equality is by Theorem 14.18. Now F'oX e I4%(K+ x Q A*[M]) by Proposition 15.5. Now by Theorem 14.19, we have (F' o X)lRn]
•M = (F'o J Q T " A S ' . " A " A S ' . « • M = ((F' o X) • M) fl " A
Thus ( F o X H " A ) • M*" A = {{F' o X) • M) R " A . For w G Ac, since R„(UJ) A t = t for sufficiently large n g N, for such n we have ((F' o X*" A ) • JWH"A)(i,u;) = ((F' o X) • M)R"\t, Therefore we have shown that for u £ A (26)
c
LJ) = ((F' o X) •
M){t,u).
for sufficiently large n £ N we have
( 7 {F'oXR"A)(s)dMR"\s))(uJ)=([ \J[0,t] /
\J[0,t]
(F'oX)(S)dM(s))(.oj). I
For 5 £ [0, t] for sufficiently large n G N we have ii„(u>) A s = s so that VR"\S,UJ)
= V(iZ„(u,) A *, w ) = V(s,u>),
and similarly [M fi " A ](s,u,) = [M]R"\S,OJ)
= [ M K i ^ M A s,u) = [M](5)u>).
Thus by these equalities and by (25) we have for sufficiently large n e N (27)
(F'oIft'A)(J,u)^s"A({,u)= /
/ ./[0,<]
(foI)(s,u)jy(s,u),
-^[0,(1
and (28)
/
(F"oX^)(s,u)d[MR^](s,uJ)=
f
(F"oX)(s,u)d[M](s,u).
J
J[0,t]
[0,t]
Using (24), (26), (27) and (28) in (20), we have (1) holding on Ac
■
As an application of Ito's Formula, we have the following. Example 15.7. Let B be an {5(}-adapted null at 0 Brownian motion on a standard filtered space ( Q , 5 , {5<},.P). Then (1)
/ -'[0,4]
B(s)dB(s)
\{B2(t)-t},
= ^
CHAPTER 3. STOCHASTIC
332
INTEGRALS
and (2)
[ i f B(u)dB(u)}dB(s) J[0,t) Uio.s] J
=
±-{B3(t)-3tB(t)}.
3!
Proof. By Proposition 13.31, B G M^(Q,5, {$t},P) with [B]t = t for t G K+. For n G N, let F(x) = a;" for a: G R. Then F G C 2 (R) so that by Theorem 15.3 applied to X = X0 + M + V in which X 0 = 0, M = B, and V = 0 we have (3)
Bn(t) = n [
Bn-l(s)dB(s)+U(-n~
Jio.t]
1}
2
B n - 2 (s)m L (ds).
/ J[o,t]
In particular for n = 2 we have B2(t) = 2[
(4)
B(s)dB{s) + t,
J[0,t]
and therefore (1) holds. For n = 3, we have by (3) and (4) (5)
B3(t)
= 3 /
B2{s) dB(s) + 3 /
J[0,t]
= 3! /
B(s)mL(ds)
J[0,t]
(/
J[0,«] U [ 0 , s ]
S(u)d5(u)}dB(s) J
+ 3< [ sdB(s)+ [ B(s)mL(ds)\ U[o,t] J[o,t] J Since the integrand in / [01] s dB(s) is a continuous function, (/ [01] 5 dB(s))(cu) is in fact equal to the Riemann-Stieltjes integral /0' 5 dB(s,oj) for every u> G Q. Now /o 5 dB(.s, w)+/„* £ ( s , u) ds = [sB(s, w)]*0 = tB{t) by integration by parts for the RiemannStieltjes integral. Using this in (5), we have B\t)
= 3! / ( / B(u)dB{u)\ dB(s) + 3tB(t). J[0,t) U[0,s] J
From this last equality we have (2). I The equalities (1) and (2) in Example 15.7 make interesting contrast to the formulas /o s ds = j<2 and /o{/ 0 5 u du}ds = jjt 3 in ordinary calculus.
§75.
LTO'S FORMULA
333
[II] Stochastic Integrals with Respect to Quasimartigales Definition 15.8. For A G A' oc (£2,5, {&}, P), let LfafBU x f l , ^ , P ) fee fte linear space of equivalence classes of measurable processes $ = { $ , : ( £ E , } on (£2, 5, P ) JMC/Z that I[o,t) \®(.s,u)\l*A(.ds,u) < oo for every t G R+/or almost every u G £2. Definition 15.9. Let C(R+ x £2)feef/jg /wear space of equivalence classes of measurable processes <E> = {O t : < e R+} on a probability space (£2,5, P) s«c/i tfza/ <£(•, w) is continuous on R+for almost every u G £2. Let B(R+ x £2)feef/ie /mear space of equivalence classes of measurable processes
/ ./[CM]
|*(s,w)|rfj4(s,w) < oo.
J[0,t]
Consider a quasimartingale X = X 0 +M+V on a standard filtered space (£2,5, {5<}, P). Let Q> be a predictable process on the filtered space such that O G L ^ ( R + x Q.,pm,P)
n Lfc(R* x £ 2 , ^ , K | , P ) .
S i n c e * G L£»(R+ x £2, W W J ,P),
I
f Aa.t)
\®(s,u)\dlVl(s,u)<00,
so that f[0t] <1>0, w) dV(s, u ) £ t for (i, w) G R+ x £2. The stochastic process If O(s) d l ^ s ) : t G R+} is adapted and its sample functions are of bounded variation on
CHAPTER 3. STOCHASTIC
334
INTEGRALS
every finite interval in R+ by the same argument as in the proof of Proposition 15.5. Thus the process is in V c '' oc (£2,5, {&}, P). Definition 15.11 Let X be a quasimartingale given by X = XQ + Mx + Vx with Mx G Mc2''oc(n,$,{$t},P)andVx G V C ' ,0C (Q,5, {St}, PI Let <J> be a predictable process on the filtered space such that
(i)
o> G i4%(K+ x Q, t e ) ,P)nLi;(i t x ntfilVxlP).
By the stochastic integral of <J> with respect to X we mean the null at 0 quasimartingale {fm]<$>(s)dX(s) : t G R+} defined by (2)
/
<5>{s)dX{s)= I
•no,*]
0(s)dMx(s)+
./[o,<]
f
®(s)dVx(s)
forteR*.
-Ao,!]
We also use the notations O • X, $ • Mx, and O • Vx /or the processes {/[0ij] ®(s) dX(s) : t e R+}, {/ [0!] $(s)(iMjf(s) : t 6 R + }, and {f[ot]&(s)dVx(s) : < 6 R+j respectively. Thus <£> • X = O • Mx +
J[o,t]
(F' o X)(s) dX(s) + ~ f
2 J[o,t]
(F" o X)(s)
d[Mx](s),
or in the alternate notations, (F o X)t - (F o X)0 = (F'oX)»X
+ ]-{F" OX)»
[MX].
Contrast the first of the two expressions above with the formula F(t) — P(0) = Jjj F'(s) da forPGC'(R). Theorem 15.12. Let X and Y be two quasimartingales on a standard filtered space ( A S , {&}, P) given by X = X0 + Mx + Vx and Y = Y0 + MY + VY where MX,MY
G
335
§75. TTd'S FORMULA
M?°c(n,$,{$t},P)andVx,VY 6 \c-loc(Q,$, {&},P). Then the stochastic integral of Y with respect to X,Y • X, a/ways exists as a n«H af 0 quasimartingale with martingale part MY.x =Y • Mx and bounded variation part Vy.x =Y • Vx. Proof. Since Y is a quasimartingale, it is in C(R+ x Q) which is contained in L vio( R + x "> Pv*xl> ■P)nL',°^(R+ x n , / i | V j r | , P ) . Thus by Definition 15.11, F « X exists and My.x = F • M^ and VY.X = Y • V*. Theorem 15.13. Lef X = X 0 + Mx + Vy and Y = Y0 + MY + VY be quasimartingales on a standard filtered space (£2, ^, {iJt}, .P) and let 0 tv/W *F be predictable processes on the filtered space such that (1)
<S,r
6
L^(R+xO,/i[Mx],P)nL'1%(R+xii,/i,^,,P)
n L ^ ( R + x a, K ^ ] , P) n LfeaL. X n, ^ , yy ,, P). Then for a, 6, a, /3 G R we /lave (2)
(aO + &¥) • ( a X + ,SF) = a a $ • X + a/?* • Y + i a ^ • X + fc^T • F
Zrc particular when <&, ¥ e B(R+ x £2), (2) w satisfied. Proof. By the linearity of stochastic integrals with respect to local L2 martingales and Lebesgue-Stieltjes integrals we have the linearity of stochastic integrals with respect to quasimartingales. I
[DI] Exponential Quasimartingales If X is a quasimartingale then since the exponential function has continuous derivatives of all orders Theorem 15.3 implies that ex = {exm : t e K+} is a quasimartingale. Definition 15.14. For a quasimartingale X, ex is called the exponential quasimartingale ofX. Theorem 15.15. Let X be a null at 0 quasimartingale given by X = Mx + Vx where Mx £ Mj' o c (£2,5,{&},P)a/KfFx G V c -' o c (n,5, {&},P). Thenfor the null at 0 quasi martingale defined by Y = X - \[MX], we have (1)
eK«>=l+/ J[0,t]
tY^dX{s)
336
CHAPTER 3. STOCHASTIC
INTEGRALS
on R+ x Ac where A is a null set in (£2,5, P)Proof. Since Y = X - ^[Mx] = Mx + Vx - \[MX], Y is a null at 0 quasimartingale with its martingale part My and bounded variation part Vy given by MY
(2)
=
MX
VY = Vx - tlMxl
By Theorem 15.3, we have (3)
e ™ - ey(0> = /
J[0,t]
e y w dMY{s) + f
J[0,t)
e™ dVY{s) + \ [
2 J[o,t]
ey(s> d[MY](s).
Now en0) = e° = 1. Substituting (2) in (3) we have (1). ■ Corollary 15.16. Let M e M%Ioc(£l, 5, {& }, P ) and let 4> be a predictable process such that <£> £ L^CR-f x £2, /U[M), P). If we define a null at 0 quasimartingale by setting (1)
Y(t)=
f ®(s)dM(s)-J[0,t)
[ ®2(s)d[M](s) 2 J[o,t]
fort£R+,
then its exponential quasimartingale eY satisfies the condition (2)
eYm=\+f
eYU)0(s)dM(s) J[0,t]
on R+ x Ac where A is a null set in (Q, 5, P). In particular the exponential quasimartingale ez of the quasimartingale Z defined by Z = M — \[M~\ satisfies the condition (3)
em
= 1+ /
eZ(s) dM(s)
J[0,t)
on R+ x Ac. Proof. Let X =
337
%15. ITO'S FORMULA
X - \[MX] satisfies the condition eYm - 1 = / [01] e y(s) dX(s). Thus for stochastic integrals with respect to M the quasimartingale e* ~ i [ M x l corresponds to the exponential function in Riemann integrals. Observation 15.18. Let us define the Hermite polynomials Hn(t, x), (t, x) 6 (0, oo) x R for n 6 Z+ as the coefficients in the power series expansion of exp{7x - ^ } in 7 £ R. Thus (1)
n
exp{7x - 2 - U = £
7
-ffn(t,x)
for 7 g R.
Differentiating the power series n times, we have (2)
Hn(t,x)
=
1
9"
drexpVx~ (x2)
1
= ^T
exp
7H
2
7=0
[
[ <9"
i2^)[a^
exp
t (
x\
\-2^-7;
-,=0
where the second equality is by completing square for 7 in yx — &. Now with y = 7 — ~, we have f^ = 1 and thus
5 expj97
4( T - -f)
_ 9 exp|- * 2 <9y -2*
}
By iteration we have gr,
If we set7 = 0, then y = — f and dy = —\dx so that
9"
1
4(- T)"}
» 3n
= M)^exp
1 •7=0
f v
4}-
Using this in (2), we obtain (3)
"■"
!=L
x ex ex i? forn€Z+, n! p{?7}£ [ 2 t j dx P^-^[ It
CHAPTER 3. STOCHASTIC
338
INTEGRALS
which is the customary definition of Hn(t, x). It follows from (3) that Hn(t, x) is a polyno mial in the two variables t and x of mixed degree 4 with leading term (n!)_1x™ for n G Z+. For instance we have H0(t,x)=l, Hl(t,x) = x, H2{t, x) = \x2 - | t ,
(4)
Hi(.t,x)=lzxi-12-xt, Hi{t,x) = ^ - \ x H + \t2, H5(t, x) = ^-0x5 - ±xH + \xt2
For small values of n, Hn(t, x) can be obtained more easily by writing
4 }.■{£*?}{*(=?)"n!1 )J
exp < -yx -
and then by long multiplication of the two power series on the right side and by equating the coefficients of 7 to Hn(t, x) according to (1). Theorem 15.19. Let X = Mx +Vx be a null at 0 quasimartingale on a standard filtered space. Then for n 6 N, we have
I
dX(U) [
J[0,t)
J[0,*i]
dX(t2)■■■ I
1 dX{tn) =
Hn([Mx]tlXt),
J[0,t„_i]
that is, (• ■ • ((1 . X) . X) . ■ ■ ■) . X =
Hn([Mx],X).
Proof. For 7 6 R , let (1)
Y = 7 X - ^Mx]
=7X-
^-[Mxl
With the quasimartingale -yX replacing the quasimartingale X in Theorem 15.15, we have (2)
ey(i)=l+7/
ey^dX(s).
A0,t)
By (1) above and (1) of Observation 15.18, we have
(3)
e y( " = exp f 7Xt - ^-[Mx]t\
= £
jnHn(.[Mx]t,Xt).
§/5.
ITO'S FORMULA
339
Note that since Hn(t, x) is a polynomial in t and x and since [Mx] and X are quasimartingales, Hn([Mx], X) is a quasimartingale by Theorem 15.3. For brevity, let us write Ln for Hn([Mxl X). Then by (2) and (3), we have £
n 7
L„(t) = 1 + 7 /
nSZ+
E 7"-Ms) dJf(a) = 1 + £
•'("■'Inez,
7"
+
nlz.
' /
£„(«) <*X(s).
^.'l
Equating the coefficients of 7" of the two sides of the last equation for n G Z+ we have / £o(t) = 1 \ £»(*) = /[0,t] i n - i W dX(a)
for n £ N
and therefore by iteration L„(i) = =
/
£„_,(*!) dX(t,)
/
{/"
£„_2(t2)dA-(i2)}dX(t,)
./[o,<] U[o,ti]
=
f
If
J
{[
L_3(ij)(Ur(*3)}«W(t2)}dX(ti)
J[o,fl U[0,ti) U[o,t 2 ]
= /•••(/ y[o,d
J
(/
J
lrfX(u}^(t,-i)}---^«,),
|/[o,t„_ 2 ] [^[o,( n _,]
j
j
that is, £„(*)=/
dX{ii) f
J[0,t]
dX(t2)-~
I
J[0,til
\dX(tn).
J[0,t„-,]
Since Ln = if„([Mx), X ] by definition this completes the proof. ■ Example 15.20 To continue with Example 15.7, let B be an {5,}-adapted null at 0 Brownian motion on a standard filtered space (Q,5, {&}, P). By Theorem 15.19 and by (4) of Observation 15.18, we have t
I
J[0,t]
r dB{U) I
./[0,
r dB(h) /
S dB{W /
J[0,t2]
B*(t) tB2(t) t2 = - ~ - ~ 1 + ^,
1 dB(U) = H4([B]t,Bt)
^4
J[0,t3]
and similarly / J[0,t]
=
dB(U) I J[0M]
dB(t2) I
-/[0,*2] 3
dB(t3) f
•/[0,t3]
B\t) tB (t) ?B(t) ffs([5L,.B,)--J2o---J2+ —g—■
dB(U) f JlO.U]
1 dB(ts)
4
8
CHAPTER 3. STOCHASTIC
340
INTEGRALS
[IV] Multidimensional Ito's Formula Definition 15.21. By a d-dimensional quasimartingale we mean a d-dimensional stochastic process X = {Xt : t G R+} on a standard filtered space (Q, 5, {$t}, P) whose components X ( 1 ) , . . . , X w are quasimartingales on the filtered space, that is, X ( , ) = X$ + M w + V® where XQ1' is a real valued ^-measurable random variable, M w £ MJ ° c (£2,5, {5«}, -P). and 7 ® G V C ''-(Q, 5, {&}, P),/or i = 1 , . . . , d. Let C 2 (R d ) be the collection of all real valued functions F on Rd whose first order partial derivatives Ff, i = 1 , . . . , d and second order partial derivatives i^"-,», j = 1,... d, are all continuous on Rd. Theorem 15.22. (Multidimensional Ito's Formula) Let X = {Xt : t G R+} be a d-dimensional quasimartingale on a standard filtered space (Q, 3 , {5t}> P) with compo nents given by X ( i ) = Xf + M (i) + V(i> for i = 1 , . . . , d. Let F G C 2 ^ ) . Twera F o X is a l-dimensional quasimartingale given by (F o X)t - (F o X) 0 = £
;=i
I
(Ft o X)(s)
dM{%)
J
\M
+ 32 [ (F^oX)(3)dV^(s)+1-32 I
(i^oX)( 5 )d[M ( '\M«]( 5 )
o/i R+ x Ac wnere A is a null set in (Q., 5, P). Proof. Theorem 15.22 can be proved in the same way as Theorem 15.3 using Taylor's Theorem for F G C^R*). that is, for any a, b G R d , F(b) - F(a) = 32F!(a)(b, - a,)+ i £ i=i
^ = 1 ( c ) ( 6 ; - a,)(6,- - aj),
i,j=i
where c = (cu ■ • •, cd) G Rd with Ci G (a,- A &,-, a,- V 6,)for i ~ 1 , . . . , d, ■ Let us note that if we write MXc> for the martingale part of the quasimartingale X ( i ) in Theorem 15.22 then in the notations introduced in Definition 15.11 the multidimensional Ito's Formula takes the following form: (F o X)t - (F o X) 0 = 3JF! o X) . X w + \ £ ( F £ o X)(s) . ;=i
* i,j=i
[MxV»tMxw].
%15. LTO'S FORMULA
341
Theorem 15.23. Let B = ( P ( 1 ) , . . . , P
X«\t) = Xf + j^f k=l
■/[0,*]
Y^k\s)dBik\s)+
I
Z«\s)mL{ds).
J[0,t]
Then X is a d-dimensional quasimartingale and for any F € C2(R
(Fo X)t - (Fo X)0 = E E / + W fc? •'[0,0
(l!?°J0«5'ft*)M<*Bwto (^'°^)(s)Z W (5)m L (d 5 )
+ J E E / (fI>i)(S)rW)(S)ytti»(S)mL(
»,i=l t=i •/[°'i]
c
on R+ x A where A is a null set in (Q, 5, P). Proof. Let us show first that XM is a quasimartingale. Now since Y{l'k) 6 L2°L(R+ x Q., mL x P ) , the stochastic integral y(i'*> • B(*> is defined and is in M;;''0C(Q, 5, {&}, P). Then (3)
M w = E^(',A:)»BWeM2'°c(«,5.{Srt},-P)
fori = l , . . . , d .
ifc=l
For i = 1 , . . . ,
Z (,) (s, w)mi(ds)
for (i, w) e R+ x Q.
•/[o,<]
Every sample function of V{*> is absolutely continuous and in particular of bounded varia tion on every finite interval in R+. The continuity of the sample functions also implies that F ( , ) is a measurable process. By Theorem 10.11, V (,) is an adapted process. Thus (4)
7 ( , ) 6 V ^ ( I 2 , 5 , { 5 , } , P ) far » « ! , . . . , *
342
CHAPTER 3. STOCHASTIC
INTEGRALS
By (3) and (4), X ( l ) = Xf + M(i) + V ( 0 is a quasimartingale for i = 1 , . . . , d and therefore X is a d-dimensional quasimartingale. By Theorem 15.22, we have (5)
(F!oX)(s)dM(i)(s)
(FoI),-(FoI), =W {F'ioX){s)dV&{s)+l-Jz
+ y-j
[
(P/>X)(s)d[M('\Mw](s),
on R+ x Ac where A is a null set in ( 0 , 5 , P). Now by Theorem 14.21, we have
(Ff o X) • M (,) = J2(F! o X) . {Y{i'k) • P w ) = £ ( * ? o X)F (i ' 4) . B w
(6)
Next by the Lebesgue-Radon-Nikodym Theorem we have (7)
/
(F;oX)(s)dV-'\s)=
[
J[0,t]
(F! o
X)(s)Z{i)(s)mL(ds).
J[0,t]
Finally
[Af®, M°>] = f ) £[y<«» . B«, y W> . B<«] i=l «=1
= E E /
y(,'1*)(a)y°'A(a)d[Bw,B{0](s)
= T f i*• \s)Y k
(jA
(s)mL(ds),
where the second equality is by Theorem 14.20 and the third equality is by Proposition 13.33. Therefore (8)
/
■/[0,t]
(F!'i3oX)(s)d[M^\M^](s)
= J2 I
Using (6), (7), and (8) in (5) we have (2).
(F^oX)(s)Y^k\s)Y^%)mL(dS).
J[0,t]
k=l
|
Theorem 15.24. Let X be a d-dimensional quasimartingale on a standard filtered space (Q, $, {& }, P) with components given by X^ = X 0 0 ) +M 0 ) where X00> is an do-measurable
%15.
343
ITO'SFORMULA
real valued random variable and MU) G M^0C(Q,5, {$t},P)forj = l,...,d. Then X is a d-dimensional {$t}-adapted Brownian motion if and only if every sample function of X is continuous and [M(j\Mm]t
(1)
= Sjikt
forteR+andj,k
=
l,...,d.
Proof. If X is a d-dimensional {5(}-adapted Brownian motion then every sample function of X is continuous and M = (Mm,...,Mw) is a d-dimensional {5<}-adapted null at 0 Brownian motion so that (1) holds according to Proposition 13.33. To prove the converse, according to Proposition 13.25 it suffices to show that for every s , t e R t such that s < t and y G Rd we have E[e .
_ e-^C-s)
a .e.
on(Q,3s,P),
or equivalently E l y f o ' * ' - * ' ^ ] = P(A)e-^v-s)
(2)
for A G £,.
Consider the function F(x) = Sv'^ for x G Rd We have F G C2(Md) with Fj(x) = iy^'*) and F?k(x) = -yjykeiM for j , k = 1,..., d. Thus by Theorem 15.22 and by (1) we have
(3)
ei<**)
_ e '(^> = i V y, / ~^
e'<^<*» dM0)(«) - ^
J(S,(]
2
Let yl G 5 S . Multiplying both sides of (3) by e~'^'x'hA
(4) E \e'^x'-x'h y
A - P(A) = zJ2y^U"{yMs))^A '
. j
L
/
e^^W**)-
7( s ,t]
and integrating we have
I e'^M»» dM«\u) J(s,«]
_ Ji^ E \1A f e^XM-x^mL(du) 2
L
.
J(s,t]
By (1), [M 0 ) ] ( = i and then E([M 0 ) ] ( ) = t < oo for every t G R+. Thus by Theorem 12.36, M 0 ) G M£(Q,5, {&}, P ) for j = 1,... ,d. The process e'"<s,:?> is continuous and hence predictable. It is also bounded so that it is in la,c<,(M+ x £2, M[MW], -P) by Observation 11.19. Then the stochastic integral e'<*'x'«Af °"' is defined and is in M|(£i, 5, {3<}, P). Thus
CHAPTER 3. STOCHASTIC
344
INTEGRALS
we have E [/{s t] e*
E {e^x'-x°hA}
- P(A) = - ^
j
E [e'^W-^lx]
mL(du).
Now if we define a real valued function tp on [s, oo) by setting ip(t) = E [e^y-x,~x'hA]
(6)
for t G [5,00),
then by (5) we have (7)
V(t)-P(A)
=- ^ - f 2
J(s,t]
The unique solution of this integral equation is given by (8)
By (6) and (8) we have (2). I As a particular case of Theorem 15.24, we have Corollary 15.25. Let X e M 2 (£2,5,{fo},P). Then X is an {^-adapted motion if and only if
Brownian
1°. every sample function of X is continuous, 2°. {X? —i:t€
R+} is a right-continuous null at 0 martingale.
§16 Ito's Stochastic Calculus [I] The Space of Stochastic Differentials For a real valued function / with a continuous derivative / ' on R, we have /(i 2 ) — f(tj) = ft? f'(t)dt for any t\,t2 € K such that i] < t 2 according to the Fundamental Theorem
§/6". LTO'S STOCHASTIC
CALCULUS
345
of Integral Calculus. We write df for / ' dt and thus f(t2) - }{tx) = //* df(t). Every antiderivative g of / ' satisfies the condition g(t2) - g(U) = ffi f{t)di = f(t2) - f(h) for any t\,t% G R such that t\ < t2. Also every antiderivative g of / ' has the form g = / + C for some C G R. For a quasimartingale X on a standard filtered space (£1,5, {$t},P) the derivative is undefined. Instead we define the stochastic differential dX of X as the collection of every quasimartingale Y on the filtered space satisfying the condition that Y(t2) - Y(ti) = Y(t2) - Y(ti) any tut2 g E+ such that t\ < t2 almost surely. Equiva lent^, dX is the collection of every the quasimartingales Y on the filtered space of the form Y = X + C where C is a real valued 50-measurable random variable on (D, 5, F). Note then that X £ dX andifY E dX then dX = dY Definition 16.1. Let Q(Q,5, {5t},P) fee £/ie collection of all quasimartingales X on a standard filtered space (Q,5, {3i},P), tfiaf js, X = X0 + Mx + Vx where Mx G M j ' o c ( n , 5 , { 5 t } , P ) , Vx 6 V c - , o c (n,5,{5 f },PX anrfXo G (&,), the collection of all real valued ^Q-measurable random variables on (Q, 5, P)Observation 16.2. By Proposition 15.2, the decomposition of X € Q(fi, 5, {5t}, P) as a sum of the random variable X0 G (5o), the martingale part Mx G M2lcc(Q.,$, {&},P), and the bounded variation part V* G VC,,0C(Q, 5, {5<},P) is unique. Let us call this decom position the canonical decomposition of a quasimartingale. Since M2 °c, Vc''oc, and (#0) are linear spaces, so is Q. Furthermore for X,Y e Q and a, b G R, if we let MaX+bY and Vax+bY be the martingale part and bounded variation part of the quasimartingale aX + bY, that is, aX + bY = (aX0 + bY0) + (aMx + bMY) + (aVx + bVy), then by the uniqueness of decomposition we have MaX+bY = aMx + bMY VaX+bY =aVx + bVY. Let X e QiQd, {dt}, P) given as X = X0 + Mx + Vx. For a predictable process <£ on the filtered space which is in B(R+ x Q), we have by Remark 15.10 and Definition 15.11 O . Mx
={[
<5(s) dMx(s) : t G R + } G M£' oc ,
J[0,t]
®,VX
={[
*(«) dVx(s) : t G R + } G Vc'loc,
J[0,t]
®,X
=
CHAPTER 3. STOCHASTIC
346
INTEGRALS
and as an alternate notation for (O • X) t for t G K+, we have / J[0,t]
3>(s)dX(s)=
[
®(s)dMx(s)+[
J[0,t]
0(s)dVx(s). J[0,t]
If X, Y G Q and if we assume that every sample function of Y is continuous then Y is a predictable process which is also in B(K+ x Q) so that Y • X exists in Q. Definition 16.3. We say that X, Y G Q ( A 5, {&}, P ) are equivalent as quasimartingales and write X ~ Y iff/iere ex^te a WK/Z .ye* A in (Q, 5, P ) such that for w G Ac we have X(i,w)-X(s,w) = Y(t,w)-Y(s,w)
for every s,teR+,
s
For X G Q(Q, 5 , {&}, P), we write dXfor the equivalence class in Q(Q,#, {&}, P ) vWtfi respect to the equivalence relation ~ fo w/ii'c/i X belongs and call it the stochastic differ ential ofX. We write AQfor the collection of the equivalence classes in Q(£2, 5 , {3t}j P) with respect to ~. For two stochastic processes X and Y on a probability space (Q, 5 , P ) we write X = Y to indicate their equivalence in the sense of Definition 2.1, that is, there exists a null set A in ( Q , 3 , P ) such that X(t,u) - Y(i,w) for (i,w) G R+ x Ac. If £ is a real valued random variable on (Q,§, P), then we write X = f to indicate that X(i,u>) = £(u>) for ( i , u ; ) € R + X Ac. Remark 16.4. Let X, Y G Q(Q,3,{5 4 },P) given by X = X 0 + Mx + Vx and Y = Y0 + My + Yy respectively. Then (1) (2) (3)
X tY & X-Y = X0-Y0, X^Y & MX=MY and VX = VY, X X Y <* X - Y = Z0 for some Z 0 G (&,)•
Proof. 1) (1) is immediate from Definition 16.3 by considering 5 = 0 . 2) If X £ Y then by (1) we have X - Y = X 0 - Y0 G (So)- Since X - Y g Q this implies M*_y = 0 and VX-Y = 0 by Proposition 15.2. But according to Observation 16.2, MX-Y = Mx -MY and VX-Y = Vx - VY. Thus Mx = MY and Vx =VY. Conversely, if Mx = MY and Vx = VY, then X - Y = X0 - Y0 and thus X ~ Y by (1). This proves (2). 3) By (1) the implication =4- in (3) holds. Conversely if X - Y = Z0 G (j 0 ) then by Proposition 15.2 we have MX-Y = 0 and VX-Y = 0. Then by Observation 16.2, we have Mx = MY and VX = VY. Thus by (2) we have X ~ Y This proves (3). ■
§ 16. LTO'S STOCHASTIC
347
CALCULUS
Observation 16.5. 1) Since (5o) contains the identically vanishing random variable 0 and M2'' oc (Q,3, {&}, P ) and VC''°C(Q,5, {&}, P) contain the identically vanishing stochastic process 0, M = 0 + M + 0 G Q ( n , 5 , {&},P) for any M G M^^Q, £, {&},P) and similarly V = 0 + 0 + V G Q(Q,3, {&},P) for any V G V c ''° c (n,3,{S ( },P). Thus M ^ C Q ^ - f & ^ P ) and VC''°C(Q,5, {3 ( },P) are linear subspaces of Q(Q,5,{5,},P). Note a l s o M ^ o c ( n , 5 , {5,},P) D V c '' o c (n,5, {&},P) = {0} by Proposition 15.2. 2) Let X G Q(£2,5, {5<},P) be given in its canonical decomposition as X = Xo + Mx + Vx . If X & M for some M G M=''oc(n, 5, {&}, P), then Mx = M and V* = 0 by (2) of Remark 16.4. Similarly if X ~ V for some V G V c '' oc (Q,5, {&},P), then M* = 0 and Vx = V. 3)SincelVf 2 '' oc (Q,5,{5 t },P) C Q«2,5, {&},P), for M G Mj' DC (Q,5,{5<},P) the stochastic differential dM is defined and consists of all X G Q(£2,5, {dt},P) such that X = Z0 + M where Z 0 £ (3o) according to (3) of Remark 16.4. Similarly for V G V c '' o c (Q,5,{5 ( },P), dV is defined and consists of all X G Q(fi,5, {5 ( },P) such that X = Z0 + V where Z 0 G (5o)Definition 16.6. Wfe define dMj' 0 0 as f/ie subcollection of dQ consisting of dM for M G M2ioc(Si, 5, {5t}, P)- Similarly we define d\c,loc as the subcollection of dQ consisting of dVforV e\c-'°c(Q.,3,{dt},P). Definition 16.7. We definefour operations in dQ. ForX,Y G Q ( A 5 , {3t},P)andc Zer addition, scalar multiplication and product be defined respectively by (1) (2) (3)
dX + dY = d(X+Y), cdX = rf(cX), dY = d[Mx,MY],
andfor a predictable process ® G L ^ R * x Q, p[Mxh •-multiplication o/O fry c/X fee defined by (4)
G K,
<6«cfX
=
P) D L ^ R * X Q, ^ ^Vx |, P), to
d(*.X).
Sometimes dX ■ dY is written as dXdY and <E> • dX is written as OdX. Regarding (3) and (4), note that since [Mx, MY] G \cM and since $ i l £ Q , d(<£ • X ) is defined.
C Q, d[Mx, MY] is defined
CHAPTER 3. STOCHASTIC
348
INTEGRALS
With addition and scalar multiplication as defined above dQ is a linear space over R. We show next that dQ is a commutative ring with respect to addition and product. If we define addition and product in B(R+ x Q.) by pointwise addition and product of its elements on R+ x Q, then B(R+ x Q) is a commutative ring with identity. We show below that with respect to addition and •-multiplication, dQ is a commutative algebra over the ring B(R+ x fi). Observation 16.8. For X,Y,Z (1)
£ Q(fi, 5, {&}, P), we have
(dX + dY) ■ dZ = dX -dZ +
(2)
dX-dY
dY-dZ,
= dY ■ dX.
Thus dQ is a commutative ring with respect to addition and product in Definition 16.7. Proof. To prove (1), note that by (1) and (3) of Definition 16.7 we have {dX + dY) ■ dZ = d(X + Y)-dZ = d[Mx+Y,Mz) = d[Mx + My, Mz] = d([Mx, Mz] + [MY,MZ]) = d[Mx,Mz] + d[MY, Mz] = dX-dZ + dY ■ dZ. To prove (2), note that by (3) of Definition 16.7 we have dX ■ dY = d[Mx,MY] Theoreml6.9. LetX,Y filtered space such that
6 Q(£l,$,{$t},P)
= d[MY, Mx] = dY ■ dX.
■
and® and ¥ be predictable processes on the
L^(R+xa,WMxl,P)nLi%(R+xQ,^|Vx|,P)
n L^(R + xn,fHMYhP)nL[°c00(R+ x Q l M | ^ , , P ) . Then (1) (2) (3)
Q>»(dX + dY) =
In particular, (1), (2), and (3) hold for ,¥ <= B(R+ x Q). Thus dQ is a commutative algebra over B(R+ x £2) with respect to addition and •-multiplication in Definition 16.7. Also for 0>, *F 6 B(R+ x O) we have (4)
O T • dX =
§ 16. ITO'S STOCHASTIC
CALCULUS
349
Proof. 1) To prove (1), note that by (1) and (4) of Definition 16.7 and by the linearity of the stochastic integral we have
= d(9»X)+dQ¥»X)
=
^dX+VmdX.
3) To prove (4), note that by (3) and (4) of Definition 16.7, we have on one hand (5)
= d ( * • [Mx, My]),
and on the other hand (6)
(O • dX) ■ dY = d(0 *X)-dY
= d[M^x,MY]
= d[<5 • Mx,
My],
where the last equality is from the fact that the martingale part of the stochastic integral of G> with respect to X is the stochastic integral of <£ with respect to the martingale part of X. By Theorem 14.20, we have [<5 • Mx, MY]t = fm
4>(sms)dVx(s) J[0,t]
=
f J[0,t]
®(s)d\ [ I-'[0,5]
V(.u)dVx(u)\
= (
by the Lebesgue-Radon-Nikodym Theorem so that <J>Y • Vx = $ • 0? • Vx)- Thus O)^ • X =
CHAPTER 3. STOCHASTIC
350
INTEGRALS
This proves (4). ■ Corollary 16.10. Under the same assumptions as in Theorem 16.9, we have (1)
(2)
d(<X> • X) ■ d(¥ • Y) =
and for predictable processes <£],..., O n G B(E+ x Q), we fcave « ! . . . 4»„ • dX = * , • ( ■ • ■ (*»_i • (*„ • dX)) ■ • •).
(2)
Proof. The first equality is by (3) of Theorem 16.9. The remaining equalities are from the fact that the product in dQ is commutative and the fact that O • dX, O • dY € dQ. To prove (2), note first that by (3) of Definition 16.7 and by Definition 15.11, we have d(* • X) ■ dQ¥ • Y) = d[M^tX, MWmY] = d [ 0 • Mx, V • MY]. Now [<S • Mx, ¥ • My] = O T • [Mx, My] by Theorem 14.20 so that by (4) and (3) of Definition 16.7 we have d[<& • Mx,¥
• My] = d(OY • [M x , My]) = 0»P • d[Mx,MY]
= <£¥ • (dX • dY).
Therefore d(<£ • X) • d(»F • Y) =
dQ dQ C dV c ' 0C , dV c ' / o c -dQ={dO}, dQ dQ dQ = {dO}.
§ 16. FT&S STOCHASTIC
CALCULUS
351
Proof. To prove (1), note that if X, Y g Q then by (3) of Definition 16.7 we have dX ■ dY = d[Mx,MY] € dVc''0C. To prove (2), note that if V £ dVc''DC, then Mv = 0 so that [MV,MX] = 0 for any X e Q and thus dV ■ dX = dO. Finally for X,Y,Z eQwe have dX-dY £ dV c ' ,oc by (1) and then dX ■ dY ■ dZ = dO by (2). This proves (3). ■ Definition 16.12. Let X € Q(Q, 5, {& }, P). For the stochastic differential dX € dQ and for s, t 6 R+ SMC/I f/iaf s < ( w define (1)
/
dX(u) = X(t) - X(s).
J(s,t]
For dX, dY edQ and a,be K, we
/
(adX + bdY)(u) = a (
J[0,t]
dX(u) + b f
J[0,t]
dY(u).
J[0,t]
Remark 16.13. 1) Recall that the stochastic differential dX consists of all quasimartingales Y e Q ( i i , 5 , {5(},P) such that Y ~ X, that is, Y(t,u) - F{s,w) = X(t,ui) - X(s,u) for every s, t e R+, s < i, and u £ Ac where A is a null set in (Q., 5, P). Thus if Y is an arbitrary representative of dX, then we have L t] dX{u) = Y(t) — Y(s) = X(t) — X(s). This shows that / (s t] dX(u) in (1) of Definition 16.12 is well defined. 2) According to Definition 15.11, for the stochastic integral of the process 1, that is, the process 4>(i,w) = 1 for (t,uj) 6 R.+ x Q, with respect to a quasimartingale X S Q(«, 3, {&}, P) g^en by X = X0 + Mx + Vx, we have /
lM(u)dX(u)=
J(s,t]
= {Mx(t)
I Jte.t]
lM](u)dMx(u)+
I
\{s,t](u)dVx{u)
J(s,t]
- Mx(s)} + {Vx{t) - Vx(s)} = X(t) - X(s).
Thus / [01] dX(u) in Definition 16.12 is equal to/ [ 0 ( ] 1 dX(u) in Definition 15.11. In partic ular for the stochastic differential d(& • X) where 3> is a predictable process in L ^ C K t x ii,/i [ M x ],P)nL' 1 %(lR + x a , M | V x | , P ) , wehave / J(.s,t]
(O . dX){u) = ( J(s,t]
®(u) dX(u).
J(.s,t]
Observation 16.14. Let X = (X ( 1 ) ,... ,X<«) where X ' 1 ' , . . . ,X^ e Q(Q,$,{ft},P) andletP € C 2 (R' i ). According to Theorem 15.21 there exists a null set A in (Q, 5, P ) such
CHAPTER 3. STOCHASTIC
352
INTEGRALS
that on Ac we have for any s, t G K+ such that s < t (F o X)(i) - (F o X)(s) = £ { ( ( J V o X) . X (i) )(i) - ((Jf o X) . X » ) ( 5 ) } i=i
=
J2 li((F"j .',j=i
° X > • Wfcw, AfjcwDM - « J ^ o X ) . [M*<.„ Afjr«])(a)}.
2
By Definition 16.3 and Theorem 16.9, we have i
d
d
d(F o X) = £
i=\
.,j=i
[II] Fisk-Stratonovich Integrals L e t X , F G Q ( Q , 5 , { 5 J , P ) . ThenF G BtR.xfl) C L ^ ( R t x « , / i [ M j r ] , P ) n L ' , % ) ( R t x Q, ^ | V x | , P ) so that F • X exists in Q(Q,5, { 5 J , P) by Definition 15.11 and d(Y • X ) = F • dX by Definition 16.7. Lemma 16.15. 7 / X , F G Q(Q,3, {&},P),tfiercX Y G Q(Q,$,{3t},P)
and
d(XY) = X»dY + Y»dX + dX-dY. Proof. Consider the 2-dimensional quasimartingale (X (1) , X (1) ) = (X, Y) and F G C 2 (R 2 ) defined by F(x, y) = xy for (x, y) G K2. By the multidimensional Ito's Formula we have F o (X (1) , X (2) ) G Q, that is, XY G Q, and furthermore since F{{x, y) = y, F±(x, y) = x, F[\(x,y) = F£2{x,y) = 0, and F;[2(x,y) = Fi'A(z,y) = 1 for (x,y) G R2, we have according to the multidimensional Ito's Formula as given in Observation 16.14 d(XY)
= X (2) . dXm + X(1> . dXm + \{dXm = Y*dX
+ X»dY
+ dX-dY.
■ dXm + dXi2) ■ dXm}
■
Definition 16.16. Let X, Y G Q(Q, 5, {&}, P). The symmetric multiplication ofY by dX is defined by YodX
= Y*dX
+ \dY ■ dX. 2
§/6~. ITO'S STOCHASTIC Note that YodX
353
CALCULUS
e dQ.
Lemma 16.17. LetX,Y
£ Q(Q, 5, {&}, P). ffcen d(Xr) = X o d F + F o d X
Proof. By Definition 16.16, we have Y o dX = Y • dX + \dY ■ dX and X o dY = X*dY + \dX ■ dY. Adding the two equalities side by side and recalling Lemma 16.15 we have X o dY + Y o dX = X • dY + Y • dX + dX ■ dY = d(XY). ■ Theorem 16.18. For X,Y,Z (1) (2) (3) (4) (5)
e Q(Q,5, {&}, P), we have
Xo(dY + dZ) = XodY+XodZ, (X+Y)odZ = XodZ + YodZ, XYodZ = Xo(Yo dZ), X o (dY ■ dZ) = X • (dY ■ dZ), Xo(dY ■ dZ) = (Xo dY) ■ dZ.
Proof. To prove (1), note that X o(dY + dZ) = X od(Y + Z) = X • d(Y + Z) + -dX ■ d(Y + Z) = X»dY
+ X»dZ
+ -(dX
dY + dX -dZ) = XodY
+
XodZ,
where the first equality is by (1) of Definition 16.7, the second equality is by Definition 16.16, the third equality is by (1) of Definition 16.7 and Observation 16.8, and the last equality is by Definition 16.16. The equality (2) is proved likewise. To prove (3), note that (6)
XY o dZ = XY • dZ + -d(XY) = XY»dZ
■ dZ
+ -(X • dY + Y • dX + dX ■ dY) ■ dZ
354
CHAPTER 3. STOCHASTIC
by Lemma 16.15. Since YodZ (7)
Xo(YodZ)
= XY»dZ
e dQ, we have by Definition 16.16
= X*(YodZ)
+
)-dX-(YodZ)
+ ]-dY ■ dZ) + X-dX (Y»dZ
= X»(Y»dZ
INTEGRALS
+ X-dY ■ dZ)
+ -(X • dY) ■ dZ + )-{Y • dX) ■ dZ + jdX
-dY ■ dZ
by (4) of Theorem 16.9 and (1) of Corollary 16.10. By (6), (7), and the fact that dX ■ dY dX = dO, we have (3). To prove (4), note that since dY ■ dZ £ dQ we have by Definition 16.16 and (3) of Theorem 16.11 X o (dY • dZ) = X • (dY ■ dZ) + -dX ■ (dY ■ dZ) = X • (dY ■ dZ). To prove (5), note that (X o dY) -dZ = (X»dY = (X»dY)-dZ
+ ^dX ■ dY) ■ dZ
=X» (dY ■ dZ) = X o (dY ■ dZ),
where the second equality is by (3) of Theorem 16.11, the third equality is by (3) of Theorem 16.9, and the last equality is by (4). ■ Theorem 16.19. Let X = (X<«,... , I < « ) where X^ G Q(fi,£, { & } , P ) / o r i = l , . . . , d and let F e C 3 ^ ) . Then FoX e Q(Q, 5, {&}, P ) and d
d(F oX) = J2(Fl o X) o dX{i) i=\
Proof. By Definition 16.16, we have (1)
(F! o X) o dX{i) = (F[ o X) • dX(i) + l-d(Fl o X) ■ dXm.
Since F- £ C2(Rd), by Ito's Formula as given in Observation 16.14 we have
d(F; o X) = £(*& o x). ax™ +l~£ Wkk ° X). dx& ■ dx«l
§ 16. LTO'S STOCHASTIC
CALCULUS
355
Substituting this in (1) and recalling that dX(i) ■ dXu) ■ dX{k) = dO by Theorem 16.11, we have
£(F;oi)odx" = Yfift°x)• «ur w +JE^°^)' d x & -dx^ = d{FoX). ■ Definition 16.20. Ler X,Y € Q(£2,3, {&},P) fee given by X = X0 + Mx + Vx and Y = YQ + My + Vy. The Fisk-Stratonovich integral ofY with respect to X,Y o X, is the quasimartingale Y o X defined by (1)
YoX
= Y •X +
hLMy,Mx}
and thus (2)
d(Y oX) = d(Y»X)
+ ^ d[MY ,Mx] = Y»dX
+ -dY-dX
=
YodX.
We also use the notation J[ot] Y o dX for (Y o X)t, that is, f[ot] Y o dX = / [0 (] d(Y o I ) ( s ) forteR+. Example 16.21. Let B be an {g(}-adapted null at 0 Brownian motion on a standard filtered space (£2,5, {5,}, P). Then / BodB= Jio.t]
f B(s)dB(s) + - f d[B,B](s) J[o,t] 2 J[o,t]
Since / [0 (] B(s) dB(s) = \{B2(t) — t) as we showed in Example 15.7 and since [B, B]t = t, we have /
BodB
=
)-B\t\
which resembles /0' sds = \t2 Theorem 16.22. Let X € M|(£2, #, {St}, P) and let Ybea bounded continuous martingale on (£2,3, {5i}, P). For n G N, let t^bea partition o/E+ into subintervals by a strictly increasing sequence {tn,k : k e Z+} in R+ such that tnfi = 0 and tn
356
CHAPTER 3. STOCHASTIC INTEGRALS
lim |A„ | = 0 where |A„ | = sapk^(tn,k — tn,k-1 )■ Let t 6 K+ befixed.For each n E N, Ze?
F o dX = P • lim £ ^{F((„,i) + K(<„1*_i)}{A-(tBii) - X(t n , t _,)}.
Proof. We have £ ^{F(i„,fc) + F(tn,,_,)}{X(in,A) - *(*„,*_,)} = Y.Y{in,k-i){X{tn,k)
- X(tn,k_,)}
k=\
+ \ Y.{Y(tn,k) - Y(tn,k-i)}{X(tn:k)
- *(*„,*_,)}.
By Proposition 12.16 P ■ lim £ r(i n ,n){X(i n ,i) - X(tn^)}
= (Y . X)(t).
By Theorem 12.27 p
>)" - F(*n>-■t)}W*«.,*)- - -X" (*«,*-■.)} == [y,x](i).
Thus
*=1
•i)H*(*M>-Jrcw- .)}
Z
= (Y . X)(<) + - [F, X](i) = / 2
J[0,t]
F o dX.
Chapter 4 Stochastic Differential Equations §17
The Space of Continuous Functions
[I] Function Space Representation of Continuous Processes Let W d be the collection of all Revalued continuous functions w on R+. Our objective is to introduce a cr-algebra 2Urf of subsets of Wd and an increasing system of sub-a-algebras of 13d, {2U? : t E R+}, with the following properties: 1°. For every t e R+, the mapping qt of Wd into Rd defined by qt(w) = w(t) for tii 6 Wrf is a 2Ud/*BMd-measurable mapping so that g = {qt : t g R + } is a d-dimensional {iHJ^-adapted stochastic process on the filtered measurable space (Wd, Wd, {Wd}). 2°. For an arbitrary continuous d-dimensional stochastic process X = {Xt : < G M+} on an arbitrary probability space ( 0 , 5 , P), the mapping X of Q into W d defined by A"(a;) = X(-,u>) for w € Q is S/aU^-measurable. Under the assumption of 1° and 2° let P% be the probability distribution of X on the measurable space (Wd,
358
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Definition 17.1. Let Wd be the collection of all Rd-valued continuous functions on R+. For t e R+, let qt be a mapping of\Vd into Rd defined by qt(w) = w(t)for w £ Yfd. Let T = { i , , . . . , i„} be a finite strictly increasing sequence in R+ and let E\,...,En £ 55B
= {we\Vd:
(iu(*i), ■ ■ ■, w(tn)) £ £ , x ■ ■ ■ x E n }
Let 3 be the collection of all cylinder sets in Wd, and for t £ E+ let 3[o,«] be the collection of all cylinder sets with indices r = {t\,... ,<„} preceding t, that is , tn < t. We write 3t for the collection of all cylinder sets with index r = {t}, that is, 3t = qt (®]g<<)Lemma 17.2. 3 is a semialgebra of subsets o / W d semialgebra of subsets ofVfd.
For every t £ K+, 3[o,«] is also a
Proof. Since 0 = g~l(0) and Wd = g("'(Rd) for an arbitrary t £ R+, 0 and W* are in 3Clearly 3 is closed under intersections. Therefore it remains to show that if Z £ 3 then there exists a finite disjoint collection {Z* : k = 0 , . . . , n } in 3 such that Zo = Z and U|=0Zj £ 3 for k = 0 , . . . , n with U]^Zj = W Let Z = g^'CEi) n • • ■ n q^(En) with 0 < t\ < ■ • ■ < tn and Ej £ 33Md for j = 1 , . . . , n. Let
zk = qrkl(BQ n qrkl(EM)
n • ■ • n ?,-„'(£„) for k = l , . . . , n
and in particular Zn = q^ (E„). Then {Zk : k = 0 , . . . , n} is a disjoint collection in 3- Also
Zo u • • • u ^ i =ft-'CJ?*)n • ■ • n q;n\En) e 3 so that in particular
Zo u • • ■ u z„_, u z n = g^CE,,) u 3 ( ;'(^) = 9r„'(Rd) = w d This completes the proof that 3 is a semialgebra of subsets of Wrf. The fact that 3[o,<] is a semialgebra of subsets of W d can be shown in the same way. ■ Since 3 is a semialgebra of subsets of Y?d, the algebra of subsets of Wd generated by 3 is the collection of all finite unions of members of 3- Since 3[o,s] C 3[o,t] C 3 we have ff(3[o,»]) C
§77. THE SPACE OF CONTINUOUS FUNCTIONS
359
Definition 173. Let W1 =
(2) a(uielE+3f) = arr*. (3) <x(U,e[0,t]3.) = 2nf /or every t 6 R+. Lef (fl,g) fee a« arbitrary measurable space and let T be a mapping ofil into Wd. Then (4) T is g/W*-measurable if and only ifqt o T is 3793^ -measurable for every t G R+, (5) For fixed t G R+, T is g/W*-measurable if and only ifq, o T is ^/^d-measurable for every s g [0, t\ Proof. To prove (1), note that for t G R+ we have 3[o,«] C 3, c(3[o,*]) C cr(3), and thus Ut€]Rto-(3[oi(]) C
air'. To prove (2), note first that Utej^5t C 3 so that
^ 1
x • • • x
x • ■ ■ x QS.,))
=
■
360
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Vfd is a linear space over R if we define addition and scalar multiplication by (u>i + w2)(t) = wi(t) + w2(t) for t G R+ and wuw2 G Wd and (Xw)(t) = Xw(t) for t G R+, A G R, and w G W Consider the translate of a subset E of W* by an element w0 of Wrf defined by E + w0 = {w + w0 : w G E} and the A-multiple of E defined by XE = {Xw :w G E). Remark 17.6.
w(ti) G £,•} + wo = {w G Wrf : w(U) eEt + w0(U)} G 3
since JEJi + «)Q(<,) G 25md. Therefore Z + w0 G 3 C cr(3). This shows that Z E & and thus 3 C 65. Since <S is a d-class containing the 7r-class 3, we have
361
§77. THE SPACE OF CONTINUOUS FUNCTIONS
3[o,i] is the collection of all finite intersections of members of Use[o,f]g71(931
on(W,arr',{aD?}). 3) We have g(t, tu) = g((tu) = tu(t) for (t, w) G R+ X W*. Thus g(-, w) = w. ■ Theorem 17.8. Let X = {Xt : t G R+} be a continuous d-dimensional stochastic process on a probability space (Cl,$,P). Then the mapping XofQ. into W d defined by X(UJ) = X(-,u>)for Lj 6 O. is 51'2tT-measurable. Let Px be the probability distribution of X on (Wd,'Wd). Then the d-dimensional {2U*}-adapted process q = {qt : t G R+} on the filtered space (Wd,2Bd, {W*},Px) and the d-dimensional process X on (fi, 3 , P ) have identical finite dimensional probability distributions, that is, for anyfinitestrictly increasing sequence r = { f j , . . . , tn} in R + the probability distribution (Px)qT on (Knd, 23and) of the random vector qT = (qtl,..., qt„) and the probability distribution PXT on (Knd, 331"d) of the random vector XT = (Xtl,...,Xin) are identical. Also, for every E G 23ffind and the cylinder set Z = {w 6 W : (iu(
X~\Z)
= A - , ( n ^ g - | ( £ , ) ) = nr =1 A- 1 (g t - 1 (£,))
= ntiCft, o *)"'(£.) = n?=1x-'(£,-) 6 5. This shows that r ' ( 3 ) C j . To show that (Px)qr = PXr on (R'"', ® r * ) , let £ G ! 8 r * . Then (P*), r (S) = P^(g71(JE)) = - P ° ^ " 1 ° g T " , ( ^ ) = P o (,T o # ) - ' ( £ ) = P o JCT-'(E) =
PXr(E).
To show that P ^ C E ) = Px(Z), note that P X r ( £ ) = P o X 7 ' ( £ ) = P{u G a : Xr(E) G E}
362
CHAPTER 4. STOCHASTIC DIFFERENTIAL = P{u e i2 : X(-,w) £ Z} = Po X~\Z)
EQUATIONS
= Px(Z).
When X is adapted to a filtration {& : i € R+}, we show that X is &/2U?-measurable by starting with Z G 3[o,<] in (1). I Definition 17.9. Let B be a d-dimensional null at 0 Brownian motion on a probability space (£2,3, P). Let B be the mapping ofQ. into XVd defined by B(OJ) = B(-,u>)for iv <E £1 We call the probability distribution PB ofB on (Wd, Z0d) the Wiener measure on (Wd, %3d) and write mwfor it. We call (Wd, Wd, m ^ ) the d-dimensional Wiener space. For a cylinder set Z in W J with index r = {tu..., we have by Proposition 13.16 mdw(Z) = PB(Z) = PB{weV/d: (w(*i),...,io(tJ) eE}=
tn } where ti > 0 and base E € 9 3 ^
P{(B(ti),...,B{tn))
72
|g
e E}
l|2
= {wfi^-^o}-* / exp(-^E ;~^- }m2^(g,,...,^)) j=i
-'■E
[
z
j=i
'J
r
j-i
J
where to = 0. Theorem 17.10. Let W be a stochastic process on (Wd, 2Ud, mfy) defined by setting W(t, w) = w(t)
for (t, w) e R+ x Wd
Then W is a d-dimensional null at 0 Brownian motion on (Wd, 2Ud, m^). Proof. Let us verify that W satisfies the conditions in Definition 13.14. Clearly every sample function of W is an Rd-valued continuous function on R+. Let us show that for every s,t £ R+, s < t, the probability distribution Q on (Rd, 93Kd) of the random vector Wt — Ws is the d-dimensional normal distribution Nd(0, (t — s) ■ T). Let x = T(y) be the nonsingular linear mapping of R2d onto R2d defined by x ( = y\ and x2 = y\ + 3/2 for y = (.V\,yi) 6 ^2d Then for any E e Q3md we have Q(E) = mdw 0 (Wt - WST\E) = mdw o {Ws, Wt - Wsy\Rd = mfyiw eWd :(w(s),w(t)-w(s))£Rd xE} d d d = m v{w€'W :(w(s),w(t))eT(R xE)}
x E)
§77. THE SPACE OF CONTINUOUS FUNCTIONS = {(2n)h(t-s)}-^
exp(-^-^f^}m2M^,x2))
f JTVXE)
{(2^) 2 .<*-
=
s)}-^[d
[
{2TT(<-
2s
2 expj- M
2s
d
Jm xE =
363
2(t - s) J
llbl2 1
2(t-
*>J'
n2i{d{yu
yi))
2(t - .•
This shows that Q = JVd(0, (t - s) • / ) . To show that W is a process with independent increments, we show that if { t i , . . . , *„} is a strictly increasing sequence in R + , Q is the probability distribution of the random vector (W t „ Wtl - Wt>,..., W,„ - Wtn_,) on (R™*, «8m„d), and QuQt,...,Qn are the probability distributions of the random vectors Wt,, W<2 — W t | , . . . , Wtn — Wtn_, on (Rd, SB^) respec tively, then Q(E) = (Qi x ■ • • x Qn)(E) for every £ G ©K»<<- For this it suffices according to Corollary 1.8 to show Q(Ei x • • • x En) = Qi(-Bi) ■ • • Qn(En)
for £,- e SB,*, j = 1,... ,n.
Let i = T(y) be the nonsingular linear mapping of Rnd onto R nd defined by x\ = y\, x2 = yx + y2,...
,xn = yx + • ■ ■ + yn
for y = (t/,,... ,y n ) e R"
so that 2/i = * i , 2/2 = £2 - x : , ...,!/n = i » -xn-i
f o r i = ( i i , . . . , i „ ) G R"
Let us write E = Ei x ■•■ x En. Then Q(E, x ■ • • x En) = rniy o (W„, W,2 - Wt = m^{u; G W d : (w(t,),io(i 2 ) - w(t{),...,w(tn)
,Wu - Wtn_xy\E) - u>(t„_,)) G E]
= m^{u) G W : (w(*i),..., «»(*„)) G T(E)}
= {(2^-n^-*,--,)}-^/ exp ( - 1 £ |T; " *J~l''1 *#(*»„ ■..,«.)) ■ - , & ) )
m ) =ft{2^ - *-.>r* jEj ^{-\T^T-;} n{2^-*i-.)}-^/ ^ Bj «p{4J-l,}<^ 3=1
364
CHAPTER 4. STOCHASTIC DIFFERENTIAL
This proves the independence of increments for W
EQUATIONS
■
[II] Metrization of the Space of Continuous Functions Our next objective is to introduce a metric to Wd with respect to which Wd is a complete separable metric space and the Borel cr-algebra of this metric space is equal to the cr-algebra Wd = (7(3). Definition 17.11. For every t € R+, let us define a seminorm vt on Wd by setting vt(w) = max |ui(s)| A 1 for w e Wd s€[0,t]
where \-\is the Euclidean norm on Rd, and let pt(wuw2)
= i/t(wi-w2)
forwuw2£\Vd.
Let us define a quasinorm v^ on Wd by ^oo(u0 = Yl 2~mvm(w)
forw 6 Wd
mgN d
and define a metric p^ on W by pxi(wi,w2)
= ^ooOi - w2) = ^2 2~mpm(wuw2)
for wuw2
G W*.
mgN
Lemma 17.12. Let {wn : n 6 N} be a sequence and w be an element in Wrf. Then lim poo(wn, w) = 0 if and only if lim wn(s) = w(s) uniformly in s £ [0, i] for every t G n—*oo
n—*oo
R+. Proof. It suffices to show that lim poo(wn, w) = 0 if and only if lim pm(wn, w) = 0 for 71—i-OO
ri—»-oo
every m G N. Clearly lim p^Wn, W) = 0 impl ies that lim pm{wn. w) — 0 for every m G N. Conversely suppose lim pm{wn, w) = 0 for every m G N. Given e > 0 let N G N be n—^oo
so large that Em>Ar+i 2 _ m < e. Then since pm(wn, w) < 1 for every m, n G N, we have Em>N+i 2" m p m (iu„, u>) < e for every n e N so that lim sup £ } 2~"Vm(i/)n,u>) < e. n
-"x
m>N+l
§ ; 7. THE SPACE OF CONTINUOUS FUNCTIONS
365
Then limsupp 0o (u) n ,u;) = l i m s u p i ^ 2 - m p m ( u ; „ , u ; ) +
T
2~mpm(v>n,w)\
N
<
J|i^£2- m /> m (u> n ) u>) + lirnsup S
2 _ m /) m (w n ,«;)<e
since by assumption lim pm(wn, tu) = 0 for every m e N. By the arbitrariness of e > 0 we have lim sup Poo(wn, w) = 0 and thus lim ^ ( u i , , w) = 0. ■ n->oo
n-too
Theorem 17.13. (W"', p^) is a complete separable metric space. Proof. To show that (W d , p^,) is a complete metric space, let {wn : n G N} be a Cauchy sequence in (Wd, p^). By Lemma 17.12, it suffices to show that there exists w € W1 such that lim p m (w„, w) = 0 for every m 6 N. Let e € (0,1). Let m € N be fixed. Since n—KX>
{wn : n e N} is a Cauchy sequence there exists N 6 N such that Poo{wi,wn) < 2~me when £,n > N. Then 2~mpm(wt, wn) < 2~me when £,n > N and thus pm{wi,wn) < e when £,n > JV. Thereforemaxse[o,m] \w((s)—wn(s)\/\l < e, orequivalently,maxse[o,m] \w((s)— tw„(s)| < e, when £,n> N. This shows that the restrictions of wn to [0, m] for n 6 N is a Cauchy sequence with respect to the metric of the uniform norm on the space of continuous E.d-valued functions on [0, m]. By the completeness of this metric space there exists a continuous R''-valued functions / ( m ) on [0, m] such that the restrictions of w„, n 6 N, to [0,m] converge uniformly on [0,m] to / ( m ) . For mi,m2 6 N, mi < m,2, we have / ( m i ) = / ( m 2 ) on [0, mi] by the uniqueness of the limit of the convergence. Thus there exists a continuous Rd-valued functions / on R+ such that / = / <m) on [0, m] for every m € N. Since lim max \wn(s) - / ( m ) (s)| = 0 and / = / ( m ) on [0,m] for every m £ N, n—*oo s€[0,m]
we have lim max \wn(s) - / ( s ) | A 1 = 0, that is, lim pm(wn, f) = 0. This proves the n-wo s6[0,m]
n
-*°°
completeness of the metric space (W*, Poo)To show the separability of the metric space (Wd, Poo) let us show that it has a countable dense subset. For every k G N let Vk be the collection of all Rd-valued polygonal functions v on R+ with vertices occurring at £2~k where £ G Z+ and with v(£2~k) equal to rational points in R"*. Then Vk is a countable subset of W d and so is V = Ujt6N Vi. To show that V is dense in Vid we show that for every w € W and e > 0 there exists some o e V such that Pooiwjv) < e. LetiV € N be so large that Em>/v+i 2 _ m < 2 _ l £ - By the uniform continuity
366
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
of w on [0, N] there exists v G V such that max(e[o,/v] \u>(t) - v(t)\ < 2 le. Then AT
= V ; 2 - m { m a x \w(t) - v(t)\ A 1}+ V 2" m { max > ( t ) - v(*)| A 1} t€[0 m] ^ 'el0'"1' ' m iN + i
Poo(w,v)
N
< ^ 2 - m 2 - ' e + Y. m=l m=\
2" m <e.
m>N+l m>N+\
Lemma 17.14. LetOMdbe the collection of all open sets in Rd and let Dy/d be the collection of all open sets in (Wd, p^). Then q^\0Md) C Oy/dfor every t G R+. Proof. LetG G DBd. Consider ql'(G) for an arbitrary t G R+. If G = 0, then g ( - '(G) = 0 G Dy/d. Consider the case where G =f $ and thus q^l{G) j - 0. To show that qTx(G) G Ow« we show that for every w0 G ft-'(G) there exists an open sphere in Wd with center wQ and radius 77, S(w0,r]), contained in gt~'(G). Now since wQ G q7l(G), we have u>o(£) G G. Then there exists 6 G (0,1) such that the open sphere in Rd with center u>o(t) and radius 8, S(w0(t), 6), is contained in G. Let TV G N be so large that t G [0, N]. We proceed to show that S(w0,2~N8) C gt_1(G). Let w e S(w0,2~N6). Then P^W^WQ) < 2"N<5 and thus 2-7V{max36[o,w] \w(s) - w0(s)\ A 1} < 2~NS. Since 6 G (0,1) we have maxse[o,N] \w(s) — w0(s)\ < 6. Since t G [0, N] we have \w(t) — w0(t)\ < S. Thus w(t) G S(wo(*);^) C G. Therefore u; G q^i(G). Since UJ is an arbitrary element in S(w0,2~NS), we have S{w0,2~N8) C q^(G). ■ Theorem 17.15. Let Dy/d be the collection of all open sets in (Wd, p^). Consider the Borel a-algebra 55Wd = CT(DWM) in Wd. Then
i)3c<8w«. 2) D w „ C <J(3), 3)
gr'CV) =
367
§77. THE SPACE OF CONTINUOUS FUNCTIONS
2) To show D W J C CT(3), let us show first that for every w0 g W* and e > 0 the closed sphere in Wd with center w0 and radius e, ir(tuo,e) = {w £ Wd : poo(iu,u>o) < e}, is in cr(3)- Now N
Poo(u), w0) < e O 5 3 2~mpm(w, w0) < e
for every TV € N.
m=l
Thus X"(wo,e)= p |
L e W ^ ^ - ^ K ^ ^ E l
W6N I
m=l
J
Since tr(3) is closed under countable intersections, to show that K{w0, e) G <J(3), it suffices to show that | w S W* ; Yl 2-mpm(w, wo) < e l 6 0-Q)
(1)
for every TV e N.
Note that (2)
{w e W : pm(u>, u>0) < e} = {u> £ W : max |u;(s) - w0(s)\ A 1 < e}. sG[0,m]
If e e [1, co), then the last set is equal to W d , which is a member of 3- On the other hand if e 6 (0,1) then the set on the right side of (2) is equal to {io € W : max \w(s) - w<,(s)\ < e} = Se[0,ml
D
f]
{w eWd:
\w(r) - w0(r)\ < e]}
r6[0,m]nQ
q;\K(wo(r),e)),
r6[0,m]nQ
where Q is the collection of all nonnegative rationals and K(wo(r), e) is the closed sphere in Rd with center wo(r) and radius e, by the continuity of wo and w. Since q~} (K(wo(r), e)) is a member of 3 the last countable intersection above is in o-(3)- Thus we have shown that {w e W d : pm(w, wo) <e} e
for any w0 £ W , e > 0, and m 6 N.
Consequently we have {w G V?d : pm(w, wo) < e} = \J{w 6 W : pm(u;,u>o) < e - r 1 } e
368
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
and then (3)
{w e W : 2-mPm(w,
wo) <s} = {weWd:
Pm(w,
w0) < 2 m e} G
Let (ri,..., rN) be an N-tuple of positive rational numbers r i , . . . , r^ such that n + • • • + r w < e. Let Q be the countable collection of all such iV-tuples. Then \w G Wd : £ 2-> m (w,u>6) < e l
(4)
N
U
( 1 { » e W ' : 2 - > m ( u ; , wo) < rm } G oQ)
(ri,...,rN)6Qm=l
by (3). From this we have \weWi:Yi^mpm(w,wo)<e} =
H \w G W : Y, 2-mpm(w,w0)
< e + r 1 } G aQ).
This proves (1) and therefore K(wo, e) GCT(3)for every u>o G W d and e > 0. For an open sphere in W* with center wo G W* and radius e > 0, we have S(wo, e) = Uee^K(w0, e — l~l) G cr(3). Since (Wd, p^) is a separable metric space, it has a countable dense subset D. Every open set in Wd can be given as the union of open spheres with centers in D and with rational radii. Thus as a countable union of open spheres, an open set in Wd is always in CT(3). Therefore Oy/d C c(3). 3) By 1) we have <x(3) C 23w<<- By 2) we have 35 wd =
§18 Definition and Function Space Representation of So lution [I] Definition of Solutions Consider stochastic differential equations, that is, equations involving stochastic differen tials, of the type T
(1)
dX\=Y<x){t,X)dB1(t)
+ pi(t,X)dt
far»
= l,...,d,
§ 18. DEFINITION AND FUNCTION SPACE REPRESENTATION
369
where B = (J5 1 ,..., Br) is an r-dimensional {5t}-adapted Brownian motion, X = (X1,...,Xd) is a d-dimensional continuous {5«}-adaptedprocess on a standard fil tered space ( 0 , 5 , {5(}, P), and the coefficients aj and ft for t = 1 , . . . , d, j = 1 , . . . , r, are real valued functions on R+ x Wd satisfying certain measurability and integrability con ditions to ensure the existence of the stochastic integrals {/[01] a)(s, X) dB'{s) : t G R+} and {J[01] /3'(s, X)mL(ds) : t G R+}. The first and the second term on the right side of (1) are called the diffusion term and the drift term respectively of the stochastic differential equation. Note that at any ( £ R t the coefficients a) and /?' depend on X(-, u), not just X{t, UJ), for w G Q. The particular case of (1) where at any t G R+ the coefficients depend only on J ( ( , UJ), that is, a stochastic differential equation of the type r
(2)
dX\ = J2 a){t, X{t)) dB\t) + b\t, X(t)) dt for i = 1,..., d,
is called Markovian. Here aj and 6' are real valued functions on R+ x Rd satisfying some measurability and integrability conditions. As a further specialized case we have T
(3)
dX\ = Ytfi(X{t))dBi{f)
+ g\X{t))dt
far»
= l,...,d,
where /j and g' are real valued functions on Rd satisfying some measurability and integra bility conditions. This type of equation is called time-homogeneous Markovian. In (2), if a!- = 0 for i = 1 , . . . , d, j = 1 , . . . , r, then we have dX't=b\t,X(t))dt
fori =
\,...,d,
which is a randomization of a dynamical system. Definition 18.1. Let M.d ® R r be the collection of all d x r matrices of real numbers. We identify Rd ® RT with Rdr and consider the measurable space (Rd ® RT, ©m'*')Definition 18.2. Let MdXT(\Vd, Wd, {Wd}) be the collection of all Rd ® KT-valued {Wdt}progressively measurable processes a on the filtered measurable space (Wd,Wi, {Wd}), that is, every a in this collection is a mapping ofR+ x Wd into Rd ® Rr such that for every t G R+ the restriction of a to [0,i] x Wd is aff(Q3[o,ox Wd)/<8Wdr-measurable mapping of[0,t]x Wd into Rd®RT. Remark 18.3. According to Observation 2.12, a progressively measurable process is al ways an adapted measurable process. rThus if a G M d x r ( W , 2Bd, {Wd}) then
370
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
1°. a is aCT(Q3mtx Wd)/
a(t, w) = a(t, w(t))
for (i, w) G R+ x W*,
d
then a G M ^ ^ W ' ' , 2U , {2Hf }). In particular, if / is a SS^/OS^-measurable mapping of Rd into Rd ® R r , then the mapping a of R+ x W into Rd
a(t, w) = f(w(t))
is a member of Mdxr(Wd,Wd,
for (t, w) G R+ x Vfd
{Wd}).
Proof. 1) With t G R+ fixed, consider the restriction of a to [0,t] x R d . Since 5 H-> S for s G [0,£] is a 2$[o,t]/23[o,4]-measurable mapping of [0, t] into [0, t] and w i-> w(t) for w G W d is a QU^/SS^-measurable mapping of W* into R d , (s,w) >-» (s,iu(i)) is a o-(S3[0,t) x 2D^)/o-(©to,t] x <8ld)-measurable mapping of [0, t] x Wd into [0, *] x Rd. Then since the restriction of a to [0, i] x Rd is a cr(Q5[0,t] x S S ^ ) / ^ * - -measurable mapping of [0, t] x R^ into Rd ® RT, (s, w) h-> a(s, u>(<)) is a cr(<8[o,<) x 3Df )/25 B *-measurable mapping of [0, t] x W* into Rd
§18. DEFINITION AND FUNCTION SPACE REPRESENTATION
371
defined by X{ui) = X(-, u) for u> e fl is an 5/2U''-measurable mapping, and if X is adapted then X is &/20?-measurable for every t G R+. Let us show that for every t G R+, the restriction of a * to [0,t] x Q is a
(1)
dX't = Y, a K<> X ) dB'V) + £'(*, * ) <*<
fori=l,...,d.
(Ve 5ay fnaf fne stochastic differential equation has a solution if there exists a standard filtered space (fl,5, {5<}, P ) wM a pa;> of stochastic process (B, X) on it satisfying the following conditions. 1°. B is an r-dimensional {$t}-adapted null at 0 Brownian motion on (fl,5, {3t}i P)2°. X is a continuous d-dimensional adapted process on (fl, 5, {St}, P)3°. 77K: {$t}-progressively measurable processes (ax)) for i = l , . . . , d , j = l , . . . , r d e finedby(ax))«t,w) = aft, X{;ui))for(i,u) G R+x flare in L j ^ R + x f l . n n x P ) , and fne {5 ( } -progressively measurable processes (fix)' for i = 1 , . . . , d, defined by (/3xy(t,ui) = P\i,X(-,L>))for(t,u)) G R+ x flare m L,,00(R+ x fl,mL x P). 4°. 77iere exisfs a n«W «tf A G (fl, 3 , P) Jucn fnaf on Ac we have for every t G R+ X ' « ) - X'(0) = V / /
(ajr)JW <*PJ(s) + /
(fo)*(a)m L (ds)
372
CHAPTER 4. STOCHASTIC DIFFERENTIAL
In this case we say that (B,X)
EQUATIONS
is a solution of the stochastic differential equation.
Note that the fact that X' satisfies condition 4° above implies that X, is the sum of a process in MSj(Q,5, {&},P), a process in V C (Q,5, {5 ( },P), and an Jo-measurable real valued random variable and is therefore a quasimartingale on (£1, J , { 5 J , P). Hence the stochastic differential dX' is defined. Remark 18.7. If X satisfies condition 2° of Definition 18.6, then by Lemma 18.5, (a*)} and (fix)' are {&}-progressively measurable processes on (Q, J , {&}, P). 1) If (otxjj are in L2,oo(R+ X £2, m j x P) as required by condition 3° of Definition 18.6, then by Theorem 13.38 there exist equivalent processes which are {foj-predictable. Let us understand (ax)) to be {5(}-predictable versions so that the stochastic integrals {/[0]1](ax)j(s)cZPJ'(s) : t € R + } exist in M£(Q, 5, {5 ( },P). Condition^ in Definition 18.6 refers to these stochastic integrals. Note also that by Theorem 12.8 there exist sequences {(o»)j : n 6 N} in L 0 (fi,5, {5,}, P) such that lim ||(a")j - ( « x ) j f | ^ x P = 0. 2) If (/3x)' are in L loo (R+ x Q,TTIL X P ) as required by condition 3° of Definition 18.6, then the stochastic integrals {Jm](fix)'(s)mL(ds) : t G R+} exist in VC(Q, 5, {&}, P). By Theorem 13.38, (fix)' have {&}-predictable versions. Let us understand (fix)1 to be {&}predictable versions. Note also that by Theorem 12.8 there exist sequences {(&")' : n g N} in Lo(Q,5, {&}, P ) such that Jim 11(6")' - (fix)'\\?£F = 0. Whether or not a solution exists for the stochastic differential equation (1) in Def inition 18.6 depends entirely on the given coefficients a G Mdxr(Wd,Wi, {Wd}) and dxl d d d fi e M. (W ,W , {W }). We shall show this by showing that whenever the stochas tic differential equation has a solution on some standard filtered space it has a solution on the function space Wr+d
[II] Function Space Representation of Solutions Our objective here is to show that if the stochastic differential equation (1) in Definition 18.6 has a solution (B,X) on some standard filtered space (Q,5, {&}, P), then it has a solution (W, Y) on a standard filtered space constructed on the function space Wr+<J and furthermore (B, X) and (W, Y) have the same probability distribution on ( W + d , T3r+d). Proposition 18.8. Let Wr+d = W x Wd. LetWf^, W, and<md be the a-algebras gener-
8. DEFINITION AMD FUNCTION SPACE REPRESENTATION
373
ated by the collections of the cylinder sets, 3, 3', and 3" in WT+d, W , and Wd respectively. Then
(i)
zcr+d = a(w x 2nd).
Similarlyfor the a-algebras Wt+d, Wv and 2Df generated by the collections of the cylinder sets with indices preceding t G R+, 3[o,t], 3'[0,(]. and 3"0,i] in Wr+d, W , and Wd, we have (2)
Wl+d = a(Wt x Wd).
Proof. To prove (1), it suffices to show that (3)
To prove the first equality in (3), note that 3'x 3" C 3sothato-(3'x3") C <x(3). To show the reverse inclusion, let w 6 Wr+(* be given as w = (w1, to") with w' G W and tt>" 6 W*. For t e R+, let qt, q[, and gj' be mappings of WT+d, W r , and Wd into R1* defined by qt{w) = w(t) for w € Vfr+d, q't(w') = to'(i) for w' G W r , and q't'(w") = w"{t) for to" G W* Then for £ ' € QSm' and £ " G <8K„, we have ft-'C-E' x E") = (q'trl(E') x (tfr'C-E") 6 3 ' x 3 " Thus g( _1 (®s r x 93E<<) C 3 ' x 3 " and then by Proposition 13.2 and Theorem 1.1 we have ? r , 0 8 i ~ 0 = 9t~ V ( » r x
for (t, to) G R+ x W + d for (*, to) G K, x Wr+d.
374
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Suppose the stochastic differential equation (1) in Definition 18.6 has a solution (B, X) on a standard filtered space (£2,5, {&}, P). According to Theorem 17.8, the mapping (B, X) of i2 into W+d defined by (B, X)(u) = (£(•, w), X(-, w)) for w € £2 is ^/ST*''-measurable and 5(/23;+''-measurable for every t G K+. Let P(B,x) be the probability distribution of (B, X) on (W+d,Wr+d). We define a standard filtered space on the probability space (W+d, <mr+d, P0iX)) as follows. Definition 18.10 Consider the probability space (Wr+d, W+d, P(B,X)) where ( 5 , X) is a so lution of the stochastic differential equation (1) in Definition 18.6 on some standard filtered space. Let 1°. WT*d'* be the completion ofW*d with respect to P(p,xy 2°. Wrt+d'° = <j(Wrt+d U 91) where 9t is the collection of all the null sets in the complete measure space (W+d, STT^'*, P{B,X)l 3°- Wt+d'' = nE>02UI^'°. We then have a standard filtered space (Wr+d,W*d'*, {Wt*d''}, P{B ,X)) in which the aalgebra, the filtration, as well as the probability measure, depend on (B, X). Let us call this standard filtered space the standard filtered space on (WT+i, W+d) generated by P(B,X))Our aim is to show that with W and Y defined in Observation 18.9, (W, Y) is a solution of the stochastic differential equation on (yVT+d,W+d'*, {WC%+i>*},PiB,xy)~ This is done in Theorem 18.13 below. For this we show in Lemma 18.11 that W is a Brownian motion satisfying condition 1° of Definition 18.6. In Lemma 18.12 we show that ay and/?y satisfy condition 3° of Definition 18.6. Lemma 18.11. The r-dimensional continuous adapted process W on the filtered space {W+d,W+d, {W+d}) defined in Observation 18.9 is an r-dimensional {2Brt¥d'*}-adapted null at 0 Brownian motion on the standard filtered space (Wr+d, 2XT+,i'*, {'Wrt+d'*}, P{B,x))Proof. As we noted in Observation 18.9, W is an r-dimensional continuous {Wrt*d}adapted process on (W+d,Wr+d, {TT^}). Since Wl+d is a sub-cr-algebra of 2IJJ+d'*, W is a 2UTrf'*-adapted process on (W +rf , WT+d-*, {W^'*}, PlB,X))- To show that W is null at 0, let g0 be the arT^/B^-measurable mapping of W+d into Rd defined by q0(w) = w(0) for w e W + d . Then (1)
P(B,X){W0 = 0} = P{B,X){(W, F)(0) e {0} x Rd) = P{qo o (B, X) e {0} x Rd}
§18. DEFINITION AND FUNCTION SPACE REPRESENTATION
375
= P{(B, X)(0) G {0} x Rd} = P{B = 0} = 1 by the fact that B is an r-dimensional {5t}-adapted null at 0 Brownian motion on the stan dard filtered space (Q, #, {J ( }, P). Thus W is null at 0. To show that W is an r-dimensional {2rrT'i'*}-Brownian motion on ((W r+,i , 2DT+<''*, {Wt+d'*},P(B,x)\ it remains according to Theorem 13.26 to verify that for any s, t e R+, s < t, and y G W we have E[e^'w'~w^
(2)
\W;d-']
= e-^(i-s>
a.e. on (W+d, Wrs+d', P(B,X)).
Since the right side of (2) is W*d>* -measurable, it remains to verify (3)
/ jW-«U
dP(BX)
=
for A G 2^ + "'-,
P^X){A)e-^^
or equivalently according to the Image Probability Law (4)
ei^B'-B^dP
/
= P((B,X)-\A))e-il^it-s)
forAG2n! +
J(B,X)-*(A)
To verify (2), consider first the case where A G %8rs+d. Since B is an r-dimensional continuous {5(}-adapted process, B is an 5t/2UJ-measurable mapping of Q. into W for every t G R+ by Theorem 17.8. By condition 2° of Definition 18.6, X is an 5
E[e
'(^-^)|2n;^0] =e-^-s»
a.e.on(W+d,w;dfi,PiB,X)).
Now W;d'" = n £ > 0 2U^'° = n n6N 2n;;^°„. Let n G N be so large that s + l/n < t. Then (6)
E[e
' ( ^ - " W » > |2D^'*] = E t E t e ' ' ^ - 1 ^ ' / " ) | f f l C # , ] | 2 D ^
= E[e-T i(t - (s+1/n)) |23J: +
a.e. on (Wr+
where the second equality is by (5). Letting n —> oo in (6) and applying the Conditional Bounded Convergence Theorem we have (2). I
376
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Lemma 18.12. For the d-dimensional continuous adapted process Y on the filtered space ( W + d , W+d, {Wrt+d}) defined in Observation 18.9 and for the coefficients a and j3 in the stochastic differential equation (1) in Definition 16.8 for which (B,X) is a solution on a standard filtered space (£2,5, {&}, P), the {Wt+d,*}-progressively measurable processes (aY))fori = l,...,d,j = l , . . . , r , defined by (aY))(t,w) = a){t,Y{-,w))for (t,w) 6 R+ x WT+d are in L2]00(R+xWT+d,mLx P{B,X)), and the { Wt+d'* } -progressively measurable processes {^YY forl = l,...,d, defined by {ftyj% w) = f3\t,Y{-,w))for (t,w) £ R+ x W+d are in L, ,«,(]&+ x W+d, mL x P{B,X))Proof. We noted in Observation 18.9 that Y is a d-dimensional {2UJ+rf}-adapted process on (WT+d,Wr+d, {Ztrt+i}). Thus Y is a { 2 ^ ' ^ - a d a p t e d process on the standard filtered space (WT+d, W+d-*, {Wt+d-*}, PiB,X))- Then the mapping of y of W+d into Wd defined by y(w) = Y(-, w) for w g V/T+d is Wt+d/Wd-measurable for every t € R+ by Theorem 17.8. Thus y is WTt+d'/Wd-measurable for every t <E R + . From this we have the {Wrt*d'r}progressive measurability of (ay)) for i = 1 , . . . , d, j = 1 , . . . , r and ((Hyf for i = 1 , . . . , d by Lemma 18.5. To show that (aY)) are in L2|00(R+ x W+d, mL x P(B,xy), we show that for every t £ R+ we have (1)
j
\a)(s, Y(; w))\\mL
x P{B,X))(d(s, w)) < oo.
Consider the measure space ([0, t] x Q., o-(*8[o,i] x &), mL x P). Let t be the identity mapping of [0, t] into [0,t]. Since (B, X) is an fo/aiJ^-measurable mapping of Q into W + d for every t € R+ as we noted in the proof of Lemma 18.11, the mapping (i,(B, X)) of [0,t] x Q into [0,i] x W*d is
x P) = (m L x P) o (i, (5, ^ ) ) - ' ( P x P )
= (m L x P ) ( 0 _ 1 ( P ) x (B, XT\F) = mL{E) ■ P( fl ,x)(P).
= mL(E) -Po(B,
Xy\F)
This shows that the two finite measures P(lt(B,x)) and mL x P(B,X> on o-(©[o,t] x W[*d)) are equal on the 7r-class QS[0,(] x Wt+d and therefore equal on £T(33[0I(] X Wt+d)) by Corollary 1.8. Thus for any extended real valued o-(Q3[0,«] x 2UJ+ti)-measurable function
v(s,w)(mLxPiBiX))(d(s,w))
§;«. DEFINITIONAND FUNCTIONSPACE REPRESENTATION
377
in the sense that the existence of one side implies that of the other and the equality of the two. Now since w i-» to" for w = (to', to") e W*d is a W^/W* -measurable mapping of W+d into Wd for every t G R+, (5, to) i-» (5,10") for (a, to) € [0, t] x Wr+d is a <7(<8[0,t] x 2UJ+li)/(7(f8[o,(] x 2U?)-measurable mapping of [0, t] x W+d into [0, i] x Wd Then since (s,to") h-> aj(s,u>") for (s,to") G [0,f] x W d is a
/
\a)(s,X(-,u,))\\mLx
P)(d(s,uj))
J[o,t]x.n
=
j
\a](s,Y(;w))\2(mLxP{BtX))(d(s,w))
in the sense that the existence of one side implies that of the other and the equality of the two. But the right side of (3) exists and is finite since (ax)) are in L2i<:o(K+ x £2, mL x P). This proves (1). Thus (ay)) are in L2]00(R. x Wr*d, mL x P^B.X))- Similarly (/3y)' are in L^dtXF^mixPfBj)). ■ Theorem 18.13. Consider the r-dimensional continuous {WTt+d}-adaptedprocess W and thed-dimensionalcontinuous {Wt*d}-adapted process Y on(\VT+d, WT+d, {Wrt+d})defined by W(t,w) = w'(t) and Y{t,w) = w"{t) for w = (to', to") G W r+d , to' G W , to" G Wd, and t G R+. Suppose the stochastic differential equation (1) in Definition 18.6 has a solution (B, X) on a standard filtered space (Q, 5, {5<}, P). Then (W, Y) is a solution of the stochastic differential equation on (Wr+<J, 23 r+ti '*, {Wt+d'*}, P(B,X)) in Definition 18.10. Furthermore the two processes (W, Y) and (B, X) have identical probability distributions on (W+d, © w ^ ) and the two processes Y and X have identical probability distributions on (Wd, 93w*)Proof. Since (B,X) is a solution, there exists a null set A in (Q, 5, P) such that on Ac and for every t G R+ we have X\i)
- X'(0) = V /
(axfAs) dB\s) + f
(Px)\s)rnL(ds)
for i = 1 , . . . , d,
or equivalently, writing (ax)) • BJ and (/?*)' • mL respectively for the stochastic integrals {Jl0:t](ax))(s)dBj(s) : ( £ « + } and {Jl0:t](Pxy(s)rndds) : t G R + }, we have (1) «F(t) = YMx)) 3=1
. B>)(t) + ((fixf . mL)(t) + X\0) - X\t)
=0
for i = 1,... ,d.
378
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
By Lemma 18.11, W is an r-dimensional {2UJ+,i'*}-adapted null at 0 Brownian motion on the standard filtered space (W r+d ,W +d '*', {Wt+i*},PiB,x))By Lemma 18.12, Y is a ddimensional continuous {2U[+
for t = 1 , . . . , d.
Since Y, (ay)J • W3 and (/3y)' • m£, are all continuous processes, it suffices to show that for every t e K+ there exists a null set At in (W+d, W*d'*, PfB,X)) such that on A£ the equality (2) holds. Now if we show that for every i = 1 , . . . , d, the random variable
fMmllWJ- ( « * « " * * « 0, \nlirn||(^)'-tfy)"||^xP'^'=0.
Thus by Definition 12.9, we have lim |(a K )'; • W J - (a y Y. • W j l ^ = 0 in the space n—+oo
*'
j
■
r
M|(W + ' i ) aU r+ '''*, {Sa^+,l'*}JP(B,x)) and then by Remark 11.5 for every t G R + we have
§ 18. DEFINITION AND FUNCTION SPACE REPRESENTATION
379
J™, I D ° ? ) j • Wi - £(<*y )j • w' I1 = 0, in other words,
i ™ / W r ^ I E « Q y ) ; • WJM
~ fftar))
• Wj)(t)\2dP(B,X)
= 0.
Similarly we have
■S3* jCm l(('S?)i " mi)(t) ~ ((i3y)' * m i ) ( < ) l dP(fl';f) = °" Since convergence of a sequence in L 2 (W + '', W+i, P(B,X)) implies convergence in Li(W r+d , 2DT+d, P(B,jf)) which then implies convergence in the probability measure P(B,X), the last two equalities imply
(4)
P(B,X) ■ Jim
£ ( ( a j ) j • W J )t + ( ( # ) ' • ™L)* + *o " 1? I
= E ( ( a y ) j • W"")i + ((A') 1 • mL)t + Yj - Yt'. 3=1
Now since (<*£)} and (/??)'' are in L0(Wr+<<, 2rT+li'*, {ZOl**'*}, P(B,X)) we have (aJXa, w) = (/ 0 n )>)l { o}(s) + £ ( / £ ) > ) ! / „ > (s), ArgN
C8?)(a, IB) = (5o")'(^)l{o}(5) + E ( ^ ) > ) l / „ . k ( s )
for
(5> ^ ) G »U x W+d,
fceN
where /„_* = (tn,k-i,tn,k] for n 6 N; {t„ it : k £ Z+} are strictly increasing sequences in R+ such that tn 0 = 0 and lim *„,* = 0; {(/£)' : fc 6 Z + } and {(g£)* : k 6 Z + } are bounded it-+oo
sequences of real valued random variables on (WT+d,Wr+d'*, PiB,X)) such that (/£)} and (g$y are 2D£*"-measurable and (ft)) and (g£)' are 2D££,-measurable for fe 6 N. If we replace (/£)' and (
CHAPTER 4. STOCHASTIC DIFFERENTIAL
380
EQUATIONS
in,*_i by Theorem 17.8. Thus (/ t n )j((B, # ) M ) and (g£)''((B, X){u)) foru> g Q are & n measurable. Then the random variables (a£)j and (&£)' on (Q, 5 , P) defined by f(flZ)j(«) = (/?)j((B,^)(«)), I (*!)'<«) = (ff*y((B, * X « » for w e f t m e a s u r a r j l e ;. This implies that the stochastic pro< are 5 t n f c _ r - This implies that the stochastic processes (<*£)} and (/?£)' on the standard filtered space; ((£2, {&}, F ) defined by " , 5J , {&},-P) (6)
(7)
1
(o&)(a,w) = (a$(w)l{o>M + E ( a S ) 5 ( u ) l ^ M , * eN (/9J)(a,w) = (65)'(w)l {0} (s)+ X > D V ) l i „ , * ( s )
for(a,w) g R + x O ,
fceN
are in L 0 (fi, 5, {&}, P). Let us show
Um||(ai)}-(^)}||™ix^-x,=0, LXP,£
°'
\
i-
II , i n M '
V » |i| W H XxP ( B J( .fX )
yo
Tlim||O3jy-09^)
||^ ^ '=O.
Note that according to Remark 11.18, the first of these two expressions are equivalent to the condition that jirn ||(oft)j - (a^)jll2!*iXP<s'x> = 0 for every t g R+. Now for fixed t g R+, letting tnj)n = t for n g N, we have |n>iXP( B | X)
=
/
|(o5)j(w)l{0}(3) + £(aJ)j(w)l/ B 4 (a) - aj(a,X(-,a;))| 2 (mL X P)(d(a,w))
•/[O.flxJJ
=
/
fca
'
|(y?)j(w)l {0 }(s) + E ( / j f ) j ( « ) l w ( a ) - a)(s, w")\\mL ixr-
(B| = \\{^ Il(^)*-(«y)*ll^^ Y))-{ay))\\ir
x P ( s,x))(d(s, w))
x)
by the Image Probability Law. But according to Remark 11.18, the first equality in (3) is equivalent to having Vbx^ ||(o£)j - ( a y ) } | | ^ x P ( B ' x l = 0 for every t 6 1+. Thus the first equality in (8) holds. The second equality in (8) is proved likewise. From (8) we derive (9)
P • Jim I £((orJ)J • B>)t + ((/?£)< . mL)t +X*- X\ 1 £ ( ( e r x ) J • B>)t + ((fey • rnL)t +Xl03=1
X\
381
118. DEFINITION AND FUNCTION SPACE REPRESENTATION
in the same way we derived (4) from (3). We show next that the sum on the left side of (4) and the sum on the left side of (9) have identical probability distributions on (R, 58«), For brevity, let these two sums be denoted by SfWY) and S"B X ) respectively. Then by (5), we have (10)
S(V,y,(u>) = Y,((-aY))»W3)t(w)
+ ((l3Zy»mL)t(w)
+
YJ(w)-YtXw)
= ££c/D>){wt,>)-- < * ..,(»)} .7=1
i=i k=\ j = l k=\
+ X K ) » ( * * , * - t«,k-i) + Yj(w) - Yt'(w)
for w G W r+d ,
/fc=l
and similarly by (7) and (6) we have (11) SfciX)(u)
= £((<**); • BJ)t(")
+ ( ( « ) ' • mL\{u)
+ Xj(w) - X\{w)
i=i
= EE(/?)}((S, *)(")){<>) - #„,,_,(«)} j = i fc=i
+ lL(9tf(.(B,X)(uWn,k
-*»,it-i) + Xfa) - Xi(.w)
forwGQ.
fc=i
Consider two (2r + l)(p n + 1) + 2-dimensional random vectors V(w,Y) space (WT+d, mr+d'', P{B,X)) and V{B,X) on (Q, 5, P) defined by
on tne
probability
f W > = x;., x^w/ n i xj=1 x £,(/£)■ • x^o^)' x yt'n0 x Ylpn, > = X; ; =1 X £,(<$)} \[ W ViB,x) X^==11 X X C^ B B j ^^ t X J=1 fcZo«)j •• X CCftgy x JCJ^ x X(' Let us show that their probability distributions P{B,X) ° V
^ W ) ° Vr(Wly)(A) == PVBJCWU
^
Wl,kr\K,k)
■ n;=, n£„ ( W d ^ )
• nEo^)-')"'^.*)n (^.r'Cifn.o)n ( t f r ' t f l ^ ) )
CHAPTER 4. STOCHASTIC DIFFERENTIAL
382
EQUATIONS
and
p°y(tx)iA) ■--- p
n (jr;'npn )-'(#„,,„)).
Since Bfnfc(w) = W/n ,,((£, #)(w)) and X*n t (w) = r/ n k((B, X)(tv)) for w e Q. we have
(Binkr\Ei)k)
=
{B,X)-\Wik)'\Elk\
and
(x-n tr'(.ff„,t) = (tf^r'a^r'cff*,*)Also from (6), we have
( ( O i r ' c o =«/?)} o (s, A - ) ) - 1 ^ ) = (B, A-)-1 o ((/,")})-'(<,) and similarly
((&D'')_1(Gifc) = (B,Xyl o ((^)')"'(Gn,fc). Substituting these equalities in the expression above for PoV^X)(A) and recalling P(B,X) — Po(B,X)-\ wehaveP^.xjoVJ^y^A) = PoV(B]X)(A) Now since X ^" l l , " + 1 , + 2 ®m m isa 7r-class and the
X
r
\s Pn
j=l
A
j
wr
k^n.fc ■ A j=l
w Pn A
j
k=oVn,k
w Pn '
A
k=0Zn,k
X V„,0 X V n , p n
w i t h x ^ , y£_fe, £„,*, un,A G R. Let us define a mapping T of R ^ 0 ^ 1 ' * 2 into R by setting T(XTj=] X^ 0 i J „ f c • Xj = 1 X ^ 0 ^ i J ; • Xpkn=0zn,k x un,0 x vntPJ T
=
Pn
H H
pn
Vn.k&i.k -
x
i,k-l)
+ H
Zn,Ar(in,A ~
Clearly T is a *Bm(2r*L,(p„*iw)/2Sm-measurable mapping of R(2r+,)(p"+1)+2 into R. Thus T o V^w,Y) and T o T^B.XJ are real valued random variables on (WT+d, 2I^+'',*, P(B,X)) and (£2,5, P ) respectively. Furthermore by (10) and (11), we have T o V(W,Y) = S£WY) and T o V{B,x) = S^B,xy T n u s f o r t n e probability distributions of S^wy) and S ^ X) on (R, 3Sm),
383
§ 18. DEFINITION AND FUNCTION SPACE REPRESENTATION
which we denote by ^ S( v, y , and ns^X), we have ^ t Y ) = p(fl,*> ° ( T ° % ) ) " ' = ptf,x) ° ( % « ) " ' o T-1 and /uSnflx) = P o'(T o V^,*)) - 1 = P o (F(B,X)) _I ° T" 1 But P ( S , X ) o (T-W.y))-1 = P o (VJB,*)) - 1 on Q3i as we showed above. Therefore HsnWY = A's^x o n (R, 23m). Let nn, n e N, and ,u be probability measures on a measurable space (S, 53s) where 5 is a metric space and 23 s is the Borel a-algebra of subsets of 5. We say that fin converges weakly to // and write w ■ lim fxn = fj, if lim / / d^„ = j f dfj, for every real valued bounded continuous function / on 5. The limit of weak convergence of probability mea sures is unique in the sense that if v is a probability measure on (5,03s) and w ■ lim un = \i n—t-OO
as well as w ■ lim p,n = v, then yu = v on (S, Q3S). If £n, n £ N, and £ are S-valued random variables on a probability space and their probability distributions on (5,23s) are denoted by fin, n £ N, and /J respectively, then we say that £„ converges in distribution to £ if w ■ lim /i n = fj,. If £n converges in probability to £, then £„ converges in distribution to £. Now according to (4), the random variables S™WY) on (Wr+d, W+d-\P(B,x)) denned by (10) converge in probability P(B,X) to the random variable *F*(t) defined by (2) and there fore their probability distributions ^ s " , on (R, 93») converge weakly to the probability distribution of *¥'(t). On the other hand according to (9), the random variables S"B X) on (Q, 5, P ) defined by (11) converge in probability P to the random variable &{t) defined by (1) and thus their probability distributions fj,s"B on (R, Q3i) converge weakly to the proba bility distribution of O' (i). But us" v = A*S" v on (R, 03m) for every n 6 N as we showed {Vv, Y)
(fl,X)
above. Thus by the uniqueness of limit in weak convergence of probability measures the probability distributions of *F'(i) and
384
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
for A e ©vv- This shows that Y and X have identical probability distributions. ■ As a converse of Theorem 18.13 we have the following. Theorem 18.14. Let B be an r-dimensional {&} -adapted null at 0 Brownian motion and X be a d-dimensional {$t}-adapted continuous process on a standard filtered space (Q,3, {$t},P). Let PiB,X) be the probability distribution of(B,X) on (Wr+d, W+d) and let (Wr+d, <WT+d'*, {Wt+d' *}, P(B,X)) be the standard filtered space generated by P(B,x) as in Definition 18.10 and let W and Y be two processes on this filtered space defined by setting W(t, w) = w'(t) for (t, « i ) £ 8 t x W+d Y(t, w) = w"(t) for (t, w) e K+ x W+d, where we write w = (to', w") with w' 6 W and w" € Wrf. If(W, Y) is a solution of the stochastic differential equation (1) in Definition 18.6 on (WT+d, W*d'*, {W*d>*}, P{B,X)X then (B,X) is a solution of the stochastic differential equation on (Q, 5, {dt},P)Proof. To show that {ax)) € L2l00(R+ x £l,mL x P), note that since (W, Y) is a solution of the stochastic differential equation, we have (ay)* e L2,oo(R+ x W +
Jm.
Then by the Image Probability Law as in the proof of Lemma 18.12, we have /
\a)(s,X(;u>))\\mL
x P)(d(s,u:)) < oo,
and thus (ax)) € L2|0o(R+ x £2, m L x P). Similarly from (/?y)' G Lli0O(R+ x W + d , m L x P(B,X)) we have ( f e ) ' € L1|00(R+ x Q, m L x P). Then the stochastic integrals (a*)*- • Bj and (,9x)' • n»x, exist. For t £ R+, let
**'(*) = £((<*4 . B>){t) + «fixy . mL)(0 + X\0) - X\t)
for i = 1,..., d,
and r
TO = £ ( ( « r ) } • W")W + (OW* • m £ )(i) + F ; (0) - F*(i) for i = 1 , . . . , d. The fact that $*(*) and T*(<) have identical probability distributions on (R, 55K) can be verified by the same argument as in the proof of Theorem 18.13. Since (W, Y) is a solution
§ 18. DEFINITION AND FUNCTION SPA CE REPRESENTATION
385
of the stochastic differential equation we have 4"'(t) = 0 a.e. on (W+d,W+d,P(B,x)) and thus *'(<) = 0 a.e. on ( Q , 3 , P ) . This shows that (B,X) is a solution of the stochastic differential equation on (£2,5, {5t}, P). ■ Combining Theorem 18.13 and Theorem 18.14 we have the following. Theorem 18.15. Let Biq) be an r-dimensional {5j*'}-adapted null at 0 Brownian motion and Xiq) be a d-dimensional {g*5)} -adapted continuous process on a standard filtered space (£2W, 2H(,), {3[q)}, pM)for 8 = 1 , 2 . 7/(P (1) , X<1>) w a solution of the stochastic differential equation (I) in Definition 18.6 on (Q (l) , 2U 0 ', {5',1'}, P (1) ) and (B (1) , Xm) and (BG\ X<2)) ftave identical probability distributions on (Wr+d, WT+d), then (P <2) , X (2) ) is a solution of the stochastic differential equation on (Q<2), 2n(2), {gf >}, P (2) ). Proof. Let P/B W ,X<") b e t h e probability distribution of (P (9) , X<»>) on (W + l i , 2Tr+,i). Considerforg = 1,2 the standard filtered space (Wr+<J, 2Ir+'i'*'(5,, {2n?+'i'*'(')},P(B)^)) generated by P(B(,, x ( „ . . Since P/gm x'»i = MBO I B I by o u r assumption these two standard filtered spaces are identical. Let W and Y be two processes on the standard filtered space defined by W(t,w) = w'(t) and Y(t,w) = w"(t) for (f,w) G R+ x W r+,i , where w = (w',w") with to' € W r and u>" 6 W d Since (P ( 1 ) ,X ( 1 ) ) is a solution of the stochastic differential equation on (Q (1) , 3U*1*, { $ " } , P (1) ), (W, Y) is a solution of the stochastic differential equa tion on ( W + l i , 2 i r + ' w , ) , {2UJ+d,*'(,)},P((B)Hs,) by Theorem 18.13. Then by Theorem 18.14, (B (2) ,X (2) ) is a solution of the stochastic differential equation on (Q(2,,2U(2), {# 2 ) },P ( 2 ) ).
[Ill] Initial Value Problems In Theorem 18.13 we showed that if the stochastic differential equation (1) of Definition 18.6 has a solution (B,X) on some standard filtered space (£2,5, {5 ( },P), then (W,Y), where W and Y are defined by W(t, w) = w\t) and Y(t, w) = w"(t) for i € R+ and iw = (u>', w") G W r+,i with to' € W r and w" S W , is a solution of the stochastic differ ential equation on CNT+d,wr+d'*, {Wrt+d'*},P(B,x))- Let p be the probability distribution on (Rd, Q3md) of the d-dimensional random vector X0 on (Q, #, P). We shall show that for a.e. x in (Rd, <8Ed, /z), the mapping (W, F) is a solution on a certain standard filtered space (W*d, W+d'*'x, {2nj+,J'*'*}, P(B,A-)) o f t h e initial value problem dXt = a(t, X)dB(t) + P{t, X)dt = x.
XQ
386
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Observation 18.16. Consider the solution (W, Y) of the stochastic differential equation on (Wr+d,W+d'*,{Wrt*d-*},P&,x)) in Theorem 18.13. Now Y is a d-dimensional {Wr*i}adapted process on (W+d, W+d, {Wt+d}, P(B,X)) and Y0 is a 23K+,i/<81d-measurable map ping of Wr+
on 2X1 x R satisfying the following conditions according to Definition mjr+d ly-
1°. There exists a null set N\ in (R , 23]!^,^) such that f(B,x) sure on QXF+ti for every x 6 TVf.
(">a;) ' s a probability mea
2°. P ( ^ ) ' y ' 0 ( A , ■) is a 53md-measurable function on Rd for every .4 e Wr+d 3°. P C B, X) U n y 0 "'(G)) = fG P(B,7)|y°(A> *)/*(<&) for every A G 2rr+rf and G G 93K«. By Theorem C.14, for an extended real valued random variable £ on (W+d, Wr+d, P(B,x))> we have for every G G 93m
4
i, G) « dp ^>=j£ LL « ^ > | r o ^ > *> fi(dx)
°-
in the sense that the existence of one side implies that of the other and the equality of the two. According to Theorem C.15, there exists a null set N2 in (Rd, Q5md, (j.) such that •P(B7)IV0"1(W),X)=1
5°
for*eJV|.
Let NYo = Nil) N2. Let us define (i)
■P(B,Jf)(') -
P&jc) "(■.*) lO0
for x<ENYo forxeN forxeNYo.
Thus defined, P ( s, X) (A) is a <8Ed-measurable function of x on Rd for every A e 23T+d, and •PfB^jC)is
a
measure on 2IT+'i for every x eRd and in particular for x g JVfo, P £
X) (-)
equal to the probability measure P ( B i J 0 ' "(-, x) on 20r+d. From 3°, 4°, and 5°, we have (2)
P(B,X)(A n Y0~\G)) = I JG
P^X){A)ii{dx)
is
§ 18. DEFINITION AND FUNCTION SPACE REPRESENTATION
387
for every A in W*d and G G *8md
(3)
JL*e dPiB-x)=I L L « w >^ .***»>
/i(dx),
in the sense that the existence of one side implies that of the other and the equality of the two, and (4)
^WV({*}))=1
forzGJTC.
Definition 18.17. For each x G N$o, let 1°. 2Ur+d'*,x be the completion ofW"^ with respect to P,
(B,xy
+d
x
d
2°. Wt '°' = a{Wt* U 91) where V\ is the collection of all the null sets in the complete measure space (Wr+d, W+d-*'x, PfeiX)).
3°. mrt+d-*'x =
ne>0wtHAl
Thus (WT+d, <SJr+d-*-x, {Vfft*d'*'x}, PxBiX)) is a standard filtered space for each x 6 Np0. Observation 18.18. The BorelCT-algebraWd of subsets of W* for any d £ N is countably determined according to Proposition C.8 since Wd is a separable metric space. We show here that for any t G R+, the
max \vi(s) - v2(s)\
se[o,<]
forui, v2 G V*,
V* is a complete separable metric space. Let 2J be the Borel
(v(ti),...,
v(tn)) G Ex x • ■ • x £ „ } ,
where 0 < t\ < ■■■ < tn < t andEi G ©»for i = l , . . . , n . The fact that 23 =
388
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Let T be the mapping of Wd into Vd defined by letting T(w) be the restriction of to 6 W d to [0, *]. For T = {tu ..., tn) where 0 < t\ < ■ ■ ■ < tn < t and E = E\ x ■ ■ • x En where Ei G QSmforz = l , . . . , n , let ZTiE = {w € W : (to(ii),..., w{tn)) € Ex x • • • x En) € 3[0,fl CT,E = {» € V : (w(*i),.. •, »(*»)) G JS?i x ■ ■ • x £ „ } G C Clearly T(ZTtE) = CTyE and T " ' (CT,g) = ZT,£. Thus T establishes a one-to-one correspon dence between 3[o,t] and £ This implies T _ 1 (£) = 3[o,<) and then T-'(QJ) = r ' ( a ( 0 ) = <7(T-'(£)) = a(3 [0 , t] ) = 2D?. This shows that T is a 2ITf /93-measurable mapping of W* into V d . Now since \d is a separable metric space, its Borel IT-algebra of subsets 93 is countably determined according to Proposition C.8. Let CD = {Dn : n G N} be a countable collection of determining sets for 93. Let <S = { £ n : n G N} where £ „ = T~l(En) for n G N. Let us show that <E is a countable collection of determining sets for Wd. Let Pi and P 2 be two probability measures on (Wd,Wd) and suppose Pi = P 2 on (£. Let (Ji and Q2 be the probability distributions of T on CVd, 93) relative to the two probability measures Pi and P 2 on (Wd, 2Uf), that is, Qi = Pi o T" 1 and Q2 = Pi o T" 2 on 93. Now the fact that P[ = P 2 on <£ implies that Qi(D K ) = Pi o T~\Dn) = P , ( £ J = P2(En) = P 2 o T~\Dn) = Q2(Dn) for every n G N. Thus Qi = Q 2 on S3 and therefore Qi = Q2 on 93 since 33 is a countable collection of determining sets for 93. Take an arbitrary A G 223?. Since 20? = T _1 (93), there exists B e 93 such that A = T~\B). Then P,(A) = Pi(T-'(B)) = Q,(P) = Q2{B) = P 2 (T-'(B)) = P 2 (A). Thus Pi = P 2 on 2U?. This shows that £ is a countable collection of determining sets for
233?. ■
Lemma 18.19. There exists a null set N in (Rd, 9 3 ^ , fi) such that for every x G Nc, the rdimensional continuous adapted process W on (WT+d, Wr+d, {WTt+d}) is an r-dimensional {Wrt+d'*'x}-adapted null at 0 Brownian motion on the standard filtered space (Wr+d, W+d'*'x, {Wrt+d'*'x}, P
§ 18. DEFINITION AND FUNCTION SPACE REPRESENTATION
389
Thus W is a 2UJ+li'*'3;-adapted process on (W + d , SZTT^'*'1, {Wtu'*>x}, PXB,X)) for x € N$0. We show next that there exists a null set iV0 in (Rd, 53Md, jj.) such that for x G JV^, W is a null at 0 process on (Wr+d, W+i"*'*, { 2 ^ ' * ' }, Pfe,X)l Now since P £ i J f , is a ©^-measurable function of x on Rd, integrating with respect to fi we have
/Kd P{XB,X){W0 = o}M(dx) = PowW-'ao}) n y0-'(Rd))
(i)
^(fl,X)(W 0 - 1 ({0}))=l,
=
by (2) of Observation 18.16 and (1) in the proof of Lemma 18.11. Now since PxBiX){W0 = 0} G [0,1] for i 6 l ' and since JKd fi{dx) = 1,(1) implies that there exists a null set NQ in (Rd, *8Kd, ^) containing the null set jVf0 such that P^tXy{ Wo = 0} = 1, that is, W is null at 0 when x G Nfi. To show that there exists a null set JV in (R1*, 2 3 ^ , /i) such that for every x £ Nc the process W is an r-dimensional {2UJ+,i'*'x}-adapted Brownian motion on a standard filtered space (W +
E[e^'w'-w->
\Ws+d-*'x] = e-^{t-s)
a.e. on (W+d,2U^+rf-*^, P ( ^, X) ),
that is, 0)
/" e %,w,-w.) ^
=
P(^|Jf)(A)e-^(t-s)
for A G 2D;**-*,
JA
Now for A G 2 0 ^ and G G
e^w'-w'HAdP^x\^dx)=j
/ { /
,
e^'-^UdPuw
by (3) of Observation 18.16. By Lemma 18.11, W is an r-dimensional {2D[***}-adapted Brownian motion on QNr+d,Wr+d'*, {2U^**},P(/B,X))- Also since A G Ws+d and since Y0-\G) G W^, we have A D y 0 "'(G) e 23^+,i C 20;***. Thus by (3) in the proof of Lemma 18.11 and (2) of Observation 18.16, we have (5)
j
e*H''-^
JAnY-'(G)
= e-^-s)
j PxB%x)(AMdx). JG
390
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Combining (4) and (5) we have for A 6 20"'' and G € 23K<<
la { j L ^ ' " ' ^ <"$.*>} *<»*) = e-^* jQ PfrjcMMdx). Since this equality holds for every G £ 23,^ and the integrands are 23Kd-measurable func tions, there exists a null set Naj,y,A in (Rd, 9 3 ^ , A*) containing 7V0 such that for z G NZ,t,v,A we have (6)
/
e'^-w'hAdP^BX)
=
e-^-s)P^X){A).
Let 33 be a countable collection of determining sets of the countably determined a-algebra Ws+d. Let Q+ be the collection of all nonnegative rational numbers and let Qd be the col lection of all rational points in Rd. Let N = Us,ieQ*,s
= B -*JrV«)
ae. on
(W^'ZO^'0'*,?^,).
Then by same argument as in the last part of the proof of Theorem 18.11 we have (2). ■ According to Lemma 18.11, W is an r-dimensional {20J+li'*}-adapted null at 0 Brownian motion on (W+d,W+d'*,{Wt+d'*},P(B,x))According to Lemma 18.19 there ex ists a null set N in (K , 9J K d,^) such that for every x £ Nc, W is an r-dimensional {2ij^«.s}_ a daptednull at OBrownianmotion on (WT+d,mr+d'*'x,{Wrt*d"*'x},P^BiXyy. If
§ 18. DEFINITION AND FUNCTION SPACE REPRESENTATION
391
{p(B,X) ■ f[o,t]^)dWj(s): t £ R + } respectively. We write | | S f l x ' and | - | 2 a , x l for the quasinorm | !«, in Definition 11.4for the two spaces Ml(Wr+d,W+d'*, {Vtrt+d'*},P(B,x)) and MXW+d,Wr+d'*'x, {Wt+d^x},PfetX)) respectively. Note that if the process
'
f P*B,X) ■ (ay)) • W> = PiB,X) ■ (ay)) . W*, \ P?B,X) ■ (fir)1 • mL = Paw ■ (PYY • mL
on AJ.
Proof. According to Lemma 18.19 there exists a null set N{ in (Rd,*8md,fj.) such that for every x £ JVf, W is an r-dimensional {22JJ+,i'*':c}-adapted null at 0 Brownian mo tion on (Wr+d,Wr+d'm'x, {Wt*d'*'x}1P^B:X)). Since Y is a d-dimensional {23J£+
= / , [ / AI 4
=
d
\(aY))\2d(mL
x PfBiX)) fi(dx)
\(ay))\ld(mAdP^)) J
Jm Uv/'-* U[o,m] > I \i \(<XY))\2d(mL)]dP(B,X)) Jy/r+d L/[0,m] 1
fi(dx)
1
392
CHAPTER 4. STOCHASTIC DIFFERENTIAL =
\(aY))\2d{mL
/
EQUATIONS
x P(B,X)) < oo,
where the first equality is by applying the Fubini-Tonelli Theorem to the measurable process (aY)), the second equality is by (3) of Observation 18.16, and the finiteness of the last integral is by (1) in the proof of Lemma 18.12. Thus there exists a null set in (Rd, QS^, fi), whose union with iVj we call N2, such that /
\(ay))\2d(mLxP^BtX))<^
„
for all m G N when x G JVf. This shows that (ay)) is in L2]0o(lR+ x Wr+d, mL x P*B%X))
forxeNi. According to Lemma 18.12, (aY)) is in L2|00(R+ x Wr+d,mL x P(B,xy) SO that there exists a sequence {(ay)) : n G N} in L 0 (W + , i , 2Dr+
(2)
Recall that (aY)) can be chosen from L 0 (W r+ti , snT"*, {Ttrt+d},P{Btx)) as we showed in the proof of Theorem 18.13. By Definition 12.9, forthe stochastic integral P{B,xy(ay))»W:lof (ay)) with respect to the {Wt+d'*}-adapted null at 0 Brownian motions W3 on the standard filtered space ( W + d , 2nr+
Jim \(aY)) . W - P(B<X) ■ (ay)) . Wi |2 f l ^' = 0, or equivalently, lim \(aY)) • Wj - P{B,x) • (ay)) • W-7' |£<*•■*> = 0 for every m g N by Remark 11.5, that is, lim / n
—°°J[0,m]
(aY))(s) dW3(s) = Pm X) ■ f '
'
(ay)Us) dWJ(s) in J
J[0,m]
L 2 (W r+ '',2n r+d, *,P (B ,x)) for every m G N. Now convergence in L2 implies the existence of a subsequence which converges a.e. Note that a null set in (W + ( i , 2D r+i, * ) P(S,jf)) is always a subset of a null set in (W r+d , 2n r+,i , P (S ,x)) since W+d'* is the completion of W+d with respect to P(B,X))- Thus there exist a null set A in (W r+d , Wr+d, P(B,XI) and a collection of subsequences {(m, fc) : k £ N} of {n} for m G N in which {(m + 1, k) : k £ N} is a subsequence of {(m, k): k G N} for m G N such that (3)
lim/
(4 m '' : ) )K5)^W(5) = P(B,X)- /
fc—oo J[0,m]
J[0,m]
(aY))(s)dW3(s) '
§ 18. DEFINITION AND FUNCTION SPACE REPRESENTATION
393
for every m € N on Ac. On the other hand (
(4)
k)
U<x ?' 11(4"
■p (B,X) ))-{ay))\\l£ ■i<*Y)%Zy ™
{/ L-'ro.n
i i d LAv"i
P
\{af'% - ( a y ) } | : dm,L \
U[0,r>
JW1 U[0,m] x W "
^ | ( « ^-
dP{B,X)
{aY))\2dmi<\dP*B,X) M<&0
(a<^>)' --(aY))\2d(mL
xP(B,X))
Htfx),
where the second equality is by (3) of Observation 18.16. Since (2) is equivalent to Jirn^ \\{aY)) - (aY))\\™n*FiB,X' = 0 for every m 6 N by Remark 11.18, (4) implies that there exists a null set in (Rd, V8Md,(i), whose union with N2 we call N3, and a subsequence {(m,e) : £ G N}of {(m,k) : £ G N} for m G N chosen so that {(m + l,£) : £ G N} is a subsequence of {(m, £) : £ G N} such that
& L w - |(a^(s) - (a^2dimL x ^ = ° for every m G N when i G JVJ. If we take the diagonal sequence {(£, ^)} and write {£} for this subsequence of {n}, then we have lim /
«—oo J[0,m i W " '
|(4)}(S) -- ( ^ • ^ ( r r i i X J
-P(B,X)) = = 0,
that is, lim \\{aY)) - ( a K ^ I I ^ ^ ' ^ = Ofor every m G Nwhenz G N$. Thus by Remark 11.18 we have
(5)
Hm ||(4)" - MC"*"*' = °
for
* e AJ.
Recall that our (a?-)" are in Lo(WT+i, W*d, {WTt+d},PlB,x))- Since the spaces L 0 do not actually depend on the probability measure in the underlying filtered space, (a y )} are also in L0(Wr+d, Wr+d, {W*d},P*B,X)) f o r x e Ni- T h u s b y Definition 12.9, (5) implies lim | ( 4 ) 5 . W1 - FfBXy ■ (ay)} • Wi \PIB-X) = 0
for x G N^
I—*oo
and then by Remark 11.5, lim / ' +d
+d
:r
L2(W ,%IF '*' ,Pm
<-H»./[0,m] r
(aY))(s)dW3(s) '
= P(^X) • / J[0,m]
(aY))(s)dW1(s)
in
J
X)) f° every m G N when i G N%. Since convergence in i 2
394
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
implies the existence of a subsequence which converges a.e. and since a null set in (W+d, W+d'*'x, P[BX)) is always contained in a null set in (W+d, WT^, Pfe.x)), f o r e v e r y x G 7V3C there exists a null set A^ in (W*d, W+d, P*B,x->> a n d a subsequence {k} of {1} such that (6)
lim /
(4)}(s) dWj(s) = P(B X) ■ I
^ W
for all m G N on (A^.)c when x g N$. From (3) we have (7)
lim/
( 4 ) k 5 ) r f ^ ( 5 ) = P (jB , X )- /
*->oo J[0,m] c
(ayKW^W
J[0,m]
for all m G N on A . Now by (2) of Observation 18.16, we have
(8)
J^ P{%X}(A)fi(dx) = P{B,X)(A) = 0.
Thus there exists a null set N4 in (Rd, <8Md, fi) such that P*BtX)(A) = 0 for a; G JV|. Let N0 = N3U N4. For every z G iV0c, let A , = A ^ U A. Then for every x G iV£, Ax is a null set in (W r+d , 22r+
(*Y)){S) dW\s)
= P(B>X) ■ I
(ay)'(s)
dW^s)
for all m G N on Acx, that is, Pfe<X) ■ (ay)} • Wj = PiB,x) • (ay)) • W* on Acx. Following similar line of argument we have P*Btx)' (PY)' • mi = P(B,x> ■ (ft-)' • m-h on A^. ■ Theorem 18.21. Suppose the stochastic differential equation (1) in Definition 18.6 has a solution (B, X) on some standard filtered space (Q, 5, {St}, P)- Let p. be the probability distribution ofXo on (Rd, QSjjd). Then there exists a null set N in (Kd, S8ffid, fi) such that for every x G Nc the pair ofprocesses (W, Y) on (WT+d, Wr+d) defined by Wit, w) = w'(t) and Y(t, w) = w"(t)for t € R+and w = (to', w") G W+d with to' G W and w" G Wd is a solution of the stochastic differential equation (1) in Definition 18.6 on the standard filtered space (WT+d,W+d-*'x,{Wrt+d'*-x},P{IBX)) satisfying the condition Y0 = x a.e. on (B,Jf)) (w^.arr^.p&jn) D
Proof. Let N0 and A, for x G N$ be as specified in Lemma 18.20. For x G N§, let
(1)
"¥*/ = Jl(P5w ■ ( " 4 • Wj)t + {P^X) ■ {M1 • mL)t + YJ - Yj.
§ 19. EXISTENCE AND UNIQUENESS OF SOLUTIONS
395
Let r
(2)
V\ = Yl(PiB,X) ■ {<*Y)) • Wj)t + (P{B,X) • (ft,)< . mL\ + Y* - Yt\ 3=1
According to Theorem 18.13, (W, Y) is a solution of the stochastic differential equation on (Wr+<<, W+d;*, {aBI^'''}, P^,*,) and thus there exists a null set A in (W+d,fflT^, P (B ,;o) such that »P; = 0 on Ac. Since a null set in (WT+d, W+d'", P(B,X)) is always contained in a null set in (Wr+d, W+d, PiB,xy), we can choose A to be a null'set in (Wr+d, %r+d, P(B,X))Then as we saw in (8) in the proof of Lemma 18.20, there exists a null set Nt in (Rd, 95K<<, /*) such that A is a null set in (V/d,Wd,PfB X}) for every x e Nf. Let N = N0 U Nt and A; = A* U A for x e Nc. Applying (1) of Lemma 18.20 to (1), we have %{ = *FJ = 0 on (A')c for x e 7VC. This shows that (W,Y) is a solution of the stochastic differential equation on (XVr+d, SZT^'*'1, {Wt*d,*'x},P*B%X)) for every x € Nc. Finally according to (4) of Observation 18.16, we have Pfe X)(YQ~l({0}) = 1 for x G 7VC, that is, Y0 = x a.e. on
{vr+*,mr+d,p?B:X)).
§19 Existence and Uniqueness of Solutions [I] Uniqueness in Probability Law and Path wise Uniqueness of Solutions Let (Biq\X{q)) be a solution of the stochastic differential equation (1) in Definition 18.6 on a standard filtered space (Cliq\&q\ {Sr(5)},P(«,) for q = 1,2. Consider the standard filtered space (W+d, mT+d-'M, m+d-*M},P$wx
396
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
identical probability distributions on (Wd,'Wd). We say that the solution is unique in the sense of probability law under deterministic initial conditions if whenever X$ = x a.e. on (Qm, 5 ( , ) , P(1>) and X^ = x a.e. on (Q ( 2 \ g<2), Pm)for some x G Rd, then Xm and X™ have identical probability distributions on (Wd, 2Urf). Lemma 19.2 Uniqueness of solution in the sense ofprobability law for the stochastic differ ential equation (1) in Definition 18.6 is equivalent to uniqueness in the sense of probability law under deterministic initial conditions. Proof. Clearly uniqueness in the sense of probability law contains uniqueness in the sense of probability law under deterministic initial conditions as particular cases in which the initial probability distributions are unit masses. Conversely assume that whenever XQ" = x a.e. on (£2(1), 5 ( I ) , P (1) ) and XQ 2) = x a.e. on (Q (2) , 3<2), P (2) ) for some x G Krf, then X (1) and Xm have identical probability distribu tions on (Wd, Wd). Suppose that XQ1) and X™ have identical probability distribution fi on (Rd, tBjfd). We are to show that Xm and Xl2) have the identical probability distributions on (Wd, Wd), that is, (1)
P(1)o(Xwyl(A)
= Pwo(X(2)rl(A)
for A G 2 ^
where (X^) is the mapping of Q (?) into Wd denned by (A,(9,)(u>) = X<9)(-, w) for w G Q<5). For the mapping (#<*>, # « ) ( « ) = (B(«»(-, w), X(«»(-, w)) for w G Q into W + d we have (2)
P ( ? ) o (X{q)y\A)
= P ( ? ) o (B(5), Xiq)y\W
x A)
for A e ^
Now since the process (W(«>, Yiq)) on (WT+d, W+d'*M, {W^*-^}, P^',,,,* <„,) and the pro cess (B (9) ,X (9) ) on (Q (,, ,5 ( » ) , {$'>}, P(«>) have identical probability distributions on (W+d, W*d) according to Theorem 18.13, we have (3)
P ( 9 ) o (B<«>, X™yl(W = ^ ( W x A )
x A) = P$qKX(q)) o (W ( 9 \ y 5 , ) - ' ( W r x A) forAG2U d ,
where the last equality is by the fact that (W(-, i»), F (9) (-, to)) = to for it) G W r+ti , that is (W (,) , yq)) is an identity mapping of Wr+d into W + d . By (2) and (3), to prove (1) it suffices to show (4)
p (
< B<»,xa>)(wr * A) = P,^,,* Q ) (W r x A)
for A G 2 ^ .
§ 19. EXISTENCE AND UNIQUENESS OF SOLUTIONS
397
Now since (W<»>, y ( '») and (B{q}, X^) have identical probability distributions on (W r+[/ , W+d), F <9) and X ( " have identical probability distributions on (Wd,ZOd). This implies according to Theorem 17.8 that Y"0(,) and X^ have identical probability distributions on (Rd, 93 K J). Therefore both yo(1) and yo(2) have ^ as their probability distributions on (Rd,(8mj). According to Theorem 18.21, there exists a null set N in (Rd,fB^,fj.) such that for every x e Nc, {W{q\ Y(q)) are solutions of the stochastic differential equation on (W*d,Wr+d'*^\{^t+d^^},P^KXW)) satisfying the condition Y*> = x a.e. on (\Vr+d, Wr+d, PffiKX(q))). Therefore Y ( , ) and Y(2) have identical probability distribution on ( W , Wd). Thus for x e Nc and A € 2Urf, we have P
(X<") ° O^V'W) = i ^ r a , ° (J(2))-'(A),
and consequently ^ o . ,
° (W(1>, y ' T ' C V T x A) = P^xa))
o (Wt2>, y 2 ')">(W r x A).
Since (W ( , ) , ^ ( , > ) are identity mappings, the last equality reduces to (5)
"(Bt'»,X<»)(
x A) = P(ga)xa))("
x
•"•)■
But according to (2) of Observation 18.16, for every x € iVc we have
(6)
J # » W W x A) = X. pw*,x<4w x ^)M^)
for A
e an*
Using (5) in (6), we have (4). ■ Definition 19.3. We say that the solution of the stochastic differential equation (1) in Defi nition 18.6 is pathwise unique if whenever (B, Xil)) and ( P , Xm) are two solutions on the same standard filtered space (£2,#, {&}, P) such that X^ = X™ a.e. on (Q, 3 , P) then there exists a null set A in (Q, 3 , P) such that Xm(-, w) = Xm{-, m)for u <E Ac. We say that the solution satisfy the pathwise uniqueness condition under deterministic ini tial conditions if whenever X^ = Xf] = x a.e. on (fi,3, P) for some x € & then there exists a null set A in (Q,3, P ) such that Xm(-, w) = Xm(-, uS)for u g Ac. Unlike uniqueness in the sense of probability law, pathwise uniqueness assumes that (B, Xm) and (B, X (2) ) are two solutions on the same standard filtered space with the same Brownian motion B. Nevertheless pathwise uniqueness implies uniqueness in the sense of probability law. This will be shown in Lemma 19.19 below. Also the equivalence of
398
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
pathwise uniqueness and pathwise uniqueness under deterministic initial conditions will be shown in Theorem 20.20. Let us define two conditions on the coefficients a and B of the stochastic differential equation. The first is a Lipschitz condition which will imply the pathwise uniqueness of solutions. The second is a growth condition which together with Lipschitz condition will imply the existence of solutions. Definition 19.4. We define two conditions on the coefficients a and B in the stochastic differential equation (1) in Definition 18.6 asfollows: There exists a measure A on (R.+, Q5mt) which is finite for every compact subset of R+ and satisfies the condition that for every T G K+ there exists LT £ R+ such that for t G [0, T] and wuw2 e\Vd we have \a(t, w,) - a(t, w2)\2 + \B{t, wx) - 3(t, w2)\2
(1) <
Lj ■ <
1
|w,0) - w2(s)\2X(ds) + |«n(i) - w2(t)\2} ,
f /
'[o,*] KJ[0,t]
)
d
and for t G [0, T] and w £\V we (2)
have |u;(a)|2A(da) + |u>(t)| 2 +l}.
\a(t,w)\2 + \B(t,w)\2
Note that in the definition above, | • | is the generic Euclidean norm for all finite di mensional Euclidean spaces. Thus \a(t,w)\ = { £ i i EJ=i \a){t,w)\2yi2 and \8(t,w)\ =
{Eti \B'(t,w)\2y/2. Lemma 19.5. Let Z bea nonnegative left-continuous adapted process on a standard filtered space (Q, 5, {3t}, P) which is increasing in the sense that Zs < Ztfor any s,t £ E + such that s
A? = {weQ.:Z{t,u)<
N}.
The process IN on (Q, 5, {& }, P) defined by (2)
IN(t,u)
= 1AH(U>) far(t,u>) G K+ x Q
isa{0,1} -valued decreasing left-continuous adapted process. For every w G Q, the sample function IN(-,u>), if it is not identically equal to 0 or 1, is given by (3)
I
(i W)
'
-\0
/0rt€(rHoo)
§ 19. EXISTENCE
AND UNIQUENESS
OF
399
SOLUTIONS
for some T(OJ) G R+. Proof. Since Z is an increasing process, A f decreases as t increases. Since Z is an adapted process, A? G fo for t G R + . Thus l^jv is an immeasurable random variable and IN is an adapted process. Since Af O A f for 5 < i, we have 1 A N > l^w and thus IN is a decreasing process. To show that every sample function of IN is left-continuous, let OJ G Q. and t G R+ be fixed. If IN(t,uj) = 1, then l ^ M = 1 so that ui G A f . This implies that OJ G A^ for every 5 G [0,*] so that 1 A W(CJ) = 1, that is, IN(S,OJ) = l f o r s G [0, i ] , proving the left-continuity of IN(-,u) at t. If on the other hand 7 w ( i , w) = 0, then I^N(OJ) = 0, that is, OJ g A?. This implies by (1) that Z(t,oj) > N. Then by the left-continuity of Z(-,OJ) there exists 6 > 0 such that Z ( s , w) > TV for s G (t - 5, t]. Thus w £ A f and l,4w(u>) = 0, that is, 1^(5, OJ) = 0 for 5 G (i — <5, i ] , proving the left-continuity of 7 W (-, OJ) at t. This shows the left-continuity of every sample function of IN. Now since every sample function of IN is a {0, l}-valued, left-continuous and decreasing function on.R+, it is given by (3) unless it is identically equal to 0 or 1 on R+. ■ L e m m a 19.6. Let J be an adapted process on a standard filtered space (£2,5, { & } , P) such that every sample function is of the form
[0
fort
e (T(U>),CX>)
for some T(OJ) G K+ unless J(-,OJ) is identically equal to 0 or 1 o« R+. 77ien (2)
' J{t,Ljf
= J(t,oj)
' J(s,oj)J(t,oj)=
fort J(S,OJ)
fors,t
eR+and
OJ eQ
G R+, s
G £2.
F o r M G M 2 ( f i , 5 , { & } , P ) , Jf G L 1 ] 0 0 (R + x £2, W M ) , P ) and t G R + , we have (3)
J ( < ) | ( * • m L )(<)| < \{JX • m L ) ( i ) |
o« £2.
IfX is a predictable process and is in L2 i00 (R+ x £2, fj,[M], P)> in (£2,5, P ) such that (4)
J ( i ) | ( ^ • M)(t)\
< \(JX • M)(t)|
tnen
tnere
exists a null set A
on Ac.
Proof. The equalities in (2) are immediate form the properties of the sample functions of J . Regarding (3) and (4), let us note that since J is a left-continuous adapted process, it
400
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
is a predictable process and in particular a measurable process. Thus if X is a process in Lii0O(R+ x £l,filM],P), then the measurabihty and the boundedness of J imply that JX too is in Li^fR* x Q , ^ ( M ] , P ) . Similarly, if X is a predictable process and is in L2i0O(R+ x Q, fi[M], P), then so is JX. To prove (3), note that if J(t, u>) = 0, then J(t,OJ)\ j
X(s,uj)mL(ds)\
J[0,t]
< | /
J(s,u>)X(s,w)mL(ds)\
J[0,t]
and if J(t,u) = 1, then J(s, w) = 1 for s G [0, t] so that J(t,w)\
(
X(s,cj)mL(ds)\
J[0,t]
<| /
J(s,v)X(s,u)mL(ds)\.
J[0,t)
Therefore (3) holds. To prove (4), let us consider first the case where X is a bounded left-continuous adapted process. In this case JX too is a bounded left-continuous process. Thus by Proposition 12.16 and by the fact that convergence in probability implies the existence of a subsequence which converges almost surely, for t G R+ there exists a null set A in (Q, 5, P) such that (5)
lim £-X-(*n,t_,){Af(*„,*) - M«„, fc _,)} = (X • M)(t)
on Ac
fel
and (6)
lim £ J(tn,k^)X(tn^){M{tnik)
- M(i„,,_,)} = ( J X . M)(t)
on Ac
where {tUik '■ k G Z + } is a strictly increasing sequence in R+ with i„,o = 0 and tn$ t co as k —> oo for each n G N and lim sup(i„i; — tnk-\) = 0, tnv = t and t„h < t for fc = 0 , . . . ,p n — 1 for n G N. From (5) we have (7)
lim J«)£x(i„, f c _,){M(i n > f c )-M(i r l , f c _ 1 )} = J ( i ) ( X . M ) ( t )
onA c .
Letw G Ac be fixed. If J(t,u)= 1, then J(i„,jt_i) = 1 for k = 1,... ,pn so that the left side of (7) and that of (6) are identical and thus we have J(t, u>)(X«M)(t, u>) = (JX»M)(t, u>) by (7)and(6). On the other hand if J(t,u>) = 0,then J(t,oj)\{X»M)(t,u>)\ < |(JX«M)(t,cu)|. Thus (4) holds in this case.
§ 19. EXISTENCE AND UNIQUENESS OF SOL UTIONS
401
For the general case where X is a predictable process and is in L2|00(R+ x Q, H\M], P) there exists by Theorem 12.8 a sequence {X(n) : n 6 N} in Lo(£2,j, {$t},P) such that Jim \\XM - X | | ^ , p = 0 and thus Jim | X w • M - X . M | o o = 0by Definition 12.9. Now Jim \\XM - X\\%2'P = 0 implies that Jlirn || JX{n) - JX\\[MJ;P = 0 as can be ver ified easily. This convergence implies lim | JX(n) • M — JX • M|oo = 0 according to Proposition 12.13. Now since Xin) are bounded left-continuous adapted processes, there exists according to our result above a null set Ao in (Q, g, P) for our t 6 K and all n 6 N such that we have J(t)\(X™ • M)(t)\ < \{JXM
(8)
• M)(t)|
on A£.
Now lim | X »M -X iMloo = 0 implies lim E(\X • M - X ; Ml 2 ) = 0 according to Remark 11.5 and thus the existence of a subsequence {nf} of {n} such that lim Xinl) • M = X • M on h\ where Ai is a null set in (£2,5, P). Similarly the converln)
M
n'—t-co
gence lim | J X < n } • M — JX • M|oo = 0 implies the existence of a subsequence {n") n'—*-oo
of {n1} such that lim ; i ( " " ' • M = JX ; M on A^ where A2 is a null set in (Q, 5, P). n"—KX>
Let A = U2=0A,-. Now (8) holds for every n" on Ac. Letting n" - » o o w e have (4). ■ Lemma 19.7. Let cbe a bounded nonnegative Lebesgue measurable function on a finite interval [0, T\. If for some a,b>0we have (1)
c(t)
+ bf
c{s)mL(ds)
forte[0,T]
J[0,t)
then (2)
c(t)
forte[0,T].
Proof. By substituting the inequality (1) into its right side and by repeating this substitution we obtain for every n e N the inequality c(i) < o f ^ - + bn+l / / ••• / c(tMl)mL(dtn) ~ £5J k\ Jlo,t] Jio,tt] J[0M
■ ■ ■ mL{dt2)mL{dh).
This inequality can then be proved by mathematical induction on n. Let M 6 R+ be a boundofcon[0,T]. Then
cit)<«±^
+^
M
for i e [ 0,T].
402
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Letting n —> oo, we have (2). I Theorem 19.8. If the coefficients a and fi in the stochastic differential equation (I) in Definition 18.6 satisfy the Lipschitz condition (I) in Definition 19.4, the solution, if it exists, is pathwise unique. Proof. Suppose (B,XW) and {B,Xm) are two solutions on the same standard filtered space (Q, 5, {&}, P) such that X^ = X^2) a.e. on (Q, 5, P) Assume that a and p satisfy the Lipschitz condition. We are to show that there exists a null set A in (Q, 5, P) such that X (1) (-,w) = X(2)(-,u) for bj e Ac. Now since X (1) and Xm are continuous processes, it suffices to show that for every ( e l , there exists a null set A( in (Q, 5, P) such that Xm = Xi2) on A£. To show this we show equivalently that for every t G R+ we have P{X\X)-X{2)\
(1)
>0}=0.
Now since (B,X(l)) and (B,Xm) are two solutions and X(0l) = X^ condition 4° of Definition 16.8 implies that for i = 1 , . . . , d we have
a.e. on
. B>)t + ({/3xm - / W
• mL)(.
xm
_ xm
=
J2({axm
- axai})
(£l,$,P),
For brevity in notation, let us define k ;
f A„(i, w) = aXm(t,u) - axm(t, to) = a(t,X(I>(-, w)) ~ <*(t,X™(; w)) \ ^ ( t , W ) = ftro>(t,w) - j9 X B (t l W ) = / ) ( ( , I < » ( , U ) ) - /3(t,X»>(-,w))
for (i, w) e Rj x Q. We write (A^- and (A^)' for the component processes of A„ and A/j. Then xm,i
_ xm,i
=
£ ( ( A J , _ B3){
+ ((A ^i B m i ) ;
i=l
For any real numbers c i , . . . , c n , we have {£" =1 Cj}2 < n £" = 1 c2. Thus
(3)
[X™ - Xf >'*'|2 < (r + 1) I £((AJJ . B') 2 + ((A^)1 . mL)2 J .
Let us define a nonnegative continuous increasing process Z by setting £ ( t , w ) = sup {|X(1)(s,u/)|2 + |X <2) (s,u/)| 2 } s6[0,t]
for(t,w) G R+ x £1.
§/P. EXISTENCE AND UNIQUENESS OF SOLUTIONS
403
By the continuity of Xm and Xa\ sups6[01] is equal to supse[0 t]nQ where Q is the countable collection of the rational numbers and thus Z{t) is 5-measurable. Actually since X(l) and X(2) are adapted processes, Z(t) is ^(-measurable and thus Z is an adapted process. The continuity of X(1) and X(2) also implies that of Z. As in Lemma 19.5, for N G N and t G R+ let Af = {u> G Q, : Z(t,u) < N} and define a {0, l}-valued decreasing left-continuous adapted process by setting IN(t,u>) = IAN(UJ) for (t,oj) G R+ x £2. By Lemma 19.6, for every t G K+ we have f IN(t)|((A„)j . B'')(t)| < \(IN(K)) • B J )«)| \ Iw(i)|((A*)' . mL)(t)| < |(/W(A,)< . mL)(t)|. Multiplying both sides of (3) by IN(t), using the inequalities above and the fact that IN(t)2 = IN(t), we have 7"(i)|X((1)'*' - * f >'f < (r + 1)
• 50? + (IN(Apy . m L ) 2 1
fel'UjJ
and hence (4). E[/ N (t)|X» w - X f ' f ] < (r +1) j J2 E[(/"(A0)- . B')2] + E[(/^(A^)' . mL)2] 1 By Observation 12.11,
E[(/ N (A4 • #)?] ■ = 1 IN{^))•B>\t = / ./[0,(]
HI^MlE?'
E[/A'(5)|(Aa)'(5)|2]mL(^) < / J
J[0,<]
E[J%)|A8(«)p]mz;(
for i = 1,...,rfand j = 1,..., r. Also, applying the Schwarz Inequality we have E[(IN(Apy • mL)2] = E[| /[0|(] Jw(«)(A,)«"(a)mL(ds)|2] < E[t/ B ) f ,/%)|(^j)''(a)| 2 m L (da)]
K[I*(«)[(A»X*)lVi;(«fa)
•'[0,*]
for i = 1,..., d. Using these estimates in (4), we have (5)
E[/"(t)|X((1) - X((2)|2] = £E[I N (t)\X™ - Xf w | 2 ] i=l
< d{r + \){T + t)j
w
E[I (*){|Aa(a)|2 + |A/J(a)|2}]mL(da).
404
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Let T G M.+ be arbitrarily fixed. By the Lipschitz condition we have for s G [0, t] C [0, T] and u> G Q
=
|AQ(5,uO|2 + |A,3(S,u;)|2 \a(t,Xw(-,u)) - a(t,X(2)(;Oj))\2
<
Lrff
+ \p(t,XW(;u))
ftt,Xm(;w))|2
-
\Xw(u,u)-X(2)(u,u>)\2\(du)+\X0)(s,u)-Xi2\s,uj)\2}
U[o,s]
J
N
Substituting this in (5) and recalling 1 % ) < I (u) for u G [0, j ] , E[IN(t)\X™
(6) <
d(r+l)(r
-
X^2]
+ T)LT f J[0,t]
+ d(r + l)(r + T)LT f
E[IN(u)\Xm(u)-X<-2\u)\2]X(du)\mL(ds)
If U[0,s] N
) w
E[I (s)\X (s)
2
-
X™(s)\ ]mL(ds).
JlO.t]
Since \Xm(u) A^ = {ueQ: (7)
- Xm(u)\2 < 2{\Xw(u)\2 + \X™(u)\2} < 2Z{u) and IN(u) = 1AN and 2Z(u) < 2N}, we have IN(u)\Xm(u) - X (2) (u)| 2 < 2N. Thus
Ef/^OIX, 0 * - X((2,|2] < d(r + l)(r + T)LT2iV{A([0, T]) + 1}T
for < 6 [0, T].
Define a function c on [0, T] by setting (8)
- X (2) (s)| 2 ]
c(t) = sup E[IN(s)\Xm(s)
for t G [0, T].
s6[0,(]
By (7), c is bounded on [0, T]. From (6) and (8) we have E[IN{t)\X\l)
- X<2)|2] < d(r + l)(r + T)L T /
< d(r + \)(r + T)LT{X([0,T])+l}
[
c(s)mL(ds)
{c(s)A([0, s]) + c(s)}mL(ds) fortG[0,T],
J[0,t]
Substituting this in the right side of (8), we have c(t) < d(r + l)(r + T)L T {A([0, T]) + 1} /
V[o,*]
c(s)mL(ds)
for i g [0, T].
Applying Lemma 19.7 with a = Oandfe = d(r+l)(r+T)L T {A([0, T])+1}, we have c(t) = 0 for t 6 [0, T]. Then by (8), sup 36[0 () E[IN(s)\Xw(s) - X (2) (s)| 2 ] = 0 for t e [0, T] and in particular E[I^(i)|X (1 '(*) - X<2>(0|2] = 0 for t G [0, T] and then for every t G K+ by the
§ 19. EXISTENCE AND UNIQUENESS OF SOLUTIONS
405
arbitrariness of T e R+. Thus for every t e R+ we have E[lAN\Xm(t) This implies
- X (2) (i)| 2 ] = 0.
|X(,)(i)-X(2>(i)|2
I„
Thus P(A» n {|X(1)(t) - X<2>(*)| > 0}) = 0 and then P{|X(1)(i)-X<2>(0|>0}
(9)
0
forTVGN.
c
Since A? | as TV -» oo, (vlf) | and P ( ( ^ f ) ) j as TV -> oo. To show P ( ( ^ f ) c ) I 0 as TV -> oo, assume the contrary, that is, there exists e > 0 such that P((A^)C) > e for all TV G N. Then P ( n W 6 N ( A f )c) = lim P ( ( ^ f ) c ) > e, that is, N—*oo
P{w G « : sup {|X (1) (s,a;)| 2 + \Xi2\s,u)\2 «e[o,t]
> TV for all TV} > e,
in other words, P{u> G Q : sup {|X(1)(6,u>)|2 + |X (2) O,u0| 2 = oo} > e, sg[0,(]
which is impossible since the continuity of |X (1) | 2 + |X (2) | 2 implies that for every w G £2 we have sup s e [ 0 ( ] {|X ( 1 , (s,w)| 2 + |X(2)(s,a>)|2} < oo . This shows that P((A?)C) J. 0 as TV —» oo. Using this in (9), we have (1). ■
[II] Simultaneous Representation of Two Solutions on a Function Space Our immediate goal here is to show that any two solutions of the stochastic differential equa tion can be represented on one function space with a common Brownian motion on it. We need this to show that pathwise uniqueness implies uniqueness in the sense of probability law. To be specific, if (P ( , ) ,X ( ? ) ), q = 1,2, are two solutions of the stochastic differen tial equation (1) in Definition 18.6 on two standard filtered spaces (Q (,) ,5 (5> , {&?}}, P l , ) ), q = 1,2, then we can introduce a probability measure and a filtration to the measurable space (Wr+2d,W+2d) in such a way that three processes W and Yiq\ q = 1,2, on the resulting standard filtered space defined by W(t, w) = v{t), Yw(t, w) = v'(t) and Ym(t, w) = v"(t) for (t, w) G R+ x W+2d where w = (v, v', v") with v G W and v', v" G Yfd are such that (W, Y{q)) is a solution of the stochastic differential equation for q = 1,2 and (W, Ym) and (W, y ( 2 ) ) have identical probability distributions on (WT+d, WT+d).
406
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Observation 19.9. Let (B,X) be a solution of the stochastic differential equation (1) in Definition 18.6 on a standard filtered space (Q,5, {&}, P)- Consider the probabil ity space (W+d,Wr+d,P(B,x)) where P(B,X) is the probability distribution of (B,X) on (W+d,W+i). Let 7T0 be the projection of W+d onto W r . The probability distribution (P(B,X))*0 of 7T0 on ( W , W) is given by (1)
(P(B,X)U
= P(B,X) O TTo"1 = P O (B, A")" 1 O TTQ-'
= P o (TTO O (S, A1))"1 = P o 6 - ' = P s = m r w by Definition 17.9. Definition 19.10. On the r-dimensional Wiener space ( W , W, mrw), we define a filtered space ( W ) 2 r r v " , {mrt'w},m\v) by letting W'w be the completion ofW with respect to the Wiener measure mw and letting Wt'w = a(Wt U VI) for every f 6 E t where 9t is the collection of all the null sets in (Wr, W'w, mrw). Lemma 19.11. {W[-w : t g R + } is a right-continuous filtration on (Wr,Wr'w,mrvy) ( W , W>w, {Vtrt'w}, mrw) is a standard filtered space.
and
Proof. The stochastic process W on ( W , W, mTw) defined by W(t, w) = w(t) for (i, w) G R+ x W is an r-dimensional null at 0 Brownian motion on ( W , 2 I F , m ^ ) by Theorem 17.10. Since the completion of (Wr,W,mTw) to (Wr,W'w,mTw) has no effect on the probability distributions of random vectors defined on ( W , <3BT, mTw), our W remains an r-dimensional null at 0 Brownian motion on ( W , 2DT,<", mTw). Thus by Proposition 13.22, a{a{Ws:s g [0, t]} U 91} for t G R+ is a right-continuous filtration on ( W , 237'"", mTw). But according to Proposition 17.7, a{Ws : 5 g [0,<]} = 2UJ for t G R+. Thus 23^'"' = o(Wt U 9t) for t g R+ is a right-continuous filtration. Since Wrt'w is augmented for every t g R + , ( W , 237'™, {23^'"'}, m ^ ) is a standard filtered space. ■ Let (W+d, Wr+d'*, {20^'*}, P(B)X)) be the standard filtered space generated by P(B,X) as in Definition 18.10. On the probability space (Wr+d, SIT""''*, PiB,x)), consider a regular image conditional probability given the projection n0 of YfT+d onto W , P{B X) °(A, v) for (A,v) g 23r+ti'* x W r . (See Definiton C.ll.) For brevity let us write QV(A) for P
(B,X)
(A,v)-
Then
1°. there exists a null set N in ( W , ! ^ ' ' " , mTw) such that Q" is a probability measure on (W +
407
$19. EXISTENCE AND UNIQUENESS OF SOLUTIONS 2°. Q(\A) is a StT^-measurable function on W for every A G W+d'*, 3°. for every A G 2Xr+,i'* and A0 € 257'"' we have P(B,X)Un7r 0 - 1 (A 0 ))= /
Q"{A)mrw(dv).
In particular if A G W+d'* is of the type A = W ' x i , with A, G Wd, then A n 7r^'(A0) = (W r x AO n (A0 x Wd) = AQ x Ax. Thus by 3° we have 4°. for Ao G W'™ and A, G 23J
Q»(WT x A,)mW(dw).
Let an element in WT+d be arbitrarily chosen and redefine Qv() to be a unit mass at the arbirarily chosen element for v E N. With this redefinition Qv() is a probability measure on W+d'* for every u G W r . Property 2° is unaffected by this redefinition. Lemma 19.12. For every Ai G W, <3(>(W x Ai) is a W,w-measurable function on W . Furthermore if A, G 2U?/or some t G R+,tfiereQ ( 0 ( W x A,) w a Wt'w-measurable function on W . Proof. If A, G 23d, then W x Ai G 2rr+,J C W+d'* so that by 2° above Q ( ) ( W x A,) is a 2nr,*'-measurable function on W Suppose A, G 2H? for some i G K+. Then A, G Wd and thus Q ( ) ( W x Ax) is a 2Hr'u'measurable function on W r . The projection TT0 of YT+d onto W is a 22T'"' x W'/aTT''"measurable mapping of W*d into W . Thus the composite mapping Q ^ ' ^ W x Ai) is a 2jjr,™ x W^-measurable function on VT*'. By 4° above and (1) in Observation 19.9 and by the Image Probability Law we have (1)
Q'»<""(Wr x A,)P ( B ,x)(^)
P(B,x)((W x A,) n (Ao x Wd)) = /
for Ao G 2rr , u and Ax G 20^. In the probability space (XVT+d, W+d>*, PiB,xy) consider the conditional probability P
Q*»«(W x AO G P(B,^)(W' x A, 1 2 3 ^ x W^),
that is, Q*°<->(W x A,) is a version of P(B,x^T
x
A, |2Ur'"' x W ) .
408
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Let us show that for our Ax £ Wd we have (3)
P(B,X)(W x Ai \Wt'w x Wd) = PlB,x)(W
x A, |2XT^ x Wd).
Now if 5li, 212, and 2l3 are sub-u-algebras of 5 in a probability space (Q, 5, P ) such that a(% U2l2) and Ql3 are independent with respect to P and if (Cl,
P{A\%2) = P(A|
foryi € 2li.
(See Theorem B.34.) To apply (4) to our probability space (W+d, W+d'*, P (B ,;n), let » i = Wt+d'* and 2t2 = aUJ''" x W C 2ti. The process W defined on the standard filtered space (WT+d, W*d'*, {2U[+d'*}, PiB,x)) by setting W(i, w) = (TT0 O w)(t) for (t, u>) £ R + x W + l i is an r-dimensional {2ZJ£+<''*}-adapted null atOBrownian motion according to Lemma 18.11. Let 2l3 = a(ay/r{Wtn -Wt,:t
=
{Wt,,-Wt,
: < < t ' < i" < oo} U 9t)) x V?d
W'wxWd
This shows that (3) is a particular case of (4). Thus Q^^iVf x A\) which is a version of P(B,X)(WT x A{ \
(i)
Q»*<\A)
= (p^xm)
u,v) f«(J4,»)eairJxwT
According to (1) of Observation 19.9, the probability distribution (P ( BI,) xwiXro of 7r0 on ( W , SET) is the Wiener measure mrw on ( W , 2Ur). By our convention Q" ,(,) is a probability measure on Wr+d for every v £ W . If we let (2)
i T ' ^ O l , ) = Q"' (?) (W x Ai)
for A, € 3Hd,
§ 19. EXISTENCE AND UNIQUENESS OF SOLUTIONS
409
then RvM) is a probability measure on (Wd, Wd) for every v 6 W r Lemma 19.14. There exists a probability measure Q on (W + 2 d , W*2d) such that for every A0xAixA2e W-a x <mdm x Wd'G) we have (1)
R"^\Ax)Rv'a\A2)mrw(dv).
Q(Ao xAlxA2)=f JAQ
Proof. For every v G W , define a set function n(-,v) on 2XTi,<1) x Wd,m by setting (2)
/i(A, x1xA A 22,,v) w) = iT'(1)(Ai).R"'(2)(A2) fi(A
for A, x i 2 e SIT'''1' x WdX2).
Clearly /i(0,v) = 0 and it can be verified readily that n(-,v) is a countably additive set function on the semialgebra 2Ud,(1) x WdX2) of subsets of Wd'(1> x W^ 2 '. Thefinitenessof the two measures if' ( ' > (W r x •) for g = 1,2 then implies that //(•, u) can be extended uniquely to a probability measure on (Wd'w x Wd'(2), a(2IJ'i•(,, x aF*'®)) for every v G W . Since Q"'(9)(A) is a arT'*"-measurable function of v on W for every A G 2Br+'', #"'<9)(A,) defined by (2) of Observation 19.13 is a SHT^-measurable function of v on W for every A\ G 2U'' Thus /i(Ai x A 2) ■) is a aXT^-measurable function on W r for every Ax x A2 G 2nd,(1) x aiT*'02'. To show that p(E, •) is a 2nr'w-measurable function for every E G a(2Dd'(1) x aU^ 2 '), let © be the collection of all members E of <J(2»''' (1) x W}d<m) such that //(£, •) is 2Ur'u'measurable. Clearly W * 0 ' x W ^ 2 ' is in <3. Let £ n G <S, n G N, and £ „ \. The fact that u(-,v) is a measure for every v G W r implies that fi,( lim i? n , t)) = lim fi(En, v) every n—*oo
n—t-oo
v G W and thus the 3Xr"'"'-measurabi]ity of //(£„, •) for every n G N implies the JHF'"'measurability of u( lim £„,«). Thus lim £„ G <5. Similarly if E,F € 0and E C F, n—*oo
n—>oo
then JF - £ G <S. This shows that (S is a d-class of subsets of W*>(1) x W*-'2'. As we have pointed above, & contains the 7r-class aU^ 1 ' x Wd'{2). Thus it contains a(Wd'm x WdX2)) by Theorem 1.7. Thus n(E, •) is a W'w-measurable function on W for every £ G
Q(A0 x E) = I fi(E, v)mTw{dv) -Mo
for A0 x E G arT'1" x aOD* 0 ' x an*'*2')
410
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
can be extended to be a measure on
Wr+2d.
Proposition 19.16. Let W, Ym, and Y{2) be as in Definition 19.15. Then W is an rdimensional {Wrt*2d'*}-adapted null at 0 Brownian motion on the standard filtered space ( W + M , 2ir +M '*, {ZtrS2'*}, Q), (W, Y<«>) and (B<«>, X<«>) have identical probability distri butions on (WT+d, WT+d) and (W, Y(»>) is a solution of the stochastic differential equation (1) in Definition 18.6 for q = 1,2. Proof. Clearly W is an /--dimensional continuous {2D£+M}-adapted, and hence {%B[+2d'"}adapted, process on ( W + M , VXr+2d'\ {2trt+2d'}, Q). To show that W is null at 0, let Z = {v G W : v(0) = 0}. Then by (1) of Lemma 19.14, we have Q{W0 = 0} = Q(Z x WdM x Wd'i2))
= J Rv'(1\V/d'm)Rv'l2\XVdx2))mTw(dv)
= mrw(Z).
Now the stochastic process V denned on the probability space ( W , W'w, mrw) by V(t, v) = v(t) for (t,v) G K+ x W is a null at 0 Brownian motion so that mrw(Z) = 1. Thus Q{ W0 = 0} = 1. This shows that W is null at 0. To show that W is an r-dimensional {2UJ'+2'i'*}-adapted Brownian motion, it suffices according to Theorem 13.26 to verify that for every s, t G R+ such that s < t and y G W+2d we have (1)
/ e^W-^dQ JA
= Q(A)e-^t-s)
for A G W3+2d'\
§ 19. EXISTENCE AND UNIQUENESS OF SOLUTIONS
411
Consider first the case where A = A0 x Ax x A2 6 W;w x 2Uf,(1) x 2Uf'(2>. In this case we have
/ e^w-w->dQ= JA
S
jWr+2d
e^w'-w'HAoxAlxA2dQ=
f
JlVr
e'^^^{vWw{dv),
where *(w) = lMRv<m(Ai)IV-®U2)
x A,)Q"'<2)(Wr x At)
= U0QvXl)(W
for v G W . Since A0 6 20?'", U„ is a 2D;''"-measurable function on W . Since A 1( A 2 6 2Uf, Q ( 1 , ' ( ) (W x A t ) and Q (2) - () (W r x A2) are W,'w-measurable functions by Lemma 19.12. Thus <3> is a aU^-measurable function on W Now if we define a process V on ( W , arr'1", m ^ ) by setting V(t, v) = v{t) for (i, u) G R+ x W r , then V is an r-dimensional {2BJ'"'}-adapted Brownian motion. Thus for any s,t € K+, s < £, the cr-algebra Ws,w and the random vector v(t) — u(s) are independent in the probability space (W r ,2U 7 ' ,u ',m^) by Definition 13.17. The 2U;-measurability of O then implies the independence of the two random variables <J> and e'(».*(«-«<*». Thus »<*)--v^(i>(v)mTw(dv)
JVfr 1
.■)
= e-^-^QiAo
*(*>} {/wr ®0)7T i ^ ( d i ; ) |
{yrtth ■«w>,
■-
x A, x A2) =
.
.2
e-^-s)Q(A)
by Lemma 19.14. This verifies (1) for the case A G 2Uf,(,) x 2Uf,(2). By applying Corollary 1.8, we then have (1) for A G W*2i = er(3Bfa) x 2Uf'(2)). By the same argument as in the proof of Lemma 18.11, we then have (1) holding for A e 2H^+M'* This completes the verification that W is W is an r-dimensional {aU^^I-adapted null at 0 Brownian motion. Let us show next that (W, Y(q)) and (5 ( ? > , X ( , ) ) have identical probability distributions on (W+d, 2tr +xm)(E) = Q ° (W, yl))~\E) for E 6 2XT+
o o (w, y'r'tAo x A,) = Q(W-'(A0) n (y'r'cA,)) = Q(Ao x Wd x W d n W x A, x W1*) = Q(A0 x A, x W d )
-L =
Q"' (I) (W r x A 1 )Q"' (2, (W r x Wd)mTw(dv) [ Q"'(1)(Wr x Ax)mrw(dv) = P$, U ( 1 ) ) (Ao x A,),
412
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
by 4° of Observation 19.9. This shows that Qo(>V,J rtl, )~ 1 and P^',,, x(1)) agree on WxW*. Then by Corollary 1.8, they agree on ^ + i =
S x S:sx
= s2}.
Ifipx v){D) = 1, then p. = v on 93 s and furthermore there exists a unique so € S such that p({s0}) = v({s0}) = 1. Proof. Let S x S be topologized by the product topology and let 93s x s be the Borel aalgebra of subsets of S x S. Let p be the metric on S. Then p(s 1, s2) for (s 1,52) € S x 5 is a continuous function on 5 x S and is thus 93sxs-measurable. This implies that the diagonal D, which is the subset of 5 x S on which the 93sxS-measurable function p is equal to 0, is a member of 93sxS- Since 5 is a separable metric space it satisfies the second axiom of countability and therefore 93s x s =
§ 19. EXISTENCE AND UNIQUENESS OF SOLUTIONS
413
otherwise we would have p.(S) > 2. Let Kn be the intersection of all those closed spheres with ^-measure 1. Then p(Kn) = 1 and the radius r(Kn) < n~l. Consider the sequence of the closed sets Kn, n G N. By the same reason as above we have Kn n Km ^ 0 for n =/ m. If we let Cn = n^ =1 iir m , then we have a decreasing sequence of closed sets Cn, n G N, with p(Cn) = 1 and r(Cn) < n _ 1 for every n G N. Since S is a complete metric space and r(Cn) ! 0 as n -» oo, there exists sQ G S such that n ne NC„ = {s0}. Then M((so}) = lim /i(C„) = 1. Since //(S) = 1 such s0 G 5 is unique. ■ Theorem 19.18. Lef i £ R
be fixed. Suppose that the initial value problem f dXt = a(t, X)dB{t) + /3(t, X)dt \XQ=X
for the stochastic differential equation (l)in Definition 18.6 has a solution on some standard filtered space and suppose that the solution of the stochastic differential equation ispathwise unique under deterministic initial conditions. Then there exists a mapping Fx of WT into ~Wd, unique up to a null set in (Wr,Wr,w, mrw), such that 1°. Fx is W['w IWdt-measurable for every t G R+, 2°. ;/ (B,X) is a solution of the initial value problem on some standard filtered space (£2,5, {5 ( }, P), then there exists a null set N0 in ( W , Wr,w, mTw) such that X(-,u) = Fx[B(-,u)]
forix> G Q. with B(-,u>) G N£,
and therefore there exists a null set A in (£2,5, P) such that the equality holds for wG Ac. Let (Q, 5, P) be an arbitrary complete probability space on which an r-dimensional null at 0 Brownian motion B exists. Let (il, 5, { j f }, P) be the standard filtered space generated by B. Let X be a continuous d-dimensional process on (£2,5, {57 }, P) defined by X{-,UJ) = FX[B(-,U)-\
forced.
Then (B, X) is a solution of the initial value problem on (Q, 5, Proof. Let (-B(,), X(,)) be a solution of the initial value problem on a standard filtered space (Q (,) , 5 ( , ) , {S^}, Pw) for q = 1,2. Let P^w.xw) b e t h e probability distribution of
414
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
(B<«», X('>) on (W+d, W+d) for q = 1,2. Let <5(0'(') and #•>><*> be as in Observation 19.13 and (W+2d,W+2d'\ {Wt+2d'*}, Q), W, Ym, Ym be as in Definition 19.15. According to Proposition 19.16, (W, F ( , ) ) is a solution of the stochastic differential equation on ( W + M , 2jr +M '*, {Wt+2d'*}, Q) and its probability distribution on (W + r f , 2DT+<') is identical to that of (B{q), X{g)) for each value of q = 1,2. Since X ( ? ) = x a.e. on (n(9,,5(»),P)}) =
/
(rH(D
x
JJ**.),®)
{D«MWw(d{ir0(w))),
where ATOM = {(TO(«0, wi(u>),*2(w)) G {TTQCU;)} x W " 1 ' x Wd'i2) : * , ( « ) = ir 2 («0} .
Since the integrand in the last integral is nonnegative and bounded by 1, the fact that the integral is equal to 1 implies that there exists a null set TVo in ( W , Wr'w, mTw) such that for "o(«0 G NQ we have ( P W ( , ) x r w ' ( 2 ) ) ( A ^ , ) = 1.
(1)
If we let Ni = {we W r + M : 7r0(io) G /V"0} then Q(/V,) = /
JJV0
R^wW\WdM)R^wU2\Vfd'm)mrw(d^oM))
= /
,w0
mrw(dfr0(w)))
=0
so that TV", is a null set in (W r + M , 2rr +:w , Q) and (1) holds for w G JVf. Thus by Lemma 19.17, for every w G JVf we have (■ ^ „ ( u , ) , ( I ) _ i J7r„ (w ),(2) (2)
Qn
(W^
gjj^
< TT^U)) = -K2{w)
{ ^0(w),(D({7ri(u;)J) = iP0(»W)({^2(u,)}) = 1. Then for every t £ J\f{ there exists a unique ^?(t>) G W such that R"'0)({
*AV)
10
for^GiVo.
119. EXISTENCE AND UNIQUENESS OF SOLUTIONS
415
Since only one singleton can have probability measure 1, the last equality in (2) implies that
FlMw))=l^w)
(4)
= w)
^
f w€
" %
10 for w e Ni. Let us verify that the mapping Fx defined by (3) satisfies conditions 1° and 2° To show that Fx is W[M/Wdt -measurable for every t e R+, note that since Wt'w contains every null set in ( W , W'w, mrw) and in particular N0 and since Fx(v) = 0 for v £ N0, it suffices to show that the restriction of Fx to N§ is Wt'wIWdt -measurable. Now for every Ax g 2Uf, we have F~\A,)
= {ve
JV"0C
n N£ = { » E N£ : V(v) e A,} = {v g JV0C: iT ( I ) (Ai) = 1}
: <2"'(1)(W x A , ) = 1} e W[w,
since according to Lemma 19.12, Q"'(1)(Wr x Ai) is a 2U^-measurable, and hence 2D[,Wmeasurable, function of v. To verify 2°, let ( 5 , X) be a solution of the initial value problem on a standard filtered space, say our ( P ( 1 \ Xw) on (ii (1) , 5 (1) , { $ " } , P (1) ). Recall that the probability distribution PgJ, of P ( l ) on (W r , fflT) is given by mfo- by Definition 17.9. Let A = (B^-^iVo). Then p(D ( A )
=
pm
w l
0 {B
r (N0)
= P^,(N0)
= m^(iVo) = 0,
that is, A is a null set in (£2(,), $m, P (1) ). For w £ Ac, F . ^ - . w ) ) = P*(B(1>M) = Fr(7r0 o (B(1), *(1>)(u,)) = TT, o (S<",X m )(u) = * ( 1 V ) = #'>(•, w), where the third equality is by (4). Similarly for w G A, we have F r (B (l) (-, u>)) = 0 by (4). Let (Q, 5 , P ) be an arbitrary complete probability space on which an r-dimensional null at 0 Brownian motion B exists. Let (£2,5, {#f }, P) be the standard filtered space generated by P. as in Theorem 13.23, that is, j f =
an S/2XT -measurable mapping of £2 into W and furthermore it is gf/23^-measurable for every t € R+ by Theorem 17.8. The probability distribution of B on ( W , 2 i r ) is given by mrw by Definition 17.9. Consider Wt'w as in Definition 19.10. If E0 is a null set in ( W , 2ZT, roM then 0 = mrw(E0) = P o B^C^o) so that B-'(-Bo) is a null set in (Q, 5, P).
416
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
If Ei c E0, then as a subset of the null set B'X(E0) in the complete probability space (£1,5, P), B~l(Ei) is a null set in ( A 5, P) and thus B~\EX) e # f for every t £ R + since § f is augmented. This and the fact that B~\Wt) C $f implies that H - ' ^ ' " ' ) C tff, that is, £ is #f/SB^-measurable for every t € R+. The mapping Fx is W^^/Wf -measurable. The mapping qt of W d into Rd defined by 5i(w) = w(t) for to e W* is Wdt /©^-measurable as we saw in Proposition 17.7. Thus qtoFxoB is an 57/93 E d-measurable mapping of Q into a*. But qtoFxo
B(u) = qto FX[B(-,OJ)]
= qt o X(-,u) = X(t, u>) for u e f i .
Thus X( is jf/QSjjj-measurable, that is, X is an {5, }-adapted process. Let us show next that the process (B, X) on (Q, 5, {$f } , P ) and the process (Bm, Xm) on (Q ( , \ 5 ( 1 ) , { # " } , P (1) ) have identical probability distributions on (WT+d, W+d). For the probability distribution P^B,X) of (B, X), writing I for the identity mapping, we have by Definition 17.9 for mrw P(B,X) = P o (B, Xyl = P o (B, Fx o B)"' = P o (J o B, Fx o B)" 1 = P o ((J, JFX) O B ) - ' = P o B" 1 o (I, P J - 1 = mW o (I, Fx)~\ On the other hand the probability distribution PfJw Xw\ ' s given by P[B\m
= Pm ° C5 W
= P<» o (B' 1 ')- 1 o {I,Fxr
= m' w o ( I . P , ) - ' ,
by 2° Thus P^\,KXW) = P(B,xy This shows that ( B ( " , I ( 1 ) ) and ( P , X ) have identical probability distributions. Then since (Bm,Xm) is a solution of the stochastic differential equation on (Q (1) , 5 ( 1 ) , { $ " } , P (1) ), (B, X) is a solution of the stochastic differential equa tion on ( 0 , 5 , {&},P) by Theorem 18.15. Also X ^ ' = x a.e. on ( Q ^ . S ^ P " ' ) implies that XQ = x a.e. on (Q, 5, P). Thus ( B , X ) is a solution of the initial value problem on (£2,3,{&},P). ■ Lemma 19.19. Pathwise uniqueness under deterministic initial conditions of solution of the stochastic differential equation (1) in Definition 18.6 implies uniqueness of solution in the sense of probability law. Proof. According to Lemma 19.2, uniqueness in the sense of probability law is equiv alent to uniqueness in the sense of probability law under deterministic initial conditions.
§19. EXISTENCE AND UNIQUENESS OF SOLUTIONS
All
Therefore it suffices to show that pathwise uniqueness under deterministic initial condi tions implies uniqueness in the sense of probability law under deterministic initial condi tions. Now let (P ( 1 ) , X (1) ) and (P (2) , X (2) ) be two solutions on two standard filtered spaces (£2(1\ &l\ {&»}, ? ( " ) and (Q (2) ,5 (2) , {&2)}, P (2) ) respectively and suppose X<" = x a.e. on (Q (1) , # » , P<") and X<2) = x a.e. on (S2(2), 5<2), P<2>) for some x G Rd Let Fx be the mapping of W into Wd defined in Theorem 19.18. Then by 2° of Theorem 18.20 there exists a null set A (,) in (£2(,), &q\ P<«') such that Xiq)(;u>) = FJB ( "(-,uO]
foru; e ( A w ) '
for q = 1,2. Now for the probability distribution P$q) of X (9) on (W*, 2ITd) we have pW, = P<«> o ( ^ ( 9 ) ) - ' = P(«> o (P x o /5 (?) )-' = P(<> o (6 ( 5 ) )-' o P " 1 = mTw a F'K Thus P^o, = P^Q) on (W d , 2U''). This proves uniqueness of solution in the sense of proba bility law under deterministic initial conditions. ■ Theorem 19.20. For the stochastic differential equation (1) in Definition 18.6, pathwise uniqueness of solution and pathwise uniqueness of solution under deterministic initial con ditions are equivalent. Proof. Since the former contains the latter as a particular case, it suffices to show that the latter implies the former. Let us assume the latter. Let (B, X (1) ) and (B, X (2) ) be two solutions of the stochastic differential equation on a standard filtered space (£2,5, {5t},P) such that X^ = Xf> a.e. on (Q, 3 , P). The representation {W("\ YM) of (P, X<«>) on (Wr+d, Wr+d'*M\ {Wl^'^^^^n) is a solution of the stochastic differential equation, (W < ' ) , F c?) ) and (B, X ( , ) ) have identical probability distributions on (Wr+d, W+d), and F ( ? ) and X (9) have identical probability dis tributions on (Wd, Wd) for each of q = 1,2 according to Theorem 18.13. Since X^1' = X^2) a.e. on (£2,5, P), X^1' and X^2) have identical probability distributions on (Rd,
for {A, x) € WT+d x Rd
418
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Now pathwise uniqueness under deterministic initial condition implies pathwise unique ness in the sense of probability law according to Lemma 19.19. Therefore y0<5) = x im plies that Ym and Y{2) have identical probability distributions on (Wd, 2D''). As we noted above, the probability distribution P{B,XW) of (B, X (?) ) is also the probability distribution of (WCq\ y ( ? ) ). Thus P (B ,^w)(W r x Ai) for Ax g SD^ is the probability distribution of F(«> on (Wd,2Bd). By (2) and (4) of Observation 18.16, we have x
A
p
?B,xw)(Wr
(1)
JW<*>(W
(2)
PSBJCW) ( O t f V t f s } ) ) = 1
^ = Jd
x AI)/I(JT)
for A, € 2Dd
foriGiVc.
Thus for x g iVc, P ( B ^„, ) (W r x A,) for A, g Wd is the probability distribution of F when y 0 (,) = x. Therefore we have (3)
P ( B,X<»)(W x Ay) = P ( B,*<2>)(W x AO
for A, g 2Drf when a g
w
Nc.
To show that pathwise uniqueness holds, we show that there exists a null set A in (£2,5, P ) such that X (1) (-, w) = X (2) (-, w) for w g Ac. Suppose no such null set in (£}, £, P ) exists. The continuity of the sample functions of X (1) and X (2) then implies that there ex ists some t 0 g R+ such that P{u g Q : X%\u) 4 Xt(2)(w)} > 0. Then the probability distributions Pxm and Pxm on (Rd, %Smd) are not equal. Thus there exists E g 93Ed such that Pxm{E)4Pxm{E). S°ince Pxm(E) = P ( f t Jf(i))(Rr X £ ) , we have P
(4)
(Bl0,x«>)(Kr x E) 4 P(Bt0,x«')(Kr x E).
Let 9(„ be the mapping of Wr+d onto R r+,i denned by gto(tu) = io(i0) for iw g W r + f Then P
(B10,<>> = -P ° ( ^ „ , X"))- 1 = P o (9<0 o (B, X™))-1
= Po^,^'1)-
'•tf
- P(B,XW) ° %
so that P(B,o,x^(W
XE) = PlB,xm
o q~\Rr x £ ) .
Thus from (4), we have (5)
iW«>)(<7*;W x £)) y P ^ ^ ' C R ' x £ ) ) .
§20. STRONG SOLUTIONS Since q^\W
419
x E) is a set of the type W r x A, with A, £ 2D"', (1) and (5) imply
/ K r ^ T B , * . . . ) ^ ' ^ x -E))Mdz) ^
J ^ e ^ ' O r
x
E)Mdx).
Thus there exists F G <8Ed with /i(F) > 0 such that for x 6 F we have r
-P(B,X('))(9(0 '(R
xE))i
■f(B,XO))(9t0 \W
xE)),
contradicting (3). This shows that pathwise uniqueness holds. ■
§20 Strong Solutions [I] Existence of Strong Solutions Consider the product measure space (Rd x Wr,cr(?81d x 2XT),/i x mrw) where p. is an arbitrary probability measure on (Rd, 25^). Let c ^ ® ^ x 2ZFy ""vf be the completion of
Vary™™/Wd-measurable,
2°. for every x 6 Rd, F„[x, •] is a Wt,w /W*-measurable mapping efW into Wdfor every t € R+, 3°. X(-,LV) = F^Xoi^Bi^fora-e.
u in (£2,g,P).
In Theorem 20.5 below we show that if the coefficients a and /3 in the stochastic dif ferential equation (1) in Definition 18.6 satisfy the Lipschitz condition (1) and the growth condition (2) in Definition 19.4 then a strong solution exists on any standard filtered space (£2,#, {St}-, P) on which an r-dimensional {5,}-adapted null at 0 Brownian motion exists. Definition 20.2. Let L2£,(R+ x Q., {&}, mL x P),or briefly 1%%, be the collection of all d-dimensional continuous {&} -adapted processes X on a standard filtered space (£2, $, {&}, P) satisfying the condition that sup, e[(M] E(|X(s)| 2 ) < oo for every t e R+.
420
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Lemma 20.3. Let (Q, 5, {St}-, P) be a standard filtered space on which an r-dimensional {■St}-adapted null at 0 Brownian motion B exists. Assume that the coefficients a and j3 in the stochastic differential equation (1) in Definition 18.6 satisfy condition (2) in Definition 19.4. With fixed x G Rd, let us define a mapping r o / L 2 ^ by setting for X £ L 2 £, (1)
(rX)(t) = x+ [
a(s,X)dB(s)+
J[0,t]
f
/3(s,X)ds
Then TX G L ^ . If we define a sequence {X(t,) :qeZ+}
in l$% by letting
[X®\t) = x fort€R+ \ X (9) (i) = (rX ( »-")(0 for t£R+andq£
m
/orteR+.
J[0,t]
N,
then for every T € R+ there exists Kj € K+ such that sup E(|X<"(*)|2) < KT teio.T]
(3)
forq£Z+.
Proof. To show the existence of the stochastic integrals in (1), we show that a* (•, X ) and /?'(-, X) are in L2,oo(K+ x ^ , TTIL x P) and L1|00(M+ x Q, mj, x P) respectively. Now since a and /3 satisfy condition (2) in Definition 19.4 and since X G L ^ , for every T G K+ we have
{\a(t,X)\2 + \P(t,X)\2}mL(dt)
E\[ U[0,T]
<
LT f E\[ IX(s)|2A(d5) + IX(i)| 2 + l mL(dt) Jm.T] [0,T] Um.t] Ul0,t]
< LT
sup E(|X(t)| 2 ){A([0,T])+l} + l T < o o . L*e[o,T]
This shows that aJ(-,X) and /?'(•,X) are all in L2,oo(R+ x Q.,m,L x P) and hence the stochastic integrals in (1) exist. To show that r X is in L2'^,, it remains to verify that for every t G R+ we have sup E(|X(s)| 2 ) < oo.
(4)
»e[o,«]
Letc 0 = max{|x| 2 ,1}. ForX G L ^ and T G R+ let A(<;X) be a real nonnegative valued monotone increasing function for t G [0, T] such that (5)
max{ sup E(|X(s)| 2 ), 1} < A(t;X) s6[0,(]
fort G [0,T].
§20. STRONG SOLUTIONS
421
Since supsg[0_,, E(|X(s)| 2 ) < oo for every t 6 M+ such function A{-\ X) always exists. Let us show that with A{-\ X) we have for TX defined by (1) and T G R+ the estimate E(|(rX)(t)| 2 ) < 3 fc 0 + CT f
(6)
I
A(s;X)mL(ds)\
J[0,i]
J
fort G R+,
where (7)
C T = (l+r)L T {A([0,T]) + 2}.
Note that (6) and (7) imply E(|(rX)(t)| 2 ) < 3 {c0 + CTA(T;X)T] < oo which is (4). To prove (6), note that since {£"=i aj } 2 < n £" = 1 a) for any real numbers a{,..., an, we have from(l) E(|(rZ)(t)|2)<3fc0+Ef| / a(s,X)dB(s)|2l+E[| / P(s,X)ds\2\\. I L J[o,t] J L J[o,t] J) Now by (3) of Proposition 13.39 E[|/
= /
E[|a(«,X)| 2 ]m L (ds),
E[\/l(s,X)\2]mdds).
a(s,X)dB(s)\2}
and by the Schwarz Inequality X)ds\2]
E f| / fa, L J[o,t]
J
Thus for t G [0, T] we have by (2) of Definition 19 A and (5) E(|(rX)(t)| 2 ) < 3 {co + (1 + T) /
E [|a(a, X)| 2 + |/?(s, X)| 2 ] m L (ds)}
|X(u)| 2 A(du)+|X(s)| 2 +l]m£,(ds)l
<
3(co + ( l + T ) L T /
<
3 | c o + (l+T)L T {A([0,T]) + 2 } y o ( A ( s ; X ) m L ( d s ) | ,
I
j[o,<]
E | 7
t
U[o.s]
J
J
proving (6) and in particular (4). Consider the sequence defined by (2). Clearly X{0) € L ^ , . Then since r is a mapping of Lj£, into L ^ we have X<«> e L^L, f o r ? G N. To prove (3), let us show first that for every T G R+ we have for t G [0, T] and g G Z+ the estimate (8)
Pi(3CrO* E(\X^(t)\2)<3c0\gJ:^^+3-
,„-i(Qtty
422
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Now for q = 0, we have E(|X (0) (i)| 2 ) = \x\2 < c0 so that (8) holds. Suppose (8) holds for some q e Z + . Let A(t;X(q)) be defined to be equal to the right side of the inequality (8) fori 6 [0,T]. Then maxfsup^^EdXWCs)! 2 ), 1} < A(i;X<«>) fort G [0,T] so that A(-;X(q)) satisfies (5). Therefore by (6) we have E(|X" + »(t)| 2 ) = E(|(rX<")«)| 2 ) < 3 {CO + CTJ
J i c0 ^ n = 3 + 3CQCJ
A(s;
Xw)mL(ds)}
{<jl (3CTs) ,,-,(Cr£) + 3»- r L J[0,t] «I J k
I
',3CTt)h k\
= 3co{t-
+l
y(cTty
\
forte[0,T],
+
that is, (8) holds for q + 1. Thus by induction (8) holds for all q e Z+. From (8) we then have for t e [0, T] and 9 G Z+ the bound E(|X<"(i)|2) < 3C0S ^Tp-
±
3c fi3CTT
o
-
With KT = 3c0e3Cl-T we have (3). ■ Lemma 20.4. Suppose a and f3 satisfy both (1) and (2) in Definition 19.4. Then for every T G R+ there exists MT G R+ such that sup | ( T X ) ( 5 ) - ( T Y ) ( S ) | 2
(1)
s6[0,(]
< Mr /
E
<MT
I
sup E(|X(u) -
J[0,t] u6[0,a]
Y{u)\2)mL(ds)
sup [ X ( t * ) - y ( u ) | : m L (ds) /or t G [0, T] and X, F G L^
u6[0,s]
Proof. For X, Y G L^L we have rX, TY G L ^ by Lemma 20.3. By (1) of Lemma 20.3 we have \(TX)(S)-(TY)(S)\2
<
2| /
+ 2\[
{a{u,X)-a{u,Y)}dB(.u)\2
{/3(u,X)-/3(u,Y)}du\2
and thus (2)
E
sup |(rX)(3) -
'S[0,i]
(rY)(s)\2
< 2E sup I /
>e[o,t] ~'[o,s]
{a(u,X)
- a(u,Y)}
dB(s)\:
%20. STRONG
423
SOLUTIONS + 2E sup | ,/ seto.t] - (o,s]
{/?(«,X)-P{u,Y)}du\ 2
N o w E [ s u p s e [ 0 (] M 2 ] < 4E(M(2) for a continuous L2-martingale M on (12,5, {&},-?) according to (2) of Theorem 6.16. Since | /
{«(«,X) - o(u,F)}rfB(u)| 2 = J2Y,\I
{ « K « . * ) - <*;(«> y )l<W'(«)| 2
311(1
{/[o s ]{ a j( u >-^) — aj(u,y)}t/B'(u) : 5 G R+} is a continuous I/2-martingale on (n,5,{5(},P),wehave (3)
E
{a(u,X)-a(u,Y)}dB(s)\2
sup I / .j>e[o,(] J[o,s]
{a(u,X)-a(u,Y)}dB(s)\2
< 4E\\ f L J[Q,t]
= 4E[/
L/[o,(]
\a(u,X) - a(u,Y)\2mL(du)
.
Similarly by the Schwarz Inequality {p{u,X)-p{u,Y)}du\2<s
\[
\/3(u,X)-P(u,Y)\2mL(du)
I
JlO,s]
J[0,s]
so that {0(u,X)-P{u,Y)}du\2
sup I /
(4)
e[0,t] J[o,s]
\P(u,X)-/3(u,Y)}\2mL(du)
7 < * E [U[o,s]
J
Using (3) and (4) in (2) and applying (1) of Definition 19.4 we have E <
sup
-
\(TX)(S)
(TY)(S)\:
s£[0,i]
(8 + 2 T ) /
E[{\a(s,X)-a(s,Y)\2
+ \0(s,X)
-/3(s,Y)\2}]mL(.ds)
E f|X(«) - Y(u)\2\(du)
+ \X(s) - Y(s)\]
J[0,t]
<
(8 + 2T) / J[0,t]
< <
L
(8 + 2 T ) / sup J[o,t] ue[0,s] (8 + 2T){A([0,s])+l} /
J
mL(ds)
E(\X(u)-Y(u)\2){\([0,s])+l}mL(ds) E(|X(u) -
Y(u)\2)mL{ds).
424
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
WithM r = (8+2T){A([0,s])+l), we have the first inequality in(l). The second inequality is immediate from E(|X(u) - Y(u)\2) < E[sup u e [ 0 s ] \X(u) - Y(u)\2] for u € [0,s] and then suPu€[0,s]E(\X(u) - Y(u)\2) < E[sup u6[(M , \X(u) - Y(u)\2]. ■ Theorem 20.5. Suppose the coefficients a and /3 in the stochastic differential equation (1) of Definition 18.6 satisfy conditions (1) and (2) of Definition 19.4. Let ( Q , ^ , {St}, P) be a standard filtered space on which an r-dimensional {$t}-adapted null at 0 Brownian motion B exists. Then there exists a mapping F ofR.d x W into Wd such that (1) F is (7(
(4) 1 ;
foriGK+ for t e K+ and q e N,
where r is a mapping of L j ^ into L 2 ^ defined by setting (5)
{rX){t) = x+f
a{s,X)dB(s)+
J[0,t]
[
forteR+
/3(s,X)ds
J[0,t]
for X G L 2 '^. By iterated application of (1) in Lemma 20.4 we have E <
sup |X ( « +1) (i) - X ( ? ) (t)| 2 < 610,7) T
MT I' JO
<
AiUT JO
<
M£ j JO
sup E [|X (9, (i) - X{q-l\t)\2}
<6[0,(,]
f*
L
(6[0,t,_,]
T q
■■■ t
f
dtq
sup E f|X{«-»(t) - X ( '- 2 '(i)| 2 l A g di,
JO
JO
J
JO
L
J
sup E [|X(1>(i) - X<°>(t)|2l < W i . •. &, t € [0,(,]
L
J
425
120. STRONG SOLUTIONS By (3) in Lemma 20.3 we have sup E [\Xm(t) - X (0) (t)| 2 ] < sup E[2|X (1 >(i)| 2 + 2|X (0) (i)| 2 l < 4KT
*eto,ti]
(6[o,i|]
'
and therefore (6)
sup |X (?+1) (t) - X w ( t ) | :
E
*6[0,T]
/■T ,«,
,(2
(Mrr)«
< 4KTM$.jg J " ■ ■ ■ p dt,dtq^ ■ ■ ■ dti = 4KTFor q G Z+ let sup |X (9+1) (i) - X —}.
X, = { w e f l :
«6[0,T]
2?
By the Chevyshev Inequality we have
P(Aq)<(Tf4KT{-^ Since EqezS4MTT)qm~l
=
q\
=
4KT^pi. q\
< oo, we have P(liminf A=) = 1 by the Borel-
Cantelli Lemma. Thus for a.e. w e f i w e have sup J€[0TJ |X (?+1) (t,cj) - X{q){t,u)\ < 2~q for all but finitely many q G Z+. Then for an arbitrary e > 0, for a.e. u> G Q there exists iV(aj) G Z + such that m— 1
m—l
sup |x(m)a,uo-x(n)(t,uoi < £ sup I X ^ ' C ^ U O - X ^ ^ I < y; 2-' < £
<6[0,T]
, = n ie[0,T]
g=„
(?)
for m > n > N(u>). This shows that {X (-,o;) : q G Z+} converges uniformly on [0,T] for a.e. oi G £2. Considering T = n for n G N, we have a null set A in (£2,5, F ) such that {X (,) (-, u>) : g G Z+} converges uniformly on every finite interval in R+ for u G Ac. Let us define a process X on (£2,5, -P) by (7)
v/
,
,
/ lim X ( ? , (i,^)
for^GR+xA'
X ( t , w ) = { 9—°°
(.0
for(t,u;)GR + x A.
By the uniform convergence of {X (,) (-, CJ) : q G Z + } to X(-, u) on every finite interval for UJ G Ac and by the continuity of {X
426
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
continuous process. The uniform convergence also implies according to Lemma 17.12 that for w e A ' X(q){-, w) converges to X(-,ui) in the metric px on W d .
(8)
By the fact that A e J i for every t G R+ since 5 ( is augmented, and by the fact that X^\t, •) is 5t-measurable, X(t, •) is ^-measurable, that is, X is an {& }-adapted process. Regarding the convergence of X ( , ) to X we have furthermore
lim I sup E L\\Xw(t) - X(t)\2]J 1 = 0 .
(9)
«^°° [teio.T]
J
To prove (9) note that , ~ (5+1) * - i (i) w - ^Xw{q).(t)\ . , n2 V 2 sup E f|X (m, (t) - X(n)(<)|2]i>/2 < TV V sup „Er f|X telo.T] l ' ,=„ *e[0,T] L 1/2 , m-l
< £E
SU
P l*(5+1)w-xw(t)\:
<2^r£
»
by (6). Since ^ , 6 z + \J(MTT)i(q\)-^ < oo, the Cauchy Criterion for uniform convergence implies that there exists X*(t) G ta(Q, 5, P) for i G [0, T] such that
M^*^"**®1*'}*0, Since convergence in L2 of X{q)(t) to X*(i) implies the existence of a subsequence which converges a.e. to X*(t) and since Xiq)(t) converges a.e. to X(t), we have X*(<) = X(t) a.e. on (Q, 5, P) for every i 6 [0, T\. Thus (9) holds. To show that X e L2'^0 note that by Fatou's Lemma and by (3) of Lemma 20.3 E(|X t | 2 ) < liminfE(|X ( (,) | 2 ) < liminf sup E(|X ( (,) | 2 ) < KT so that sup ( 6 [ 0 T ] E(|X ( | 2 ) < KT < oo. Thus X G L ^ and r X is defined. Let us show that TX = X for our X defined by (7). Now for every T G R+ we have sup | ( r X ) ( i ) - X ( i ) | 2 < 2 sup | ( r X ) ( i ) - X ( ' + 1 ) ( i ) | 2 + 2 S up |(rX) (?+1) (i) - X(t)| 2 <6[0,t] (£[0,4] t6[0,(]
§20. STRONG
SOLUTIONS
All
for an arbitrary q 6 Z + . Since {X{*\-,ui) : q 6 Z + } converges uniformly on [0,T] to X ( - , « ) f o r w e Ac we have ,_l lim sup |(rX) (,+1) (t,a;) - X(i,u;)| 2 = 0fora; G AC. Thus °° ie[o,(] sup |(rX)(f) - X(t)\2 < 21imirrf ( sup |(rX)(t) - X ( ' + 1 ) ( t ) | 2 j . <e[0,t] te[o,t] Then by Fatou's Lemma and Lemma 20.4 E
sup |(rX)(t) -
X(i)f
te[o,t]
<21iminfE q—»-co
/ < 2M T liminf 9^00 y
sup |(rX)(t)-X<' + 1 ) (0| 2 *e[o,t]
sup E[|X(s) - X (,) (s)| 2 ]mi,(di)
[ 0 ,r] s 6 [ 0 ,i]
and therefore (10) E sup | ( r X ) ( t ) - X ( i ) | 2 < 2M r limsup f i€[0,i]
sup E[|X(s)-X w (s)| 2 ]m I ,(dt).
Now since X, X (9) g L ^ , we have by (3) of Lemma 20.3 sup E [ | X ( s ) - X ( 9 ) ( s ) | 2 ] < 2 J sup E(|X(s)| 2 ) + sup E(|X ( "(s)| 2 )l
»e[o,<]
<
Ue[o,<]
»e[o,t]
J
2 I sup E(|X(s)| 2 ) + Kr \ < oo. Ue[o,(] J
This shows that the integrands on the right side of (10) are bounded on [0, T] uniformly in q £ Z+. Thus Fatou's Lemma for limit superior applies and we have sup
(6[0,t]
<
\(rX)(t)-X(t)\'
2MT f lim sup { sup E[|X(s) - X(q)(s)\2] \ mL{dt) = 0 J[0,T) 9—<x. [ss[0,t] )
where the last equality is by (9). Therefore sup t6(0 (] |(rX)(t) - X(t)\ = 0 a.e. on (£2, g, P). Thus (rX)(t) = X(t) for * e [0, T] a.e. on (Q,5, P). By considering T = n for n € N, we have a null set A in (Q,5, P ) such that (rX)(i, w) = X(i, w) for (f, w) € R+ x Ac.
428
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Recalling the definition of TX by (5) we have (11)
Xt = x+f
7(0,*]
a(s,X)dB(s)+
[
foriGR+,
P(s,X)ds
J[o,t]
that is, X is a solution of the differential equation (1) in Definition 18.6 with XQ = x. Let us show that there exists a mapping F of Rd x W into Wd satisfying (1) and (2) and for X defined by (4) and (7) there exists a null set A^ in (£2,5, P) such that we have (12)
forwGA^.
X{-,UJ) = F[X,B{-,UJ)}
For this we show first by induction that for every q G Z+ there exists a mapping F ( , ) of Rd x W into Wd satisfying (1) and (2) and for X(9> defined by (4) there exists a null set A, in (Q, 5 , P ) such that X ( ' ) (-,o;) = F ( « ) [i,B(-,aj)]
(13)
for^GA;;.
Now according to (4), X(0)(£,u;) = x for (t,Lo) G R+ x Q. Let uix be an element in W d defined by wx(t) = x fort G R+. If we define a mapping Fm of R* x W into W d by setting F (0) [x, w] = » , for (x, w) G Rrf x W then F (0) [x, #(•, w)] = wx = Xm(-,u>) for every w G £2, verifying (13) for F ( 0 ) . To show that F ( 0 ) satisfies (1), note that since Fi0) maps the entire Rd x W into a single point wx G W , for any A G 2XT* we have ( F ® ) " ^ ) = R d x W or 0 according a s ^ e i or wx G ^4C- Thus Fi0) is
=x + I
a{s,X{,l))dB(s)+
J[0,t]
f
P(.s,X^})ds
fortGR+.
J[0,t)
For brevity let us use the notations <*(•, X<") . B = [£J =1 a j ( - , X « ) • B< : i = 1 , . . . , d] M;XW) • mL = W\-,X^). m L : t = 1 , . . . ,d] Then X (9+1) = x +a(-,X ( ? ) ) . B + (-,X (?) ) • m L . Since x =X (0) (-,o;) = F ( 0 ) [X,B(-,CJ)] for a; G £2, if we show that there exist mappings G(?) and #<«> of Rd x W into W* satisfying (1), (2) and U
*
J
r ( a ( - , X < " ) . £ ) ( u / ) = G('>[x,B(-,u/)] \(/9(-,Jr f a ) )«m L )(w) = Jff«'>[j;,B(->w)]
foru,GA< forweAf,
§20. STRONG
429
SOLUTIONS
where A, is a null set in (£2,3, P), then by setting F*-q+l) = F ( 0 ) + G{q) + Hlq) we have a mapping Fiq+1) of R* x W into W satisfying (1), (2) and F (9+1) [:r,B(-,a,)] = i + (<*(•, X<<>) • B)(w) + (/3(-,X(?)) . m L ) M = X ( ' +1, (-,w) for a; 6 A,, verifying (13). Thus it remains to show the existence of G(q) and H{q). Let <J>(,) be a mapping of R + x Q into Rdr defined by (15)
0{q\t1u)
= a(t,X{q\-,uj))
for(i,uOeR+xQ.
Since Fiq) satisfies (13) we have (16)
&q\t,u)
= a(t,F<-q)[x,B(-,(v)])
for(t,u>)eR+xAcg.
By Lemma 18.5, 0 ( , ) is an {3t}-progressively measurable and in particular {3t}-adapted measurable process on ( 0 , 3 , {$t},P)- Also since X(q) g L ^ , we have a}(-,X ( , ) ) g L2,cx>(R+ x Q., mi x P) as we saw in the proof of Lemma 20.3. Let ipiq) be a mapping of R t x R ^ x W into R * defined by (17)
for(f,i,ki)€EtxR''xWr
Let us consider first the case where the components (fl>(,))} of <£(,) are in Lo(0,3, {3t}, -P)In this case we have for (t, u) g R+ x A, r
(18)
(*<*>• B)''(t,w) =
m
^^(^(^.^{BXi^^-S*^-!^)} j=\ r
k=\ m
= Y.Y.0i)^F(q)[x,B{-,u)]){B\tk,uj)j=\
B\tk^M}
k=\
where {ifc : ik g Z+} is a strictly increasing sequence in R+ with t0 = 0 and lim tk = co with t m = t. Define a mapping (v?(»> • W)'' o f R+ x Rd x W into R by r
(19)
m
(v ( , ) • W)\t, x, w) = £ £ «$(**. F ( , ) [ x ' w»{«','(*t) - w*'(*t-i)}
for (t, x, w) g R+ x Rd x W where w* is the i-th component of w. Let us show first that ( • W0*'(-,x,iy) as a mapping of Rd x W into W is a(«8]Ed x 2ir)/2n-measurable. According to (4) of Proposition 17.4, it suffices to show that for every t g R+ the mapping
430
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
(<«> • wy(t, ■, •) of Rd x W into R is ff(«8K(,) • WO'O, x , w ) is a
G{q)[x,w] = (ip{q)»W)(-,x,w)
f0T{x,w)eRdxWT,
then G (,) satisfies conditions (1) and (2) and also for u> G Acq we have G ( "[x,B(-,u,)] = (<^> . TVX-, x, B(-,w)) = (<J>(?) . B)(-,u>) = ( a ( - , X w ) . B)( w ) by (20), (18), (18) and (15). This verifies (14) for the particular case where (
(&f • B)(-,u>) converges to (lD(,, • B)(-, w) in the metric Poo on W*
Let Giq) be the mapping of Rd x W into W d corresponding to ^ G(q) satisfies (1) and (2) and (20)
G<,'>[x,B(-,w)] = («j» . BX-,w)
as defined by (20). Then
§20. STRONG
431
SOLUTIONS
for w G AJ_n where A,,n is a null set in (fi, 5, P). Let A, = U neZ ,A,, n and let Eq = {(i, u ) e l ' x W : Jiirn G? 5 [x, w] exists in W*}.
(23)
Since Giq) is cr(55ld x Wr)/fmd-measurable a mapping G(?) of R d x W into W* by
for every n € N, E„ G cr(Q3ffi« x 2TT). Define for (x, ID) G .E, for (x, u?) G E\.
Gw[i,u;] = ^ « - » J 0£W
(24)
Since G ^ satisfies (1) and (2) for every n G N so does G(9). Let K, be the subset of Rd x W covered by the mapping (x, B(-,u>)) of Acq C £2. Then for every (x, to) G if, there exists some u G AJ such that (x, IU) = (x, £(-, w)) so that by (22) and (21) for u g A J w e have lim G™[x,w]=
lim G<5,[x,-B(-,u>)] = lim (<&} • S)(-,w) = (
This shows that Kq C -E,. Then by (24) we have lim G{q)[x,B(-,uj)} = G{q)[x,B{-,u)]
(25)
forw G A'.
By (15), (21), (22) and (25) we have («(■, X<") • B)(w) = (* (,) • S)(-, w) = lim (4#> • 5)(-, w)
(26) =
limG^[i,B(-,o;)] = G ( ' ) [x,B(-,a;)]
foruGAJ,
verifying (14). This proves the existence of G (,) . The existence of Hiq) is proved likewise. With this we have shown that the existence of Fiq) implies that of F(q+i). Therefore by induction we have a sequence of mappings {Fiq) : q G Z+} where F(q) satisfies (1), (2), and (13). To show the existence of a mapping F of Rd x W into Wd satisfying (1), (2), and (12), let (27)
E = {(x, w)eB.dxW:
lim Fiq)[x, w) exists in Wd}.
The (7(93]^ x 2ir)-measurability of Fiq) for q G Z+ implies that £ is in CT(93KJ x Define a mapping of Rd x W into Wd by
(28)
lim F ( , ) [x,w]
F\xM = \V_„ M 0GW l-KXJ
for(x,u))G£ for (x, w) G £ c
W).
432
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Since Fiq) satisfies the measurability condition (1) for q G Z+, so does F. For fixed x G E , F (5) [x, •] is 2nj'"72D?-measurable for every t G R+ for q G Z + so that {w G W : lim F (?) [x, u>] exists in Vfd} £ SOT?'1" for every £ G K+. Then by (28), F[x, ■] is Wt'w/Wdt-measurable for every t e R+. This shows that F satisfies (2). To show that F satisfies (12), let A^ = (U,6z*A,) U A where A is the null set in (8). For u G A" lim F(q)[x,w] exists in W by (8) and (13). Thus by w
q—»-co
(28), (13), and (8) we have for u £ A ^ F[X,B(;LJ)]
= limFiq)[x,B(-,u)]
= lim X(q\-,cj)
q—>oo
= X(-,u>).
q—>oo
This proves (12). Finally if Z is an 5o-measurable d-dimensional random variable on (£2, #, -P) and if we let X(-,LO) = F[Z(UJ),B(-,UJ)] for a> G fi, then ( 5 , X ) is a solution of the stochastic differential equation on (Q, 5, {3<}, P) satisfying Xo = Z. ■
[II] Uniqueness of Strong Solutions Definition 20.6. Let S(E1' x W )feetfzecollection of all mappings F ofRd x W into W d satisfying the following conditions: 1°. /or every probability measure p on (M.d, QSKd) there exists a null set Nu in (Rd, 53 K J, p) such that F[x, •] = F„[x, •] a.e. on ( W , 2IT»*°, m ^ ) wnerc z G iV=, where F^ is a mapping ofRd x W into W1* such that 2°. .FM w
Wry*™*/Wd-measurable,
3°. /or every x G E*, i^[x, ■] is a Wt'w/V0d-measurable t GK+.
mapping ofW
into Wdfor every
Definition 20.7. We say that the stochastic differential equation (1) in Definition 18.6 has a unique strong solution if there exists F G S(Rd x W ) such that 1°. if(B, -X") is a solution of the stochastic differential equation on a standard filtered space (£1,5, { 5 J , P) and p is the probability distribution ofX0 on (Rd, 2$]^) then X(.-,u) = F„[X0(.u>),B{;u>)]
fora.e.um(a,$,P),
120. STRONG
433
SOLUTIONS
2°. if(£l, 5, {$t},P) is a standard filtered space on which an r-dimensional {$t}-adapted null at 0 Brownian motion B exists, Z is an arbitrary d-dimensional ^-measurable random variable on {Q., 5, P) with probability distribution Pz on (Rd, QSmd), and if we define a stochastic process X by setting X(-lu>) = FPz[Z(uy,B(;«>)]
forueQ,
then ( 5 , X ) is a solution satisfying the condition Xo = Z a.e. on (£2,5, P)-
Observation 20.8. Suppose the stochastic differential equation (1) in Definition 18.6 has a solution {B, X) on some standard filtered space (Q,5, {3<}, P)- Let P(B,X) be the prob ability distribution of (B, X) on (W r+d , 2ir +d ) and p. be the probability distribution of X0 on (Rd,S8m«). Let (W+d,<2rT+d'*, {2UJ+d,*},P(BiJn) be the standard filtered space gener ated by P(B,X), that is, W*d'* is the completion of 2TT+d with respect to Pm,xy, 2IF+d'0 = a(Wt+d U 91) where 91 is the collection of all the null sets in (WT+d,Wr+d'*',P(BtX)), and Wt+d-' = n e>0 2UJ«'° Let 7r0 and TT, be the projection of W r+d onto W and W d respec tively and let W and F be two processes defined on (WT+d,W+d'\ {2trt+,i'*},P{B,x)) by setting W(t,w) = (7r0(io))(t) and Y(t,tu) = (7r,(u>))(i) for (t,w) g R+ x W + d . We showed in Theorem 18.13 that (W, F ) is a solution of the stochastic differential equa tion on (W + d ,2ir + d '*, {Wt+d'*},P(B:X)\ (W,Y) and (B,X) have the identical probabil ity distribution P(B,X) on ( W + d , 2XT+d), and Y0 and X 0 have the identical probability dis tribution p on (Rd, ®md). Let Q = p x P(B,;O and consider the product measure space (Rd x W + d , (7(*8Ed x W+d), Q). Now let n2 be the projection of Rd x W r+d onto Rd x W and Qn2 be its probability distribution on (Md x W , cr(<8md x W)). Then for ExA0e <8ad x W we have Q„2(E x A0) = (px PiB,X))(E xA0x = p(E)P0,X)Ua
Wd)
x W d ) = p(E)PB(Ao) = p{E)mrw{A0).
From this it follows that Qn2= px mw on CT(251<1 X 2TT+d). Let Q(x-V\A) for A e
is a
2°. Q () (A) is a er(!8ad x 2ir)-measurable function on Rd x W for A G <7(93m.i x 2Ur+d),
434
CHAPTER 4. STOCHASTIC DIFFERENTIAL
3°. Q(A n 7r2-'(G)) = fa Q<x-"\A)(n x mrw)(d(x, v)) for A G a^B^ a(<8m* x 2TT).
EQUATIONS
x a U " ^ and G G
If we define Q&rt(Ai) = Q{x'v)(Rd x W x Ax)
(1)
for ,4, G STr*
then #<*•"> is a probability measure on (W
Q(E xA0xAl)=
f
CpdiAxXp
x
mTw)(d(x,«))
JEXAQ
for J5 x A0 x A, G 93Kd x W x 2 ^ when (a, u) G iVc. On the probability space (Rd x W r+d ,ff(95B<<x 2IF+''), Q) consider the projection TT3 of Rd x WT+d onto Rd. The probability distribution Q„, of TT3 on (Rrf, 93md) is equal to (i since for every E G 9$Bd we have = Q o ^~\E)
Qn(E)
= (ft x P(B I X ) )(JS x 2F +
Let (^(A) for A G c(23md x 2nr+
is a 93m
6°. Q(A n 7r3(G)) = Ja Qx(A)n(dx) for A G
Q7(A) = Qx(Rd x A)
foTA€Wr+d,
then Q* is a probability measure on (Wr+
Q(E x A0 x A{) = f Q^(A0 x
Ai)n(dx).
JE
Consider the probability space ( W ^ , W+d, Q*) for x G iVc By Theorem 18.21, (W, Y) is a solution of the stochastic differential equation on (Wr+'J, 2n r+d '*' 1 , { a u ^ ' * ' 1 } , Q*) sat isfying the condition Y0 = x and in particular W is an r-dimensional {2UJ+'i'''''a;}-adapted
§20. STRONG SOLUTIONS
435
null at 0 Brownian motion. Thus for the projection 7r0 of WT*d onto W , the probability distribution (Q1)^ of 7r0 on ( W , W) is that of the process W which is the r-dimensional Wiener measure mTw by Definition 17.9. Let (Q*)V(A) for A G W+d and v G W be a regular image conditional probability of Q* given n0 when x G Nc. Then 7°. there exists a null set Nx in ( W , 21T, mTw) such that (Q 7 )" is a probability measure on
(yvr+i,mr+d)foTveNi,
go (Q?)()(J4) j s
a
2IT-measurable function on W for every A G 2Ur+<',
9°. (W)(A n 7r0-'(G)) = / e ^ r W r n ^ C d w ) for A G 2Ur+<' and G G 2TT Thus if we define (5)
WYUi)
= ( ^ ) " ( W x A,) for A, e 2nd
then (Q 1 )" is a probability measure on (5) we have for every x € Nc (6)
every v € JV°. Furthermore by 9° and
Q^(Ao x A,) = / Wf{Ay)mrw{dv)
for A0 x A] G 2TT x Wd
•Mo
Lemma 20.9. There exists a null set N in (Rd, © j ^ , ^) and a collection {Nx : x G 7VC} o/ null sets in ( W , 2IT, m ^ ) iwcft ffeaf Q<«=.") = (Q*)«
on (W*, 2JT) w/iew v £ Ncxandx
e Nc.
Proof. L e t £ x A ( , x A i 6 93md x SIT x 23J'i. Then (1)
/ ( / W*KAi)mTw{dv))
i £ l.Mo
fi(dv) = / J
JEx/lo
Q5W(A,XAI x m r w )(d(x,t,))
= Q ( £ x A0 x A,) = / Q^CAo x Aj)^(dx) = J U
(Q*HAi)mrw(dv)}
fi(dx),
where the second equality is by (2), the third equality is by (4), and the last equality is by (6) of Observation 20.8. Thus corresponding to A0 x AjjEjEF x Wd, there exists a null set NAoxAl in (Rd, 9 V > Hi s u c h t h a t f o r t h e t w 0 f u n c t i o n s Qlx'vKM) and (Q*)"(Ai) o f i e R J we have f Q^(Ai)rnrw(dv)
= f Wr(Arirnrw(dv)
forz g
NAoxAr
436
CHAPTER 4. STOCHASTIC DIFFERENTIAL
Consider two measures j/f and i/f on (WT+d, W f i/f (A0 x A,) = fAo \ vl(A0 x A,) = SM
EQUATIONS
) determined by Q<^(Ai)mrw(dv) {QxY(Ai)mUdv)
for A0 x A, e 227 x 22T\ Let <S be the collection of all A g 227+<< such that i/f (A) = i/f (A) fora.e. xin(R t i ,«8 l d ,(3y 0 ). Then 227x227* C 0 and it is easily verified that © is a d-class. Thus by Theorem 1.7 we have 227+t' =
Q^^{Ai)mrw{dv)
= / T&fiAxWwidv) •'>lo
for A0 x A, G 227 x 2n d when x G ATC
Thus corresponding to every x e Nc and A) G 227^ there exists a null set iVIij4l in ( W , 2 2 7 , m ^ ) such that Q ^ C A , ) = (Q^RA,) when i> G NcxA]. Since 227 is countably determined, corresponding to our x G Nc there exists a null set iVr in ( W , 227, mrw) such that <2<*'">(Ai) = (Q^RAi) for Ai G 22Jd when u G A^. ■ Lemma 20.10. Suppose the stochastic differential equation (1) in Definition 18.6 has a solution (B,X) on some standard filtered space (£2,5, {5t}, P)- Let p be the probability distribution of XQ on (R.d, 23n<0- Assume further that the stochastic differential equation satisfies the pathwise uniqueness condition under deterministic initial conditions. Then there exists a null set N^ in (M.d, 2Jmd, p.) and a collection {Nx : x G N*} of null sets in ( W , 227,m^,) such that on (Wd, Wd) when v e Ncx and x € N<,
Q&d = 6FM
where Fx is the mapping ofW into Wd defined in Theorem 19.18, that is, the probability measure Q(x'v) on (Wd, 20d) is the unit mass at Fx(v) G W d . Proof. According to Theorem 18.21 there exists a null set N in (Rd, 93K
XQ
§20. STRONG
437
SOLUTIONS
has a solution on the function space Wr+d- If we assume further that the stochastic differ ential equation satisfies the pathwise uniqueness condition under deterministic initial con ditions then by Theorem 19.18 the mapping Fx of W into W* exists for every x G Nc. Let N^ be the union of this null set and the null set N in Lemma 20.9. Then for every x G N^ there exists a null set N'x in ( W , 2TT, mrw) such that Y(-, w) = Fx[W(-,w)\ for w G W+d with W(-, w) G {N'x)c. In other words, for the projections 7r0 and ir\ of W*d onto W and Wd we have (1)
JTi(tu) = FX[TTO(W)]
for u> G VT+d with 7r0(™) G (iV*)".
Now for x G iV=, there exists a null set N'J in ( W , 2 i r , m ^ ) such that for v G (i\^') c , (Q 1 )" is a regular image conditional probability of Qx given the projection no of W + ' ' onto W , but restricted from W*d to Wd by (5) of Observation 20.8 . But according to (1), for every w G W+d with 7r0(u>) G (iV^)c, w\(w) is uniquely determined by wo(w) and thus ( Q ^ = 6Fx{v). Let NX = N'XU N%. Then ( ^ p F = * Fl(u) for v G JVJ and x G 2V£. I Lemma 20.11. Under the same assumptions as in Lemma 20.10, let A = {(x, v) G Rd x W : £<*■»> is wof a unit mass on (Wd, Wd)}. Then A is a null set in (Rd x W , a(JB%4 x 2TT), p. x mTw). Proof. Since Wd is a separable metric space, for every n G N there exists a countable collection of closed spheres {5 n ,, : i G N} in W d , each with radius r(S n ,;) = n _ 1 , such that Wd = UigN'S'n,,. Let us show that for an arbitrary probability measure v on (W d , 233^) we have (1)
i/ is a unit mass O v(Snj) = 0 or 1 for every n and *
Now if v is a unit mass then clearly i/(Sn,,) = 0 or 1 for every n and i. Conversely assume that v(5„,,) = 0 or 1 for every n and i. For fixed n, no two closed spheres S„,; with v measure 1 can be disjoint for otherwise we would have v(Wd) > 2. Let Kn be the closed set which is the intersection of those closed spheres in the collection {Sn%i : i G N} which have v measure 1. Then v(K„) = 1 and the radius r(Kn) < n~l. For the collection of closed sets {Kn : n G N} we have Kn l~l Km ^ 0 for otherwise we would have i/(Wrf) > 2. If we let C„ = n^ = 1 Km for n G N, then Cn, n G N, is a decreasing sequence of closed sets with v(Cn) = 1 and <5(C„) < n~l. Since Wd is a complete metric space and since 6(Cn) J. 0 as n —> co, there exists a unique w G W such that n„6NC„ = {w}. Then i/({to}) = lim v(Cn) = 1. Thus i/ is a unit mass. This proves (1). n—*oo
438
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
By (1) we have Ac = {(x, v) G Rd x W : ~Q^ is a unit mass on (Wd, &}d)} = {(x, v) G Rd X W : Q&rt(Sn,i) = 0 or 1 for every n and i}. Since Sn,{ G Q3wd = 2Hd, Q^C^n,;) is a a-(Q5Md X 2ir)-measurable functions of (x, v) G Rd x W" by 2° and (1) of Observation 20.8. Thus Ac G a^* x ST). To show that Ou x m^)(A) = 0, let N„ be a null set in (Rd, 2 3 ^ , p) and {iV^ : a; G Nfi be a collection of null set in (WT,Wr,mTw) as specified in Lemma 20.10. Then for x G N£ and v G 2VJ, we have Q^'^ = SFl{v) so that (x,v) G Ac and thus v G (A%,. which is the section of Ac C Rd x W at x G R* defined by (A%,. = {v e W : ( i , u) G A c } G S T . Therefore a; G N° implies iV£ C (A'),;,., or equivalently, A,,. C Nx. Thus (p x m U A ) = / j mw(A I .)/i(di) = / 7ld
^W=
mw(Ax.)/j,(dx)
< /
»
mrw{Nx)p{dx)
= 0,
since mTw(Nx) = 0 for x G N°. This shows that (p x m^)(A) = 0. ■ Lemma 20.12. Under the same assumptions on the stochastic differential equation (1) in Definition 18.6 as in Lemma 20.10, let F^ be a mapping ofRd x W into Wd defined by (1)
%,[*,,„] = W^
(2)
F„[xt v]=0eWi
(3)
F J s , u] = 0 G W d
for (x, v) G (Nl x W ) n Ac for(x,v)£(N°xW)nA for (x, v) G JV„ x W r .
77iera FM satisfies conditions 2° and 3° of Definition 20.6, that is, (4) F„ is ff(»md x WY*mw/W1-measurable, 1 (5) /or every as G R *, F^x, ■] w a 52HJ'1"/2Uf-measurable mapping o / W info W ' / o r every i G R+. Assume further that for every x G R+ ine stochastic differential equation has a solution (B, X) with Xo = x on some standard filtered space so that the mapping Fx o / W into W d defined in Theorem 19.18 existsfor every x G Rd. Let F be a mapping ofRd x W into Wd defined by (6)
F[x, v] = Fx(v)
for (x, v) G Rd x W
TTien F^ and F satisfy condition 1° of Definition 20.6, that is, there exists a null set N^ in (Rd ,Q3md, fi) such that (7)
F[x, ■] = FIM[x, •] a.e. on (W r , 237'"', mrw) when x G Nc
439
%20. STRONG SOLUTIONS
Proof. To prove (4) note that by Lemma 20.11, Ac €
F"1 n (Ni x W) n AC = {(*,v) e (iv; x W) n AC : F^X,v)eA,} = {(x, u) £ (N° x W ) n Ac : Q^>(A,) = 1} G ( r ( » ^ x W) by the definition of A in Lemma 20.11 and by 2° of Observation 20.8. Thus F„ is
= {ve(Ac)x,:WY(Al) = i}e<Wrr
while i^[x,-]"'(Ai) n A,,. = A*,, or 0 according as 0 6 Ay or 0 G A\. Thus -FJx,-] is 2UJ',"/2Uf-measurable when x 6 JV=. When x G JV„, i ^ x , •] = 0 so that F^x, ■) is Wt'w/Wd -measurable. To prove (7), note that x G N; and u G Ncx =► Q ^ > = tfRW => (x, » ) 6 A C 4 % t I , u ] = Q55T so that i ^ O ) = FM[x, u] for a G N£ and t> G N%. Since iVJ is a null set in this verifies (7). ■
(WT,WT,mrw),
Theorem 20.13. The stochastic differential equation (1) of Definition 18.6 has a unique strong solution if and only if (1). for every probability measure /i on (Rd, Q3sd) there exists a solution (B, X) of the stochastic differential equation on some standard filtered space (Q, 5, {& }, P) with /j, as the probability distribution ofXo, (II). the stochastic differential equation satisfies the pathwise uniqueness condition.
440
CHAPTER 4. STOCHASTIC DIFFERENTIAL
EQUATIONS
Proof. 1) Necessity. Suppose a unique strong solution F exists. To verify (I), let ft be an arbitrary probability measure on (Rd, 23m.0. Consider the product measure space (Rd x W , a(^BMd x W), fj, x mTw) and consider the standard filtered space generated by fi x mrw, (Rd x W , < T ( 3 V x Wry*mw, {
Y(;W) =
FPz[Y0(w),W{;W)]
for a.e. w in (WT+d,Wr+d't, P / ^ , , ) . If we define (2)
X(;u)
=
FPx[Z(.u),B(;u>)\
§20. STRONG
441
SOLUTIONS
for w € £i, then (B,X) and (W,Y) have an identical probability distribution P[B',X') o n (Wr+
X(-,UJ) =
F[Z(LJ),B(;UJ)]
for Z(LO) € Nc where N is a null set in (Rd,
Appendix A Stochastic Independence Definition A.l. With an arbitrary index set A, let (£„ be a subcollection of$ in a probability space (Q, #, P)for a £ A. We say that system { { „ : < « £ A} is stochastically independent with respect to P, or simply independent when P is understood iffor every finite subset { « i , . . . , a „ } of A and for arbitrary Eaj 6
P(r\]=1Eai) =
l[P(Eaj).
In particular {Ea : a 6 A} where Ea 6 5 '* said to be independent iffor every finite subset {on,..., an} of A we have we have P(n^=lEQ]) = n"=i P(Eaj). Note that for a finite index set A, say A = { 1 , . . . , n } and we have {l£j : j ~ 1 , . . . , ra}, the assumption that P(nj =1 -E J ) = ETjU P(-Ej) for arbitrary^ € £jforj = 1,... ,ndoesnot imply that P(n} = ] Ej) = JTJ=i P(Ej) for arbitrary Ej € <Ej for j = 1,... ,k for k < n. For instance consider the particular case {Ej : j = 1,..., n} where i?, = £ for j = 1 , . . . , n — 1 for some E 6 ? with P(.E) = 1/2 and En = 0. In this case we have P(n]=]Ej) = 0 = Il"=i P(£j), but PCn^i'-Bj) = P(E) = 1/2 ^ 1/2""1 = n ^ l 1 P(Ej). There is an exception to this when O G (5j, and in particular when <£,- is a
P(n%lEj) = 443
f[P(Ej)
444
APPENDIX A. STOCHASTIC
INDEPENDENCE
for arbitrary Ej € <&jforj = 1 , . . . , n, then {<Ej : j = 1 , . . . , n} is an independent system. Proof. Take an arbitrary subset of { 1 , . . . , n}, say { 1 , . . . , k} where k < n. Then for arbitrary Ej G
P(El n---nEh)
= P{EX n • ■ ■ n Ek n nM n ■ ■ ■ n Q„)
np(^)}(n p(a4= n w ;
[j=l
J l;=yfc+l
-I
(./=1
by (1). This shows that {(£,-: j = 1 , . . . ,n} is an independent system. ■ Observation A.3. As immediate consequences of Definition A. 1 we have the following. 1) If {<£<, : a G A} is an independent system, then for any subset A0 of A, {<£„ : a G ^40} is an independent system. 2) If {<8a : a £ A} is an independent system, and
P(n%1Fj) =
f[P(Fj).
We prove (1) by induction. Thus let us establish first that for any F\ G
P(Fi n n%2E3) = PCF,)
f[(Ej). 3=2
With fixed Ej e <£j for j = 2,...,n, setting (X.
()
define two finite measures \i\ and v\ on
j /ii(*i) = P{F, n H J ^ )
\ux{Fl) = P{Flm%1P{Ej)
for ^ G <7(<£0
forFGa^).
445
APPENDIX A. STOCHASTIC INDEPENDENCE
Then for Ej G CT(<£J) we have AM(PI) = P(Tlj=1Ej) = Il"=i -P(-Bj) = ^IC^I) w h e r e t h e second equality is by the independence of {<£Q : a G A). Thus ^i = v\ on the 7r-class (Hi. We have also px{Q) = P(£2 n n? =2 Pj) = P(Q) n" =2 P(£y) = ^i(Q), where the second equality is by the independence of {<Ea : a G A}. Thus by Corollary 1.8 we have p.\ = vx on
(4)
P(n*=IF; n nj=t+I£,) = n PW ft P(*i)
for Fj G ff(€j) for j = 1 , . . . , k and E} € €> for j = k + 1,... ,n. With fixed P,- G
f /«*ti(f]tn) = P ( n £ j J J n n ^ a B i ) \ •/*+,(Pt+1) = Yl-ll P(Fj)YlU+2P{Ej)
for P i + I G a(<Et+1) forF M G <7((£,+1).
Then for Ek+\ G (£k+i we have f,k+l(Ek+l) =
P(n%lF3nn]=k+lE]) = f[P(F]) ft P(E3) = uM(EM) J=\
j=k+i
where the second equality is by the induction hypothesis (4). Thus p,k+\ = vk+i on
/ifcri(fl) = P(nkjStFjnflnn]=k+2Ej) = f[PiE^PW j=l
n
H P^) = VM(.a) j=k+2
where the second equality is by (4). Therefore by Corollary 1.8 we have /ik+i = vk+l on a(j£.k+\). Therefore by (5) we have fc+I
n
Pirt&Fi n n%k+2Ej) = n P{F3) n P&J) forP, G a(
446
APPENDIX A. STOCHASTIC
INDEPENDENCE
Proof. Let <Ea = {Ea} for every a 6 A. Then <Ea is trivially a 7r-class. Thus the in dependence of {Ea : a G A} implies that of {a(£a) : a e A} by Theorem A.4. Now cr(<£a) = {Ea, Eca, 0, Q.}. Thus {JF*a : a G A} is an independent system by 2) of Observa tion A.3. ■ Let X be a d-dimensional random vector on a probability space (£1,£, P ) , that is, X is an 5/QSjd-measurable mapping of Q. into R d . Let a(X) be the a-algebra generated by X, that is, the smallest sub-cr-algebra of g with respect to which X is measurable. In other words o(X) = X-\fBm<). Definition A.6. Let {Xa : a G A} be a system of random vectors on a probability space (Q, g, P), the dimension ofXa being da G N. We say that {Xa : a £ A} is an independent system if{a(Xa) : a G A} is an independent system in the sense of Definition A. 1. Remark A.7. A system {Ea\a € A] of members of 5 in a probability space (Q, g, P) is independent if and only if the system of random variables {lsa : a G A} is independent. Proof. By Definition A.6, { 1 ^ : a G A] is an independent system of random variables if and only if {O-(1E 0 ) : a G A} is an independent system of sub-c-algebras of g. But vi^Ea) = {Ea, Ea, 0, £2} = <j({Ea}) and according to Corollary A.5 the independence of {a{{Ea}): a G ^4} is equivalent to the independence of {Ea; a G A}. I Theorem A.8. Let {Xa : a € A} be an independent system of random vectors on a probability space (Q, J , P), the dimension of X being equal to da G N. For each a G A, let Ta be a 53mdQ/QSjjite, -measurable mapping ofRda into Rk" where ka G N. Then {Ta o Xa '■ a G A} is an independent system of random vectors, the dimension ofTa o Xa being kaProof. Since Xa is an g/23 i; i -measurable mapping of O. into Rda and Ta is a Q3mda /2$ffi*„ measurable mapping of R*" into Rk" ,TaoXa is an S/93K*<» -measurable mapping of Q. into Rh", that is, a ka-dimensional random vector on (Q, 5 , P). To show that {Ta oXa : a G A} is an independent system of random vectors we show that {a(Ta o Xa) : a G A} is an independent system of sub-cr-algebras of 5 according to Definition A.6. Now (1)
a{Ta o Xa) = (T„ o J r a ) - , ( » 1 » a ) = X - ' C T - ' O B ^ ) ) .
APPENDIX A. STOCHASTIC
447
INDEPENDENCE
Since Ta is 03^0,/nSji*,,-measurable, we have T ^ ' O B E * ) C © B **- Thus from (1) we have (2)
Now since {Xa) : a £ ,4} is an independent system of random vectors, {a(Xa : a £ A} is an independent system of sub-cr-algebras of euf. Then (2) implies that {o(Ta o Xa) : a £ A} is an independent system of sub-u-algebras of 5 by 2) of Observation A.3. ■ Theorem A.9. Lef Xj be a dj-dimensional random vector on a probability space (£2,5, P) for j = 1,...,n. Consider the d-dimensional random vector X = ( X i , . . . , X n ) where d = d\ + ■ ■ ■ +dn. Let Pxl, ■ ■ ■, Px„, and Px be the probability distributions ofX\, . . . , Xn and X on (R
{Xi, . . . , X„} is an independent system -» Px = Px, x • ■ ■ x Px„ on (Rd, 23E<<).
Proof. Since 55Kd, x • ■ • x 93^,, is a semialgebra of subsets of Rd and since ^(05^^, x ■ • ■ x 551d„) = QSjd by Proposition 13.2, we have (2)
Px = Px, X • • • x P*„ on (R',
<* PxCBi x ••• x P „ ) = n ^ J ( S J ) f o r B j £ » , « , , j = l , . . . , n j=i
«■ ( P o X - ' ( P i x ■ ■ • x Bn) = f[(P o X-l)(Bj)
for B} £ Q S ^ , j = 1 , . . . , n
i=i
On the other hand
&
{X|, . . . , X„} is an independent system {o-(Xi), . . . , o-(Xn)} is an independent system n
*> P(n? =1 £j) = I I p (- E i) f o r ^ «
P{^(X-\Bj)))
€ ff
(*•>), i = 1 , . . . , n by Proposition A.2
= f[PCXf'CBy)) for Bj £ B B «,, j = 1,...,n
448
APPENDIX A. STOCHASTIC INDEPENDENCE
since o(Xj) = XJ1^*,). Now the fact that V^{Xjl{Bj)) = X~\BX x • • • x Bn) implies l that Ptf%y(Xj {Bj))) = (Po X-l)(B{ x ■ • • x Bn). Thus (3)
{X\, . . . , Xn} is an independent system n
<* ( P o J f - ' ) ( 5 , x • • • x B J = n P{Xj\Bjy)
for ^ e » , - , , j = 1, • • • , n
By (2) and (3) we have (1). ■ The characteristic function of a probability measure p. on (Rd, 2$md) is a complex valued function on Rd defined by ¥>(»;; /0 = / . exp{i (e, T?) }/i(dO
for ^ e l
d
where (£, T?) = Y?j=\ £.jVj f° r £>»7 £ ^ . It is well known that the characteristic function of a probability measure on (Rd, QSJJJ) is unique, that is, if p and v are two probability measures on (Rd, Q3Md) and (•; v) on Rd then p, = u on (Rrf, Q3 M J). Let ^ i , . . . ,/<„ be probability measures on (Rdl, 9SB
for r\ = (r)h ... ,r)n) £ Rd' x ■ ■ ■ x E
4
Theorem A.10. (M. Kac) Consider a system of random vectors {X\,..., Xn } on a prob ability space (Q, 5, P), the dimension of X, being dj g N. Let Px, be the probability distribution ofXj on (Rd', SBa
(i)
v(.n\Px) =
for T) = (»?i,..., »7n) € Rd' x • • • x Rdn, or equivalently (2)
E f f t e x p ^ {*;,%)}] = flEfexplj ( X ^ ) } ] 3=1
forrj = (riu---,Vn) £ Rd] x ■ ■ • x R*«
i=i
449
APPENDIX A. STOCHASTIC INDEPENDENCE Proof. Note that (1) can be written as
(3)
Lnex P {^ >J?i )}p*(do=n L «?(*<&. »y}}fti(«*o
and (3) is equivalent to (2) by the Image Probability Law. Now suppose { X i , . . . ,Xn} is an independent system. Then by Theorem A.9 we have Px = Px, x • ■ ■ x Px„- Fubini's Theorem then implies (3). Conversely suppose (3) holds. To shows the independence of {Xt,..., Xn} it suffices to show that Px = Px, x • ■ ■ x P*„ according to Theorem A.9. Let Q = PX] x ■ • ■ x PXn. If we show that ^(77; P x ) = ¥>(T?; <2) for 77 G Rd = Rdl x • ■ • x Rd", then Px = Q and we are done. Now ^n\Px)=
fexV{i{(,n)}Px(dO
= J^exV{i{£,r,}}Q(dO
= f[fd
exp{i(fc,ifc)}iVdfc)
=
where the second equality is by (3) and the third equality is by the fact that Q = Px, x • • • x Px„ and Fubini's Theorem. ■ Theorem A.ll. Consider a system of random vectors on a probability space (Q, 5, P) given by (1)
ixk,qk : s * = l , . . . , n t ; f c = l , . . . , m } ,
rte dimension ofXk,qk being dk,qk £ N. For each k = 1,..., m, consider the subsystem
(2)
{X M ,...,**,„,}.
Lef X4 be a dk,\ + ■ ■ ■ + dk,nk -dimensional random vector defined by X = (Xk,l, ■ •• ,Xk,nk)
and consider the system of random vectors (3)
{Xk:k
=
l,...,m}.
1) If the system (1) is independent, then so is the system (3). 2) Conversely if the system (3) is independent and furthermore for every k = 1 , . . . , m the system (2) is independent, then the system (1) is independent.
450
APPENDIX A. STOCHASTIC
INDEPENDENCE
Proof. Suppose the system (1) is independent. Then m
m
E [ g e x p { i (Xk,r,k)}]:= E[exp{i^(X,, J ? A : )}] ErJJexpi; (X ,r, )}] = E[exp{i £ (Xk,T)k)}] m nkk k
= E[« P {»XE(^**. »?*,»*>}] : = IHlE[exp{i(Xfe,ti»•?*,!»}}]
= E [ c a p { * X E ( * * * . ^ . « ) } ] = 1 1 I I E[exp{i(Jrt^,i7t, w )}] k=\ 9i =l t = l 9t=l = Y[E[tx P{i(Xk,nk)}] =
m k=\
Y[E[txP{i(Xk,nk)}]
k=\
where the third equality is by the independence of the system (1) and Theorem A. 10 and the last equality is by the independence of the system (2) as a subsystem of the independent system (1) and by Theorem A. 10. According to Theorem A. 10, this shows the independence of the system (3). Conversely suppose the system (3) is independent and for every k = 1 , . . . , m the system (2) is independent. Then 771
Tlfc
Etn n
ex
k=l qqk=l k=\ k=\
771
7lt
k=\
qk=\
)}]=E[exp{i'£ E(X kMT,kM)}] z ££(X t , ? * % , ' u»3 Pi i (**..*»& : ,,J}]=E[exp{ 1
m
m
= £ (Xk, Vk}}] = (Xk, r,k}}] E[nexp{:(Jr*,i/*)}] = E[exp{i E[exp{i^(X,,^)}] == Etflexp{i m
771 m
Tit Tlfc
= l[E[ex Y[E[ex {i(X ,Vk)}]-.==n IY[Y[E[cx (X I E t e x P {VJ {i{X (Xk, )}] V k qk,Vk,qk,}}] V{i k,Vkk)}] fc=l
k=\ k=l *=1 q9k=l
where the second from last equality is by the independence of the system (3) and Theorem A. 10 and the last equality is from the independence of the system (2) and Theorem A. 10. This shows the independence of the system (1) according to Theorem A. 10. I Definition A.12. Let <£„ be a subcollection of $ for a g A in a probability space (Q., 5, P) and Xp be a dp-dimensional random vector on (Q., g, P) for fi 6 B. We say that {(£„, Xp : a £ A,f3 € B} is an independent system if{£a, a(Xp) : a G A, 0 £ B} is an independent system in the sense of Definition A. 1. Theorem A.13. Let (Sj be a sub-o-algebra o / 5 in a probability space (Q, #, P) for j = 1 , . . . , m and Xk be dk-dimensional random vector on (Q, 5, P)for k= 1 , . . . , n. Then the following three conditions are equivalent:
451
APPENDIX A. STOCHASTIC INDEPENDENCE
!"• {®ji-^t '■ j = l,...,rn;k = I,... ,n} is an independent system in the sense of Def inition A. 12, that is, {(Sj,
(i)
PlriJUGj n nnk=lFk) = fl P(G,) f [ P(Fk). j=\
k=l
Now 1 G is a (Sj-measurable random variable and lpk is a crCJGO-measurable random vari able. Thus by 3°, {1Q, lpt : j = 1 , . . . , m; k = 1,... ,n} is an independent system and thus by Definition A.6 the system {cr(lG;),
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Appendix B Conditional Expectations [I] Definition and Basic Equalities Let © be a sub-er-algebra of 5 in a probability space (£2, 5, P). Two ©-measurable extended real valued random variables X\ and X2 are said to be ( 0 , P)-a.e. equal if X\ = X% on Ac where A £ 0 with P(A) = 0. In the collection of all ©-measurable extended real valued random variables on (£2,5, P), ( 0 , P)-a.e. equality is an equivalence relation. Proposition B.l. Let & be a sub-a-algebra of$ in a probability space (£2,5, P). Con sider the equivalence relation o / ( 0 , P)-a.e. equality in the collection of all 0 -measurable extended real valued random variables on (£2,5, P). For every integrable extended real valued random variable X on (£2,5, P) /Aere arista a unique equivalence class consisting of all &-measurable extended real valued random variables Y on (£2,5, P ) such that [YdP=fxdP Ja Ja
foreveryGe®.
Proof. For an arbitrary integrable extended real valued random variable X on (£2,5, P), let us define a set function p. on 0 by setting p(G) = JGX dP for G G 0 . Then p is a signed measure on 0 . Since p(G) = 0 for every G G 0 with P(G) = 0, p is absolutely continuous with respect to P on (£2,0). The Radon-Nikodym derivative of p with respect to P on (£2, 0 ) is then an equivalence class with respect to the ( 0 , P)-a.e. equality consisting of all 0-measurable extended real valued functions Y on £2 such that JaY dP = p(G), that is, / G YdP = JGXdP, for every G G 0 . ■ Definition B.2. Let X be an integrable extended real valued random variable on a proba453
454
APPENDIX B. CONDITIONAL
EXPECTATIONS
bility space (Q, 5, P) and let <& be a sub-a-algebra o / J . By the conditional expectation of X given 0, denoted by E(X | 0 ) , we mean the unique equivalence class of all 0-measurable extended real valued random variables Y on (Q, 5, P) such that (YdP=j Ja
XdP
for every G € <5.
JG
The members of the equivalence class E(X\<3) are called versions of the conditional ex pectation of X given 0. Thus an extended real valued random variable Y is a version of E(X 10) if and only if it satisfies the following two conditions: 1°. Y is &-measurable. 2°. fG Y dP = fa X dPfor every G € 0. For convenience the notation E ( X | 0 ) is also used for an arbitrary version of the con ditional expectation. For an arbitrary i 6 5, the conditional probability of A given 0, denoted by P(A\<8), is defined as the conditional expectation of the random variable 1A given 0 , that is, F ( A | 0 ) = ECU |®). Note that every version of Y of E(X | 0 ) is an integrable random variable since O g 0 and thus fnY dP = / n X dP £ R. This implies that every version is finite a.e. on (Q, 0 , P) and thus there exists a real valued version of E(X | 0 ) . Since the same notation E(X 10) is used for both an equivalence class and an arbitrary representative of it, what is meant by the notation E(X 10) should be determined from the context. For instance in expressions such as " JG E(X | <8)dP " and "E(X 2 10) > E(X] 10) a.e. on (Q.,®,P)", E ( X | 0 ) , E ( X i | 0 ) , a n d E ( X 2 | 0 ) are arbitrary versions of the con ditional expectations, and in expressions such as " Z £ E ( X | 0 ) " and "ECX^©!) C E(X2102)", the notations E(X \ 0 ) , E(X, 10,), and E(X 2 10 2 ) are for equivalence classes. For a random variable Z the expression " E ( X | 0 ) = Z" is also used to indicate that Z is a version of E(X 10). The notation ZE(X \ 0 ) may mean the collection of all versions multiplied by Z of an arbitrary version multiplied by Z. Observation B.3. Let X\ and X2 be two integrable extended real valued random variables on a probability space {SI, 5, P) and let 0 be a sub-cr-algebra of J. If there exists a 0 measurable extended real valued random variable Y0 on (Q, 5, P) which is a version of both E(Xi 10) and E{X2 \ 0 ) then E(X, 10) = E(X 2 10), that is, if E{X^ 10) n E(X 2 10) 4 0 thenE(X,|0) = E(X2|0).
APPENDIX B. CONDITIONAL EXPECTATIONS
455
Proof. Let Yj be an arbitrary version of E ( X j | 0 ) for j = 1,2. Then Yj = Y0 a.e. on (£2, 0 , P ) for j = 1,2 so that yj = y 2 a.e. on (Q, (5, P). Thus the two equivalence classes with respect to ( 0 , P)-a.e. equality, E(Xj 10) and E(X 2 10), are identical. ■ Observation B.4. Let 0 be a sub-cr-algebra of g in a probability space (£2,g, P). Then for any c 6 R, we have c G E(c|0), that is, {c} C E(c|0), but in general {c} 4 E ( c | 0 ) since E ( c | 0 ) consists of all ©-measurable extended real valued random variables which are ( 0 , P)-a.e. equal to c on Q. Theorem B.5. Let X, X\, and X 2 be integrable extended real valued random variables on a probability space and let & be a sub-a -algebra of$. Then 1) X, = X 2 a.e. on ( G , 3 , P ) => E(X, | 0 ) = E(X 2 |0), 2) X is 0-measurable => X G E(X 10), 3) X = ca.e. on (Q, g, P ) => c G E(X 10), 4) X >Oa.e. on (Cl,$,P) ^ E ( X | 0 ) > 0a.e, o « ( Q , 0 , P ) , 5) E ( c X 1 0 ) D cE(X | <S)for c G R an
3) A constant c is always 0-measurable. If X = c a.e. on (Q,, g, P), then for any G G 0 C g we have JacdP = faX dP. Thus c satisfies conditions 1 ° and 2° in Definition B.2 and is therefore a version of E(X | 0 ) . 4) Suppose X > 0 a.e. on (Q, g, P). Let Y G E(X 10). Then / G y rfP = JG X dP > 0 for every G G 0 and therefore y > 0 a.e. on (£1, 0 , P). 5) Let y G E ( X | 0 ) . Then Y is 0-measurable and so is cY. For every G G 0 we have JacYdP = cfGY dP = cfaX dP = / G cX dP and thus c y G E ( c X | 0 ) . This shows that c E ( X | 0 ) C E ( c X | 0 ) for any c G R. Suppose c 4 0. To show that in this case we have E ( c X | 0 ) C c E ( X | 0 ) , let Z G E ( c X | 0 ) . Then Z is 0-measurable and JaZdP = JGFcXdP = cfaXdP = cfGYdP for every G G 0 and thus Z = cY a.e.
456
APPENDIX B. CONDITIONAL
EXPECTATIONS
on (fi, 0 , P). This shows that Z/c = Y a.e. on (Q, 0 , P ) and thus Z/c G E(X | 0 ) , that is Z 6 c E ( X | 0 ) . Therefore E ( c X | 0 ) C c E ( X | 0 ) and thus E ( c X | 0 ) = c E ( X | 0 ) when 6) Let y, 6 E(X, | 0 ) and F 2 g E ( X 2 | 0 ) . Then c,X, + c 2 X 2 is 0-measurable and for every G e S w e have / ( c F , + c 2 y 2 )dP = ci / YxdP + c2 I Y2 dP
JG
JG
= cx f XxdP JG
+ c2 f X2dP= JG
JG
f (CJXI + c2X2)dP.
■*G
Thus c{Yx + c2Y2 G E(ciX, +c 2 X 2 |(S). This shows that c , E ( X , | 0 ) + c 2 E ( X 2 | 0 ) C E ^ X j + c 2 X 2 | 0 ) . Suppose at least one of cx and c2 is not equal to 0, say c2 i- 0. To showE(cjXi+c 2 X 2 |©) C c 1 E(X 1 |0) + c 2 E(X 2 |0), let 2 G E C d X , + c 2 X 2 | 0 ) . Let y G E(X! |(8) and y = l/c 2 {Z - ciYi}. Since Z and y are ©-measurable so is Y2. Also for every G G 0 we have \ Y2dP = - { \ ZdP-cx Ja c2 {JG
\ YxdP~\ J
JG
= — / {c,Xi + c 2 X 2 }dP - - / X , d P = / X 2 dP. c2 JG C2 7G ./G This shows that Y2 G E ( X 2 | 0 ) . Thus we have shown that for an arbitrary version Z of E(cxXx + c 2 X 2 | 0 ) we have Z = cxYx +c2Y2 where Y\ is a version of E(Xi |(S) a n d y is a version of E(X 2 10). Thus Z G c,E(X 1 |0) + c 2 E(X 2 |0) and then E f a X i + c 2 X 2 | 0 ) C c,E(X, | 0 ) + c 2 E(X 2 |0). Consequently E( C l X, + c 2 X 2 | 0 ) = c,E(X, | 0 ) + c 2 E(X 2 |0). 7) If X 2 > Xi a.e. on (Q.,3,P), then X 2 - X, > 0 a.e. on ( Q , 3 , P ) so that by 2) we have E(X 2 - X, | 0 ) > 0 a.e. on (Q, 0 , P). By 6) we have E(X 2 - X, | 0 ) = E(X2|0)-E(X1|©).ThusE(X2|0)>E(Xi|0)a.e.on(n,0,P). I Remark B.6. Regarding 5) of Theorem B.5 note that c E ( X | 0 ) = E ( c X | 0 ) does not hold in general when c = 0 since in this case we have 0 • E ( X | 0 ) = {0} while E(0 • X | 0 ) = E(O|0) which consists of all ©-measurable extended real valued random variables on (Q, j , P ) which are ( 0 , P)-a.e. equal to 0. Observation B.7. Let X be an integrable extended real valued random variable on a prob ability space (Q, 5, P ) and let 0 be a sub-
457
APPENDIX B. CONDITIONAL EXPECTATIONS that is, for every version Y of E(X | ©) we have E(Y) = E(.X").
Proof. If Y G E(X 1©), then Ja Y dP = JG X dP for every G G © and in particular with £2 G © we have JnYdP = fnX dP, that is E(y) = E(X). ■ Theorem B.8. Let & be a sub-a-algebra o / 5 in a probability space. Let X be an integrable extended real valued random variable and Z be a © -measurable extended real valued random variable on (£2, 5, P ) such that ZX is integrable. Then E(ZX\<S) = ZE(X\<8). Proof. If we show that for an arbitrary version Y of E(X | 0 ) , ZY is a version of E(ZX 10) then E(ZX | 0 ) D ZE(X 10). The fact that ZY is a version of E(ZX 10) also implies that for an arbitrary version V of E{ZX | 0 ) we have V = ZY a.e. on (£2, ©, P) and thus V is in ZE(X 10) so that E{ZX10) C ZE(X 10) and therefore E(ZX | 0 ) = ZE(X | ©). Thus it remains to show that for an arbitrary version Y of E(X \ 0), ZY is a version of E ( ^ X | ©). Since ZY is ©-measurable, it remains to show that (1)
I ZY dP = I ZX dP
for every G e ©.
To prove (1), consider first the case where Z = 1G0 where Go £ ©• In this case we have f ZYdP= JG
=
/ JGnGa
I lGoYdP
= f
JG
XdP=
YdP
JGnGa
f lGoXdP= JG
f
ZXdP
JO
where the third equality is by the fact that G n Go € ©. This verifies (1) for the particular case Z = 1G<) where Go G ©. If Z is a simple function on (£2,0, P) then (1) holds by the linearity of the integral with respect to the integrand. Next consider the case where Z > 0 and X > 0 on £2. Since Z > 0 on £1 there exists an increasing sequence {Zn : n G N} of nonnegative simple functions on (£2,0,P) such that Zn | Z on Q. Since Zn is a simple function on (Q, ©, P ) we have by our result above (2)
/ ZnY dP= f ZnX dP
for G G 0 .
Since X > 0 on Q we have Y > 0 a.e. on (£2,0, P ) by 4) of Theorem B.5. Thus Zn\ Z on £2 implies ZnY T Z ^ a.e. on (£2,0, P). Since X > 0 on £2 we have also Z„X | ZX
458
APPENDIX B. CONDITIONAL
EXPECTATIONS
on £2. Letting n -> oo in (2) we have (1) by the Monotone Convergence Theorem for nonnegative functions. Thus we have shown that for the case where Z > 0 and X > 0 on £2, (1) holds and therefore we have (3)
Z > 0 and X > 0 on Q =» E(ZJC|<5) = ZE(X|<S).
Let us remove the condition that Z > 0 on £2 but retain the condition that X > 0 on £2. In this case we have E(ZX\®) +
= E({Z+ - Z-}X\<S)
= Z E{X\&)-Z-E{X\&)
=
E(Z+X\<$)-E(Z-X\<3)
= ZE{X\<&)
where the third equality is by (3). Finally if we remove the condition that X > 0 on £2, then E(ZX\<8) = E(Z{X+ - X-}\0) = E(ZX+\<£>) = ZE{X*\<&)- ZE(X-\<8) = Z{E(X+\®)-E(X-\&)}
E(ZX~\e) = ZE(X\&)
where the third equality is from the result above for nonnegative X and the last equality is by 6) of Theorem B.5. ■ As a particular case of Theorem B.8, let us observe that if X and Z are two extended real valued random variables on a probability space (Q, 5, P), Z is 0-measurable where 0 is a sub-c-algebra of 5, X € LP(Q, 5, P) and Z 6 £,(£2,5, P) where p, g € (1, oo) with i + i = 1, then Z X is integrable so that E(ZX|(S) = ZE(X|<8). Definition B.9. Lef X be an integrable extended real valued random variable on a proba bility space (£2,5, P) and let & and 9) be sub-o-algebras o / J . With an arbitrary version YofE(X\<£>), we define E[E(X|(S)|fi]=E(y|S). An alternate notation E{X \ <S | ft) is used for E[E(X 10) | Sy]. The fact that the equivalence class E[E(X 10) | ft] of random variables does not depend on the choice of the version Y of E(X|(5) in Definition B.9 is shown in the following proposition. Proposition B.10. Let X be an integrable extended real valued random variable on a probability space (£2,5, P) and let <& and ft be sub-o-algebras of$. Let Y\ and Y2 be two versions of E{X\@). Then E(Yi\ft) = E(Y2\ft)
APPENDIX B. CONDITIONAL EXPECTATIONS
459
as equivalence classes. Proof. For j = 1,2, E(Y, \Sj) consists of all immeasurable extended real valued random variables Z such that JH Z dP = JH Y3 dP for every H £ ft. Now since Yi and Y2 are versions of E(X | 0 ) , we have Y{ = Yz a.e. on (£1, 0 , P) which implies that Yi = Y2 a.e. on (Q, 5 , P ) so that JA Y, d P = fA Y2 d P for every A £ 5 and in particular fH Y, d P = JH Y2 d P for every H £ S. If W, is a version of E(Yt | £ ) and W2 is a version of E(Y2\ft), then for any H £ ft we have JH W, dP = fH Y, dP = fH Y2 dP = fH W% dP and thus Wi = W2 a.e. on (Q, 3 , P). From this we have E(Yi \ft) = E(Y2 \ft). ■ Observation B.ll. For E[E(X 10) | £ ] defined above, note that E [ E [ E ( X | 0 ) | S ] ] = E[E(X|0)] = E(JQ. Note also that if Z £ E[E(X\<S)\Sj] and W is an ^-measurable extended real valued function such that W = Z a.e. on (Q.,ft,P), then W £ E[E(X\&)\ft]. Theorem B.12. Let X be an integrable extended real valued random variable on a proba bility space (£2, 5, P ) and let ®\bea sub-a-algebra o / 5 and 0 2 be a sub-a-algebra of<5\. Then E ( X | 0 , | 0 2 ) = E(X|02). Proof. Let Y be an arbitrary version of E ( X | 0 ! ) and let Z be an arbitrary version of E ( Y | 0 2 ) . To show E(Y|(S 2 ) = E ( X | 0 2 ) , it suffices according to Observation B.3 to show that there exists an 02-measurable extended real valued random variable which is a version of both E ( Y | 0 2 ) and E ( X | 0 2 ) . Since Z is a version of E ( Y | 0 2 ) , it suffices to show that Z is also a version of E ( X | 0 2 ) . Now Z is 02-measurable. Also for every G2 £ 0 2 we have fa Z dP = JQ2 Y dP = JGz X dP where the first equality is from the fact Z is a version of E(Y 10 2 ) and the second equality is from the fact that Y is a version of E ( X | 0 i ) . Thus we have shown that for an arbitrary version Y of E ( X | 0 t ) we have E ( Y | 0 2 ) = E ( X | 0 2 ) . Then E [ E ( X | 0 , ) | 0 2 ] = E ( X | 0 2 ) by Definition B.9. I Theorem B.13. Let X be an integrable extended real valued random variable on a proba bility space (£2,5, P ) and let <&\bea sub-a-algebra of$ and 0 2 be a sub-a-algebra of (Si. Then (1)
E(X|0 2 |0i)DE(X|0 2 ).
460
APPENDIX B. CONDITIONAL
If every version o / E ( X | © 2 | ® i ) is ^-measurable, (2)
EXPECTATIONS
then
E ( X | © 2 | © , ) = E(X|© 2 ).
Proof. 1) To prove (1), let F be an arbitrary version of E(X | ©2). Then F is ©2-measurable and hence ©i-measurable. According to Observation B.ll, to show that Y is a version of E ( X | © 2 | © j ) it suffices to show that there exists a version Z of E ( X | © 2 | © 0 such that Z = Y a.e. o n ( n , © i , P ) . Now let Z bean arbitrary version of E ( X | © 2 | © i ) . Then Z g E(W|©,) for some W g E(X|© 2 ). Thus for an arbitrary G, g ®i, we have / G | Z d P = / Gl W d P . Since W and F are versions of E(X |© 2 ), we have W = Y a.e. on (Q, © 2 , P ) which implies W = F a.e. on ( 0 , 5 , P ) and consequently / G | W dP = fGi Y dP. Thus / Gi ZdP = / G i F d P . Since both Z and F are ©i-measurable, this implies that Z = F a.e. on E(X |©,). This proves (1). 2) Suppose every version of E ( X | © 2 | © i ) is ©2-measurable. To prove (2), we show that an arbitrary version of Z of E(X|(S 2 |©i) is aversion of E(X|<S 2 )- Since Z is <S2measurable, it suffices to show that there exists a version F of E(X | <S2) such that Z = Y a.e. on (fl, (S2, P ) according to Observation B.ll. For this it suffices to show that fG2 Z dP = / G2 F d P for every G2 g 0 2 . Now if G 2 g <82, then G2 g <Si so that
/ ZdP= I E(X|<32)dP= / XdP=
[
YdP.
This proves (2). ■ According to (1) of Theorem B.13, we have E ( X | © i | © 2 ) D E(X | (S2) if ©2 C ©1 C 5. For an example of E(X | ©, | ©2) i E(X | ©2), see Example B. 14 below. Example B.14. Let (£2,5,P) = ((0, l],9K«),i],"U) where Sm(0>i) is the or-algebra of all Lebesgue measurable subsets of (0,1] and TUL is the Lebesgue measure. Let f(x) = x for x g (0,1]. Consider ©1 = 5 = M(0:1] and ©2 = {0,(0,1]}. Since the only © 2 measurable extended real valued functions on (0,1] are the constant functions on (0,1] and since fm]cdmL = / ( 0 1 ] / d m i , = \ for c £ E implies that c = | , E ( / | © 2 ) consists of the constant function \ on (0,1]. Thus E ( X | ® 2 | ® i ) consists of all Lebesgue measurable extended real valued functions g on (0,1] such that / G[ g dmL = fGi j drriL = \mL{G\) for every G\ g 9Jt(o,i]. The constant function \ is such a function but it is not the only one. Thus we have E(XI©, I©2) D E(X|© 2 ) a n d E ( X | © , | © 2 ) ^ E ( X | © 2 ) . Since E ( / | © , ) consists of all Lebesgue measurable extended real valued functions g on (0,1] such that
APPENDIX B. CONDITIONAL
EXPECTATIONS
461
/G, 9 dmL = JGi f dmt for every G\ G 9H(0,i] and since the constant function \ does not satisfy this condition, we have E(X | 0 2 ) n E(X 101) = 0. The condition 0 2 C 0 i implies that every version ofE(X|0 2 )is0i-measurable. How ever in general there is no inclusion relation between E(X | 0 2 ) and E(X 101) since every version Y of E(X 10 2 ) satisfies the condition fG Y dP = Ja X dP for G G 0 2 but not nec essarily for G G 0 1 . Example B. 14 above presents a case where E(X 10 2 ) n E(X 101) = 0. See Example B.15 for a case where E ( X | 0 2 ) C E ( X | 0 , ) b u t E ( X | 0 2 ) ^ E ( X | 0 i ) . Example B.15. Let (£2,5,P) = ((0, l],9H(o,i],"U) and let / be an integrable Lebesgue measurable but not Borel measurable function on (0,1]. Let 0 i = 5 = 9K(o,i] and 0 2 = 53(o,i], the a-algebra of Borel measurable sets in (0,1], We have 0 2 C 0 i . Let us show thatE(X\<St) CE{X\®X)butE(X|©2)^E(X|®i). Now E(X 101) consists of all 9Jt(o,i]-measurable extended real valued functions g such that (1)
/ gdmL= Ja,
Ja,
for G] G OT(0,i]
f dmL
and E ( X | 0 2 ) consists of all 55(0,i]-measurable extended real valued functions h such that (2)
/
Ja2
hdmL=
for G2 G 35(o,i]-
I fdmL
JGi
Let h be an arbitrary member of E(X 10 2 ). Then h is 93(0,i]-measurable and hence 9H(o,i]measurable. To show that ft is a member of E(X 10i), it remains to show that (3)
/
hdmL=
Ja,
I fdmL
for Gi G 2K(o,u-
Ja,
Let SDtm be the cr-algebra of all Lebesgue measurable sets in E. Recall that the measure space (R, SDTE, mL) is the completion of the measure space (R, ©m, mL). If d is in aTt(o,i] it is in 9 % and thus can be given as a disjoint union Gi = B n iV0 where B G 23m and N0 C N where TV G 93a with mL(N) = 0. Note that since B G 23m and 5 C (0,1] we have B G «8(0,i] and that 7V0 G 3Jt(0,i] with mL(N0) = 0. Thus / h dmL = h dmL + h dmL = f dmL + 0 Ja, JB JN0 JB =
I fdmL+
JB
JN0
fdmL=
Ja,
f dmL,
462
APPENDIX B. CONDITIONAL
EXPECTATIONS
probing (3). This shows that an arbitrary member h of E(X | ©2) is a member of E(X | ©1) and therefore E(X|© 2 ) C E(X|©i). Now since / is 9Jl(0,i]-measurable and since ©i = g =OT(o,i],/ is a member of E(/1 ©i). On the other hand since / is not <8(0,i]-measurable, / is not a member of E(/1 ©2). This shows that E(X | ©2) i E(X | © i). Theorem B.16. Let X be an integrable extended real valued random variable on a proba bility space (Q, g, P) and let <&\bea sub-o-algebra ofS and ©2 be a sub-o-algebra of<8\. 7JE(X|© 2 ) D E ( X | © 0 if and only if every version o / E ( X | © 0 is <S2-measurable. 2) E(X I ©2) C E(X I © 1) if and only if for every version Y ofE(X | 0 2 ) we have (1)
/ YdP= JG,
f XdP
/orGig©,.
JG,
3) If (Q, ©1, P ) is a complete measure space, ©2 contains all the null sets in (Q, © 1, P), and condition (1) is satisfied, then E(X | ©2) = E{X | ©1). Proof. 1) If E(X|© 2 ) D E(X|©i), then every version of E ( X | © 0 is ©2-measurable. Conversely, assume that an arbitrary version Y of E ( X | © 0 is ©2-measurable. To show that Y is a version of E(X |© 2 ), it remains to verify that JGl Y dP = fG2 X dP for every G2 G © 2 . Now if G2 £ ©2 then G2 G ©1 so that JGl Y dP = JGi X dP and thus Y is a version of E(X|© 2 ). ThereforeE{X\© 2 ) D E(X|©,). 2) Since every version Y of E(X|© 2 ) is ©2-measurable and thus ©1-measurable, Y is a version of E(XI©1) if and only if (1) holds. ThusE(X|© 2 ) C E ( X | © , ) if and only if (I) holds. 3) If (1) is satisfied then E(X|© 2 ) c E ( X | © , ) by 2). If we show that an arbitrary version Y\ of E ( X | © 0 is ©2-measurable, then by 1) we have E(X|© 2 ) D E ( X | © 0 and therefore E(X \ 0 2 ) = E(X | © 1). Thus it remains to show the ©2-measurability of Y\. Let Y2 be an arbitrary version of E(X | ©2). Since Y2 is a version of E(X 10 ]) b y 2), we have Y2 = Fi a.e. on (Q, ©1, P). Thus there exists a null set A, in (Q, ©1, P ) such that Y2 = Yx on K\. Since ©2 contains all the null sets in (Q, ©1, P), A* is a null set in (Q, © 2 , P ) and thus A^ G © 2 . The equality of Y\ and Y2 on A[ then implies the ©2-measurability of Y\ on \\. Since (Q, ©1, P) is a complete measure space and since ©2 contains all the null sets in (Q, © 1, P), (Q, ©2, P ) too is a complete measure space. Thus Yj , which is ©2-measurable on A,, is indeed ©2-measurable on Q. ■ [II] Conditional Convergence Theorems Theorem B.17. (Conditional Monotone Convergence Theorem) Let Xn, n G N, and X
463
APPENDIX B. CONDITIONAL EXPECTATIONS
be integrable extended real valued random variables on a probability space (Q, 5, P) and let & be a sub-a-algebra of 3. If Xn | X (resp. Xn i X ) a.e. on (£2,5,P), then E(X„ | ©) T E(X 10) (rcwp. E(X n 10) 1 E(X 10); a.e. on (£2, 0 , P). Proof. Consider the case where Xn f X a.e. on (£2,5,P). Let F„, n e N, and y be arbitrary versions of E(X„ | 0 ) , n G N, and K(X | 0 ) respectively. Let us show that Yn\Y a.e. on (£2,0,P). Now since Xn+i > Xn a.e. on ( f i . & P ) , we have E ( X n + , | 0 ) > E ( X n | 0 ) a . e . on (£2,0, P ) by 7) of Theorem B.5. Thus Yn t a.e. on(£2,0,P). Since Xn] X a.e. on (£2, 5, P ) and since X\ is integrable we have by the Monotone Convergence Theorem for an increasing sequence (1)
lim / XndP= "->°° JA
for A e 5
[ XdP
JA
and similarly since Yn | a.e. on (£2,5, P ) and since Y\ is integrable we have (2)
lim f YndP=
n—oo 7 G
f lim F„ d P
for G 6 0
y G n—oo
Since F n is a version of E ( X n | 0 ) we have JaYndP we have
= JGXndP
(3)
forG e 0 .
/ lim YndP= f XdP JG n—°° ./G
Thus from (1) and (2)
Since F is a version of E(X 10) we have fa Y dP - fa X dP for G e 0 . Using this in (3) we have / lim YndP= f YdP forG e 0 .
/ G n-»cx> yG Therefore lim F„ = F a.e. on ( £ 2 , 0 , P ) . This shows that E ( X „ | 0 ) | E ( X | 0 ) for an n—K>O
increasing sequence. By a parallel argument we have E(Xn | 0 ) J. E(X 10) for a decreasing sequence. I Theorem B.18. (Conditional Fatou's Lemma) Let Xn, n 6 N, be integrable extended real valued random variables on a probability space (£2,5, P ) and & be a sub-a-algebra of$. 1) If'lim inf Xn is integrable and if there exists an integrable extended real valued random n—*oo
a.e. on (£2,5, P)for n g N, then
variable X such that Xn>X (1) x
'
E(liminfX„|0) < liminfE(X n |0) v
n-.oo
'
n-*oo
a.e. on (£2,0,P).
464
APPENDIX B. CONDITIONAL
EXPECTATIONS
2) If km supX n is integrable and if there exists an integrable extended real valued random n—*oo
variable X such that Xn < X a.e. on (Q, 5, P)for n G N, then (2)
E(limsupX n |0) > limsupE(X„|0) n—t-oo
a.e. on (£2,0,P).
n—too
Proof. To prove (1), recall that liminf Xn = lim {inf Xk}. Since X < inf Xk < Xn a.e. n—t-oo Kk>n
n-»oo
k>n
on (£1,$,P) and since X and Xn are integrable, so is inf Xk. Thus inf Xk, n G N, is /:>n
fc>n
an increasing sequence of integrable random variables. Since inf Xk t lim inf Xn on Q as n - t o o , Theorem B.17 implies that E ( l i m i n f X „ | 0 ) = lim E(inf,t>„ Xk\<3) = liminf n—KX>
n—*oo
n—t-oo
—
E(.mfk>nXk\&) —
a.e. on (£2, 0 , P). Since infk>„ Xk < Xn on O, we have E(inf,t>„ Xk | 0 ) < E(X„ 10) a.e. on (Q, 0 , P ) by 7) of Theorem B.5. Thus (1) holds. By a parallel argument we have (2).
■
Theorem B.19. (Conditional Dominated Convergence Theorem) Let Xn, n G N, and X be integrable extended real valued random variables on a probability space (£2, 5, P ) such that lim Xn = X a.e. on (Q, 5, P ) and let & be a sub-o -algebra o/5- If there exists an integrable extended real valued random variable Xo such that \Xn\ < X0 a.e. on (£2, #, P ) for n e N , then \imE(Xn\<3) = E(X\<3) a.e. on(Q,&,P). n—*oo
Proof. The condition \Xn | < X 0 a.e. on (Q, 5, P ) for n G N implies that I liminf XJ < X0 n—too
a.e. on ( & , # , P ) . The integrability of X 0 then implies that of liminf Xn. We also have n—too
-X0 <X n a.e. on (£2, ^ P ) for n G N. Thus the conditions in 1) of Theorem B.18 are satisfied and consequently we have E(liminfX n |0) < liminfE(X„|0) a.e. on(£2,0,P). n—*oo
"
n—too
■
'
'
/
Now since X = lim inf Xn a.e. on (£2,5,P) we have E ( X | 0 ) = E(liminf X n | 0 ) a.e. on n—t-oo
(£2,0, P ) by 1) of Theorem B.5. Thus we have E ( X | 0 ) < luninfE(X„|0)
n—*oo
a.e. on (£2,©,P).
APPENDIX B. CONDITIONAL
465
EXPECTATIONS
Similarly by 2) of Theorem B.18, we have E ( X | 0 ) > limsupE(X„|0)
a.e. on ( 0 , 0 , P )
n—*oo
and therefore E(X | 0 ) = Jikn E(Xn | 0 )
a.e. on (Q, 0 , P). ■
Remark B.20. Let Xn, n G N, and X be integrable extended real valued random variables on a probability space (Q, 5, P ) and let 0 be a sub-a-algebra of g. Convergence of X n to l o n f i alone does not imply lim E(Xn10) = E(X10). See the following example. Example B.21. Let (ft, g, P ) = ((0,1], <8(0,i], m L ). For n 6 N, let Xn be defined by L;
J
2" fore 6 ( 0 , ^ ] -'0 force (i,l]
and let X be identically equal to 0 on Q. We have lim Xn(uj) = X{u>) for every o> G Q. 71—►CO
Consider a sub-cr-algebra 0 of g given by 0 = {0, (0, | ] , (^, 1], (0,1]}. For n 6 N, let Y„ be defined by ' 2 f o r c e (0,±] 0 forwe(|,l] and let Y be identically equal to 0 on Q. Since Y„ is 0-measurable and JG Yn dP = JaXn dP for every G G 0 , Y„ is a version of E ( X n | 0 ) . Furthermore since 0 is the only null set in (£2, 0 , P), any 0-measurable extended real valued function that is equal to Y„ a.e. on (Q, 0 , P) is actually equal to Y„ everywhere on Q. Thus Yn is the only version of E ( X n | 0 ) . Similarly Y is the only version of E ( X | 0 ) . We have however lim Yn{u>) 4 Y(CJ) for u> G (0, }].
w=i:
n—*oo
™": ?!
z
[III] Conditional Convexity Theorems Let
466
APPENDIX B. CONDITIONAL
EXPECTATIONS
and m G [ ( £ » ( & ) » (£>»(&)] we have
V(0 = s u p { a „ e + ^ }
foreeR.
Since X is integrable, X(UJ) € R for a.e. u> in (£2, 5, P). Since a convex function is a continuous function, y> is Borel measurable and thus
¥>W^)) = s u p K * ( w ) + /?„}> a n X(u;) + /?„ n€N
for all n £ N for a.e. w in (£2,5, P). Applying 7) of Theorem B.5 to (2) we have E [ ^ ( X ) | 0 ] > a „ E ( X | 0 ) + /3„ for all n G N a.e. on (£2,0, P). Recalling that every version of E(X 10) is real valued a.e. on (Q, 0 , P ) , we have by (1) E | > ( X ) | 0 ] > S up{a„E(X|0) + /?n} neN
=¥>(E(X|0))
a.e. on (£2,0, P). ■ Remark B.23. Let X be an integrable extended real valued random variable on a proba bility space (£2,5, P)- Since , + =
467
APPENDIX B. CONDITIONAL EXPECTATIONS
Example B.24. Let ( Q , 5 , P ) = ((0, l ] , 5 , m L ) where? = {0,(O,f],(|, 1],(0,1]} and let 0 = {0, (0,1]}. Define an integrable random variable X by setting I
( U A W
) = ( 2 for « e (0,1] \-\ forW€(|sl].
If we define Y(u>) = fnXdP = jforo; G Q, then as a constant function Y is ©-measurable. Y also satisfies the condition fG Y dP = Ja X dP for G = 0 and G = Q. The fact that 0 is the only null set in (Q, <9, P ) then implies that Y is the unique version of E(X |G5). From the nonnegativity of Y we have E(X 10) + = F + = F . If we define Z(u) = fa X+ dP = 1 for u> £ Q. then by the same reason as above Z is the unique version of E(X+1 <S). Thus we h a v e E ( X | © ) + < E(X + |(S)onQ. Corollary B.25. Let X be an integrable extended real valued random variable on a prob ability space (Q., 5, P) and let & be a sub-a-algebra o/J. Then (1)
|E(X|0)| <E(|Jf||0)
a.e.on(Q,&,P).
IfX E £ P (Q, 5 , P ) for some p€ [l,oo), tfzen (2)
E(|X||0)<E(|X|p|0)^
a.e. on(Cl,<S,P)
and (3)
||E(X|0)||p<||E(|X||©)||p<||X||p.
Proof. Since £ >-» |£| for £ 6 R is a convex function on It, (1) is a particular case of Theorem B.22. Suppose X € LP(Q., 5, P) for some p g [1, oo). Since £ >-> [f | p for £ € R is a con vex function on R for p € [1, oo), we have by Theorem B.22 the inequality |E(X |<5)|p < E(|X| P |<3) and then | E ( X | 0 ) | < E(|X| p |<S)r a.e. on ( 0 , © , P ) . Applying the last in equality to \X\ E LP(Q., ?, P) we have (2). By (1) and (2) we have / \E(X\<S)\pdP < f E(|X||®) P < / E(\X\"\&) = I
\X\pdP.
Taking the p-th roots of the members in these inequalities we have (3). ■
468
APPENDIX B. CONDITIONAL
EXPECTATIONS
Theorem B.26. Let ( A 5 , P ) be a probability space, Xn G Lv(Q,3,P)for n G N and X G LJQ., 5, P)for some p G [1, oo). / / lim Xn = X in L p (£2,5, P), tfie/J lim E(X„ 10) = E(X 10) m L„(Q, g, P) /or an arbitrary sub-a-algebra 0 o/5Proof. By (3) of Corollary B.25 we have ||E(X„ 10)|| p < \\Xn ||„ < oo so that E(Xn 10) € I„(Q, 5, -P) for n 6 N. Similarly E(X 10) € L „ ( 0 , 5 , P). By 6) of Theorem B.5 and by (3) of Corollary B.25 we have | | E ( X n | 0 ) - E ( X | 0 ) | | p = ||E(X n -X\<S)\\p
< \\Xn - X\\p
Thus if lim \\Xn - X\L = 0, then lim | | E ( X n | 0 ) - E(J\T|Q5)!|P = 0. ■ n—*oo "
"p
n—t-oo "
Theorem B.27. (Conditional Holder's Inequality) Let 0 be a sub-a-algebra of$ in a prob ability space (Q,5, P). IfX G L P (Q, 5, P) and F 6 £,(Q, 5 , P ) wnere p, q G (1, oo) and i + i = 1, tfzerc E(|Jfy||0)<E(|X|P|0)pE(|y|«|0)'
a.e. o n ( Q , 0 , P ) .
Proof. For n G N, let X n = \X\ + \ and Yn = \Y\ + ±. Then Xn, Yn > £ on £2, X n G i p ( Q , 5 : , P ) a n d r n G L , ( Q , 5 , P ) . By 7) of Theorem B.5 we have a = E ( | X n | p | 0 ) f > £ and/? = E ( | K | ' | © ) ' > ± a.e. on (£2,0, P). By the inequality |£T?| < p _ 1 |£| p -+-«—* I77I9 for £,77 e R , we have
^<W
+
1 ?
,e.on(£2,0,P).
Note that since a, (3 > £ > 0 a.e. on (£2,0,P), the divisions in the last inequality is possible a.e. on (£2,0, P). By 7) and 6) of Theorem B.5,
E
<;>)^M'">K (>)
a.e. on(£i,0,P).
The fact that a and ,8 are ©-measurable implies according to Theorem B.8 1 E(|X n |»|0) E f l y n | ' | 0 ) 1 1 —-T < + = - + - = 1 a.e. on (£2, 0 , P ) a/3 pa>> qfr p q
469
APPENDIX B. CONDITIONAL EXPECTATIONS and thus E ( | X n y „ | | ® ) < a ^ = E(|X n |P|®)pE(|F n |9|0)i
a.e. on
(Q,6,P).
Letting n —► oo and applying Theorem B.17 (Conditional Monotone Convergence Theo rem), we complete the proof. I [IV] Conditioning and Independence Theorem B.28. Let X be an integrable extended real valued random variable on a prob ability space (Q, 5, P) and let <& be a sub-a-algebra of$. If {X, 0 } is an independent system, then E ( X | 0 ) = E(X) a.e. on (£2,0, P). Proof. Let {X, 0 } be an independent system, that is, {cr(X), (8} is an independent system of sub-cr-algebras of 5. We are to show that every version of E(X | 0 ) is equal to the constant E(X)a.e. on(£2,0,P). Let E e
JG
where the second equality is by the independence of {a(X), <S}. Thus we have shown that for an arbitrary E £ c(X) we have (1)
E ( 1 E | 0 ) = E(1 £ ) a.e. o n ( n , 0 , P ) .
Let us write X = X+ — X~ for the integrable extended real valued random variable X on (£2,
470
APPENDIX B. CONDITIONAL
EXPECTATIONS
Remark B.29. The converse of Theorem B.28 is false, that is, E(X | 0 ) a.e. on (Q, 0 , P ) does not imply the independence of the system {X, 0 } . See Example B.30 below. See also Corollary B.33. Example B.30. L e t ( Q , 5 , P ) = ((0, l],Q3(o,i],mL). Consider G u = (0,±], G,,2 = (±,±1 Gi = (0, | ] , G2 = ( i 1], and let 0 =
{
P(G,tl)P«?i) = H = 1 s o that p ( G '.' n G>) ^P(Gi,,)P(G,).
Theorem B.31. Let X be an integrable extended real valued random variable on a proba bility space (Q, 5, P ) and let 0 be a sub-a-algebra o/5- Then the following two conditions are equivalent: 1°. E(X\&)
= E(X)a.e.
on(Q,&,P).
2°. E(XZ) = E(J*QE(Z)/or every bounded 0-measurable random variable Z. Proof. Assume 1°. Then E(XZ) =
E[E(XZ\&)]=E[ZE(X\(S>)]=E{X)E(Z)
where the second equality is by Theorem B.8 and the third equality is by 1° This proves 2°. Conversely assume 2° Then for every bounded 0-measurable random variable Z we have (1)
E[ZE(X | ©)] = E[E(ZX 10)] = E(ZX) = E(X)E(Z)
= E[ZE(X)]
where the first equality is by Theorem B.8 and the third equality is by 2°. Now for an arbitrary G 6 0 , 1G is a bounded 0-measurable random variable so that by (1) we have
APPENDIX B. CONDITIONAL EXPECTATIONS
471
fG E(X 10) dP = JG E(X) dP. Since both E(X | ©) and the constant E(X) are ©-measurable, the arbitrariness of G G © implies that E ( X | 0 ) = E{X) a.e. on (£1,(8, P). This proves 1°
Theorem B.32. Let ©i and ©2 oe two sub-a-algebras of$. Then the following two condi tions are equivalent: 1°. {©i, ©2} is an independent system. 2°. E(X\ |© 2 ) = E(Xi) a.e. on (Q, <82,P)for every integrable <&\-measurable extended real valued random variable X\. Proof. Assume 1°. Let Xi be an integrable ©1-measurable extended real valued random variable. Since cr(X\) C ©1, the independence of {©(,©2} implies the independence of {a(Xi), © 2 }. Then by Theorem B.28 we have E(X, | 0 2 ) = E(X,) a.e. on (Q, ©2, P). Conversely assume 2° Then for every integrable ©1-measurable extended real valued random variable X\ and bounded ©2-measurable random variable X2, X\X2 is integrable and E(XtX2)
= E[E(XiX 2 |© 2 )] = E[X 2 E(X, |© 2 )] = E[X 2 E(X,)] = E{XX)E(X2)
where the second equality is by Theorem B.8 and the third equality is by 2°. Let G\ 6 ©1 and G2 e 0 2 - Then with Xx = 1 G , and X2 = \Gl we have P(G\ n G2) = E(l G l l G 2 ) = E(l G ,)E(lo 2 ) = P(G\)P{G2), proving the independence of {©,, © 2 }. ■ Corollary B.33. Let X be an integrable extended real valued random variable on a proba bility space (£2,3, P) and let & be a sub-a-algebra o/S- Then the following two conditions are equivalent: 1°. {
472
APPENDIX B. CONDITIONAL
EXPECTATIONS
Theorem B.34. Let %, 2l2, and 2l3 be three sub-a-algebras o / 5 in a probability space (Q, 5, P). //(T(21I U 2l2) and 2l3 are independent, then for every At e 2li we have (1)
P ( A , | 3 W c P ( A , \crQXi U2l 3 )),
tfwffc,every veraow o/P(A, |2l 2 ) is a version o/P(Ai |cr(2l2 U 2l3)). //(O, a(2l 2 U 2l3), P ) & a complete measure space and i/2l2 contains all the null sets in (Q, <J(Q12 U 2l3), P), then (2)
P(A,|2t 2 ) = P ( A , | ( 7 ( a 2 u a 3 ) ) .
Proof. Since 2l2 is a sub-a-algebras of cr(2l2U2l3), to prove (1) it suffices to show according to 2) of Theorem B.16 that for every version Y of P(A\ |2l 2 ) we have (3)
I YdP = P(AX n B)
for B e
To prove (3), let us prove first that for Ai e 2li and A3 € 2l3 we have
(4)
P(Ai n A3 |a 2 ) = p(Ai |a 2 )P(A 3 |a 2 ).
Now for A2 6 2t2 we have
(5)
/ P(A{ n A3 |a 2 ) dP = P(A, n A2 n A3) = P(A, n A2)P(A3) JA2
= P(A3) I P(A 1 |a 2 )dP=/ P(A1|a2)P(A3)dp! JA2
JA2
where the first and the third equalities are by the definition of conditional probability, and the second equality is by the independence of a^\ U 2l2) and 2l3. Since P(A\ n A3 |2l 2 ) and P(A] |2t 2 )P(A 3 ) are both 2l2-measurable, (5) implies P ( A n A3 |2l 2 ) = P(Ai |2l 2 )P(A 3 )
a.e. on (0,2t 2 ) P ) .
Now the independence of 17(21] U 2l2) and 2l3 implies that of 2l2 and 2l3 and consequently we have P(A 3 |2l 2 ) = P(A3) a.e. on (Q,2l 2 ,P) by Theorem B.28. Therefore we have P(A! n A3 |2l 2 ) = P{AX |2l 2 )P(A 3 |2l 2 ) a.e. on (Q, 2l2, P). This proves (4). Next let us observe that
E[P(A,|a 2 )i^|a 2 ] = PUi ia 2 )E(U s |ai 2 ) = P(A, n A 3 |SI 2 ) = E[P(A, n A 3 |a(a 2 u 2i 3 ))|a 2 ] = EUU.ECU. |
413
APPENDIXB. CONDITIONAL EXPECTATIONS
where the first equality is by the 2l2-measurability of P(AX |2l 2 ) and by Theorem B.8, the second equality is by (4), the third equality is by Theorem B.12, and the last equality is by the fact that lA} is CT(212 U ^-measurable. Thus we have (6)
/
P(i4,|Sl2)dP= /
JA^nAj
=
f
E[l Al |<x(2l 2 U2l 3 )]rfP
JA2nA-i
lAldP = P(Ai
r\A2nA3),
JA2nA)
where the second equality is from the fact that A2 (1 A3 G cr(2l2 U 2l3). Now 2l2 (~l 2t3 is a 7r-class of subsets of Q. and cr(Ql2 n 2l3) =
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Appendix C Regular Conditional Probabilities Given a probability space (Q, 5, P). Let <S be a sub-cr-algebra of 5, A G 5 and consider the conditional probability of A given 0 , that is, P(A|(S) = E(l,t|(S). Let {An : n G N} be a countable disjoint collection in 5- By the linearity of the conditional expectation and by the Conditional Monotone Convergence Theorem, there exists a null set A depending on {An : n e N} in (Q, <S, P ) such that P(Un^An | <5)(u>) = £ „ 6 N P(A„ |<S)(u;) for w G Ac. Since there may be uncountably many countable disjoint collections of members of 5, we may have to exclude uncountably many null sets whose union may not be an 5-measurable set. Thus a subset E of Q such that P(-1 &)(u>) is a measure on 5 for w G E may be an empty set or it may not be an 5-measurable set. To overcome this difficulty, let us consider regular conditional probabilities. Definition C.l. Let (Q, 5, P) be a probability space and let <S be a sub-a-algebra o/J. By a regular conditional probability given <S we mean a function P s l e on 5 x £2 such that 1°- there exists a null set A in (Q, 0 , P) such that P 5 I 0 ( - , u>) is a probability measure on "Sfor every u> G Ac. 2°. P 5 l e ( y l , •) is a (25-measurablefunction on Q.for every A e $. 3°. P(A n G) = / G P5l®(yL, oj)P(dw)for every A G ? and G G 0. Hfe raj rtar f/ze regular conditional probability given 0 is unique if for any two functions p andp' on 5 x Q. satisfying conditions 1°, 2°, and 3° there exists a null set A in (Q, 0 , P) such that p(-,tx>) = p'(-,ui) ongforco G Ac. 475
476
APPENDIX C. REGULAR CONDITIONAL
PROBABILITIES
We show in Theorem C.6 below that if Q. is a complete separable metric space, # is the Borel cr-algebra of subsets of Q and P is an arbitrary probability measure on #, then the regular conditional probability given an arbitrary sub-cr-algebra <S of 5 exists uniquely. Regular conditional probability is known to exist for a standard measurable space, of which a complete separable metric space with its Borel cr-algebra of subsets is an example. See [26] K. P. Parthasarathy. Observation C.2. Conditions 2° and 3° in Definition C.l are equivalent to the single con dition P s l 0 G V ) e P G 4 | < S ) for every A e 5, that is, for every A 6 5, the function P 5 ' ® ^ , ■) on Q is a version of P{A\<5). Proof. By 2°, P f f l e (A, •) is a ©-measurable function on Q. By 3°, we have for every G 6 <5 the equality JG P:g\@(A,iv)P(cL>) = P(A n G) = fG lA(u)P(du>). This shows that P5l's(^,-)isaversionofP(yl|©). ■ Proposition C.3. IfP^l® is a regular conditional probability given a sub-a-algebra <£> ina probability space (£2,5, P), then for every integrable random variable X on the ( 0 , 5 , P) we have (1)
E(Jf|0) = | AV)P5le(dwV)
a.e.
on(n,&,P),
that is , the right side of {I) as a function on Q. is a version o/E(X |<S). In particular for every G e (S, we have (2)
f XdP=
f U
X(a/)P5|e(
Proof. Consider the case where the integrable random variable X is nonnegative. Let us show first that the right side of (1) is a (5-measurable function on Q,. Now there exists an increasing sequence of nonnegative simple functions {ipn : n £ N} on (£2,5) such that (fin T X. Let ipn = Y%Z\ cn,k1-Ank where An
APPENDIX C. REGULAR CONDITIONAL
All
PROBABILITIES
which, being a finite sum of ©-measurable functions according to 2° of Definition C.l, is ©-measurable. Then by the Monotone Convergence Theorem,
/ X(u/)P 5 I S (^V) = lim I ¥ > n (c/)P s l c W, ), which, as the limit of a sequence of (S-measurable functions, is ©-measurable. To show that fa X ( u ) ' ) - P 5 , e ( ^ ' , •) is a version of E(X |<9), it remains to show that for every G € <S we have y M X(oj')P^&(duj',u)\
(4)
P(duj) = / X(w)P(dw)-
Now by the Monotone Convergence Theorem, (3), and 3° of Definition C.l, we have / If
X(a.')P 5 , < 9 (dw',u;)}p(dw)= / Jim | J
V
,(W')P?IV,W)}F(4I)
=
lim / £c„, A ; P ! 5 l , 8 ( J 4 n ,,,^)P(du;)= lim £ ^ ( ^ 0
=
lim /
0
Similarly for X " . Therefore (4) holds. This proves (1) for a nonnegative integrable random variable X . For an arbitrary integrable random variable X, let X = X + — X~. Since E(X|<8) = E(X + |©) - E ( X " |<S) a.e. on (£2, &, P) and since (1) holds for E ( X + | 0 ) and E ( X ~ | 0 ) it holds forE(X|<8). Finally (2) is an immediate consequence of (1). ■ Proposition C.4. Let &bea sub-o-algebra of$ in a probability space (Q, 5, P). Suppose a regular conditional probability P s l e exists. Let (Q, 3*, P ) be the completion o/(Q, 3) P). Then a regular conditional probability P s * I s exists. Proof. 3* consists of subsets of Q of the form E = A U TV0 where A € 3 and iV0 is a subset of a null set TV in (Q, 3 , -P)- Let us define a function ip on 3* x £2 by setting (1)
<^(£,a.) = P 5 l e ( A , a ; )
for (72, w) £ 3* x£2,
where A € 3 is as specified above for the given set E g 3* To show that c^(75, •) is uniquely defined up to a null set in (Q, <S, P ) for each 75 e 3*. suppose 75 is given as E = A1 U N(, as well as 75 = A" U 7V£ where A', A" e j ^ c TV', TV^' c TV" and TV', N" are null sets
478
APPENDIX C. REGULAR CONDITIONAL
PROBABILITIES
in (Q, 5, P). We assume without loss of generality that A' n TV^ = 0 and A" n Ng = 0 and thus A' = E-N'Q and A" = £ - JVJ\ By (1) and condition 3° of Definition C.l, we have / P 5 l e ( A » P ( < M = P(A' n G) = P((E - M ) D G)
./G
= P ( ( £ - JV?) n G) = P(A" n G) = f
P^6(A",uj)P(dw)
JG
for every G G <S. Thus P 5 I®(A', •) = P 5 I 6 ( A " , •) a.e. on (Q, 0 , P). Let us show that y> satisfies the conditions in Definition C.l. Now since P 5 !® satisfies condition 1° of Definition C.l there exists a null set A in ( Q , 0 , P ) such that P 5 l®(-,a>) is a probability measure on 5 for w G Ac. From this and from (1) it follows immediately that y>(-,w) is a probability measure on #* for w G Ac. By (1), v(-E, ) is a (S-measurable function on ft for every £ G 5*. For 73 G 5*, writing £ = A U TV0 with A G $ and TV0 C TV where TV is a null set in (Q, 5, P), we have P(E n G) = P(A n G) = j P 5 l®(A,w)P(dw)= / JG
ip(E,Lj)P(duj)
JG
for every G G <S by condition 3° of Definition C.l satisfied by P 5 I® This shows that tp satisfies all the conditions in Definition C. 1 and is therefore a regular conditional probability on the probability space (Q, 3"", P ) given (5. ■ Lemma C.5. Let p be a finite, nonnegative, finitely additive set function on an algebra 53 of subsets of a Hausdorff space S. Let ^ be the collection of all compact sets in S and let 21 be a sub-algebra of 3). If p satisfies the condition that (1)
p(A) = sup{p(K) : K d A,K G £ n 5 3 } for every A G 21,
Jtoen /J is countably additive on 21. Proof. Since 21 is a subalgebra of 5), p is a finite, nonnegative, finitely additive set function on 21. A finite, nonnegative, finitely additive set function p on an algebra is countably additive if and only if for every decreasing sequence {A n : n G N} in the algebra such that An I 0 we have p(An) J. 0. Thus if p is not countably additive on 21, then there exist 6 > 0 and a decreasing sequence {An : n G N} in 21 such that An J. 0 and p(An) > 6 for every n G N. Let us show that this contradicts (1). Now according to (1) for every n G N there exists Kn e ^ n S such that Kn C An and p{An - Kn) < <5/3n. Then we have An - n,"=, Ki C U?=1(An - j £ ) C U?=1(A,- - #,■)
APPENDIX C. REGULAR CONDITIONAL
PROBABILITIES
479
so that n(n^Ki) > n(An) - Eli <5/3* > 6/2. Then Fn = n^Ki 4 0. Since 5 is a Hausdorff space, the compact set jFn is a closed set. Consider the decreasing sequence of closed sets {Fn : n G N}. Let us show that DneN-Fn ¥ 0- Suppose n „ 6 N F n = 0. Then U„ e N i^ = 5 so that in particular {F£ : n G N} is an open covering of the compact set Kt and thus K\ C U?=lFf for some N G N. Since {F£ : n G N} is an increasing sequence we have K\ C Ffr and thus K\ n FN = 0. This contradicts the fact that F w ^ 0 and FN = n£,tf,- C K\. Therefore nnexFn 4 0. But Fn C KnC An and n„ 6N A„ = 0 so that n„6N.F„ = 0. This is a contradiction. Therefore p. is countably additive on 21. ■ Let (5,03, p) be a measure space in which S is a topological space and 03 is an arbitrary (7-algebra of subsets of S. Let M. be the collection of all compact subsets of S. We say that p is regular if for every A G 03 we have p(A) = sup{fi(K) : K C A, if G ^ n 03}. According to Ulam's Theorem if 5 is a complete separable metric space, 03s is the Borel a -algebra of subsets of 5, and /J is a finite measure on (5,03s), then \x is regular. For a proof of Ulam's Theorem we refer to [4] R. M. Dudley. Theorem C.6. Let (5,03s, P) be a probability space in which S is a complete separable metric space and 03s is the Borel a-algebra of subsets of S. Then for every sub-a-algebra <S o/03s, the regular conditional probability given <£>, P ^ l 8 , exists uniquely. Proof. 1) Let us show first that there exists a countable algebra 21 of subsets of S such that 03s = c(2l). For an arbitrary collection £ of subsets of 5 let us write a(£) for the algebra generated by C. Let O be the collection of all open sets in S. Since 5 is a separable metric space it has a countable dense subset. Let Do = {On : n e N} be the countable collection of all open spheres in 5 having centers in a countable dense subset of 5 and having rational radii. Every O € D is then a union of members of Do- Let 21 = a(£>o). Then O C
480
APPENDIX C. REGULAR CONDITIONAL
PROBABILITIES
Since a finite union of compact sets is a compact set we can choose Kn,m so that Kn,m f as m —> oo. Let £> = a({A n ,/<„,„, : m g N , n g N } ) . Since 2) is an algebra generated by a countable collection, it is a countable collection by the same argument as in 1) in showing the countability of 21 = a(Oo)- Now 1°. P ( D | < S ) > 0 a . e . on (S,<S,P) for every D e D . 2°. P ( S |<8) = 1 a.e. on (S, 0 , P). 3°. P(U*L,D,■ | 0 ) = Ef=i P(-D. | « ) a.e. on ( 5 , 0 , P) for A g 2), * = 1, -- -, k, disjoint. 4°. P(A' n , m |(S) T P(An\<5) a.e. on ( 5 , « , P ) . Let p be a function on 53s x 5 defined by letting p(A, ■) be equal to an arbitrarily fixed version of P(Aj&) for every A g 23s- Now 2) C 53 s is a countable collection and the collection of all finite combinations of members of 23 is countable. Therefore there exists a null set A^ in (5, 0 , P) such that conditions 1°, 2°, 3°, and 4° hold at x for x £ A ^ . Thus for a; g A^, p(-, z) is a nonnegative, finitely additive set function on the algebra 2) with p(S, x) = 1 by 1°, 2°, and 3° Writing J? for the collection of all compact sets in S, we have by 4° p(An,x)
= P{An\<S)(x) = sup{P(A|0)(z) : K C An,K
= sup{p(A,K) : if c An,K
g #n23}
g #n£>}.
Thus by Lemma C5,p(-,x)is countably additive on thealgebra 21 when x g A^. Therefore when x g A^, the restriction of p(-, x) to 21 can be extended uniquely to be a measure q(-, x) on 53s = c(2l) which is a probability measure since p(S, x) = 1 by 2°. Let q be a function on 53 x 5 defined by setting q(-,x) to be equal to the probability measure on 53s obtained by the extension of the restriction of p(-, x) when x g A^, and by setting g(-, x) = 0 when x g AJO. Thus defined, q satisfies condition 1° of Definition C.l. To show that q satisfies 2° and 3° of Definition C.l we show equivalently according to Observation C.2 that q(A, ■) is a version of P(A|<3) for every A g 53 s . Let <£ be the collection of all members A of 53 s such that q(A, ■) is a version of P(A|(S). Now if A g 21 then q(A, •) = p(A, •) on A ^ and q(A, •) = 0 on A^,. But p(A, •) is a version of P(^4|(S). Thus q(A, ■) is a version of P(A\<&). This shows that 21 C <E. Let us show that <£ is a d-class of subsets of S. Now since S g 21 we have S g (£. Let ^4 and B be two members of <£ such that A
APPENDIX C. REGULAR CONDITIONAL
481
PROBABILITIES
and q(B, x) = P{B | 0 ) for x e ACB. Also there exists a null set AAtB in ( 5 , 0 , P ) such that P ( S - A | 0 ) ( x ) = P ( P | 0 ) ( x ) - P ( A | 0 ) ( x ) f o r x e A ^ B . LetA = A^UA B UA yM3 UA 0O . Then for i £ A c w e have, recalling that q(-,x) is a measure on 93 s for x e A^, g(jB - A, x) = q(B, x) - q(A,x) = P(B10)(x)
- P(A|<8)(x) = P{B -
A\0)(x).
This shows that B - A £ <£. Next let {A„ : n € N} be an increasing sequence in <£. Then for every n e N there exists a null set A„ in (5, 0 , P) such that g(j4n, x) = P04„ | ©)(x) for x G A^. By the Conditional Monotone Convergence Theorem there exists a Ao in (5, ©, P ) such that P ( lim An\<8){x) = lim P ( A n | 0 ) ( x ) for x 6 AS. Let A = (U Cj A„) U A^. n—too
n—too
Then for x 6 Ac we have g (lim 71—'oo
An,x)=
'
nfcau+
lim g(A„,x)= lim P(An 10)(x) = P( lim A„|0)(x).
n—too
n—too
n—too
This shows that lim An £ <E. Thus <£ is a d-class of subsets of 5. Then 93 s = ^"(20 = 71—»CO
d(QV) C <£ by Theorem 1.7. Therefore q(A, ■) is a version of P ( A | 0 ) for every A 6 93sThis shows that q is a regular conditional probability given 0 . 3) To prove the uniqueness of regular conditional probability given a sub-c-algebra in our probability space (S, 93s, P)> let gi and q2 be two regular conditional probability given a sub-cr-algebra 0 of 03s- Then there exists a null set Ao in ( 5 , 0 , P) such that q\(-,x) and qi{-,x) are probability measures on 93s when x g AJj. For every A 6 93s and in particular for every An in the countable algebra 21, q\{An, •) and q2(An, •) are versions of P ( A n | 0 ) by Observation C.2 and thus there exists a null set A„ in (5, 0 , P ) such that q\(An,x) = q2(An,x) when x £ Acn. Let A = U„<=2+A„. Then for x 6 Ac, <7i(-,x) and q2(-,1) are probability measures on 93s and are equal on 21. For fixed x £ AC, let € be the collection of all members A of 93s such that qi(A, x) = q2(A, x). As we saw above, 21 C 2. By the fact that qi(-,x) and qi(-,x) are measures it is easily verified that <E is a tf-class of subsets of S. Then by Theorem 1.7, 93 s C <E. Therefore giL4,x) = q2(A, x) for A £ 93s when x € Ac. This completes the proof of the uniqueness of the regular conditional probability. ■ Let (Q, 5, P ) be a probability space and let 0 be a sub-a-algebra of 5- Consider P 5 l e , a regular conditional probability given 0 . For every i e j w e have according to 3° of Definition C.l the equality fG pt\0(A,w)P(du) = P(A 0 G) = Ja lA(u)P(
482
APPENDIX C. REGULAR CONDITIONAL
PROBABILITIES
the existence of a null set A in (Q, (S, P), not depending on A, such that for every A in a sub-a-algebra fj of© we have P$\@{A,w) = 1^(01) for every A 6 Sj when a> G A c . Definition C.7. Lef (Q, §)feea measurable space. We say that 5 is countably determined if there exists a countable subcollection 33 o / 5 such that whenever two probability measures P and Q are equal on 53 then they are equal on 5- We call 33 a countable collection of determining sets for $. Proposition C.8. Let S be a separable metric space. Then the Borel cr-algebra of subsets ofS, 93s, is always countably determined. Proof. Let D be the collection of all open sets in S. We have 93s = a(Q). Since S is a separable metric space, there is a countable subcollection Do of D such that every member of D is the union of some members of Do- The collection S3 of all finite unions of members of Do is a countable subcollection of 95s- Suppose P and Q are two probability measures on 5 such that P(D) = Q(D) for every D G S3. Let O G D. Then O = U n 6 N O n where O n G D 0 for n G N. Let Gn = lS^=iOk G S3 for n G N. Then Gn T and O = U n e N G n = lim Gn. Since G n G 33, we have P(G n ) = Q{Gn) for every n G N n—*oo
and therefore P(O) = lim P(G n ) = lim Q{Gn) = Q(0). Thus P = Q on D. Since D is a n—*oo
n—*oo
7r-class, this implies that P = Q on <J(D) according to Corollary 1.8. Thus the equality of P and Q on 33 implies the equality of P and Q on 93s- Therefore 33 is a countable collection of determining sets for 93s- ■ Theorem C.9. Let (£2,5, P ) be a probability space, let <& be a sub-o-algebra of 5 and suppose a regular conditional probability given (3, P>M® exists. Let Sj be a countably determined sub-a-algebra of <8. Then there exists a null set A in (Q., <S, P ) such that P 5 l®(A,uO = 1A(w) for every A G £ wfren w G A c .
Proof. Let 33 C .f) be a countable collection of determining set for fj, that is, any two probability measures on fj that are equal on 53 are equal on Sj. Let A e 53. Then since A G 0 , there exists a null set A^ in (Q, ®, P) such that P 5 I ( 8 (A, w) = 1^(OJ) for u> G A^. Let Aoo = LUgjjA,!. Since 53 is a countable collection, A is a null set in (£2, (8, P). Then (1)
P 5 l®(A,o)) = l^(w)
for every ,4 G 53 when u g A ^ .
APPENDIX C. REGULAR CONDITIONAL
483
PROBABILITIES
Now by 1° of Definition C.l there exists a null set Ao in ( Q , 0 , P ) such that for every w G A£, P 5 l e ( - , a ; ) is a probability measure on 5 is hence a probability measure on the sub-cr-algebra Sj. Let A = AQ U A*,. With fixed w G Q, the set function l A (w) for A G f> is a probability measure on fj which assigns the value 1 to every A & Sj which contains w and the value 0 to every A G Sj which does not contain w. For w 6 Ac, the two probability mea sures P 5 I S ( - , O J ) and l(.)(w) on Sj are equal on 35 according to (1). Since 5) is a countable collection of determining sets for Sj, the two measures are equal on Sj for w 6 Ac. ■ Corollary C.IO. Ler (Q, J , P ) fee a probability space, let <& be a sub-a-algebra o / 5 and suppose a regular conditional probability given 0, P 5 ! 1 9 , exists. Ler(S, 93) be a measur able space such that 23 is countably determined and {i} G 93/or every x G 5. Lef £ 6e a 0/'23-measurable mapping ofQ. into S. Then there exists a null set A in (£2,0, P) JMCH f / i a / P 5 l s ( r ' ( { ^ ) } ) , w ) = 1/oru 6 Ac, that is, P 5 ' ( 8 ( K G Q : £(u/) = £M},u>) = 1 / o r u G Ac. Proof. Note that since £ is 0/95-measurable and { £ M } G 33, we have £"'({CM}) G 0 for every u> G fl. Let £> be a countable collection of determining sets for 93. Let Sj = £~'(23). Since f is 0/93-measurable, Sj is a sub-a-algebra of 0 . Let <E = £_1(3}). To show that (£ is a countable collection of determining sets for Sj, let ^ and v be two probability measures on Sj which are equal on (£. Let n^ and U( be two probability measures on 93 defined by fi({B) = M ( £ _ 1 ( # ) ) and v^B) = v{£-l(B)) for B G 93. For D G 3 , we have p.^{D) = //(£~'(-D)) = v(£~'(-D)) = u((D) since /^ = v on <£. Thus /^ = v^ on £>. Since ID is a countable collection of determining sets for 93, we have ^ = i/f on 93. Thus Mf~'CB)) = v<Jt\B)) for P G 93, that is, (i{H) = v(H) for P G Sj. This shows that Sj is countably determined. Then by Theorem C.9 there exists a null set A in (Q, 0 , P ) such that for every A G £ we have P 5 '^(A, u>) = \A{w) for u £ A c . Now for every u e Q w e have {u/ G Q.: £(w') = c » } = r ' ( £ M ) e £ since { c » } G 93. Thus P 5 I « ( { W ' £ n : « u ' ) = « ^ ) } , W ) = l{u.6si:{w)=«„)}^) = 1 foru,GA c
I
Let (Q, 5, P ) be a probability space and let (S, 93) be an arbitrary measurable space. Let X be an 3/23-measurable mapping of Q. into S. Then X~'(93) is a sub-cr-algebra of 5 so that a regular conditional probability given X, P 5 I X " ( S ) , is defined as a function on 5 x £1 We can also define a regular image conditional probability given X as a function on 5 x 5 as follows.
484
APPENDIX C. REGULAR CONDITIONAL
PROBABILITIES
Definition C.U. Let (Q, J , P) be a probability space and (S, 23) be a measurable space. Let X be an g/VB-measurable mapping ofQ. into S and let Px be the probability distribution ofX on (S, 23). By a regular image conditional probability given X we mean a function P^\x on 5 x S satisfying the following conditions. F . There exists a null set A in (S, 03, Px) such that P$\x(■, x) is a probability measure on $for every x 6 Ac. 2°. P*\X(A,
■) is a 03-measurable function on S for every A 6 $. = fB P*\X(A,
3°. P(A n X~\B))
x)Px(dx)for
every A € $ and B € 23.
We say that the regular conditional probability given X is unique iffor any two functions p andp' on$ x S satisfying 1°, 2°, and 3° there exists a null set A in (S, 23, Px) such that p(A, x) = p'{A, x)for every A £ 5 when x 6 Ac. Remark C.12. The regular image conditional probability defined above exists uniquely when the probability space (Q, 5, P) is a complete separable metric space with the Borel cr-algebra of subsets. This can be shown by the same argument as in the proof of Theorem C.6. Proposition C.13. Let (Q, 5, P) be a probability space, (5,23) be a measurable space, X be an 5 /SB -measurable mapping ofQ, into S, and Px be the probability distribution ofX on (S, 23). Assume the existence of a regular image conditional probability given X, P^\x Let f be an extended real valued 23 -measurable function on S. Then for the extended real valued random variable f o X on (Q, J , P) we have for every B 6 23 /
f(x)pZ\x(X-\B),x)Px(dx),
f°XdP=[
JX-^B)
JS
in the sense that the existence of one side implies that of the other and the equality of the two. Proof. Consider first the particular case where / = lg where E £ 23. Then I JX
foXdP=[ (B)
•>X (S)
= P(X-\B)nX-\E))=
l B (Jf(w))P(dw) = /
lx-'(B)(w)lj r -. (B) (w)P(dw)
J£l
j P*lx(X-\B),x)Px(dx)
by 3° of Definition C. 11
JE
= JslE(x)P*\x(X-\B),x)Px(dx)
=
Jsf(x)P*\x(X-l(B),x)Px(dx).
APPENDIX C. REGULAR CONDITIONAL
PROBABILITIES
485
The general case follows by writing / = / + - / " and by the fact that there exist two increasing sequences of nonnegative simple functions on (5,33) which converge to / + and / ~ respectively on 5. I Proposition C.14. Let ( 0 , 5 , P) be a probability space, (S, 53) be a measurable space, X be an $/^-measurable mapping of£l into S, and Px be the probability distribution ofX on (S, 93). Assume the existence of a regular image conditional probability given X, P 5 \x Let f be an extended real valued random variable on (£2, 5, P). Then for every B G 03 we have i
, n t
d P =
l
au)PSlX(du>,x)]pX(dx),
\l
Jx-'iB) JB Un J in the sense that the existence of one side implies that of the other and the equality of the two. Proof. Consider first the particular case where £ = 1A with A G $. Then / (,dP= f lA(iv)P(du) = P(AnX-\B)) JX-^B)x Jx-'(B) = J P*l (A,x)Px(dx) =J ^lAu)P*lx{duj,x) Px(dx) = J U £M-P5|*(^,z)]
Pxidx),
where the second equality is by 3° of Definition C.ll. The general case follows by writing £ = £+ — f ~ and by the fact that there exist two increasing sequences of nonnegative simple functions on (Q, #) which converge to £+ and £ - respectively on Q. ■ Theorem C.15. Let (0,3) P) be a probability space, (5,03) be a measurable space, X be an 5/93 -measurable mapping ofQ. into S, and Px be the probability distribution of X on (S, 03). Assume the existence of a regular image conditional probability given X, P$\x. Assume further that 03 is countably determined and {x} G 'Bfor every x G S. Then there exists a null set A in (5,03, Px) such that (1)
Pdlx(X~l(B),x)
= lB(x)
for every B G 03 when x G Ac
In particular we have (2)
P 5 l * ( X - ' ( { i } ) , x) = 1 for every x G Ac.
486
APPENDIX C. REGULAR CONDITIONAL
PROBABILITIES
Proof. Let £> = {Dn : n g N} be a countable collection of determining sets for 23. By choosing A = X~\Dn) and B = Dn in 3° of Definition C.ll, we have Px(dx) = P(X-l(Dn))=
/ JD„
f
Pslx(X-\Dn),x)Px(dx).
JD„
By 1° of Definition C.ll, P 5 l * ( X - ' ( A . ) , x ) G [0,1] for x g 5 and in particular for x g Dn. Thus the last equality implies that there exists a null set A„ in (S, 23, Px) such that P?>\x(X-\Dn),x) = lDn(x) for x g Acn. LetA = U neN A. Then we have (3)
P*\x{X-\Dn),x)
= lD„(x)
for every n when x g A c .
Now for every x g S, Pdix(X~\B), x) as a function of B g 05 is a probability measure on 35 by 1° of Definition C.ll. For fixed x g S, 1B(X) as a function of B g 23 is also a probability measure on 23. According to (3), when x g Ac these two probability measures are equal on 53. Then since 3D is a countable collection of determining sets for 23, these two probability measures are equal on 23 when x g Ac. This proves (1). When B = {x} for some x g S, we have l{x}(x) = 1 and thus (2) holds. ■ Definition C.16. Let (Q, 5, P) be a probability space and (S, 23) be a measurable space. Let X be an 3/23-measurable mapping ofQ. into S. Let (8 be a sub-a-algebra of$. By a regular conditional probability distribution of X given <& we mean a function Px\® on 23 x Q. satisfying the following conditions. 1°. There exists a null set A in (Q, (S, P) such that Px I® (•, u») is a probability measure on 23 for every u> g Ac. 2°. Pxle(B 3°. P{X~\B)
, ■) is a (8-measurable function on Q.for every B g 23. f\G)=Sa
Px i0(B,uj)P(dio)for
every B
e<8andGe<S.
We say that the regular conditional probability distribution of X given & is unique if for any two functions p and p' on 23 x Q satisfying 1°, 2°, and 3° there exists a null set A in (Q, 0 , P) such that p(B, u>) = p'(B, w)for every B g 23 when to g Ac. Remark C.17. The regular conditional probability distribution defined above exists and is unique if the measurable space (S, 23) is a complete separable metric space with the Borel cr-algebra of subsets. The proof of this parallels that of Theorem C.6.
Appendix D Multidimensional Normal Distributions Definition D.l. Let Q? be a probability measure on (Rd,*8Kd) where d € N. The mean vector of O, namely M(<X>) = (M(O)j, j = 1 , , . . , d), is defined by (1)
M(*),-= j
xjtydx)
forj =
l,...,d,
provided all the d integrals exist and are finite. The covariance matrix o/O, namely V(
V ( * ) A * = f{xj-Mmj}{xk-Mmk}^dx) forj,k = l,...,d, Jwd provided all the jk integrals exist and are finite. The characteristic function <pof® is defined by (3)
Observation D.2. The covariance matrix V() of a probability measure <£ on (R , <8md) if it exists, is always a nonnegative definite matrix. Indeed for every y £ M.d we have d
d
(V(0)y,y) = E E y ( % K W 7=1 fc=l d
d
.
= E E / Ax> - MWj}vi{** - M(*)t}»«(«fe) j = l 4=1 - 7 1
= / K J E { ^ ' - M(4»),-}W]2*(dx) > 0. 487
488
APPENDIXD. MULTIDIMENSIONAL NORMAL
DISTRIBUTIONS
Under the assumption that JMd | n |" ■ • • \xd\qd®(dx) < oo, for some qu...,qd have
(1)
W---W {y) = Liix,r'''^x^deHy,x)^dx^
for G R
y
6 N, we
•
for j>! = 1,... ,f t ;... ;pd = 1 , . , , , gj, and in particular,
(2)
( 5 ^ ) ( 0 ) = JP+, ^L<---^W-
Formula (1) is obtained by differentiating the integral in (3) of Definition D. 1 under integral sign which is justified by the Dominated Convergence Theorem. Definition D.3. A probability measure 5> on (Rd, 93^) is called a d-dimensional nor mal distribution if it is absolutely continuous with respect to the Lebesgue measure mdL on (R'', QSJJJ) with the Radon-Nikodym derivative given by
-7-7(z) = (27r)-i(detVT* exp \-\{V~\x
- ™),x - m)\
for x €Rd,
where m E K ' and V is a dx d positive definite symmetric matrix. We write Nd(m, this probability measure. Observation D.4 From the well known formula /
oo
,
e " du = v ^ , /J—oo e~u du = y ^ ,
(1)
-oo
we have by a simple substitution -00 m) 2 11 ii zr°° I( (u — mY (2) (2)
(2TTO)-5 / (2TTO)-5 /
exp < ~ — - — exp ^ - i — — -
4 4
= 1 = 1
for m € R and v > 0. for m € R and t> > 0.
We obtain for a > 0 and 6b real or imaginary
/:«-—-£->{=}• by completing square for uu and, in case 6b is imaginary, by contour integration. Also OO
/
,
|u|"e-" cfo
-co
u
forpeN.
V)for
APPENDIXD. MULTIDIMENSIONAL NORMAL
489
DISTRIBUTIONS
Theorem D.5. Letfl>be the d-dimensional normal distribution Nd(m, V) and let
— m), x — m\ = (ClC{x — m), x — m) = (z, z) ,
and the Jacobian of the inverse mapping x = C~xz + m is given by IdetCT'l^detCr'^detV)*. Thus we have ®(Rd)
= (27T)-7(dety) _ 5 f
expl--(v-'(x-m),x-rn)\mdL{dx)
=
(27r)-*jfiiexp{-i(z,z>}mi(dz)
=
n(27r)-5^exp|-^-imL(d2_;)
= 1, by (2) of Observation D.4. 2) Let p i , . . . , pd € N. Then j
d
|x,| p i • • ■ \xd\Pd
z)^mdL{dz).
Now since x = C~lz + m, the component XJ is a linear combination of z\,... ,zd andmj. Thus \xi\"' ■■■ \xd\Pd is bounded by a polynomial in | z , | , . . . , \zd\. But for any qt,..., qd G N, we have
-ww
j^d\z^---\zd\«tx.v[-\{z,z)}m z) I mdL(dz) expj- dL2{dz) V I'll* '
= n,tW d
7= 1
J
,J
2?
1 L(dz ,) < oo, exp < --i^>m
490
APPENDIX D. MULTIDIMENSIONAL NORMAL
DISTRIBUTIONS
by (4) of Observation D.4. Therefore /Hd |xi|« ■ ■ ■ \xd\Pd®(dx) < oo. 3) For the characteristic function
f
exp{i(y,x)}®(dx)
= (27r)-^y =
dexp{j(y,a;)}exp|--(z,z)|m^(dz)
(27r)-Texp{i(y,m)}j|Bijexp{-^(2,z)+2((C-1)'y,z)}mi(^).
For brevity let us write w = (C~lYy. Then f{y)
= (2n)-^exp{i(y,m)}flj^txp^--z^
+ ex
= (2ir)~^ exp{z (y, m)} J[(2n)*
iw]z:ijmL(dzj) b
P j —Zw) \
=
exp{i(y,m)}exp|--(iy,w)j
=
exp{2(y,m)}exp{-^((C-1)ty,(C-1)'y)}
=
exp{j(y,m)}exp|--(Fy,y>j
y ( 3 ) o f Observation D.4
4) Let m = {rrij : j = 1 , . . . , d) and V = (»/,* : j,k = 1 , . . . ,
d
i
d
d
y(y) = exp < i Y, rn?yp - - ^ X l p=I
"MSPVS
p=l g=l
and thus 9v?
1
(y) = v(y)
im
li
U
i - ~ { £ «P,jyp + 1 ] " M V J P=I
9=1
According to (2) of Observation D.2, for j = 1 , . . . , d we have M«t>)3 = j This shows that M(O) = m.
x^{dx)
= - | ^ ( 0 ) = r?
APPENDIXD. MULTIDIMENSIONAL NORMAL DISTRIBUTIONS
491
5) Since M(O) = m, we have
=
L{x>-
V(*)ilk ■
=
-mj}{xk-
(27r)- = (detV)-5
- mk}Q>(dx) /jxj-myH**-*"*}
x
exp|--(y_l(a:-m),x-m)|m^(di)
=
(27rr^(dety)-5^xii,exp{-l(y-1x,x)}mi(dx),
by the translation invariance of the Lebesgue integral. Let *F be the d-dimensional normal distribution Nd(0, V). Then MQ¥) = 0 by 4) and therefore V(n,t
(2irr*(detVrijj^xi*fcexp|-i{v-,s,*)|mt(«fa)
=
= V(*) iJk
forj,fc = l , . . . , d .
To compute V(
9V ,.,_.,,..„ .. ,
~(y) = ip(y){-Vj,k} 5yj3y k
,„._w
foryG
By (2) of Observation D.2, we have y
m* = jniv,vMdv) =
This shows that V(*) = V.
dyjdyk
h
1
Let X i , . . . , JQ be real valued random variables on a probability space (Q, 5, P). The joint probability distribution of X\,..., Xd is by definition the probability distribution of the d-dimensional random vector X = (Xx,..., Xd) on (Rd, 93md), that is, O is the proba bility measure on (Rd,
492
APPENDIX D. MULTIDIMENSIONAL NORMAL
DISTRIBUTIONS
Theorem D.6. Let Xu... ,Xd be real valued random variables on a probability space (Q., 5, P) which are jointly normally distributed and let the probability distribution ofX = (Xu..., Xd) be given by Nd(m, V) with mean vector m = ( m i , . . . , md) and covariance matrix V = [vjik : j,k= l,...,d]. Then 1) the probability distribution of Xj is given by N\(mj,Vjj)for j = 1,... ,d, 2) {X\,..., Xd} is an independent system if and only if the matrix V is diagonal. Proof. 1) Let Px,,..., Pxd and Px be the probability distributions of X , , . . . , Xd and X respectively. Let ip\,... ,
p ( y ) = [ exp{i(x,y)}Px(dy)=
I exp{z (X,y)} dP
foryeRd,
and for every j = 1 , . . . , d we have (2)
^Cw) = / expO
fo,W)}P*,(dy)
= / exp{t (X„y,)}
dP
fory, g R.
By selecting y = ( 0 , . . . , 0, yj, 0 , . . . , 0) in (1), we have (3)
Now if the probability distribution of X = ( X , . . . , JQ) is given by Nd(m, V), then by Theorem D.5 we have
(y],...,yd)eRd
Suppose X\,..., Xd are jointly normally distributed and the probability distribution of X is given by Nd(m, V). Then according to 1), the probability distribution of X3 is given by N\{mi,Vjj)sottu&
1 - -vhJy)}
d
= JJ^(Vj)-
APPENDIXD. MULTIDIMENSIONAL NORMAL
DISTRIBUTIONS
493
Thus by Kac's Theorem, {X{,...,Xi} is an independent system. Conversely if {X\,..., Xj} is an independent system, then by Kac's Theorem we have d 1 exp{i {m,y) - - {Vy,y)} =Hexp{im]yJ
j=l
1 - -vjjyj} l
fory = (y,,... ,yd) € Rd.
Then e x p { - j (Vy,y)} = e x p { - j £^ 0 , fjjS/f} by equating the real parts on the two sides of the last equality and thus (Vy, y) = £^ = | Vjjyj. This shows that V is diagonal. ■
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Index A augmented, filtered space, 149 filtration, 149 c -algebra, 11
regular, 475 regular image, 483 countably generated cr-algebra, 10 cylinder set, 358 D d-class,4 determining sets of cr-algebra, 481 deterministic initial condition, 396, 397 discontinuity, time of, 217 discrete time, a.s. increasing process, 79 increasing process, 79 L2-martingale, 119, 120, 121 predictable process, 79 stopping time, 39 Doob decomposition theorem, 79 discrete time L2-martingale, 121 Doob-Kolmogorov inequality, 97, 100, 101 Doob-Meyer decomposition theorem, 182 downcrossig, 102, 108 number of, 102, 108 downcrossing inequalities, submartingale, 104, 107, 108 supermartingale, 106, 107, 109
B Borel cr-algebra, 1 Brownian motion, 1-dimensional, 271 adapted, 273 multidimensional, 286 quadratic variation process, 288 strong Markov property, 284 bounded variation, locally, 207 a. s. locally, 207 C class (D), 141 class (DL), 141 conditional dominated convergence theo rem, 464 conditional expectation, 454 conditional Fatou's lemma, 463 conditional Holder's inequality, 468 conditional Jensen's inequality, 466 conditional monotone convergence theo rem, 462 conditional probability, 454
E exponential quasimartingale, 335 498
INDEX F filtered space, 14, 274 augmented, 149 right-continuous, 14 right-continuous modification, 149 standard, 197 filtration, 14 augmented, 149 generated by a process, 14, 274 right-continuous, 14 right-continuous modification, 149 final element, 73, 116 existence, 116 uniformly integrable submartingale, 123 Fisk-Stratonovich integral, 355 Fubini's theorem extended, 164 H Hermite polynomial, 337 I increasing process, 158 almost surely, 158 natural, 173 with discrete time, 79 independence, 443 collections of sets, 443 events, 443 7r-classes, 444 random vectors, 446 independent increments, 268 Ito's formula, 1-dimensional, 320 multidimensional, 340 K Kac's theorem, 448
499 L Lp-bounded, 71 Lebesgue-Stieltjes measure, 157 signed, 205 Lipschitz condition, 398, 402 M martingale, definition, 71 local, 298 local L2-, 298 property, 71 regular, 187 reversed time, 124 martingale convergence theorem, continuous time, 115 discrete time, 112 submartingale with reversed time, 125 uniformly integrable submartingale, 122 martingale transform, 81 maximal and minimal inequalities, 93,98, 99 continuous case, 99 discrete case, 98 finite case, 93 measure, on a semialgebra, 155 on an algebra, 155 measurable process, 12 monotone class theorem, for functions, 7, 8 for sets, 5 •-multiplication, 348 N
500
INDEX
natural increasing process, 173 normal distribution, 488 null at 0,71 7r-class, 4
regular image conditional probability, 483 regular submartingale, 186 reversed time, 124 right-continuous modification, 149
O optional sampling theorem, with bounded stopping times, 90,131 with unbounded stopping times, 138, 140 optional stopping theorem, 87, 89 outer measure, 156
S semialgebra, 155 semimartingale, 318 seminorm, 199 separable cr-algebra, 10 cr-algebra, at stopping time, 26 augmented, 11 Borel, 1 countably determined, 481 countably generated, 10 predictable, 19 separable, 10 well-measurable, 19 signed Lebesgue-Stieltjes measure, 205 standard filtered space, 197 stochastic differential, 346 stochastic differential equation, 368 definition of solution, 371 deterministic initial condition, 396 initial value problem, 385 strong solution, 419 stochastic independence, 443 stochastic integral, of a bounded left-continuous adapted simple process, 223 of a predictable process, 238 stochastic process, 12 adapted, 14 a.s. continuous, 12 a.s. left-continuous, 12 a.s. right-continuous, 12
P pathwise uniqueness, 397 predictable, process, 21 process with discrete time, 79 rectangle, 20 a -algebra, 19 progressively measurable process, 18 Q quadratic variation process, 214 discrete time L2-martingale, 121 Brownian motion, 288 quasimartingale, 318 bounded variation part, 318 equivalent, 346 exponential, 335 martingale part, 318 quasinorm, 50 R random variable, 12 at a stopping time, 141 random vector, 262 regular conditional probability, 475
INDEX bounded, 71 continuous, 12 equivalent, 12 left-continuous, 12 left-continuous simple, 15 L„,71 Lp-bounded, 71 nonnegative, 71 right-continuous, 12 right-continuous simple, 15 measurable, 12 predictable, 21 progressively measurable, 18 simple, 15 stopped, 45 truncated, 47 uniformly integrable, 71 well-measurable, 21 stopped process, 45 stopped submartingale, 132, 134 stopping time, 25 discrete time, 39 random variable at, 141 (j-algebra at, 26 submartingale, definition, 71 property, 71 regular, 187 supermartingale, definition, 71 property, 71 symmetric multiplication, 352 T time of discontinuity, 217 total variation function, 206 total variation measure, 206
501 truncated process, 47 U Ulam's theorem, 479 uniformly integrable, 54 random variables at stopping times, 141 uniqueness of solution, in probability law, 395 pathwise, 397 upcrossig, 101, 108 number of, 101, 108 upcrossing inequalities, submartingale, 104, 107, 108 supermartingale, 106, 107, 109 W well-measurable, process, 21 CT-algebra, 19 Wiener measure, 362 Wiener space, 362
