Pseudo-Differential Operators and Markov Processes: Volume III: Markov Processes and Applications: 3

R be a convex function. Suppose that f(il) C / . Then i f o / e L ' f f l , ^ ; ! ) and it holds (2.39) In particular, when /x(fi) — 1, i.e. when fi is a probability measure, we have

(2.40)

When (fi, A, P) is a probability space and X is a real-valued (or extended real-valued) random variable which is integrable with respect to P we write E(X) := EP(X) := f X(w)P(du>) Jo.

(2.41)

18

Chapter 2 Essentials from Probability Theory

and call E(X) the expectation of X (with respect to P). Thus the properties of the expectation are those of an integral, for example we may rewrite Jensen's inequality as ip(E{X)) <E((poX).

(2.42)

Of fundamental importance is the transformation theorem, or integration with respect to an image measure. Theorem 2.2.2. Let (ft, A, /i) be a measure space and (ft1, A') be a measurable space. Further let /J,' be the image measure of (i under T, T : £7 —> Q' being a measurable mapping, and let f : Q,' —* R be a function which is integrable with respect to fj,'. Then f oT is integrable with respect to [i and it holds

I /d// = j fdT(fi) = J(fo T)d/i.

(2.43)

As shown in H. Bauer [31], Theorem 19.4, we may derive from (2.43) the standard transformation theorem for Lebesgue integrals. In probability theory Theorem 2.2.2 becomes important when recalling that the image measure is the distribution of a random variable. For example if X is an integrable real-valued random variable we find

E{X)= f xPx(dx)

(2.44)

and more generally under suitable assumptions on g E(g(X))=

f g(x)Px(dx).

Let X : $7 —> R™ be a random variable. Then the random variable w i—• is for every £ e R" bounded, hence integrable.

(2.45) elX^'^

Definition 2.2.3. The complex-valued function £^\x(O=E(eiXt)

(2.46)

is called the characteristic function of the random variable X. Note that by the transformation theorem it holds

Ax(O = E(eiX<) = f eix
(2.47)

2.2 Some Integration Theory

19

i.e. X\ is essentially the Fourier transform of Px, and therefore it has all properties of the Fourier transform of a probability measure. In particular it is a positive definite function. The next topic we have to discuss are measures with a density and the Radon-Nikodym theorem. Let (D.,A,fi) be a measure space and / > 0 an integrable function (for simplicity). For A £ Awe may define the set-function v : A -> R+ by

v(A) := jT fdfi (= I XAfd^ ,

(2.48)

and it is easy to see that v is a measure on A- We write v = f\x and call / a density of v with respect to [i. Since for any function g such that g = / /^-almost everywhere it follows that I fdfi=

JA

I gdfi, JA

a density is not uniquely determined. Clearly we have for suitable functions

(2.49)

f hdv= f hfd/j, when / is a density of V with respect to (i.

Definition 2.2A. Let {ft, A) be a measurable space and let fi and v be two measures on (Q,A). We say that v is continuous with respect to (j, or simply fx-continuous if every /i-null set of A is also a zA-null set, i.e. ^(A) = 0 implies v{A) = 0. Theorem 2.2.5 (Radon-Nikodym). Let /u and v be measures on the measurable space (Q,A). When /J, is a-finite then v has a density with respect to \i if and only if v is fj,-continuous. Let (£l,A,P) be a probability space and let Xk : Q. —> R", k G N, be a sequence of random variables. We say that (Xk)k^N converges almost surely to the random variable X : Q —> R" if there is a P-null set iV such that limXk(w)=X(v)

for allUJ e Q\N.

(2.50)

k—>oo

Equivalent to (2.49) is each of the following two statements: lim P(sup \Xi -X\>e)=Q

k—>co

l>k

for alle > 0,

(2.51)

20

Chapter 2 Essentials from Probability Theory

and P(limsup|X fc -X| >e) =0 k—>oo

foralleX).

(2.52)

The sequence (Xk)ken of real-valued random variables is said to converge in Lp to the real-valued random variable X if Xk, X e LP(Q), 1 < p < oo, and lim \\Xk

fc—>oo

-X\\LP

(2.53)

= 0.

For p = 1 we will speak of convergence in the mean, and for p = 2 we speak of convergence in the quadratic mean. Since for a probability space Lp(f2) C Lq(Q) ioi p > q it follows that convergence in Lp{£l), p > 1, implies always convergence in the mean, and for p > 2 convergence in the quadratic mean is always implied. A sequence of R"-valued random variables is said to converge stochastically or in probability to the M"-valued random variable X if lim P(\Xt-X\

fc—»oo

>e) =0

for all e > 0

(2.54)

which is equivalent to lim P(\Xk-X\

fc—>oo

> e ) = 0 for all e > 0.

(2.55)

(In case of an arbitrary measure space and measurable mappings stochastic convergence refers to convergence in measure.) For a sequence (Xk)k€N of Unvalued random variables on (fi, A, P) the convergence in distribution to X is defined as the weak convergence of the distributions (Pxfc)fceN to Px, i-e.

lim f fdPXk = f fdPx

k—»oo JW1

JRn

(2.56)

for all / € C 6 (R n ), compare Section 1.2.3. The following implications hold for K™-valued random variables (when the Z/p-norms are denned in the usual way): — Lp-convergence implies L1-convergence — L1-convergence implies stochastic convergence — almost sure convergence implies stochastic convergence — stochastic convergence implies convergence in distribution.

2.2 Some Integration Theory

21

In addition we have — stochastic convergence implies the almost sure convergence of a subsequence and therefore — L1-convergence implies the almost sure convergence of a subsequence. For doing calculations it is helpful to reformulate some "integral formulae" as formulae involving the symbol for the expectation . For example the Chebyshev inequality (compare (1.2.217)) reads as P(\X\ > e) < e-pE(\X\p)

(2.57)

P

provided X G L (Q). Further (2.56) is equivalent to lim E(f o Xk) = E(f o X)

(2.58)

fc—»oo

and X G LP(Q) means E{\X\P) < oo. When dealing with martingales and convergence results for martingales the notion of an equi-integrable family of functions (random variables) in needed. Definition 2.2.6. Let (Q,A, n) be a measure space and M a family of measurable functions / : fi —> [—oo, oo]. We call M (/i-)equi-integrable if for every e > 0 there exists a /i-integrable function g = gE > 0 on Q such that for every / £ M it holds /

|/|d/i < e.

(2.59)

J{\f\>g}

A typical characterization of equi-integrable sets of functions is given by Theorem 2.2.7. Let (Cl,A,fi) be a a-finite measure space and h a strictly positive function on fi representing an element from L1{^L). Then a set M of A-measurable functions f : fl —> [—oo, oo] is (/j,-)equi-integrable if and only if sup / |/|d/i < oo

(2.60)

f£Mj

and for every s > 0 there exists 5 > 0 such that for all A £ A and f G M / hdn < S implies JA

/ |/|d/i < e.

(2.61)

JA

In H. Bauer [31] various further necessary and sufficient conditions for a family of functions to be equi-integrable are discussed.

22

2.3

Chapter 2 Essentials from Probability Theory

Measure and Topology

The interaction of measure theory and topology is quite a difficult subject, already the "construction" of a non-Borel-measurable subset of R involves the axiom the choice or an equivalent statement. This is in some sense typical: the interaction of "measure and topology" depends on the underlying model of the set theory used. Maybe this is less surprising when we recall the fact that both set theory and modern measure and integration theory emerged from the need to understand (pointwise) convergence of Fourier series. Fortunately we need only some very basic results and we refer in addition to the monographs mentioned in Section 2.1 to G. Choquet [71]-[73], J.C. Oxtoby [283], L. Schwartz [327] and S. M. Srivastara [340]. For any topological space G we may consider the corresponding Borel <jfield B{G), i.e. the cr-field generated by the open sets. Often the topological space is a state space of a process and then assumed to be a Polish space, i.e. it has a countable base and the topology can be obtained from a complete metric. However, also spaces of functions or spaces of measures must be considered. To simplify notation, if G = Rn we will write B^ instead of B{WLn). We refer to Section 1.2.3 where we discussed regularity properties of measures. In particular, a Radon measure was defined as a locally finite and inner regular Borel measure on a locally compact Hausdorff space. Recall that /J, is locally finite if it is finite for every compact set. There is also no problem when passing from a "nice" topological space G to its one-point-compactification. Unfortunately we need to change the notation which we have introduced in Section 1.2.3. Since sometimes we need to add to [0, oo) the point +oo, and then we will consider a family of random variables (Xt)te[o,oo]froma probability space to the one-point-compactification of M" (for example), we have to use a different symbol for M^, and we use now for the point at infinity the symbol A, i.e. GA or M^ etc. denote the one-point-compactification of G or M", respectively. Typically we will extend a sub-probability measure n on (R",i?(™') to a probability measure jl on (R^,S(M^)) by setting /I(A) = 1 - M(K").

(2.62)

Various representation theorems for continuous linear functional on spaces of continuous functions have been listed in Section 1.2.3. They include also results for positivity preserving or positive linear functionals. A helpful supplement

2.3 Measure and Topology

23

is a version of the Hahn-Banach theorem for positive linear functional which we take from K. Jacobs [204], p.326. Theorem 2.3.1. Let (H,<) be a partially ordered real vector space and H+ its positive cone. Let HQ be a linear subspace of H and lo a linear functional on HQ such that u > 0, u G HQ, entails IQ{U) > 0. Then there exists a positive linear functional I on H such that 1\H0 = loLet [i be a Radon measure on a locally compact Hausdorff space G. Its support, supp fi, is defined as supp/x = G\[J{0

C G; O is open and fi(O) = 0}.

(2.63)

We say that \i has full suport if supp /x = G. In contrast to the support of /j, we give Definition 2.3.2. Let (Q,A) be any measurable space and // a measure on (Q, A)- Further let f2' C fl be a subset. We say that /i is supported or carried by Q' if M *(fi'

c

) = 0,

(2.64)

where /u* denotes the outer measure associated with /x, compare (2.33). Note that a measure /j, can be supported by a non-measurable set whereas its support (in case it is defined) is always a measurable set. As illustration let us first state the next proposition taken from H. Bauer [30], Corollary 38.5, and then indicate how it applies to Brownian motion. Proposition 2.3.3. Let E be a metric space containing at least two points and let A be any a-field on E. Consider C([0, oo), E) C E^0'00^ and take on £[°'°°) the product a-field A = a(Xt ; t G [0,oo)) where Xt : £[0'°°> -> E, t G [0, oo), denote the canonical projections. Then C([0, oo); E) ^ A, i.e. the continuous functions form a non-A-measurable subset of iJ1!0'00). In Section 3.7 we will see that the canonical process associated with the Brownian convolution semigroup on W1 gives rise to a probability measure P on (Rn)[°'°°) which is supported by C([0, oo); W1), hence it is supported by a non-measurable set.

24

2.4

Chapter 2 Essentials from Probability Theory

Independence

We want to summarize basic results on independent cr-fields and random variables. As standard references we mention H. Bauer [30], P. Billingsley [39], W. Feller [101]-[102], and D. W. Stroock [345]. Let (fi, A, P) be a probability space. For A€ A and B e A, P(B) > 0, we call

P(AIB) „ ^

1

(2.65)

the conditional probability of A under the hypothesis B. If P{A\B) = P{A) we call A and B independent events and this case occurs if and only if P(A DB) = P(A)P{B).

(2.66)

We generalize this notion of independence in two steps. Definition 2.4.1. A family (Ak)kci, 0 ^ Ak G A, is called independent with respect to P if for every finite subset {fci,..., km} C I of distinct elements it holds P(Akl n . . . D A k m ) =P ( A k l ) • . . . • P(Akm).

(2.67)

Definition 2.4.2. let (£k)kei be a family of sets £k C A. This family is called independent if for every non-empty finite set {k\,..., km} C / of distinct elements and every choice of sets Akl € £kl, I = 1 , . . . ,TO,equality (2.67) holds. Of interest are situations where the independence of a system (£k)k€i entails the independence of the generated cr-fields <j(£k)- A. typical result is the following, see H. Bauer [30], Corollary 6.4. Theorem 2.4.3. For an independent family (£k)kei of C\-stable sets £k c A the family (a(£k))keI is independent. A first application of the notion of independence is given by 0-1-laws. Let (Ak)ken be a sequence of sub-cr-fields Ak C A and define

Tk:=o-(\jAk)

(2.68)

m>k

as well as %o := f | Tfefe€N

(269)

2.4 Independence

25

It is clear that 7 ^ is a cr-field which is called the a-field of terminal events. Proofs of the following results are given for example in [30]. Theorem 2.4.4 (0-1-Law of Kolmogorov). Let («4fc)fceN be a sequence of independent a-fields, Ak C A. If AtT^ then either P(A) — 0 or P(A) - 1. Corollary 2.4.5 (0-1-Law of Borel). For every independent sequence (Afc)fegN of elements A^ e A it holds

P(limsupAfc) £{0,1}-

(2-70)

fc—>oo

Here we used the notation

limsup^fc:= P| (J Am,

(2.71)

which has the counterpart

liminf Ak := | J fl k

~*°°

A

-

( 2J2 )

k€Nm>k

Much more interesting and important is the notion of independent random variables. Definition 2.4.6. Let (Xj)jei be a family of random variables on a probability space (Cl,A, P). We call (X,)j € / independent if (cr(Xj)) is an independent family of
Theorem 2.4.7. Let (XJ)J€I, Xj : Cl —> flj, be a family of random variables on (Ct,A,P), where (Clj,Aj) is a measurable space. Further let for eachj € / a measurable mapping Yj : Clj —> Cl'j be given where (QpA'A is a further measur able space. If the family (XJ)J^I is independent then the family (Yj o Xj) j e / is independent. Let (Cl, A, P) be a probability space and for j = 1,..., m let Xj : Q —> fij, (Clj, Aj) being a measurable space, be a random variable. We define the product mapping Xi ® • • • ® Xm

: fl —> fix x • • • x fim

« ~ y ( w ) :=(*!(*),...,x m M)

(2 73)

"

26

Chapter 2 Essentials from Probability Theory

The joint distribution of X\,..., Px1®-0Xm on ®™=1Aj.

Xm is by definition the probability measure

Theorem 2.4.8. Finitely many random variables X\,..., if and only if

Xm are independent

(2.74)

Pxr<»-<sxm =Px1®---®Pxm holds.

In order to discuss products and sums of independent random variables let us recall some standard notations. Let X, Y be real-valued random variables The expectation of X is denoted by E(X), the variance is on (ft,A,P). denned as V(X) = E(X - E{X)) and the covariance of X and Y is given by cov (X, Y) = E^(Xuncorrelated.

E{X)) (Y - E(Y))Y

If cov (X, Y) = 0 we call X and Y

Theorem 2.4.9. Let X\,..., Xm be independent real-valued random variables and suppose that either they are non-negative or they are all integrable. Then we have

(

m

\

m

(2.75)

l[Xj\=1[lE(Xj).

(2.75)

Two independent real-valued random variables are always un3=1 J 3integrable =1

correlated, but the converse is not true. However we have Theorem 2.4.10 (Identity of Bienayme). For finitely many pairwise uncorrelated integrable and real-valued random variables X\,..., Xm it holds V(X1 + ... + Xm) = V{X{) + ••• + V(Xm).

(2.76)

Next we turn to the distribution of the sum of independent random variables. Theorem 2.4.11. Let X\,...,Xm be M.n-valued independent random variables. For S := Xi + • • • -\- Xm it follows that Ps = Px1+-+xm

=Px1*---*Pxm-

(2-77)

2.5 Conditional Expectation

27

The construction of a family of independent random variables uses the (infinite) product of probability space. In Section 3.1 we discuss infinite products of probability spaces and we use this result in the following. First note that Theorem 3.1.4 extends Theorem 2.4.8. Using Theorem 3.1.4 one can prove Corollary 2.4.12. For every family ((Qj,Aj,Pj))€I of probability spaces there exists a probability space (£l,A, P) and an independent family (Xj)j^i of random variables Xj : Cl —> flj such that for every j € / the distribution of Xj isPj, i.e. PXj =Pj. Finally let us give Kolmogorov's 0-1-law for independent random variables: Theorem 2.4.13. Let {Xk)keH be an independent sequence of random variables and A 6 HfceN ai^rn ; m > k). Then we have P(A)e{0,l}-

(2.78)

Remark 2.4.14. A. Independent random variables are fundamental objects when discussing limit theorems, but we will not touch this topic in our work. B. In compiling the above list of results we closely followed the presentation of H. Bauer [30].

2.5

Conditional Expectation

For this section we can rely on the same standard references as for Section 2.4. In the following a random variable X : il —> [—oo, oo] defined on a probability space (fi, A, P) is called admissible if it is either non-negative or integrable. In both cases E{X) exists but in the first case it could be infinite. We assume the reader to know the elementary notion of conditional probability and we refer for example to H. Bauer, [30] p. 110, where conditional probability and conditional expectation are related in an heuristic way. However we start our summary with Theorem 2.5.1. Let X be an admissible random variable on (£l,A,P) and C C A a sub-a-field. Then to X corresponds an admissible random variable XQ on (fi, A) such that

[ XodP = f XdP Jc Jc

(2.79)

holds for all C G C. Further Xo is P-almost surely unique, and integrable or almost surely non-negative if X is.

28

Chapter 2 Essentials from Probability Theory

Definition 2.5.2. The random variable XQ in Theorem 2.5.1 is called the conditional expectation of X under C (or with respect to C) and is denoted by E{X\C) := EC(X) := Xo.

(2.80)

Remark 2.5.3. Our formulation of Theorem 2.5.1 is close to that given in H. Bauer [30], p . l l l . In fact the whole discussion in this section follows essentially [30], §15. It is useful to extend the notation (2.80) when C is of special type, for example when it is the cr-field generated by random variables Yj : Cl —> flj, j € I. We prefer sometimes to write (2.81)

E(X\Yj , J G I ) or, if / =

{l,...,k}, (2.82)

E{X\Yu...,Yk)

when C = cr(Yj , j £ I) and C = c ( Y i , . . . , Yk) respectively. It is not difficult to see that a C-measurable random variable Xo : fi —+ R which is admissible is a version of E(X\C) if and only if

(2.83)

f ZXodP= f ZXdP

holds for all C-measurable admissible random variables Z denned on Q. Here is a list of basic properties of the conditional expectation, X and Y are admissible random variables, both being either integrable or non-negative when entering in one formula. It holds E(EC(X)) = E(E(X\C)) = E(X);

(2.84)

C

if X is C-measurable then E (X) = X X =Y

C

P-a.s. implies E {X) = E (Y)

X = a,a£R, c

C

then E {X) = a c

tt,/5sl X
P-a.s.;

P-a.s.; c

E (aX + pY) = aE (X) + f3E {Y) here

P-a.s.;

C

(2.85) (2.86) (2.87)

P-a.s.,

(2.88)

for X, Y integrable and a, /3 > 0 for X, Y non-negative; P-a.s. impliesE C (X)<E C {Y)

P-a.s.

(2.89)

2.5 Conditional Expectation

29

and \EC(X)\<EC(\X\)

P-a.s.

(2.90)

The basic convergence results for Lebesgue integrals extend to conditional expectations. For example we have: If (Xk)k£N is an increasing sequence of non-negative random variables then supEc{Xk) = Ec(supXk) fcGN

^fcGN

P-a.s.,

(2.91)

'

and further: If (Xk)keN is a sequence of random variables converging P-almost surely to X and if there is an integrable random variable Y such that \Xk\ < Y P-a.s. for all k e N, then lim Ec(Xk) = EC(X)

k—*oo

P-a.s.

(2.92)

In addition we can extend Jensen's inequality: Theorem 2.5.4. Let X be an integrable random variable on (fl,A,P) such that X(w) £ I for almost all w 6 Q where I is an open interval. Further let q : I —> M be a convex function andC a sub-a-field of A. Then EC(X) & I Pa.s., q o X is admissible and q(Ec(X))<Ec(qoX)

(2.93)

holds P-a.s. So far we extended properties of the integral to conditional expectations. Now we turn to results which have no simple analogue in case of integrals. Proposition 2.5.5. Let X and Y be either two non-negative random variables on (fl,A,P) or let X € Lp(f2) and Y £ Li(Q), ± + ± = 1. Further let C C A be a sub-a-field. Then ifX

is C-measurable then EC(XY)

= XEC(Y)

P-a.s.

(2.94)

R e m a r k 2.5.6. In Proposition 2.5.5 and in the following we are a bit unprecise when using the notation Lp(fl) etc. We always mean a random variable X : ft —> K wuch that / \X(u>)\pdP(uj) < oo, not an equivalence class of random variables.

30

Chapter 2 Essentials from Probability Theory Under the hypotheses of Proposition 2.5.5 one can easily derive EC(YEC(X))

= EC(Y)EC(X)

P-a.s.

(2.95)

Further for sub-cr-fields C\, C
P-a.s. (2.96)

Next we combine conditional expectations with independence. Theorem 2.5.7. LetCi andC2 be sub-a-fields of A and define C := o{C\,C2). Further let X be an integrable random variable on (fi, A, P). If a(X,Ci) is independent of Ci then Ec(X)

= ECl(X)

(2.97)

holds P-a.s. Clearly a(X,Ci)

denotes the cr-field generated by X and C\.

Corollary 2.5.8. If X is admissible and independent of the sub-a -field C C A then it holds EC{X) = E(X)

(2.98)

P-a.s.

Having introduced conditional expectations we may consider for a measurable set A € A the random variable PC(A):=P(A\C):=EC(XA).

(2.99)

We call P(A\C) of course the conditional probability of A under C, but note that P(A\C) is now random variable. Using properties derived before for the conditional expectation we obtain 0 < PC{A) < 1 P-a.s.; c

(2.100)

C

P ( 0 ) = 0 and P (Q) = 1 both P-a.s.;

(2.101)

C

(2.102)

Ax c A2

implies

C

P {AX) < P {A2)

P-a.s.;

and for {Ak)km being a sequence of pairwise disjoint elements from A it holds

Pc([JAk)=YJPC{Ak) fcgN fceN

P-a.s.

(2.103)

2.5 Conditional Expectation

31

There is a different, further interpretation of the conditional expectation using Hilbert space projections. In his monograph [322] R. Schilling used this as the starting point for the discussion and we refer the interested reader to his presentation. We just give following H. Bauer [30], p.121, Theorem 2.5.9. Let (fl,A,P) be a probability space and X G L2(Q,). For every sub-a-field C C A the conditional expectation E(X\C) is the almost surely uniquely determined element in L2(fl) satisfying E(\X-E(X\C)\2)
(2.104)

Note that for a sub-u-field C C A the space L2((fl,C,P)) is a closed subspace of L2((Q,,A, P)), and by Theorem 2.5.9 the orthogonal projection PTAtC:L2((Q,A,P))^L2((n,C,P))

is given b y l n prAC(X) = E{X\C). Finally we want to find a proper formulation for the following situation. Let Y : V. —> fl' be a random variable and therefore {ui £ £1; Y(u) = y E ft'} Cfl determines a sub-cr-field. Hence we can consider the conditional expectation E(X\Y = y). Now we may vary y, i.e. we may try to explore the mapping y H-> E(X\Y = y). However there is a serious problem since E(X\Y = y) for 2/ G fi' fixed is only determined P-almost surely, and for y\ ^ j/2 w e will have in general different exceptional sets. To overcome these difficulties we need first what is called the factorization lemma which we state for general measurable spaces, compare H. Bauer [30], p.62. Proposition 2.5.10. Let 0,^=0 be a set and {Of, A') be a measurable space. Further let T : fi —> fl' be a mapping and / : ( l - * I o function. The function f is a(T) -measurable if and only if there exists a measurable function g : fi' —> M. such that f = goT

(2.105)

holds. The relation to the above mentioned problem is given when we try to "factorize" E(X\Y) according to E{X\Y)=goY for a suitable mapping g.

(2.106)

32

Chapter 2 Essentials from Probability Theory

T h e o r e m 2.5.11. Let (Sl,A,P) be a measure space and (SI1, A') be a measurable space. Further let X : Q —» M be an admissible and Y : il —» fi' be a random variable. Every A' -measurable real-valued function g : Q' —> M which satisfies P-a.s. (2.106) is Py-integrable and satisfies

XdP for all A' € A'.

f gdPy = I JA>

(2.107)

J{y€A'}

The function g is Py-a.s. unique. Conversely, if g is a real-valued A' measurable mapping g : 0 ' —> R which is also Py-integrable and satisfies (2.107), then goY is a version of E{X\Y). If {y} e A' for some y £ fi' then (2.107) yields g(y) = P({Y = y})=

[

XdP

J{Y=y}

and for P({Y = y}) > 0 we find

which we can interpret as a version of the conditional expectation of X under the hypothesis {Y = y}. Therefore we give Definition 2.5.12. Let (ft, A, P) be a probability space, (Q', A') a measurable space, X : Q —> E an integrable random variable, and g : fi' —* E an .A'measurable mapping satisfying (2.107). Then for every y € fi' we call
E(X\Y = y) := g{y)

Remark 2.5.13. Note that E(X\Y = y) be definition is a real number whereas E(X\Y) is a random variable. If g is given then we can recover E(X\Y) V ~* 9(y) '•— E{X\Y — y) by considering the

from

the function

l

E{X\Y)(UJ)=E(X\Y

= Y(CJ))

for P-a.s., w e SI.

(2.109)

Sometimes we write also E(X\Y) = E(X\Y = y)\y=Y{u)

(2.110)

As a consequence of Theorem 2.5.11 and the above considerations we find

2.6

Maxtingales and Stopping Times

33

Lemma 2.5.14. If Y is a a(X)-measurable random variable then there exists a measurable function g :ffi—> M. such that Y = g(X) P-a.s.

2.6

Martingales and Stopping Times

We compile a list of results from martingale theory and refer for proofs to standard references including H. Bauer [30], R.M. Dudley [90], St. Ethier and Th. Kurtz [98], J. Neveu [276], Ph. Protter [292], D. Revuz and M. Yor [299], Chr. Rogers and D. Williams [301] and D. Williams [371]. (These are the main references used in [180] and these lecture notes we have used drafting this section.) Martingales had been introduced by J. L. Doob in [87] and soon became an indispensable tool in probability theory as well as in analysis, for the latter we only need recall Section II.3.4. Still a good reference work for martingale theory is of course the monograph [88] of J. L. Doob. We will discuss here essentially continuous parameter martingales, a discussion of the discrete parameter case can be found in H. Bauer [30] or Chr. Rogers and D. Williams [301], as well as D. Williams [371]. In the following (Cl, A, P) is a fixed probability space on which the random variables under consideration are defined, often we do not mention (Q, A, P) explicitly. Further we assume on (fi,«4, P) a family (Xt)t>o of real-valued, integrable random variables Xt : fl —> K to be given, i.e. £(|X t |) < oo for all t > 0. In the language of Section 2.7, (Xt)t>o is just a real-valued stochastic process. In addition let (^t)t>o be a filtration in (£l,A), i.e. Tt C A and Tt C Ft+sit, s > 0, and suppose that (Xt)t>o is (.^-adapted. Definition 2.6.1. A. We call {Xt)t>o an [Tt)-martingale if E{Xt+s\Tt)

= Xt

P-a.s. for a l l t , s > 0 .

(2.111)

B. The process (Xt)t>o is called an (Ft)-sub-martingale if E(Xt+a\Ft)

>Xt

P-a.s. for alii,x > 0.

(2.112)

C. If for all t, s > 0 we have P-a.s. E(Xt+a\Ft) < Xt then (Xt)t>o is called an (Tt)-super-martingale.

(2.113)

34

Chapter 2 Essentials from Probability Theory

Remark 2.6.2. A. Clearly, (Xt)t>o is an (.Ft)-sub-martingale if and only if (—Xt)t>o is an (.T-^-super-martingale and it is an (J^t)-niartingale if and only if it is both, an (J r t)-sub- and super-martingale. B. In case that (Xt)t>o is a (sub-, super-)martingale with respect to the canonical filtration (J-^)t>o we just speak of a (sub-, super-)martingale. C. If (Xt)t>o is an (^ r t ) t >o-(sub-, super-)martingale then it is also a (sub-, super-)martingale with respect to (^)t>oTheorem 2.6.3. A. Let (Xt)t>o be an (Tt)-martingale and q : M. —> R a convex function. Suppose that for allt>0 the expectation E(\q(Xt)\) is finite. Then (q(Xt))t>Q is an (Ft)-sub-martingale. B. If (Xt)t>o is an (Tt) -sub-martingale and q : R - > K o non-decreasing convex function with E(\q(Xt)\) < oo for all t > 0, then (g(X t )) t > 0 is again an (Tt)sub-martingale. In Chapter 3 we will use that (J"t°)-super-martingale have cadlag-modifications. The following results are needed to derive this result. Lemma 2.6.4. Let (Xt)t>o be a sub-martingale and F C [0,T], T > 0, be a finite set. For A > 0 it holds

p((maxXt > x\) < -E(X+)

(2.114)

< -x\)

(2.115)

and p(\mmXt

< -E(X+ - E(X0)).

Corollary 2.6.5. Let (Xt)t>o be a sub-martingale and F C [0, oo) a countable set. For x > 0 and T > 0 we have P({

V L

sup t6Fn[0,T]

Xt > x}) < -E(X+)

(2.116)

< -E(X±)

(2.117)

'

J

X

and p({

v L

inf teFn[o,T]

Xt < -x\)

>J

- E(Xo)).

x

Let (Xt)t>o be a real-valued stochastic process and F C [0, oo) a finite set. For a < b we define n := mm{t G F ; Xt < a},

(2.118)

o-fc := min{t G F ; t > rk and Xt > b} , k = 1,2,...

(2.119)

2.6 Martingales and Stopping Times

35

and Tfc+i := min{i e F ; t > ak and Xt < a}.

(2.120)

Further we set U{a,b; F) :=m&x{k; ak < oo}

(2.121)

and call U(a,b,F) the number of upcrossings of (Xt)t>o of [a,b] with respect to F . Theorem 2.6.6 (Doob's inequality). Let (Xt)t>o be a sub-martingale. For T > 0 andF c [0, T] a /irate sei i/ /io/ds E(U(a,b;F))<E{{XhT-a)+\

(2.122)

Corollary 2.6.7. Let (Xt)t>o be a sub-martingale, T > 0 andF be a countable subset of [0, T]. Further let F\ c -F2 C . . .feean increasing sequence of finite sets such that \JkeNFk = F. Then the limit U(a,b; F) := lim U(a,b; Fk)

(2.123)

exists and is independent of the choice of the sequence (Fk)k
(2.124)

Remark 2.6.8. If we work in Theorem 2.6.6 and Corollary 2.6.7 with a supermartingale instead of a sub-martingale we will get analogous results but we have to change on the right hand side in (2.122) and (2.124), respectively, (XT-a)+ to (XT-a)-. Using Corollary 2.6.7 (or Remark 2.6.8) one can show Theorem 2.6.9. Let (Xt)t>o be a sub-(super-Jmartingale. Then P-a.s. the two limits lim Xr(<jj) rTt

r€Q+

exist.

and

lim Xr(w) rlt

r€Q+

(2.125)

36

Chapter 2 Essentials from Probability Theory

Definition 2.6.10. A sub-(super-)martingale (Xt)t>o is called right-continuous if P-a.s. all its paths 11—> Xt(u>) are right-continuous. Theorem 2.6.11. Every right-continuous sub-(super-)martingale (Xt)t>o has almost surely cadlag paths. Proofs of these results are given for example in St. Ethier and Th. Kurtz [98] or in D. Revuz and M. Yor [299]. But it seems worth to make a few remarks to the proofs. Suppose that (Xt)t>o is a sub-martingale. Using the upcrossing result (Doob's inequality) we obtain a control on e-oscillations of t >-* Xt(uj), and this enables us to prove that (Xt)t>0 has P-a.s. right and left limits, compare for this some results in Section 3.3 on cadlag-functions. Once the existence of these limits are proved we can construct the cadlag-modification using these limits. A first, rather helpful result using Theorem 2.6.11 is Proposition 2.6.12. Let Z be an integrable random variable and let (Tt)t>o be any filtration. Then Xt := E(Z\Tt) is a martingale and for all t > 0 E{Z\TS)

-> E(Z\Ft+)

inLl(P)

(2.126)

as s —> t+.

Note that the proof of Proposition 2.6.12 uses Remark 2.6.2.C, i.e. the fact that any (.Ft)-inartingale is also an (.Ft0)-martingale. Clearly, enlarging the filtration may destroy the martingale property. However we have, compare D. Revuz and M. Yor [299], Theorem II.2.8, Theorem 2.6.13. If (Xt)t>o is a right continuous sub-martingale, then it is also a sub-martingale with respect to (Jrt+)t>o and its completion. Already the results discussed before need the notion of stopping times and uniform integrability conditions for their formulations or proofs. Therefore, and for other reasons, we will now collect basic ideas and results on stopping times. Definition 2.6.14. Let (Cl, A, P) be a probability space with filtration (!Ft)t>o. We call a random variable r : fl —» [0, oo] an (Ft)-stopping time if {T < t} G Tt for all t > 0. Clearly, constant are stopping times. If r < oo P-a.s. we call r a finite stopping time and if r < T < oo P-a.s. then r is called a bounded stopping

2.6 Martingales and Stopping Times

37

time. If the range of r is a countable set we call r a discrete stopping time. For any (.T-^-stopping time r it holds {r < s} G Ts C .F* for s < t;

(2.127)

{r
(2.128)

{T = t} = {r< t}\{r
(2.129)

and

Lemma 2.6.15. Let (J7t)t>o ^e a filtration in (Q,A,P) and as usual set Tt+ '•— f\>o-^*+e- -^ fan^om variable r : fi —> [0,oo] is on (Tt+)-stopping time if and only if {r < £} € ^-t+ / o r aM £ > 0. Here is a collection of simple operations for stopping times. In the following (Tfc)fceN is a sequence of (^rt)-stopping times and c € [0,oo) is a non-negative constant: Ti + c and ri Ac are (.Ft)-stopping times; suprfc is an (.^-stopping time;

(2.130) (2.131)

fcSN

minTfc is an (J r t )-stopping time;

(2.132)

k<m

if (ft)t>o is right continuous then liminf Tfc and limsuprfc k—>oo

(2.133)

k—^oo

are (^)-stopping times. For a fixed (^rt)-stopping time r on (Q, A, P) we define FT := {A € A; A(~) {T < t} £ ft for all t > 0}

(2.134)

and FT+ :={AeA;AD{T
Tt+ for all t > 0}.

(2.135)

Definition 2.6.16. Let (fl,A,P, (Xt)t>0) be a stochastic process with a locally compact Polish state space (E, B) and let (J-t)t>o be a filtration. We call

38

Chapter 2 Essentials from Probability Theory

the process (Tt)-progressive or (Tt)-progressively measurable if for every t > 0 the mapping X : [0,t] x f t - > £ (s,w) >-^ Xs(u) is measurable with respect to the product cr-field B([0, i\) cg> TtThe following lemma is proved, for example, in St. Ethier and Th. Kurtz [98].

Lemma 2.6.17. Let (Q, A, P, (Xt)t>o) be a stochastic process with locally compact Polish state space (E, B) and let (Tt)t>o be a filtration. Further let (Xt)t>o be (ft)-progressively measurable, a,r be two (Tt)-stopping times and Z is a non-negative TT-measurable random variable. With (XJ) := X T M and Yt := XT+t we define Qt •= J~Tt\t and Tit := Jv+t- Then the following assertions hold: Tt is a a-field;

(2.136)

r and T A a are both Tr-measurable;

(2.137) (2.138)

ifr
(X )t>0

(2.139)

is TT-measurable; T

{Gt)t>o is a filtration and [X )t>0

is (Qt)

(2.140)

- and (Tt)-progressively measurable; (Wt)t>o *s a filtration and (Yf)t>o *«

(2.141)

Ht-progressively measurable; T -V Z is an (Tt)-stopping

time.

(2.142)

In Section 3.5 we will see that if (Xt)t>o is a cadlag process with locally compact Polish state space (E, B) and if A is a Borel subset of E, then the first hitting time of A, aA := M{t > 0 ; Xt G A}

(2.143)

and the entry time aA := inf{t > 0 ; Xt € A}

(2.144)

are stopping times. For a closed set A a further stopping time is given by TA := inf{t > 0 ; Xt G A or Xt- £ A}

(2.145)

2.6 Martingales and Stopping Times

39

which is called the first contact time of A. Let (Xt)t>o be a stochastic process which is (^)-adapted and let r be an (.T^-stopping time. We may consider a new random variable XT obtained from (Xt)t>o and r by optional sampling: (2.146)

(XT)(w) := XTlu)(w).

In particular, since t A r is also a stopping time we can construct the random variables XtAT by optional stopping, i.e. (XtAT)(w)

:= XtAr{uj)(w).

(2.147)

It is natural to call (XtAT)t>o the stopped process (with respect to T ) . There are several versions of Doob's optional sampling or optional stopping result in the literature and we quote a few: Theorem 2.6.18. Let (Xt)t>o be a right-continuous (J-t) -sub-martingale and Ti,T2 two (ft)stopping times. For every T > 0 it holds E(XT2AT\fTl)

> XTlAT2AT.

(2.148)

If in addition Ti in P-a.s. finite, E(\XT2\) < oo, and lim E{\XT\X{T2
= 0,

(2.149)

T—•oo

then E(XT2\TT1)

> XTlAT2.

(2.150)

This formulation of the optional sampling result is taken from St. Ethier and Th. Kurtz [98]. We call an (^ r t )-martingale (Xt)t>o closed when there exists a random variable Y such that E(\Y\) < oo and for all 0 < t < oo it holds Xt = E(Y\Tt) Pa.s. Now we can formulate Ph. Protter's [292] version of the optional sampling theorem: Theorem 2.6.19. Let (Xt)t>o be a closed right-continuous (Tt)-martingale. Further let a, r be (J-t)-stopping times and a < r P-a.s. Then Xa and XT are integrable and Xa=E(XT\Fs)

P-a.8.

(2.151)

fO

Chapter 2 Essentials from Probability Theory

Further variants are given in Chr. Rogers and D. Williams [301] and in H. Bauer [30]. Some of these results need the notion of a uniformally or equiintegrable family of random variables, compare Definition 2.2.6 and Theorem 2.2.7. Definition 2.6.20. A family (Xj)jeI of real-valued random variables is called uniformly or equi-integrable if for every e > 0 there is a constant K > 0 such that E(\Xj\;

|X,-1 > K) < e

for all j e I.

(2.152)

Here we used the notation E(Y ; A) = fA YAP. Here is a variant of the optional sampling result taken from Chr. Rogers and D. Williams [301]: Theorem 2.6.21. Let (Xt)t>o be a uniformly integrable cadlag (Tt)-supermartingale and let a, r be {J-t)-stopping times such that a < r . Then XT ei'ffi) and EiXcolFr)

< XT

P-a.s.

(2.153)

E{XT\TS)

< Xa

P-a.s.

(2.154)

and

If(Xt)t>o ity.

is even an (Ft)-martingale then (2.153) and (2.154) hold with equal-

The random variable X^ is obtained as limit of (Xt)t>o as we will see soon. Our final version of the optional sampling theorem is taken from N. Ikeda and S. Watanabe [167]: Theorem 2.6.22. Let (Xt)t>o be a right-continuous (Ft)-sub-martingale and (ct)t>o be a family of bounded stopping times such that P(at < crs) = 1 for t < s. Then (Xt)t>o, Xt := Xat, is an (Ft)t>o-sub-martingale where Tt:=Tat. Remark 2.6.23. In writing this last section on optional sampling our idea was to allow every reader to work with his favoured book and his favoured notions (equi-integrable or uniformly integrable).

2.7 Some Remarks to Stochastic Processes

41

Before we turn our attention to martingale convergence results we give Theorem 2.6.24. Let (Xt)t>o be a cadlag (Ft)-martingale and r an (Ft)stopping time. Then the stopped process (Xt/\T)t>o is once again a cadlag martingale. A proof of this theorem is given in Chr. Rogers and D. Williams [301] where also a formulation for super-martingales is discussed. Next we collect some convergence results for (super-)martingales which we borrow from Ph. Protter [292]. Theorem 2.6.25. Let (Xt)t>o be a right-continuous (Tt)-super-martingale such that sup E(\Xt\) < oo. Then the limit Y := lim Xt exists P-a.s. 0
t—>oo

and defines a random variable with E(\Y\) < oo. In addition, if (Xt)t>o is closed by a random variable Z, then (Xt)t>o is also closed by Y and it holds Y = E(Z\a(Tt; 0 < t < oo)) P-a.s. Theorem 2.6.26. Let (Xt)t>o be a right continuous (J-t)-martingale which is uniformly integrable (=equi-integrable). Then Y := lim Xt exists P-a.s., ^ d ^ l ) < oo and Y closes (Xt)t>o as an (Ft)-martingale. Theorem 2.6.27. Let (Xt)o
gale with respect to (!Ft)o
2.7

Some Remarks to Stochastic Processes

In Chapter 3 which is devoted to Feller processes we give the Kolmogorov construction of a canonical process. Although most of the concepts needed we discussed there more or less detailed, we want to summarize here some basic material from the elementary theory of stochastic processes. Standard reference texts include H. Bauer [30], K. L. Chung [75], J. L. Doob [88], E. B. Dynkin [92]-[93], St. Ethier and Th. Kurtz [98], W. Feller [101]-[102] and Chr. Rogers and D. Williams [301]-[302]. Very brave readers may also consult C. Dellacherie and P. A. Meyer [79]-[83], and heroes are refered to M. Sharpe [329]. Let (E,B) be a locally compact Polish space equipped with its Borel
42

Chapter 2 Essentials from Probability Theory

or [0,oo], sometimes we need I = N or No, or even Q + . The set / is called the time parameter set. Definition 2.7.1. A stochastic process with state space (E,B) and time parameter set I is a quadruple (Cl, A, P, (Xt)t>o) where (fi, A, P) is a probability space and Xt : Q —> E is for each t g / a random variable, i.e. a measurable mapping. Often we will write (Xt)t€l instead of (Cl,A,P,(Xt)t>o), and (Xt)t>0 if I = [0,00) etc. Let (Xt)t>o be as stochastic process and fix u € fi. The mapping t 1—> X t (w) from J to £ is called a poi/i of the process (Xt)t>o- If / = [0, 00) then we call the process (Xt)t>o right-continuous, left-continuous, cddldg, or continuous, if almost surely every path is right-continuous, left-continuous, cadlag, or continuous, respectively. Clearly (sub-,super-)martingales are stochastic processes. Often we have to work with filtrations. Then we write (fl,A, P, (P^tei, (-Xt)te/) when (Xt)tei is .Ft-adapted, i.e. Xt is .Ft-S-measurable. Stochastic processes with state space R" (or E being a locally compact Polish space) and / = [0, 00) are discussed in detail in the following chapter. — Thus we do not go into details here. If (p,,A,P, {Xk)keH0) is a stochastic process with discrete time parameter set, then we call it a Markov chain if P(Xk+m

G B\Fk) = P(Xk+m

G B\Xk)

(2.155)

where Tk '•= o~(Xi ; I < k). If (Xt)t>o is a stochastic process and / has a minimal (first) element 0, then Px is called the initial distribution of (Xt)t£i- The distribution of the product variable Y := Xtl • • • ® Xtm, t\,... ,tm G / , is called the joint distribution Xtm, and is denoted by Py = Pxtl®-®xtm • The definition of the of Xtl,..., family of the finite distribution of a process (Xt)t>o is given in Section 3.1. We will mostly consider families of stochastic processes indexed by the state space E, i.e. (n,A,Px,(Xt)t>o)x£E

(2.156)

are the object of interest where Px is determined by the additional condition P * — ex, ex being the Dirac measure at x. In such a situation we write Ex (Y)

2.7 Some Remarks to Stochastic Processes

43

etc. Now, in many situations we may even consider for EpX (Y) = jYdPx, EXt^(Y). EXt(Y) which is a new random variable defined bywi-> If a nitration {Ttjtzi is given to which (Xt)tei is adapted, we may consider stoppiong times r related to the process, for example first hitting times or entry times, compare Section 2.6. Then new objects such as the stopped process (XtAr)tGi might be worth some considerations. As before X t A r is deFurthermore we may consider related conditional fined byw H XtAT(u)((j). expectations. We will need for example a result such as Lemma 2.7.2. Let (Xt)t>o be a real-valued stochastic process adapted to the filtration {^tyo- Further let T be a finite (J-t)-stopping time which has discrete values. If E(Xt\Jrt) > 0 for each t, then E{XT\FT) > 0. Recall (2.134) and (2.146) for the definition of XT and TT, respectively.

Chapter 3

Feller Processes In this chapter we discuss Feller processes, more precisely we discuss how to construct a stochastic process when starting with a given Feller semigroup, and we also study some of their path properties. We give in Section 3.1 the Kolmogorov construction of a canonical process and in Section 3.2 we show how to construct a projective family and a transition function when starting with a Feller semigroup. In particular we study how to move to the one-pointcompactification of the state space in case of non-conservative semigroups. A further topic is stochastic continuity of a process. Section 3.3 is devoted to the regularity of paths. Notions such as modification and version of a process are explained and most of all cadlag-functions are explored in detail. In the center of our consideration is the Skorohod space — the space of all cadlag functions equipped with the Skorohod metric. In Section 3.4 we introduce the concept of a Markov process and prove that Feller processes are Markov processes. Further we show that every Feller process has a cadlag modification. These considerations are extended in Section 3.5 to prove that Feller processes are strong Markov processes and for this we have to introduce and to investigate the shift operator. In addition certain stopping times are discussed. At the end of Section 3.5 we know that every Feller semigroup gives rise to a stochastic process, called Feller process, which is a universal strong Markov process admitting a cadlag-modification. — Hence with all operators discussed in Volume 2, Chapter 2, which generate a Feller semigroup we can associate a Feller process in the above sense. In Section 3.6 we find another link between an operator A and a stochastic process by discussing the martingale problem. We give the notion of the Vn-

46

Chapter 3 Feller Processes

martingale problem and its well-posedness, and prove that in case that the P n -martingale problem is well-posed then we can associate a Markov process with A. This prepares Chapter 4. In the first two volumes we often compared a given negative definite symbol q(x, £) with a fixed continuous negative definite function tp. The stochastic processes associated with continuous negative functions are Levy processes, processes with stationary and independent increments. These processes are discussed in Section 3.7 and many examples are given. In particular it is shown that Levy processes are Feller processes, hence universal strong Markov processes admitting a cadlag modification, and that path properties can be investigated by using ip. In Section 3.8 we summarize path properties of Levy processes and we state their Levy-Ito decomposition. Section 3.9 is central: it links the symbol of the generator to the process and justifies to introduce the notion of the symbol of a process which generalizes in a natural way the notion of the characteristic exponent of a Levy process. For Feller processes the circle will close: negative definite symbol —> Feller semigroup —» Feller process —> negative definite symbol.

3.1

Projective Limits and Canonical Processes

In the last chapter we summarized material from the theory of stochastic processes which is partly much more advanced than the topic of this section — the Kolmogorov construction of a canonical process. However a detailed understanding of this construction is needed in order to appreciate later on the efforts made to associate with an Lp-sub-Markovian semigroup a Markov process. In the presentation of this section we use some of our lecture notes [180], [187], and [188], however already for these lectures the main source and reference was the monograph [30] of H. Bauer, auxiliary results from measure and integration theory are taken from his text book [31]. We start with recalling some results on arbitrary products of probability spaces. Let / ^ 0 be a non empty index set and for i £ I let (f2j, At, Pi) be a probability space. For 0 ^ K C / we set

nK := J J iU := {a; : K -> | J SI*; u(i) € «<},

(3.1)

3.1 Projective Limits and Canonical Processes

47

and for K = / we just write n =

ftj,

(3.2)

thus fi is the Cartesian product of the family (fij)igj and a typical element w € £1 has the interpretation as a mapping w : I -* IJte/^V For 0 ^ J c i f C / w e may introduce the projection ^

• nK - » n j ,

(3.3)

where for wjf G fi^ we set (3.4)

KJ(WK)=WK\J-

In particular for K = I we set

and for J = {io} we write 7I

£ : = 7 r {
a n d

^o—^io}-

Clearly, for 0 ^ J C K C L C / we have *J=-XJ°*K,

(3-5)

especially for L = / we find TTj = Tlf 0 •KK.

(3.6)

By W = W(/) we denote the system of all finite, non empty subsets of I: H := {K C / ; K =f 0 and K is finite}.

(3.7)

For each J GTC the product a-field

A/:=(g)A

(3.8)

ieJ

and the product (probability) measure Pj:=
(3.9)

48

Chapter 3 Feller Processes

Definition 3.1.1. Let 1 ^ 0 and (Ai)i^i be a family of cr-fields Ai in Qi,i e / . The smallest u-field generated by the family ( A ) i e / in the product fi := I l i e / ^* w n i c n makes each projection 7Tj : D. —> Ct{ an ^-^,-measurable mapping is called the product of (-4i)j 6 / and is denoted by (3.10)

A:=(g)Ai. ie/

Thus we have A=(g)Ai

:=o-(7r i; ie J) = cr(7rj; J e H ) ,

(3.11)

where the last equality is trivial. Of central importance for all what follows is the next theorem. Theorem 3.1.2. Let (Q.i,Ai,Pi)i£i,I ^ 0, be an arbitrary family of probability spaces. Then there exists exactly one probability measure P on A := 0 i e / Ai such that for all J £H TTJ(P)

- Pj

(3.12)

holds.

A proof of Theorem 3.1.2 is given in H. Bauer [30], Theorem 9.2. Definition 3.1.3. The uniquely constructed probability measure P in Theorem 3.1.2 is called the product (probability) measure of the (probability) measures (Pi)isi and sometimes denoted by (2>i£/ Pi- The probability space Vie/ ie/ ie/ / is called the product (probability) space of the family (Cli,Ai,Pi)i^i. Next we turn to random variables on product spaces. Let (fi, A, P) be a probability space and 0 ^ / an index set. For i £ I let (fij, A ) be a measurable space and JQ :ft,—»fija random variable. We define the product mapping Y by

Y-.= (£)xi:n-*Y[ni ie/

iei UJ H->

au : I —> [ J fij ie/ i H-> (^(i) := A"j(w).

3.1 Projective Limits and Canonical Processes

49

From the construction it is clear that Y is a (lite/ ^i> ® i e / -4i)-random variable and we denote by Py its distribution, i.e. Py is the joint distribution of the family (JQ) z£ /. A proof of the following result is provided in H. Bauer [30], Theorem 9.4. Theorem 3.1.4. Let (Q,A, P) be a probability space and 0 ^ I an index set. Let for i £ I a measurable space (fij, Ai) be given and X% :ft—>ftibe a measurable mapping, i.e. a random variable. The family (Xi)iej is independent if and only if for their joint distribution it holds p

®^Xi=®pxt-

(3-13)

So far some preparations. Now we come to our central theme in this section: Projective limits. Recall that a Polish space E is a topological space which is completely metrizable and has a countable base. On E we will always consider its Borel a-field B = a(O) where O denotes the system of open sets. Further let 0 ^ I be an index set and put

EJ ^YlE^Ej-.^E,

(3.14)

i.e. EJ = {e : j —> E}. In addition we define BJ:=$$Bj,

Bj-.= B.

(3.15)

Since for 0 ^ J c K c / the projection ?rf : EK -> EJ

(3.16)

is continuous it is iBH-SJ-measurable. Now let (ft, A, P, {Xt)t^i) be a stochastic process in the sense of Definition 2.7.1 with state space (E,B) and set Xj:=(g)Xt,0^JcI, teJ

(3.17)

50

Chapter 3 Feller Processes

as well as

Pj := Xj(P) , 0 / J C / ,

(3.18)

i.e. Pj is the joint distribution of the family (Xt)teJ- Since Xj = TTJ" O XK for 0 ^ J C K C I we find (3.19)

PJ = ^{PK) and for K = I PJ = 7rj(P/).

(3.20)

For J = {t\,..., tm} C 'H(I) and for Bj G B,j G J, we have P,^

x ... x Bm) = P(Xj\Bt

x ... x Bm))

= P(-^ti G-Bi,...,X tm G B m ),

(3.21) ^ ^

i.e. Pj(Bi x . . . x B m ) is the probability that- a path of (fi, A, P, (Xt)tei) hits at "time" U the set Bj, 1 < i < m. In the situation just described we call (PJ)J^T-I the family of finite dimensional distributions of the process (p., A, P, (Xt)t£i)Definition 3.1.5. Let (Pj)jen be the family of the finite dimensional distributions of a stochastic process (Cl,A,P, (Xt)tej) with state space (E,B). If for all J,K G H,J C K, the relation (3.19) holds, then we call (PJ)J€H a projective family of probability measures over (E,B). T h e o r e m 3 . 1 . 6 ( A . N . K o l m o g o r o v ) . L e t E be a P o l i s h space

and 1 ^ 0

be an index set. For every projective family of probability measures over (E, B) there exists a unique probability measure Pi on (E1,BJ) such that TTJ(PI)

= PJ

(3.22)

holds. A detailed proof of Theorem 3.1.6 is given in H. Bauer [30], p.301-303. Definition 3.1.7. The measure Pj uniquely determined by Theorem 3.1.6 is called the projective limit of the family {PJ)J£UFor our purpose of greatest importance is

3.2 Semigroups of Kernels, Transition Functions and Canonical Processes

51

Corollary 3.1.8. Let E be a Polish space and 0 •£ I a non empty index set. Then for every projective family of probability measures {Pj)jeU(i) over C^i ^) there exists a stochastic process with state space E and parameter set I such that (Pj)jeH{i) l s the family of its finite dimensional distributions. Proof. We set Q, := E1, A := B1 and P := P/, P/ being the projective limit of the family (Pj)jeH- Then (f2, A, P) is a probability space. Further, for t e / we set X t : 0 -> £ w H-» Xt(w) := u(t) := 7rt(o;), where 7r4 is as before the projection n{t}- Since the projections are measurable, (Q,, A, P, (Xt)tei) is a stochastic process with state space E and parameter set /. For 0 / J e H and B G BJ we find in addition P p G G B) = P ^ J 1 ^ ) ) = T T J ( P / ( S ) ) = Pj(B), i.e. (3.22) holds.

D

Definition 3.1.9. The stochastic process constructed in Corollary 3.1.8 is called the canonical process corresponding to the projective family (Pj)jeuIt is important to note that for a canonical process all mappings u> : I —> E are paths of the process, and for modelling, this seems to be a non-satisfactory result. However, if for a given projective family we can prove that the (P)-measure of the paths we do not expect to observe is zero, our model would be more satisfactory. Thus we face two problems: • construct projective families of probability measures • prove that in "reasonable" cases, i.e. for "nice" projective families, the paths of a canonical process are (almost surely) "well behaved". From now on it will be a longer way to finally realize that in many cases pseudo-differential operators with negative definite symbols will give the tools to solve these problems.

3.2

Semigroups of Kernels, Transition Functions and Canonical Processes

The aim is now to construct projective families of probability measures. For this we need the notion of kernels and we prefer to recall in the beginning the

52

Chapter 3 Feller Processes

considerations from Section 1.2.3, especially Definition 1.2.3.19 and Theorem 1.2.3.20. Definition 3.2.1. Let (Q,A) and (fl',A') ping

be two measurable spaces. A map-

K : n x A' -» [0, oo)

(3.23)

is called a kernel from (Q,A) to (ft', A') if i) u> i-> isT(tL>, ^4') is for each A' G .4' measurable, and ii) A' >-> /f(w, A') is for each w G 0 a measure on .4.'. If K(w,Q') = 1 for all LJ £ ft we call # a Markovian kernel, if if(w,ft') < 1 for all u £ fl,K is called a sub-Markovian kernel. Denote by M(Q,,A) the space of all numerical .4-measurable functions u : f i - » R and denote by M+(Sl,A) the set of all u e M(£l,A) such that u > 0 holds. Given a kernel from (fi, ^4) to (Cl',A') we define on M + (fi',«4') the operator {Kopu){w):=

f u(u/)K(u>,du').

(3.24)

Since M+(tt,A) is the set of all increasing limits of simple functions, a monotone approximation argument yields that Kopu e M+(ft,A), i.e. Kop:M+{U,',A')^M{VL,A). Obviously we have for u, uv G M + (Q', A'), v € N, and /z > 0 : Kop(u + v) = Kopu + Kopv ;

(3.25)

Kop(/j,u) = \xKopu ;

(3.26)

u > 0 implies XopM > 0, i.e. Kop : M + (fi', ^ ' ) -> M+(fi, ^ ) ;

(3.27)

if uu 1 u then Kopuv

(3.28)

\ Kopu .

For A1 £ A' it follows that (KOPXA>)(U)

= K(U,A>)

(3.29)

3.2 Semigroups of Kernels, Transition Functions and Canonical Processes

53

and for u — xw we have KOPXQ' = Kopl = 1 if if is Markovian

(3.30)

KopXw = Kov\ < 1 if K is sub-Markovian.

(3.31)

and

Now it is easy to see that (3.25)-(3.28) give already a characterisation of kernels. Lemma 3.2.2. Let (fl,A) and (fl',A') be two measurable spaces and let Kop : M+(fl',A') -> M{Q,A) satisfy (3.25)-(3.28). Then there exists a unique kernel K from (Cl,A) to (£l',A') such that f

{KOPU){OJ)=

holds for all

u(w')K(w,du')

u&M+(Q',A').

Proof. We just have to define by inspection.

K(LJ,A')

:= KOP(XA/){U})

and the lemma follows •

As a next step we define the composition of the two kernels. For i — 1,2,3 let (fij, Ai) be measurable spaces and for i = 1,2 let Kl be kernels from (fij, At) to (Cli+i,Ai+i). Denote the corresponding operators by Kxop. Obviously Kll : M + (O 3 , A3)-+M+ (fi!,At) Kop

•=

Kop

°

(3.32)

Kop

is well denned. In fact K]£ satisfies (3.25)-(3.28) and therefore by Lemma 3.2.2 there exists a kernel K12 from (ili,Ai) to (fi 3 ,^3) such that

(tf#u)(wi) = f u{u>z)K12{uuduz)

(3.33)

holds for all u G M+(fl3, A3). For these it's we have further

{KlpoKlpu){w) = JK\wltdLJ2) =

U{LJ3)

(Ju(u3)K2(w2,du;3)) K2{w2,du3)K1(u1,dw2),

54

Chapter 3 Feller Processes

which implies (tf 12 )( Wl ,A 3 ) = jK2(u;2,A3)K1(oj1,dw2).

(3.34)

In particular, if K1 and K2 are Markovian (sub-Markovian) then K12 is Markovian (sub-Markovian) too. Suppose that K is a bounded kernel from (ft,.4) to (ft', A'), i.e. if (<*;, A') < C for all w G ft and A' G A', then we may extend Kop : M + (ft', .4') - • M(ft, .4) to a linear, bounded and positivity preserving operator from Mf,(ft',.4') to Mb(ft, .4), where we denote by M&(ft', A') the bounded, real-valued measurable functions on ft'. Whereas it was helpful to stress the "measure theoretical" aspects when introducing kernels, in order to construct projective families of probability measures and canonical processes it is reasonable to change a bit the notation. In the following we will consider kernels from a measurable space (E, B) into itself and in most cases (E, B) will be a Polish space with Borel cr-field. Further, we will denote typical points in E by x,y,z, etc., moreover we use from now on the widespread convention to write K for the kernel as well as the corresponding operator, i.e. Ku(x)= fu(y)K(x,dy)

(3.35)

will be considered as a convenient formula. More often, however, we denote the kernel with a lower case letter and the corresponding operator with the upper case letter, i.e. Ku{x)= I u(y)k(x,dy).

(3.36)

In addition, if k(x, dy) has a density with respect to a reference measure /i we write k(x,dy) = k(x,y)n(dy) and therefore Ku{x) = f u{y)k(x, y)fi(dy).

(3.37)

As usual, in this situation we will sometimes call k(x,y) a kernel instead of k(x,y)n(dy). Definition 3.2.3. Let (E,B) be a measurable space and let (pt)t>o be a family of sub-Markovian kernel from (E, B) into itself. If for the corresponding

3.2 Semigroups of Kernels, Transition Functions and Canonical Processes

55

operators (Pt)t>o it holds Ps+t = PsoPt

,s,t>0,

(3.38)

then (pt)t>o is called a semigroup of sub-Markovian kernels on E (or (E, B)). If all kernels pt are Markovian, we call (pt)t>o a semigroup of Markovian kernels. If the operator Po = id we call (Pt)t>o a normal semigroup. Corollary 3.2.4. Let (pt)t>o be a semigroup of sub-Markovian kernels on E. Then the Chapman-Kolmogorov equations

ps+t(x, B) = JPt(y, B)Ps(x, dy)

(3.39)

hold for s,t > 0, (x, B) G E x B. Further, (pt)t>o is normal if and only if Po(x, {x}) = 1 holds for all x € E. Proof. The first statement follows from (3.38) and (3.34), whereas the second statement follows from

u{x) = Pou(x) = / u(y)po(x,dy). D R e m a r k 3.2.5. The notion of a semigroup of Markovian kernels is closely related to that of a (Markovian) transition function which is unfortunately not uniquely denned in the literature. We will sometimes call a normal semigroup of Markovian kernels a (Markovian) transition function. However, we will not require additional regularity conditions on (pt)t>o with respect to t. Before we give examples we introduce the notion of (spatially) translation invariant semigroups of kernels. Definition 3.2.6. A semigroup of sub-Markovian kernels on (Rn,Bn) is called translation invariant (or spatially homogeneous) if for all x, z 6 K™

and B e £ ( n )

Pt{x,B) =pt{z + x,z + B)

(3.40)

holds. Our first example is based on our considerations on convolution semigroups, compare Section 1.3.6.

56

Chapter 3 Feller Processes

Example 3.2.7. Let (fJo be a convolution semigroup on K n . For i £ l " and B e B{n) define (3.41)

Pt(x,B):=nt(B-x).

Clearly, for each i > 0 a kernel on (Rn,B^) (A*t)t>o is a convolution semigroup, we find

into itself is denned by pt. Since

Pt+a(x,B) = (jLt+s(B-x)

= I IM(B - x - y)na(dy)

= JPt(x + y,B)Ps(0,dy) =

Pt{z,B)ps(x,dz),

i.e. (pt)t>o is a normal semigroup of sub-Markovian kernels which is translation invariant since Pt(x,B)

= IH(B - x) = t*t((B + z) - x - z) =pt(x

+ z,B + z).

Conversely, if a translation invariant semigroup of sub-Markovian kernels is given on (Rn,B^) then by (3.42)

fit(B):=pt(0,B)

a convolution semigroup is defined. In this case the corresponding operators are given by Ttu(x) = fu(x- y)fH(dy)

, u e Bb(Rn),

compare Example 1.4.1.3. Example 3.2.8. Let (Tt)t>o be a Feller semigroup on Coo(E";K) and denote by (Pt(')*))t>0 the corresponding kernels, compare Theorem 1.4.8.1. Then these kernels form a semigroup of sub-Markovian kernels. Example 3.2.9. Let (?t ( p ) ) t > 0 be a strongly Lp-sub-Markovian semigroup on LP(Rn;M), 1 < p < oo, in the sense of Definition II.3.2.3. Then for B £ B{n\ A<"'(S) < co, for all x £ Rn and t > 0 {p)

Pt(x,B):=(Tt

XB)(x)

(3.43)

3.2 Semigroups of Kernels, Transition Functions and Canonical Processes

57

is uniquely determined and by monotone approximation we may extend pt as a kernel from (R",# (n) ) into itself. However,

Tt{llxB =

(Tt{p)oT^)XB

holds only as equality in Z/p(Kn;R). Thus some work is required if we want to identify the semigroup (Tt ) t > 0 with the family of operators which we may define by Ptu(x) = / u(y)pt(x,dy). In the case of a strongly Lp-sub-Markovian semigroup we can proceed by choosing for Tj XB always its unique continuous representative. However, a more serious problem arises if {H ) t > 0 is just an Lp-subMarkovian semigroup. In this case for fixed B £ B^ by (3.43) we obtain only an Lp-function, i.e. x \-> pt{x, B) is only determined up to a set of (Lebesgue) measure zero which will in general depend on B. In the light of the above examples the following theorem gives us a first hint how to construct a stochastic process starting with certain operator semigroups. Theorem 3.2.10. Let (E,B) be a measure space and (pt)t>o o, semigroup of Markovian kernels on (E,B). Further let \ihe a probability measure on (E, B). For J := {*i,..., tm} e H(R+), ti < ...
XB{xi,...,Xm)Ptm-tm-Axm-lAXm) •••ptl(xo,dxi)tJ,o(dxo).

(3-44)

Then (Pj)jen(u+) is a protective family of probability measures over (E,B). Remark 3.2.11. The following proof is borrowed from H. Bauer [30], Theorem 3.6.4. Since the result is central for all of our considerations we believe that it is worth giving a proof, not only a quotation. Proof. Since for a sequence of mutually disjoint sets Bj £ BJ,j € N, it follows with B := U j e N Bj that oo

XB = ^2 XBj j=l

58

Chapter 3 Feller Processes

it is clear that Pj is a measure on BJ. In addition we find since

Pj{EJ) = J (Ptm-tm^

o .. . o Pt2_tl o Ptl) ld/i = J ldM = 1 ,

that Pj is a probability measure. It remains to prove the projectivity of the family (Pj)Jen. For this let 0 ^ J C H C K + be such that # \ J consists of just one element. As pointed out before, the projectivity of (Pj)jen is proved if we have shown

itf(PH) = PJSet J = {ti,...,

tm} and H := J U {*'}. We have to prove for all C e BJ that

PH((nJI)-1(C))=Pj(C).

(3.45)

Further, since the system of sets B\ x . . . x Bm ,B\,..., Bm e 6, form a D-stable generator of BJ, it is sufficient to prove (3.45) for all C = B1 x . . . x Bm. First suppose for some i = 1,... ,m — 1 that U < t' < t i + 1 . It follows that ^((^)"1(Q) = PH(B\ x ... x Bi x E x Bi+i =

/ / • • • / / JEJBi+1 JEJB!

•••/

JBm

x ... x Bm)

Ptm-tm-i(zm-i,d:rm)

• • • pu+x-t'{x',&Xi+i)pt'-u{xiAx')

••

•Pt1(xo,dxi)ij,(dxo)

/ •••/ / f{xi+i)pti+1-t'(x',dxi+i)pt,-ti(xudx') JE JB-, JE JBi+1 • • • ptl{xo,dxi)(j,(dxo), where for i = m — 1 we have / = 1 and for i < m - 1 we have =

f(xi+i):=

Ptm-tm^(xm-i,dxm)

••• JBi+2 ••

JBm •Pti+2-ti+1{xi+i>dxi+2).

Clearly / is a non-negative function and the semigroup property yields / JE

=

/

f{xi+i)pti+1-t>(x',dxi+i)pt>-ti(xi,dx')

t/Bj_j_i

Pt'-tioPti+1-t'(XBi+1f)(xi)

= Pu+i-ti(XBi+1f){Xi) = / f{xi+i)pti+1~u{xudxi+i). JBi+1

3.2

Semigroups of Kernels, Transition Functions and Canonical Processes

59

Hence we find PH{(^Y\C)) = \

\

f{Xi+i)pu+x-ti{xi,Axi+i)pti-ti^{xi-^^xi)

•••

JE JBX

JBi+1

• •• M(dzo)

=

=

/

/

JE JBI

••• / JBm

Ptm-tm-i(xm-i,dxm)---Pt1(xo,dxi)lJ'(&co)

PJ(C).

Now, if t' < ti we may argue in the same manner, and the case tm < f is trivial since JE pt> -tm (xm, da;') = 1. •

Corollary 3.2.12. In the situation of Theorem 3.2.10, if E is a Polish space and B its Borel a-field, then the family (Pj)jen form the family of finitedimensional distributions of a canonical process with state space E. The proof of Theorem 3.2.10 requires a Markovian semigroup it will not work for a sub-Markovian semigroup. But we know for example convolution semigroups that are sub-Markovian but not Markovian and these convolution semigroups lead to semigroups of sub-Markovian kernels not being Markovian. If (/x)t>o is charaterized by the continuous negative definite function tp and V'(O) > 0, then the corresponding semigroup of kernels defined by (3.41) is a semigroup of sub-Markovian kernels which is not Markovian. This follows immediately from /x t (R n )=e- t i / ' ( 0 ) . However, there is a method which still allows us to construct a canonical process associated with a semigroup of sub-Markovian kernels (pt)t>o- For this we have to extend (E,B), the state space, by a point "A" not belonging to E. For our purposes it is sufficient to consider the case where E is a locally compact Polish space and the extension is the one-point compactification E& of E, A being a point not belonging to E (and being an isolated point if E is compact). On E& we consider of course the Borel cr-field $A- NOW we extend a

60

Chapter 3 Feller Processes

kernel pt from (E, B) into itself to a kernel from (EA, BA) into itself by defining

for x G E and AG B

Pt(x,A) p[A)(x,A) = \1-pt{x>E) 0 1

^xeE&ndA={A} for x = A and A e S for a; = A and A = {A}.

(3.46)

If we denote by P t ' A ' the operator (P/V)(a:):= /

uA(y)p{tA\x,dy)

defined on Bb(EA) we find that (3.46)

where as usual P t denotes the operator associated with pt. Further we have P/A)1A=1

,

1A:EA-*R IHI.

Moreover, if (pt)t>o is a semigroup of sub-Markovian kernels on (E,B), then (Pt )t>o is a semigroup of Markovian kernels on (EA,BA). Thus we have Corollary 3.2.13. Let {E,B) be a locally compact Polish space with AlexanFurther let (j>t)t>o be a semigroup of subdroff compactification (EA,BA). Markovian kernels from (E, B) into itself. Then for every initial distribution fi on (EA,BA) there is a canonical process with state space (EA,BA). In addition for the semigroup of operators associated with (p\ )t>o relation (3.47) holds. In particular, with every Feller semigroup on Coo(K") and a given initial distribution we can associate a canonical process with state space (M.A,B(M.A)}. Remark 3.2.14. We will come back to this corollary at several occasions when studying concrete Lp-sub-Markovian or Feller semigroups and the corresponding processes. Especially we will find a reasonable interpretation of the state "A" for the process. Now, let (E,B) be a Polish space with Borel a-field B and (pt)t>o a semigroup of Markovian kernels. Then the family (Pj)j^n constructed in Theorem

3.2 Semigroups of Kernels, Transition Functions and Canonical Processes

61

3.2.10 depends on /i. The corresponding canonical process should therefore be denoted by

(n,AP^,(xt),>0), where fi = ER+, A = BR+, Xt = n[ty and PM is the probability measure on A accordingly to Theorem 3.2.1 using the projective family (Pj)jen- For J = {ii,..., i n }, 0 < t\ < ... < tm, we know that Pj = P"-Vert(X tl ® • • • ® Xtm)

(3.48)

and (3.44) implies for Bj £ B P"{XtleBi,...,XtraGBm} f /• /• Ptm-tm-1(xm-1,dxm)---ptl(xo,dxi)n(dxo). = / / ••• / JE JB-i.

(3.49)

JBm

In particular, for fj, = ex, x G £?, we find -P^l-Xti £ S i , . . . , X t m s Bm} := P£x{Xt1 = JBI

••• / JBm

£ Bi,...

,Xtm £ Bm}

Ptm-tm-1(Xm-l,dXm)---Ptl(x,dx1),

which gives the interesting formula P"{ItleBlr..,ltm£Bm} r

=

Jpx{xtleB1,...,xtmeBm}(i(dx).

(3.50)

Thus for B G A we have

P"(B) = Jp*(B)n(dx),

(3.51)

and in particular P a: {X t G J B}=p t (a:,B)

(3.52)

as well as P»{Xt G B) = y ftOc, B)M(do:),

(3.53)

62

Chapter 3 Feller Processes

respectively. Taking now the expectation with respect to P M we get for u 6

Bb(E)

E»(u o Xt) = f(Ptu)(x)n(dx),

(3.54)

which is trivial for simple functions and then follows by approximation. Especially for /i = ex we have

E*(u(Xt)) = Ju(y)pt(x,dy) = (Ptu)(x),

(3.55)

where Ex denotes the expectation with respect to Px = PEx. In particular, by (3.52) we find Pt(x,B)

= Ex(XB(Xt)).

(3.56)

For t = 0 it follows further P"{X0 GB}=

fpo(x, B)n(dx)

(3.57)

and if PQ = id, then we arrive at P"-Vert(X 0 ) = fJ~

(3-58)

For this reason we call /x the initial distribution of the process. Let us summarize where we are: Given a (normal) semigroup of Markovian kernels (pt)t>o and an initial distribution /x on a Polish space E. Then there exists a canonical stochastic process (Q.,A,PIJ',(Xt)t>o) with state space (E,B),B being the Borel Q)xeE

(3.59)

of canonical stochastic processes and this family is related to the semigroup of operators {Pt)t>o associated with {pt)t>o by (3.55). It is in fact this family of canonical processes we are interested in to study, not a single canonical process. In particular, if {Tt)t>o is a Feller semigroup on Coo(K") and Ttl = 1 for all t > 0, then for every initial distribution /x we may associate a canonical process with (Tt)t>o- Moreover, in the sub-Markovian case we may construct a canonical process on ( R ^ , B ^ ) .

3.2 Semigroups of Kernels, Transition Functions and Canonical Processes

63

In case of a convolution semigroup (^t)t>o on M" we assumed a continuity condition, namely that /zt —> eo vaguely as t —> 0, compare Definition 1.3.6.1. This vague continuity at 0 which was proved to imply the weak continuity of i > ^ t ) see Lemma 1.3.6.3, was essential to prove that the family of operat — tors u i-> fit * u form a Feller semigroup, i.e. it is strongly continuous when considered in Coo(R";IR), compare Example 1.4.13. A careful analysis of Example 1.4.13 shows that a certain uniformity with respect to x of properties of Pt(x, B) :— fit(B-x) makes the proof of the strong continuity of the semigroup Ttu = /i t * u in C 0o (R") working, compare especially (1.4.7). In the general case we need a bit more care. For t > 0 fixed, in order that for a continuous function u G Coo (E), E a locally compact space, the function x *-> (Ptu)(x) = f u(y)pt{x,dy) is continuous too (and vanishes at infinity), we i > pt(x, A), and in order to guarantee (a type should have some regularity for x — of) strong continuity with respect to the sup-norm for the family (Pt)t>o, we will need the existence of \impt(x, A) in some sense. But we should expect the t—o strong continuity of (Pt)t>o to hold in general only for continuous functions. In fact lim ||P t « - ull^, = 0

(3.60)

means of course that u is uniformly approximable by the family (Ptu)t>o, and if these functions are continuous then of course u must be. Thus for example for a strong Feller semigroup we should expect (3.60) to hold only for continuous functions u, not for arbitrary functions in Bf>. Further we should be aware on the need for an extra condition to characterize strong Feller semigroups, i.e. if we want Pt to map Bb into C&. Lemma 3.2.15. Let E be a locally compact space and B its Borel a-field. Further let k be a sub-Markovian kernel from (E, B) into itself. Suppose that for every open set A G B the mapping x — i > k(x, A) is lower semicontinuous i > k(x, C) is upper semiand that for every compact set C £ B the mapping x — continuous. Then the operator K associated with k maps bounded continuous functions into bounded continuous functions. If in addition for every compact set C it hold lim k(x,C) = 0, then K maps C^E) onto Coo(E). X—>00

Proof. For u G Cb{E) such that 0 < u < 1 and i / e N w e set 2"

T

uv := 2~u ^2 X{u>l2->} and vv := Tv ^ i=i

i=i

X{v>n-"}

64

Chapter 3 Feller Processes

to find

0 < u - uv < Tv and 0 < u - vv < 2~u which implies lim \\Ku - KuvW^ = lim \\Ku - Kv^ V—>OO

= 0.

(3.61)

V—>OO

Since each Kuu is lower semicontinuous and each Kvv is upper semicontinuous it follows that Ku is continuous. If u £ Cb(E),u > 0 but u(x0) > 1 for some xo we may consider the function w := „ ^ implying that Kw = K(,, ^ ) is continuous, hence Ku = WuW^Kw is continuous. For an arbitrary u G Cb(E) we use its decomposition into positive and negative part, and this proves the first part of the lemma. Now, if u € Coo(E) then lim vv{x) = 0 and further we find that lim Kvv(x) = 0 which of course implies by (3.61) that Ku £

too.

C^E)

•

Remark 3.2.16. Suppose that K is a positivity preserving contraction on i > k(x,A) satisfies CodE) with associated kernel k. Then it follows that x — the assumptions made in Lemma 3.2.15. This is seen easily by reworking the proof of Theorem 1.4.8.2. Thus Lemma 3.2.15 gives us an answer to the problem to guarantee that a semigroup of (sub-)Markovian kernels (pt)t>o is associated with operators Pt mapping continuous functions into continuous functions. Next we turn to the question of the continuity of the semigroup (Pt)t>o at 0. Lemma 3.2.17. Let (pt)t>o be a semigroup of sub-Markovian kernels from (E,B) into itself, E being a locally compact space with Borela-field B. Suppose further that for all A £ B and x £ A kmpt(x,A)

=l

(3.62)

no

holds. Then we have for all u £ Cb{E) and all x £ E ]imPtu{x) = u(x).

no

Conversely, if (3.63) hold then (3.62) follows.

(3.63)

3.2 Semigroups of Kernels, Transition Functions and Canonical Processes

65

Proof. For A £ B and x e Awe find |P t tx(a;)-u(i)| < / \u(y)-u(x)\pt(x,dy)

+ \u(x)\\pt(x,A) - 1| + /

|u(j/)|pt(x,dy)

< sup|w(y) - u(a:)| + 2||u||oo(l - p t (x, A)). 2/6.4

Now given e > 0, we may determine 6 > 0 such that

sup |u(y) — u(a:)| < e

y&Bs(x)

and condition (3.62) implies that 1 — pt(x,A) < e for t < to(e). Hence (3.63) is proved. In order to prove the converse let ip e Cb{E) and set A := {y 6 E; tp(y) > 0} which is clearly an open set. Define for x € A fl

\gg

,if|v(l/)l>l¥>(aOI

, if |^(3/)| < |^(x)|.

The function u belongs to Cb(E) and in addition we have u(x) - Ptu(x) — 1 - / u(y)pt{x,dy) > JE

l-pt(x,A).

Now, (3.63) implies (3.62) for A. But ranging over all

o at zero. This result is known as "weak equals strong" in semigroup theory. Our proof will be a sketch of the very polished proof of this result in K.-J. Engel and R. Nagel [97], p.40, and it covers a much more general situation. We need some preparation. Let X be a Banach space. Then it is known that the weak closure of a convex set equals its strong closure, compare W. Rudin [307], Theorem 3.12. Further, a result of M. Krein and V. Smulian states that the closed convex hull conv K of a weakly compact set K C X is weakly compact, compare N. Dunford and J.T. Schwartz [91], p.434. Finally, as it is proved in K.-J. Engel and R. Nagel [97], Proposition 1.5.3, a semigroup (Tt)t>0 on a Banach space is strongly continuous if and only if there exist S > 0 and M > 1, and a dense subset Y
for all t £ [0,8]

66

Chapter 3 Feller Processes

and limTiu = u

for all u &Y.

t|0

Theorem 3.2.18. Let (Tt)t>0 be a semigroup on a Banach space (X,\\-\\x). The strong continuity of (Tt)t>o(at 0) is equivalent to its weak continuity (at 0), i. e. the continuity of the mapping tv-Ku',

(3.64)

Ttu >

for all u G X and u>' G X*.

Proof. Of course we need only to prove that the weak continuity implies the strong continuity. By the uniform boundedness principle we may deduce from (3.64) that (T t ) t > 0 is pointwise and hence uniformly bounded on compact intervals I C M.+ . Therefore, by the remarks made preceeding the proof we need to show that Y := {ueX; is dense in (X, \\-\\x). ur : X* -> K (or C)

lim ||Tfu - u\\x = 0} For this, take u £ X and r > 0 and define ur on X* by

1 fr < w', ur > := - / < w', Tsu > ds, w' e X*. t Jo

Clearly, ur is bounded and linear, hence ur € X**. Since the set {Tsu; s G [0, r]} is the continuous image of [0, r] when taking in X the weak topology, it follows that this set is weakly compact. Now we apply the Krein-Smulian theorem stated above to deduce that conv {Tsu;s G [0,r]} is also weakly compact. Further note that ur is the limit of Riemann sums with respect to the weak-*-topology a(X**,X*) on X*. Hence it follows that ur G conv {Tsu; s € [0,r]}. Having in mind the definition of ur it is clear that Z :— {ur ; r > 0 and u G X}

3.2 Semigroups of Kernels, Transition Functions and Canonical Processes

67

is weakly dense in X. But for ur G Z we find llTttir-UrHjr 1 ft+r 1 /"r sup - / < w', Tsu > ds / < w', Tsu > ds |M| X .<1 r It r JO ( 1 ft+T 1 /"* l\ < sup - / <w' , Tsu >ds + < w' , Tsu > ds\ \\w\\x.
=

^

0<s
where \\-\\x x a s usual denotes the operator norm. The above estimate implies ||Tiu r — Urllx -^ 0 as i —» 0, hence Z CY, which yields that Y is weakly and therefore strongly dense (see again the remarks made above) and the theorem • is proved. Corollary 3.2.19. For a Feller semigroup (Tt)t>o with kernels (pt)t>o being all Markovian (3.62) holds. Finally we give a criterion for a (semigroup of) (sub-)Markovian kernel(s) in order that the corresponding (semigroup of) operator (s) maps bounded measurable functions into continuous functions. For this it is worth to recall some results for convolution operators. In Lemma 1.4.8.19 together with Lemma 1.4.8.20 it was shown that for u H-> U * \i to map J5b(K") into C&(IRn) it is necessary and sufficient that fi admits a density with respect to the Lebesgue measure A^™). Thus when longing for semigroups (Pt)t>o associated with sub-Markovian kernels (pt)t>o having the property that each pt maps Bb(E) into Cb(E) it is reasonable to assume that each pt(x,dy) has a density with respect to a fixed, i.e. t and x independent, reference measure. We will denote such a density by pt(x,y), in fact we will use the notations pt{x,dy),pt(x, (x,y) as well as p pt(x,A) if no confusion is possible. Suppose that on (E, B) a Borel measure \i is given and assume that Pt(x,dy)

=pt(x,y)ij,(dy)

and hence Pt is given by (Ptu)(x) = J

u(y)pt(x,y)fj,(dy).

68

Chapter 3 Feller Processes

Our problem is to find conditions on pt(x, y) which imply that Pt maps Bb(Mn) into Cb(M.n). Clearly, Lemma 1.2.3.22 gives an option if the constants x >-* c are integrable with respect to /z. But already in the case E = M.n,B = B^ and /J, = \(n^ this does not hold. We suppose that E is a locally compact metric space with Borel cr-field B. Further, the reference measure JJ, is assumed to be a regular Borel measure on B. In this case fi(C) < oo for every compact set C C E, and CQ{E) C Ll{E; /J.) is dense with respect to the norm ||-|| L i. Now we are in a position to prove a useful condition for a (Feller) semigroup to be a strong Feller semigroup. Lemma 3.2.20. Let E be a locally compact metric space with Borel a-field B and let \x be a regular Borel measure on B with full support. Further let (Pt)t>o be a semigroup of sub-Markovian kernels from {E,B) into itself and assume that the corresponding semigroup (Pt)t>o maps CQ(E) into Cb(E), i.e. Tt : Co(E) —> Cb(E) for each t > 0. / / in addition each kernel pt(x,dy) has a density with respect to n, i.e. Pt(x, dy) = pt(x, y)n(dy),

(3.65)

satisfying the estimate Pt{x,y)
,0
(3.66)

and if

lim

sup Pt(z,fl£(6>))=0

(3.67)

fl-+oox6Bs(x0)

holds for each fixed 5 > 0 where £o £ E is a fixed point, then the semigroup (Pt)t>o maps Bb(E) into Cb{E). Proof. The fact that TtBb{E) C Bb(E) is trivial. Fix t0 > 0 and u G Bb{E). We want to prove that for each XQ £ E the function

zi-> (Ttou)(x) := / u{y)pto(x,y)fi{dy) is continuous at XQ. For e > 0 and 5 > 0 choose R > 0 such that sup p t o (x, BCR(£O)) < 4 x€B6(x0) lMlco

3.2 Semigroups of Kernels, Transition Functions and Canonical Processes

69

and take R > 0 such that Bc~(x0) C B%(£0). Next define

U£(y) := XBaMivMy). It follows that \Ttau(x)-Ttou(x)\ = I u{y)Pto{x,y)iJ,(dy) - I u{y)pto{x0,y)fj,(dy) ^ / uii(y)Pto(x,y)n(dy) - / + IMIoo(Pto(z,-B|(zo))

u^(y)pto(xo,y)n(dy)

+pto(x0,Bc~(x0)).

For a; G Bs(x0) it follows by (3.67) that \Hoo(Pto(x,Bc~(x0))+Pt0(xo,Bcn(x0)) h\L(Pto(x,BcR(Z0))+pto(xo,BcR(Z0)))

<

Next we prove that x^

I

un(y)pt(x,y)n(dy)

is continuous. Since uR 6 L^EjS,//) and CQ{E) is dense in Ll{E,B,ii) we may approximate uR by a sequence (
IJ

^

- /
\uR.(y) -
< cto I \ujiiy) -
70

Chapter 3 Feller Processes

i.e. x H-> /

uR{y)pto(x,y)ii(dy)

is the uniform limit of continuous functions and therefore continuous. Thus, if s > 0 is given, we may find 6 > 0 such that / u^{y)ptQ(x,y)n{dy)

- I uR(y)pto(xo,y)lJ.(dy)

< |

for all x € B$(xo) implying the lemma.

•

Remark 3.2.21. A. We adapted the proof of Lemma 3.2.20 from the proof of a similar result of W. Hoh [155], Theorem 8.9, and we will come more closer to his results later on. B. The condition (3.66) should be seen in the light of the considerations in Section II.3.6, especially formula (II.3.436). Thus in many situations we are interested if condition (3.66) is fulfilled. Let us explore the meaning of (3.62) for the corresponding canonical process ((Xt)t>o, Px)xeEProm the construction it is clear that (3.68)

PZt(dy)=Pt(x,dy)

and therefore if ((Xt)t>0, Px) E is related to a Feller semigroup (Xt)t>o by then for all u e C^E) Tt = Pt\Cx{E), \Pt+,u[x) - Ptu(x)\ = \Pt(Psu(x)

- u(x))\ -» 0 as s -* 0,

i.e. P £ t ^ —> Pxt vaguely (and when all pt(x,A) weakly) as s —> 0. But we are longing for more, namely for \imPx{\Xt+s-Xt\>£}

s—0

are Markovian kernels even

= 0,

(3.69)

that is the stochastic continuity of the process. For t = 0we have indeed Px{\Xt - x\ > e] =

{TsXBHx))(x)=Ps{x,Bce(x))

and (3.62) implies (3.69). In order to prove the general case we need a deeper understanding of the process ({Xt)t>o,Px)xeE-

3.3 A First Encounter with Sample Paths and Cadlag-Functions

71

The stochastic continuity of the process will play a major role when studying path properties, i.e. (3.69) is not just a further nice property, it will become critical. In preparing further continuity results for processes we supply an additional result for Feller semigroups, compare for example K. L. Chung [75], p.49. Lemma 3.2.22. Let (Tt)t>o be a Feller semigroup on C0O(E), where E is a locally compact metric space. Then the mapping (x, t, u) H-> Ttu(x) is continuous from E xR+ x C^E) into R. Proof. Let t £ K+, s > 0, x, y € E and tt,v£ C^E). It follows that \Tt+su(x) - Ttv{y)\ < \Tt+su(x) - Tt+Su(v)\ + \Tt+su(y) - Ttu(y)\ + \Ttu(y) - Ttv(y)\. Now, for x —> y it follows that Tt+Su(x) —> Tt+Su(y). Further, since Tt+tu{v) - Ttu(y) = Tt(Tsu{y) - u{y)) and \\Tt\\ < oo we conclude that Tt+Su(y) —* Ttu(y) as s —» 0. Finally, observe \Ttu(y)-Ttv(y)\<\\Tt\\\\u-v\\oo implying that Ttu(y) —* Ttv(y) as u —> v in Coo(E). The standard modification • for the case s < 0 but £ + s > 0 yields now the lemma.

3.3

A First Encounter with Sample Paths and Cadlag-Functions

Let (Q,A,P, (Xt)t>o) be a stochastic process with Polish state space (E,B). For each u> 6 Q, we can consider the mapping X.(u) : M+ -> £ t«Xt(u)

(3.70)

which is called a path or sample path of the process. If (fi, A, P, (Xt)t>o) is the canonical process associated with a semigroup of Markovian kernels and D, = i?!0-00) and Xt(w) = w(£), then P is a probability measure on the space of all mappings from [0, oo) to E and every such a mapping is a path of the canonical process, i.e. P is a probability measure on the

72

Chapter 3 Feller Processes

space of all sample paths! However, the examples of stochastic processes arising in modelling suggest that the paths of a given interesting process are typically more specific: Brownian motion should have at least continuous paths, the paths of a Poisson process should be piecewise constant with positive jumps, etc Thus we should expect that P is concentrated on a certain subset of _g[o,oo) determined by the process. In fact, we shall aim to find a subset of fit0-00) such that the typical path of (X t ) t > 0 belongs P—almost surely to this subset. In a next step we even might be interested to substitute Cl = E^0'00^ for given process by the set of all typical paths. However the finite dimensional distributions of a given process should not change after such a substitution. This leads us to some important definitions. Definition 3.3.1. A. Let (il, A, P, (Xt)t>0) and (Q',A',P', (Xt')t>o) be two stochastic processes with the same state space (E, B) and the same family of finite dimensional distributions. Then we call (Xt)t>o a version of (X't)t>oB. Suppose that (Q, A, P, (Xt)t>o) and (ft, A, P, (lt)t>o) a r e two stochastic processes with state space (E,B). If for all t > 0 P(Xt = Yt) = l

(3.71)

holds the we call (Xt)t>o a modification of (Yt)t>oC. Two processes (Cl, A, P, (Xt)t>o) and (ft, A, P, (Yt)t>o) with state space (E, B) are called indistinguishable if there is a P-null-set N C fi such that Xt(u>) = Yt(u>) holds for all ui e Nc and all t > 0. If (Xt)t>o is a stochastic process with Polish state space (E, B) then we may ask whether a path of (Xt)t>o has certain regularity properties, i.e. t —> Xt(uj) could be continuous, or right continuous, have left limits, be of bounded variation etc. Of course we are interested in the case where almost all paths of a given process have such a property. Definition 3.3.2. Let (Xt)t>o be a stochastic process with Polish space as state space. A. The process (X t )t>o is called (almost surely) continuous if its paths are (almost surely) continuous. B. The process (Xt)t>o is called (almost surely) right continuous if (almost surely) its paths are right continuous. C. We call (Xt)t>o a cadlag process if almost surely all paths of (Xt)t>o are continuous from the right and have limits from the left. As a first result let us prove

3.3 A First Encounter with Sample Paths and Cadlag-Functions

73

Theorem 3.3.3. Let (ft, A, P, (Xt)t>o) and (ft, A, P, (Yt)t>0) be modifications of each other and suppose that they are both almost surely right continuous. Then they are indistinguishable. Proof. Let jVx be a null set such that on N^ the paths of (Xt)t>o are right continuous and let Ny be a null set such that on NY the paths of (Yt)t>0 are right continuous. Put Nt := {u> £ ft; Xt(u) ^ Yt(uj)} and set N = UteQ+ ^*By assumption N is a null-set and so is M := Nx U Ny U N. Thus it holds Xt{w) = Yt{w) for all t G Q+ and w G Mc. For t G [0, oo) n Q c take a sequence (*n)n£N, tn G Q+, such that tn [ t. Since for w G Mc we have Xtn{ui) = Ytri(cu) for all n G N the right continuity yields Xt{w) = lim Xtn(w) = lim Ytn(w) n—KX>

n—»oo

=Yt(u)

for these u> and the theorem is proved.

•

It will turn out that all the processes we are interested in are cadlag processes. Therefore we need to investigate the space of all cadlag functions in detail. For our purposes it is sufficient to restrict ourselves to the case E = K n . We start with Definition 3.3.4. A function u : [0, oo) —> Mn is called a cadlag function if for all to G [0, oo) the function is continuous from the right and has for all to G (0, oo) left limits. The space of all cadlag functions is denoted by D n ([0,oo)). Lemma 3.3.5. Let u : [0, oo) —> W1 be a function for which

u(t+) := limu(s),t > 0, and u(t—) := limu(s),i > 0, sit

sTt

exist. Then there exists a countable set T C (0, oo) such that for all t G F c it holds u(t-) = u(t) = u(t+). Further, for each finite T > 0 the set F n D [0, T] is finite where

Tn := {t G (0, oo); maxflu(t-) - u{t)\, \u{t-) - u{t+)\, \u{t) - u(t+)\) > i } . Proof. Since F = \Jn&n Tn it is sufficient to prove that F n D [0, T] is for each T > 0 finite. Suppose that F n n [0,T] has a limit point t. Then either u(t+) or u(t—) would not exist implying the finiteness of F n n [0, T]. O

74

Chapter 3 Feller Processes

Lemma 3.3.6. Let I C [0, oo) be a dense set and u : I —» Mn be a function. Suppose that for each t > 0 (3.72)

v(t) :=limu(t) sit s€/

exists. Then v is on [0, oo) continuous from the right. If for each t > 0

v~(t) :=limu(s)

(3.73)

sit s€l

exists then v~ is continuous from the left on (0, oo). If for all t > 0 both v and v~ exist then it holds v~(t) = v(t—). Proof. Suppose that (3.72) exists for all t > 0. Fix to > 0 and given e > 0 then there exists 6 > 0 such that

\v(t0) - u(r)\ <e for all r Gill (t0, t0 + S) and therefore \v(t0) - v(s)\ = lim \v(t0) - u(r)\ < e r[s rel

for all s G (to, t0+ 5) and the right continuity of v is proved. The left continuity of v~ is proved analogously. To see the last statement, recall that \v~(to) — v(to—)\ = |limu(s) - lim (limu(r))| r€/

sel

= | lim u(s) - lim(limu(r))| sfto sel

sTto rls sei rei

=0 since for s £ I we have u(s) — limu(r).

•

rei

Definition 3.3.7. A. A function u : [0,oo) —> M" is said to have a jump discontinuity at t 0 £ [0, oo) if u(t—) and u(t+) exist and u(t+) ^ u ( i - ) . B. We say that u : [0, oo) —> R" has no discontinuity of second kind in [Ti, T2] C [0,00) if for all t € [Ti,T 2 ] the function u has right limits and for all t £ ( T i , ^ ] its

3.3 A First Encounter with Sample Paths and Cadlag-Functions

75

left limits exist. C. The function u : [0, oo) —> M™ is said to have at least m G NU {oo} e-oscillations, e > 0, in [Ti,T2\ C [0,oo) if there are points

Ti
< ...
such that

|u(t f c _i)-«(t f c )|>e

(3.74)

holds for k — 1,... , m. Lemma 3.3.8. A function u : [0, oo) —> R has no discontinuities of second kind in [Ti,T2] C [0,oo) if and only if for every e > 0 the number of its e-oscillations is finite on [T\,T2\Proof. Suppose first that the number of e-oscillations in [7\, T2] is infinite for some e > 0. Then there exists to 6 P i , T2] and a sequence ( i ^ e N j tv G p i , T2], > e. This implies such that either tv j. to or tv j to and \u{tv — u(tv-i)\ that either u(t0—) or u(to+) will not exist. Conversely, suppose without loss of generality that u(to-),to € (Ti,!^] does not exist. Then there exists a sequence (tv)u&i,tv ] to,tu G [Ti,T2] such that sup |u(tM) — u(tv)\ > e implying that there are infinitely many e-oscillations.

•

Corollary 3.3.9. Let u € Dn([0,00]). Then on every compact interval I C [0,00) the function is bounded, i.e. \u(t)\ < M for all t S / , and u has at most finitely many jump discontinuities with jumps larger than e > 0. In particular u has at most countable many jump discontinuities of size larger than e > 0 on [0,oo). From now on we denote typical elements of D n ([0,oo)) by u> since later on D n ([0,00)) will become the probability space we consider the stochastic processes on we are going to construct. On J9 n ([0,oo)) we need a topology. The cr-field generated by this topology should make all projections Xt : Ai([0,00)) —> W1 , w —> Xt(u>) — u>(t) measurable, but in addition it should be a separable (preferable metric) topology since measure theoretical considerations require to involve only countable many operations. Such a topology was introduced by A. V. Skorohod [334], see also [335]. Let us try to understand what we should require from such a topology. First of all, limits of cadlag functions should be cadlag functions. Thus locally (at least) the convergence should be of uniform type. However, even intuitively,

76

Chapter 3 Feller Processes

the sup-norm will not satisfy our requirements. For this consider the following example. Let u := X[o,2)c £ -Di ([0, oo)) and have a look at the two sequences Uv = X[o,2+i)c a n ( i ^i2' = X[o,2-i)c e ^i([0,oo)). For large v both, Jp and wl should be seen to be "near" to w. But clearly for all v we have sup \cj(t) - 4^(t)\ = sup \w(t)-tjW(t)\ = l. te[l,3]

t€[l,3]

But if we run u>l and wj, with different velocities than w, i.e. if we consider t ^ uP (riP(t)) and t >-> wi 2 ) (^ 2 ) (t)) where ^ ( i ) = ^ t and ffc (t) = gyHi^i respectively, then we have in fact o> = u>£, o ?^,' = w j oi/i and as f —> oo we find that rj\}'(t) —> £ as well as r]l2'(t) —»t. Of course, when w has several jump discontinuities or wi,^ is a more general cadlag function we can not find a time change as above. But the example shows us a way how to construct an appropriate metric: For wi,o»2 S -Dn([0,oo)) consider locally the s u p - n o r m of \u>i(t) — w2{r]{t))\ where \r](t) — t\ is small. (Since elements of D n ([0,oo)) are in general unbounded, it should be in addition helpful to make the metric finite by taking a sup with 1 — but this is technical only.) A Skorohod metric on Dn(j0,oo)) is constructed along this line by following the presentation St. Ethier and Th. Kurtz [98]. We say that r) : [0, oo) -> [0,oo) belongs to the class L if rj is a strictly increasing surjective Lipschitz continuous function such that

|M|L:= sup l o g ^ - f 0 s>t>0


(3.75)

S—t

holds. Since r\ is a Lipschitz function rj'(t) exists almost everywhere, hence ||rj|| L is indeed finite. In addition for // e L it follows that 77(0) = 0, lim r](t) = 00 t->o and ||T7||L = H??"1!^- For wi,w 2 £ £>n([0,00)) and rj e L we set g(u!i,iO2,ri,u)

: = s u p | w i ( £ A w ) — u2(r)(t) Au)\ A l , i t > 0,

(3.76)

t>o and define

(

(

f°°

\\

dD(cJi,uj2) := inf max | M | L , / e u£»(u;i,ct;2,?7,u)du . (3.77) r)GL V V Jo )} In order to prove that do is a metric we need some preparations. For two sequences (u;^)i/eN and (w£2)),,eN in Dn([0,00)) it holds lim <1D{UV ,Uv ) = 0 V—>OO

3.3 A First Encounter with Sample Paths and Cadlag-Functions

77

if and only if there exists a sequence (rju)U£N, "HV € L, such that lim \\rjv \\L = 0, and for all e > 0 and uo > 0

V—>OO

lim A ( 1 ' { n e [ 0 , « ( ) ] ; J ( 4 1 ) 1 u ( 3 » ^ , U ) > 6 } = 0

(3.78)

holds which follows just from (3.76) and (3.77). In addition, since for rj e L esssupi > O\r/(t) - 1| < 1 - e^"*- < \\r)\\L,

(3.79)

we deduce from lim | M | L = 0

(3.80)

V—>OO

that for all T > 0 lim sup \qv{t)-t\

i/-»ooO
=0.

(3.81)

Now we may prove Lemma 3.3.10. By dp a metric, called the Skorohod metric, is given on Dn([0,oo)). Proof. For wi,o;2 £ ^ ( [ 0 , oo)) it follows that for all rj € L and u > 0 sup |u;i(£ A M ) - u>2 (??(£) A u)\ A 1 t>o = sup|u;i(77~1(t) A M ) - W 2 (^ A u)| A 1, t>o i.e. Q{uj\,uj2,r],u) = ^ ( w i , ^ , ^ 1 , ? * ) which together with \\r)\\L — \\f]~1\iiL implies that dD{w\,wi) = di?(w2,wi). Next suppose that 2- Thus we have proved that dD(wi,ui2) = 0 implies u>i = u>2, clearly u>\ = 0J2 implies that do{^1,1^2) = 0. Finally we prove the triangle inequality for do- For this let Wi,a>2,3 € Dn([0,oo)),r}i,r)2 € L and

78

Chapter 3 Feller Processes

u > 0. It follows that sup|wi(t Au) - wz{r)2{m{t)) A 1 ")| A l t>o < sup \u>i(t A u) — a>2(?7i(£) A M)| A 1 t>o + sup IW2 (»7i (*) A u) - w 3 (% (m (*)) A u) I A 1 t>o

v

C3 8 2 ) •

;

= sup \w\(t AM) - W2(»7i(t) AM)| A 1 t>o + sup |a>2(i A u ) - w3(r?2(i) A M)| A 1, t>o i.e. e(wi,w3,772O77i,M) < e(wi,u;2,»7i. u ) + e(w2,W3,»72,")-

However % ° ^?i € i and

U^oTnll^lML + hilL

(3.83)

and we obtain dp(^1,^3) < dr>(wi, W2) + d(tJ2-u>3) proving the lemma.

D

Definition 3.3.11. The topology induced on 2?n([0,00)) by do is called the Skorohod topology and (D n ([0,00)),dp) is called the Skorohod space. Theorem 3.3.12. The Skorohod space (Z)n([0, 00), dp) is a complete separable metric space and its topology is weaker than the locally uniform topology. Moreover the projections Xt : Dn(j0,00)) —> K " , u >-> X t (w) = w(i) are measurable with respect to the Borel a-field A — Aon on D n ([0,oo)).

This theorem as important as it is for our purposes has a rather lengthy, technical proof which is for example provided in St. Ethier and Th. Kurtz [98], Chapter 3, but this proof gives almost no new insights. Therefore we omit the proof and refer to [98]. In [206] J. Jacod and A. Shiryaev discussed in a long Chapter, Chapter VI, the Skorohod topology (with a slightly different but equivalent definition of do) and provided all proofs. After stating the main theorem they made the following comments: "... we strongly advise the reader to skip the proof, which sheds very little light on the significance of this topology." However, we want to provide two criterions for convergence in £>n([0,00)) once again taken from [98].

3.3 A First Encounter with Sample Paths and Cadlag-Functions Proposition 3.3.13. Let ( W ^ ) ^ M LJ G -Dn([0, oo)). Equivalent to

79

be a sequence in J9 n ([0,oo)) and

lim dD{ujv,w) = 0

(3.84)

I/—>OO

is t/ie existence of a sequence (fji/Ji/gm,^ € L, such that lim ||r?v||L = 0 and lim p{ujv,uj,r)v,u) = 0 /or aH u G Cu,

(3.85)

where Cu denotes the points of continuity of UJ . Proof. The sufficiency of (3.85) for (3.84) to hold is a consequence of Corollary 3.3.9. Now suppose that (3.84) holds and let u G Cu. Prom (3.78) we deduce that there is a sequence (77,,),,6N,??I/ G L, and (u^gp^Uj, £ (u, oo) such that lim H^ll^ = 0 and V—>OO

lim sup \wv(t A Uv) - oj(rjv(t) A u)\ A 1 = 0.

v—»oo t > 0

(3.86)

Observe that sup \jJv(f A u) — oj(r\v(f) A u)\ A 1 t>o < sup \wv{t A u) — cj(r]v(t A u) A uv)\ A 1 t>o + sup \u)(r)v(t A u) A uu) - u{r]u{t) A u)| A 1 t>o

< sup \wv{t Auv) — u(r]u(t) Auv)\ A 1 0
(3.87)

M<S<77^(«)Vli

V(

sup

|w(t/v(u)-«,,)-w(s)|M)

for each i^ e N. Note that the second term in the last estimate is obtained by considering the two cases t < u and t > u separately. But now (3.86) and = 0. • (3.81) and the continuity of w at u yield lim p^v,^,^^) v—too

Proposition 3.3.14. Let (iv,,),,^ be a sequence in Dn(j0,oo)) UJ G .D n ([0,oo)). Then {uJu)uen converges to UJ, i.e. lim dD{uv,u) V —»OO

= 0,

and (3.88)

80

Chapter 3 Feller Processes

if and only if there is a sequence (r)v)veH, r\v G L, such that lim ||7?^||L = 0 and lim sup \uv{t) - u(r]u(t))\ = 0.

v—»oo

(3.89)

u—>ooO
Proof. If (3.88) holds then there exists a sequence (rfv)v^,r]v £ £, and a sequence (uv)V£n, "»- G (0, oo) such that lim \\r}v 11^ = 0 and p(uju, u), r}u, uv) —> 0 K—>0O

as v —> oo. In particular it follows that

lim sup \u}v{t A uv) — uj{r}u{t) A uv)\ = 0.

(3.90)

V—>CXD t > 0

Given T > 0, we find for v sufficiently large that uv > TVrjv{T) and therefore (3.90) implies the second statement of (3.89). Conversely, if (3.89) holds then by (3.87) for every point u € Cu it follows that lim sup \uv{t A M ) - u}(Vv(t) A u)\ A 1 = 0

v—>oo t > 0

for u,, > ??y(w) Vu,i/eN, which implies (3.88).

(3.91)

•

Remark 3.3.15. Note that (3.90) tells that at points of continuity of LJ the sequences converging to u> in Dn(j0, oo)) converge also pointwise. It is not difficult to see that the results we proved for cadlag functions CJ : [0, oo) —> W1 will hold when W1 is substituted by a suitable metric space E, compare [98]. In particular we may take the one-point-compactification R^ ofM". When discussing in Chapter 4 the martingale problem for pseudo-differential operators with negative definite symbols we have to continue our discussion of (.Dn([0,oo)),dp)- Especially we will examine compact subsets and weak convergence of measures on Aon the Borel cr-field generated by P n ([0, oo)). But before, we want to show that the paths of the canonical process associated with a Feller semigroup are almost surely cadlag paths. For this we need a deeper understanding of these processes, especially we have to discuss the Markov property.

3.4

Markov Processes and Feller Processes

We want to study how the semigroup property of a semigroup of (sub-) Markovian kernels is reflected in properties of the corresponding canonical

3.4 Markov Processes and Feller Processes

81

process. For the following however we do not need to work with a canonical process, thus we start with a stochastic process (Xt)t>o on a probability space (SI, A, P) with an arbitrary (sometimes Polish) state space (E, B). In addition we assume that a filtration (Tt)t>o in A is given and that (Xt)t>o is adapted. Definition 3.4.1. Let (Xt)t>o be a stochastic process on (Cl,A, P) with state space (E,B) and suppose that [Xt)t>o is adapted to the filtration (Tt)t>o- We call (Xt)t>o a Markov process with respect to (J~t)t>o if P(Xt e B\Ta) = P(Xt e B\Xa) P-a.s.

(3.92)

for all 0 < s < t and all B e B . Remark 3.4.2. A. Property (3.92) is often called the elementary Markov property. B. Obviously (3.92) is equivalent to E(X{Xt{»)£B}\fs) = E(x{Xt(.U)€B}W(Xl)) [= E(x{xtlu)eB}\Xt)}.

(3.93)

{

To proceed further we need Lemma 3.4.3. Let {Xt)t>o be a stochastic process as in Definition 3.4.1. Then (Xt)t>o is a Markov process with respect to the canonical filtration i^rt)t>o if and only if for every B e B and any finite choice of numbers 0 < s\ < ... < sn < t we have P(Xt e B\a(XSl,.. .,XSn)) = P(Xt e B\XSn).

(3.94)

Proof. Suppose that (Xt)t>0 has the elementary Markov property with respect to (J^)t>o. For B e B and 0 < si < ... < sn < t we find P(Xt e B\F°Sn) = P(Xt e B\XSn) P-a.s. or with A := {u> € H; Xt{u) e B} we have E(XA\^n)

= E(XA\XSJ

P-z.s.

Further it follows that E{E(XA\^n)\o-(XSl,...,XsJ)

=

E{E(XA\XSn)\a(XSl,...,Xan)) (3.95)

82

Chapter 3 Feller Processes

and using the properties of conditional expectations we arrive at E(E(XA\^n)HXSl,..

.,XaJ)

= E(XA\
,...,

XaJ)

as well as E(E(XA\X.J\*(XSI,...,X.J)=E(XA\X.J

which yields P(Xt G

B\
. .,X.J)

= P(Xt G B\XSn),

(3.96)

i.e. (3.94). Conversely suppose that (3.94) holds. We have to prove that P(Xt G B\T°S) = P(Xt G B\X.) P-a.s. for all B G B and 0 < s < t. This is equivalent to show that a version Y of P(Xt G B\XS) is also a version of P(Xt G B\T°S). Clearly F is cr(Xs) measurable, hence ^ - m e a s u r a b l e and therefore we have to prove that / YAP = P({Xt EB}DQ)

(3.97)

JQ

holds for all Q G J*. Prom (3.94) it follows that (3.97) holds for all Q G cr{XSl,..., XSn) with any choice of (finitely many) 0 < s\ < ... < sn. The family £ of all these sets forms a D-stable generator of J7® and fi G £. We may interpret (3.97) as an equality of finite measures on A which on £ coincide, i.e. they have to coincide on cr(£) = J7® which proves the second part • of the lemma. Now we can prove that canonical processes have the elementary Markov property. be a stochastic process with Polish Theorem 3.4.4. Let (Q,A,P^,(Xt)t>o) state space (E, B) and finite dimensional distributions deriving from a normal semigroup of Markovian kernels {pt(','))t>0 o,nd initial distribution /i. Then (Xt)t>o is a Markov process with respect to the canonical filtration {^)t>oIn addition for B G B and 0 < s < t P(Xt G B\7?) = Pt-s{Xs,B) holds where pt~s(Xs,B) LJ^Pt_s(Xs(uj),B).

P-a.s.

(3.98)

is the random variable (3.99)

3.4 Markov Processes and Feller Processes

83

Proof. Clearly, pt-a(Xa,B) is always cr(Xs )-measurable, hence it is J^-measurable. Therefore it remains to prove that for all Q e f s ° / pt-s(XsH,B)dP = P({Xt sB}nQ)

(3.100)

JQ

holds. As in Lemma 3.4.3 it is sufficient to prove (3.100) for the fl-stable generator consisting of all Q € a(XSl,... ,XSn),0 < s\ < ... < sn < s. In addition, by Lemma 3.4.3, we only have to prove P{Xt

€ B\a(XSl,...,XSn))=

Pt.Sn(XSn,B)

P-a.s.

(3.101)

for 0 < S\ < ... < sn < t, s = sn. Once (3.101) is proved we arrive at P{Xt€B\X,n)=pt-,JX.n,B) and further for 0 < s < t P(Xt£B\J*)=pt_s(Xs,B), i.e. the elementary Markov property. Since the random variable Y := Pt-sn (XSn, B) is cr(XSl,..., XSn)-measurable we have to show / YdP - P({Xt £B}DQ)

(3.102)

JQ

for Q £ a(XSl,... ,XSn). Without loss of generality we may take Q = {XSl G J3i} n . . . n {XSn e Bn} where Bj G B since these sets form a Pi-stable generator of cr(XSl,..., XSn). Now we find

I YdP = J{XQY)dP = J

XB^XI)

= J{XB, oXSl)--•••

(XBn o XSn)YdP

XBn(xn)pt-3n{xn,B)Pj{dx),

where J = {s\,..., sn} and dx = d ( x l 5 . . . , xn). Here Pj is the joint distribution of the random variables XSl,..., XSn. Now we use the fact that the finite dimensional distributions of the process originate from a Markovian semigroup

84

Chapter 3 Feller Processes

of kernels, i.e. we find /

YdP

JQ

= / • • • / XBI (ZI) ••• Xs n (x n )p t - Sri {xn,B)pSn-Sn^1

(z n _i, dxn)

•••Psi(xo,dxi)n(dxo) = 11 J JBX =PH{BI

••• I

pt-Srl{xn,B)---pSl(xQ,dxi)n(dx0)

JBn x...xB

n

xB),

where H = { s j , . . . ,sn,t} and P# denotes the joint distribution of XS1,..., XSn, Xt. But the special structure of Q yields finally PH{BX x...xBnxB)

= P{XSl G B i , . . . , X , B = P({xt

€Bn,XteB)

eB}nQ)

proving the theorem.

D Corollary 3.4.5. Every canonical process constructed by starting with a Feller semigroup (T t ) t > 0 on Coo(Mn) and a given initial distribution fi e Mi(M.n) is a Markov process with respect to the canonical filtration. Proof. If all operators Tt are conservative, i.e. Til = 1, then nothing is to prove since the associated family of kernels are Markovian kernels. In the sub-Markovian case we first have to consider the canonical process with state space (R^, /?£) and then recall the definition of the extension of Tt to Sfc(R^), • compare Section 1.4.8. The next step is to consider instead of one (canonical) process associated with a semigroup of Markovian kernels and an initial distribution p. the family of all processes (Q,A,PX, (Xt)t>o)xeE, Px = PEx, compare (3.59). Lemma 3.4.6. Let (Cl,A, Px, (Xt)t>o) xeE be the family of canonical processes an ^ associated with the normal Markovian semigroup of kernels (pt(-,m))t>0 initial distributions £x,x G E, where (E,B) is the Polish state space. Then a Markov kernel from (E, B) to (fl, A) is given by (x,A)>-+ PX(A).

(3.103)

3.4 Markov Processes and Feller Processes

85

Proof. By construction A H-> PX(A) is for each x £ E a probability measure on fi. On the other hand, for 0 < ty < ... < tn and B\.... Bn G B we have P I (I tl eB 1 ,...,\eB n ) = / • • • / Ptn-tn-1{Xn-l,fan)---PtAx,&x) JB!

JBn

P

= ti ((• • " Ptn-1-tn^2{Ptn-tn^1XBn)XBn_1)

• • • XB1){x),

proving the measurability of x >—> Px(Xt1 £ B,..., Xtn £ Bn) which however is sufficient to get the measurability of x H-> PX(A) for all A 6 A by the construction of A. • Note that the kernel (x, A) — i > PX{A) allows us to define an operator on x Bb(n) b y F M fY(w)P (duj). Let us return to the family (fl,A, Px, (Xt)t>o)x€E of canonical processes. We have x Pt(x,B)=P (XteB)

(3.104)

and therefore by Theorem 3.4.4 Px(Xs+t € B\J*) = pt(X.,B) Px-a.s. for s,t > 0,x e E and B € B. If we denote by PX"(A), variable

(3.105) A £ A, the random

OJ^PX'^\A),

note Xa{w) e E, by (3.104) and (3.105) we arrive at Px(Xa+teB\J*)=Px-{XteB)

Px-a.s.

(3.106)

and in addition PX(XO = x) = l.

(3.107)

For the following general definitions we make us free from the assumption of dealing with canonical processes. Definition 3.4.7. Let (E,B) be a measurable space. We call a universal Markov process with state space (E,B) if (Sl,A,Px, (Xt)t>o)xeE the following conditions hold:

86

Chapter 3 Feller Processes i) (£l,A,Px,(Xt)t>o) space (E,B);

is for each x € E a stochastic process with state

ii) for every A £ A the function x i-> PX(A) is S-measurable; iii) PX(XO = x) = l for all x G E; iv) for all s, t > 0, x G £ and B G B it holds P x ( X s + t G B | ^ J ) = Px°(Xt

e S ) P x -a.s.

(3.108)

The last property, i.e. (3.108) is called the universal Markov property with respect to the canonical filtration. Remark 3.4.8. A. It is clear that if (p., A, Px, (Xt)t>o)xeE is a universal x Markov process then for each x € E the process (Q, A, P , (Xt)t>o) ls a Markov process with respect to the canonical filtration {J~t)t>o, i-e. Px(Xt G B\J*) = Px(Xt G B\Xt) Px-a.s. for t > s. Indeed we have Px'(Xt

G B) = Px(Xs+t

and since PXs(Xt Px'{Xt

G B\T°S) = £ * ( X B o X s + t | J ^ ) P ' - a . s ,

G -B) is
EB)= Ex(Px'(Xt x

E {E (XBoXs+t\^)\Xs)

= = =

G B)\XS) x

x

E (XBoXs+t\Xs) x

P (Xs+t<EB\Xs)

implying Px{Xs+t

G B\J*) = Px(Xs+t

G B|X,).

B. Of course we may replace in Definition 3.4.7 the canonical nitration {J^)t>o by any filtration {Ft)t>o provided J^ C Tt, i.e. (Xt)t>o is adapted to (.F)t>oIn this case we write {Xt,ft)t>o, or (il,A,Px, (Xt)t>o, (^t)t>o) x€E Summing up, we have proved

3.4 Markov Processes and Feller Processes

87

Theorem 3.4.9. Let (pt)t>o be a normal semigroup of Markovian kernels on a Polish state space (E,B). Then there exists a universal Markov process (Q,,A,PX, (Xt)t>o)xeE with state space (E,B) such that for all t > 0,x G E and B e B we have Pt(x,B)

= Px(XteB).

(3.109)

Note that (3.109) is equivalent to (PtXB)(x)=Ex(XB°Xt) and by the standard procedure we arrive at (Ptu)(x)=E*(uoXt)

(3.110)

for all u £ Bb(E). Corollary 3.4.10. Let {Tt)t>o be a conservative Feller semigroup on Coo(Kn). Then the family of canonical processes (il, A, Px, (Xt)t>o)xeWLn form a universal Markov process. For a general Feller semigroup a universal Markov process is obtained by (Q,A,Px,(Xt)t>o)xem.^In case of Corollary 3.4.10 it follows from (3.110) for u G Coo(M") (Ttu)(x) = Ex{uoXt).

(3.111)

Thus we have a simple, natural relation between a Feller semigroup and its associated universal Markov process. Remark 3.4.11. It is possible to prove that every universal Markov process originates from a Markovian semigroup of kernels, compare H. Bauer [30], §42. But this part of the general theory is of little interest for us and therefore we omit any further discussion. Our next more ambitious aim is to construct cadlag modifications of certain canonical processes. This requires some preparations which are partly also needed in different context. Let (E, B) be a measurable space and T C B a sub-cr-field. By B(E, T) we denote the space of all J--measurable functions on E, but as before we write B(E) for B(E,B). The proof of the following lemma is partly taken from K. L. Chung [75].

88

Chapter 3 Feller Processes

Lemma 3.4.12. Let (Q,A,Px,{Xt)t>o,(Tt)t>o)xSE be a universal Markov process with locally compact Polish state space (E,B). The Markov property Px(Xs+t

E B\FS) = Px°{Xt G B) Px-a.s.

(3.112)

is equivalent to each of the following conditions: E»{Y\Tt) = E"(Y\Xt)

for allY G Bb(E,a(Xs,s

e [t,oo)));

for all u G Bb(E) and s > t;

E»(u(Xs)\ft)

= E»(u{Xs)\Xt)

E^{u{Xs)\Tt)

= E"(u(JC s )|X t ) /or oil u e C o (£) ond s > t.

(3.113) (3.114)

and (3.115)

ifere E^ denotes the expectation with respect to P^jfi being any initial distribution. Proof. Since for every u S Bb(E),u > 0, there is a sequence (•uJ/),,eN, uv G Co(E) and u,, > 0, which increases pointwise to u, i.e. uu \ u as u —> oo, (3.114) follows from (3.115) first for u > 0 by monotone convergence, and for general u by considering the decomposition u = u+ — u~. Next we will prove that (3.114) implies (3.113). Let Y:=Ul(XSl)---Ul/(XsJ

(3.116)

where t < Si < . . . < sv and Uj G Bb(E), 1 < j < v. For v = 1 the statement (3.113) is just (3.114). Now suppose that (3.114) implies (3.113) if Y is of form (3.116) for some v - 1. It follows for v G N that

E»(j[Uj(XSi)\Ft)

.3 = 1

J=l

Now,

^K(X S J|^_J = ^K(x s j|x ai/ _j = 5 ( ^ - J

3.4 Markov Processes and Feller Processes

89

for some g e Bb(E), compare Definition 2.5.12, and we may consider instead of uv-\ the function (u^_i • g)(X3v_1). The induction hypothesis implies now


=E"(j[ui(Xtj){uv-l-9){Xtv_1)\Ft) 1/-2

=^(II

^AX.i){uv.x-g){Xtv_1)\Xt)

=^(^(f[ Uj (X s ,)|^ s _ 1 )|X t ) =E'i(j[uj(XSj)\Xt). J=I

Thus we have proved that (3.114) implies (3.113) for all Y of type (3.116), v e N. Now taking Uj = XB, we conclude by using a monotone class argument that (3.113) holds for all XA,A e a(Xs,s G [*,oo)). From this we deduce first by using monotone approximation that (3.113) holds for all e [t, oo)) and then by decomposing u = u+ - u~ it u > 0, u e B(E,a(Xs,s follows that (3.114) implies (3.113). Thus the equivalence of (3.113), (3.114) and (3.115) is established. Now taking in (3.114) the characteristic function XB ,B £ B, we arrive at the elementary Markov property P»(Xa

e B\Ft)

= P"{XS

g B\Xt)

,t<s,

for every inital distribution \i which in our case implies (3.112) by Theorem 3.4.4 and its consequences.

•

Definition 3.4.13. Let ((Xt)t>0, Px, (^Ft)t>o)xeE be a universal Markov process with corresponding semigroup of operators (Pt)t>o- Further let u : E —> M. be a measurable function, u > 0. For a > 0 we call u a-supermedian with respect to (Xt)t>o if e~atPtu < u

(3.117)

90

Chapter 3 Feller Processes

holds for all t > 0. If u is a-supermedian and u = lime-at Ptu,

(3.118)

UO

then u is called a-excessive with respect to (Xt)t>0. Lemma 3.4.14. Let u be a-supermedian with respect to the universal Markov process ((Xt)t>a,Px,{Jrt)t>o)xeE and suppose that for each t > 0 the random variable u(Xt) is integrable with respect to every measure Px. Then (e~atu{Xt),!Ft)t>Q is a supermartingale. Proof. We use the Markov property in the form (3.112) (in its obvious extension to integrable functions) to find for s > 0, t > 0 that u(Xs) > e-atP?u{Xs) = =

e-atEx°(u{Xt))

e-atE»(u(X3+t)\Fs),

or e-asu(Xs) >

E^(e-^s+^u{Xs+t)\Ts)

proving the lemma.

•

With every strongly continuous contraction semigroup (Tt)t>o on a Banach space X we associate its resolvent, compare Section 1.4.1, by

Rxu= f Jo

e-XtTtudt=(\-A)-1.

Now, if (Pt)t>o is the semigroup of operators associated with a universal Markov process with state space (E,8) and if u e P>b(E), then Uxu:= f Jo

e-XtPtudt,X>0,

makes sense. In addition the resolvent equation Ux-Ull = ((Ji- \)VXU» holds and we have

(3.119)

3.4 Markov Processes and Feller Processes

91

for u € Bb(E). Moreover, if E — R n and (•Pt|coo(Kn))t>0 *s a Feller semigroup {Tt)t>o, then J/A|COO(R") = ^ A , where (J?A)A>O denotes the resolvent associated with (Tt)t>o. It is convenient to call the family (U\)\>0 the family of X-potential operators associated with the process {Xt)t>obe a universal Markov Proposition 3.4.15. Let ({Xt)t>Q,Px,{Jrt)t>o)xeE process, a > 0, and u e Bb(E),u > 0. Then Uau is a-excessive with respect to (Xt)t>o and {z~atUau(Xt),Tt)t>Q is a supermartingale. Proof. First we observe that

e-atPt(Uau) = e-at

[Xe-asPt+suds

Jo e-asPsuds<

/ e-asPsuds = Uau, Jo

thus Uau is a-supermedian and since />OO

OO

/

e- Q S wds= / Jo

e-asPsuds,

it follows that Uau is a-excessive. Finally, since Uau is bounded we may apply Lemma 3.4.14 to Uau instead of u to find that (e" Qt f/ Q u,^ r t)t>o is a supermartingale. • (Ft)t>o)xpE Definition 3.4.16. A universal Markov process ((Xt)t>o,Px, with locally compact Polish state space (E,B) is called a Feller process if (Tt)t>o,Tt := Pt\c^{E) is a Feller semigroup. In order to prove that Feller processes admit cadlag-modifications we need Lemma 3.4.17. Let (D,,A, P) be a probability space and let (E,B) be a locally compact Polish space. Two random variables X,Y : Q —> E are almost surely equal if and only if E(u{X)v(Y))

= E{u{X)v{X))

(3.120)

holds for all u, v S Cb(E). Proof. Clearly, X = Y a.e. implies (3.120). Now, suppose that (3.120) holds. By a monotone class argument we derive that E(u(X, Y)) = E(u(X, X))

(3.121)

92

Chapter 3 Feller Processes

holds for all UJ G B(E x E),w > 0. Since X{{x,y)eExE;x^y} Borel function on E x E we find

is a non-negative

P(X ^Y) = P(X ^ X) = 0 and the lemma is proved.

•

Remark 3.4.18. Clearly, the above lemma remains valid if Cb(E) is substituted by Coo(E). Now we can prove Theorem 3.4.19. A Feller process ((Xt)t>o,Px)x£Rn tion.

has a cddldg modifica-

The proof of Theorem 3.4.19 requires two results on supermartingales, Theorem 2.6.9 and Theorem 2.6.11, which we recollect here for convenience. The first theorem states that if (Xt)t>o is a supermartingale on (p., A, P, (^rt)t>o) then for almost all w G O the following two limits exist

lim Xr{w),t > 0 and lim Xr(u),t > 0. rtt

rit

rGQ+

r€Q+

The second theorem states that a right-continuous supermartingale has almost surely cadlg paths. Now, following D. Revuz and M. Yor [299] we give the Proof of Theorem 3.4.19. Let {UV)VQ^,UV € Coo(R™) and uv > 0, be a sequence which separates points, i.e. for x,y £ Rn,x ^ y, there exists VQ such thatu t , 0 (x) 7^ uUa(y). Since foru G Coo(Rn) wehaveC/ Q u = Rau it follows that for v G N the family (aUauu)a>o converges uniformly to uu as a —> oo, hence H := {Uauv;a G N and v e N} is a countable set of functions in Coo(Rn) which also separates points. Using the fact that {e~atUauv)t>o is a supermartingale with respect to (JFt)t>o and all Px as well as all P^ — j Pxfi(dx), the first result mentioned above, i.e. Theorem 2.6.9, yields that for every h G H the limits lim h(Xr(cj)) exist almost surely. Since H separates points rit r£Q+

it follows that for all w G Q\Af, P(M) = 0, P = P M for some initial distribution .122) Xt{w) := lim Xr{w) rit re®+

(3

3.4 Markov Processes and Feller Processes

93

exists. For u> £ Af we just define

Xt{uj) := 0 for t > 0. For u, v e Coo(Rn) we find for any initial distribution pt using the Markov property

E»{u{Xt)v{Xt)) = lim E»{u{Xt)v{Xs)) sit

SGQ+

= lim E»(u{Xt)Ex*{v{Xs-t))) seQ +

- lim E»(u{Xt){T3_tv){Xt)) sit

seQ+

= E»(u{Xt)v{Xt)), where we used that lim ||Ts_tw — v ^ = 0 since (Tt)t>o is a Feller semigroup. Thus Lemma 3.4.17 implies for each t that Xt = Xt a.s. Since (Xt)t>o is right continuous we have constructed a right continuous modification of (Xt)t>o- It follows further from the second theorem mentioned above, i.e. Theorem 2.6.11, that for every h G H the process (h(Xt))t>Q has almost surely cadlag paths because it is a right continuous supermartingale. Again we may use the fact that H separates points to conclude that (Xt)t>o has almost surely cadlag paths, i.e. we have constructed a cadlag modification of (Xt)t>o• Corollary 3.4.20. Let (Xt)t>0 be the canonical universal Markov process with respect to the canonicalfiltrationassociated with a (conservative) Feller semigroup (Tt)t>0,Tt : Coo(R") -+ Coo(Kn). Then it admits a cadlag modification. Remark 3.4.21. Using Remark 2.6.2.C and Theorem 2.6.13 we may also deduce that Corollary 3.4.20 holds for the nitrations (Jf+)t>o as well as the augmentation of (J^+)t>oRemark 3.4.22. A very detailed, clear and complete proof that every Feller process admits a cadlag modification with respect to a complete and right continuous nitration is given in the forthcoming book [323] of R. Schilling. We may give Corollary 3.4.20 a different, very important interpretation. Let (Tt)t>o be (for simplicity) a conservative Feller semigroup on Coo(Rn).

94

Chapter 3 Feller Processes

Further let fi be an initial distribution. The Kolmogorov construction yields a canonical process ((R")[o>°°), (BW)[o,oo) )PM) (Xt)t>o), where Xt{w) = w(t) is a projection. Since this process has a cadlag modification we might try to prove that P M |p n gives a probability measure on Z?nn(jB(n))[0>o°). The problem is that in general T>n is not measurable. Definition 3.4.23. Let (E,B) be a Polish space and {Pj)j^u(i) be a projective family of probability measures over E. We call a set Q. C E1 essential with respect to {Pj)j£U(i) if there exists a stochastic process with state space E, parameter set / , finite-dimensional distributions (Pj)jsn(i) a n d path-set Q. The next theorem due to J. L. Doob [88] gives a characterization of essential sets. Our formulation is taken from H. Bauer [30] where the reader may also find a proof. Theorem 3.4.24. Let E be a Polish space and [PJ)J^H(I) be a protective family over E. Denote by P the protective limit of {Pj)jen(i) and ^he outer measure determined by P is denoted by P*. Then a set Cl C E1 is essential with respect to (Pj)jeH(i) if and onMi if P*{h) = 1. Proof. See [30], p.328.

(3.123) •

Note that (the proof of) Theorem 3.4.24 yields that

(fi, n n B1, P, (Xt)t>o), P(fi n Q) = P(Q), is an equivalent process to the canonical process {E1 ,BI, P, (Xt)t>o) • Thus we find that to a given (conservative) Feller semigroup on Coo(K") there exists a process equivalent to the associated canonical process given by

(vn,vnn(Bny,P,(xt)t>0), in fact we may replace I ^ n ^ " ) 7 by a{Xt,t £ [0,oo)). This process leads again to a universal Markov process, in fact a Feller process, and all considerations made so far do hold (essentially) also for this process. Thus, determining a universal Markov process associated with a Feller semigroup (T t ) t >o on Coo(Rn) can be reformulated:

3.4 Markov Processes and Feller Processes

95

On the measurable space (Vn,Vn n (S(n))[°>°°)) find probability measures P ,x£Rn, such that for u G Coo(Kn) x

Ex{u(Xt))=(Ttu)(x) holds. We will come back to these considerations when discussing the martingale problem for pseudo-differential operators with negative definite symbols, see Chapter 4. Finally in this section we will prove that Feller processes are stochastically continuous. The proof of the following theorem is taken from K. L. Chung [75]. Theorem 3.4.25. Every Feller process ((Xt)t>o, Px)x€&n continuous, i.e.

is stochastically (3.124)

\imP(\Xs~Xt\>e)=0.

s->t

Proof. Take u,v € Coo(R") and for t, r > 0 we find using as before the Markov property E»{u(Xt)v(Xt+r) = E»(u(Xt)Ex*{v(Xr))) =

E»(u(Xt)(Trv)(Xt)).

Again, the fact that (T t ) t >o is a Feller semigroup implies for r J. 0 that lim E» (u(Xt)v{Xt+r)) = E" (u(Xt)v{Xt)).

(3.125)

rlO

As in the proof of Lemma 3.4.17 we may extend (3.125) to hold for all LO G C o o (M n x]R"), i.e. (3.126)

WmE»(uj{XuXt+r)) =E(w(Xt,Xt)) r|0

for all u) G Coo(M" x l " ) . Taking now for w a sequence approximating the function (x, y) H-> \X — y\, then (3.126) implies the L1 -convergence of Xt+r to Xt, hence the stochastic convergence. Next let t > 0 and 0 < r < t. For each x G M" it follows with u, v as above that Ex(u(Xt_r)v(Xt))

=

Ex(u(Xt_r)EXt-r(v(Xr)))

= Ex{u(Xt-r){Trv){Xt-r))

= Tt-r(uTrv)(x).

96

Chapter 3 Feller Processes

Now, as r —> 0 it follows that

lim Tt-T{uTvv){x)

= Tt{uv){x) = Ex

(u(Xt)v(Xt)),

thus lim Ex(u(Xt-r)v(Xt))

r—>0

=

Ex(u(Xt)v(Xt))

and with the same argument as before we arrive first at lim Ex(cj{Xt-r,Xt))

1—>0

= Ex(uj{Xt,Xt))

, w £ C ^ R " x Rn),

and then we get lim E*(w(Xt-r,Xt))

r—»0

= E"(u>(Xt,Xt))

(3.127)

for all w G Coo(K" xR"). Thus we find that X t _ r -> X t stochastically implying the theorem. • Remark 3.4.26. The reader might have noticed that in some of the results in this section we need to assume that the underlying Feller semigroup is conservative. But all results continue to hold for an arbitrary Feller semigroup when passing from the state space M"(or E) to the state space R^(or E&).

3.5

The Shift Operator and the Strong Markov Property

Let (p., A, Px, (Xt)t>o, (^7t)t>o) eKn De a universal Markov process. We want to give a different formulation (and interpretation) of the Markov property. Assume for a moment that the process under consideration is a canonical process, i.e. Q, = (K")!0'00). Then w € f i i s a mapping w : [0, oo) -> Mn and for each s > 0 we may consider the new mapping UJS : [0, oo) —> K", t H-> w{t + s). Since the random variables Xt are the projections Xt : £1 —> R n ,t >—> u(t) we find Xt+s(Lj) = w(t + s)=Xt(ujs),

(3.128)

i.e. shifting the argument of w by s is compatible with a shift of the "time" parameter by s. Back in the general case we give

3.5 The Shift Operator and the Strong Markov Property

97

Definition 3.5.1. Let (Q,A, Px, (Xt)t>o, (^)t>o)xeRn be a universal Markov process with state space (M.n,B^). A family 6S : Cl —> Q,s > 0 of measurable transformations is called a family of shift operators (with respect to the process (Xt)t>o) if for all w G ft and s, t > 0 it holds (3.129)

Xt+,(w)=Xt(0t(u>)).

Thus in case of a canonical process a shift operator is given by (3.128), but (3.128) yields also an example of a shift operator in case of the cadlag modification of a canonical Feller process with state space K n . This follows from the choice of the a-field AonT>n. Let (6s)s>o be a family of shift operators with respect to (Xt)t>o- From (3.129) we derive for r, s, t > 0 Xt+s+r(cj)

= Xt+s{6T{u))

=

Xt{0.(Or(w)))

as well as Xt+s+r{u) =

Xt(9s+r(v))

which gives 6s+t = 8so6r

for all r, s > 0.

(3.130)

P r o p o s i t i o n 3.5.2. Let (9s)s>o be a family of shift operators with respect to the universal Markov process (Q,,A,Px,(Xt)t>o,(^'t)t>o)xeRnThen the (universal) Markov property Px(Xs+t

G B\TS) = Px°(Xt

G B)

= Px(Xt+seB\Xs)=pt(Xs,B)

Px-a.s.,B£B(n\

(3 131) V ' '

is equivalent to P^iej^F)^)

= ^(B-^F^Xs)

Px-a.s.

,F&A.

(3.132)

Proof. The sets of all F such that (3.132) holds form a n-stable Dynkin system and therefore it is sufficient to prove (3.132) for cylinder sets of the form F = {u G fi;-Xto(w) G Bo,.. .,Xtn(w)

G Bn}

where 0 = t 0 < *i < • • • < tm and Bt G B ( n ) . In this case (3.132) reads as Px(Xs€B0,...,Xs+tmeBm\fs) = P*(XS G Bo,...,

Xs+tm G Bm\Xs)

Px-a.s.

(3 133)

'

98

Chapter 3 Feller Processes

We are going to prove (3.133) by induction. For m — 0 nothing is to prove. Suppose (3.133) holds for m - 1. Then for all ip G B b (M n m ) we have almost surely

Ex(
(3.134)

Now, (3.131) implies that PX(XS £ Bo,... ,Xs+tm_1 G Bm-i,Xs+tm =

EX(X{Xs£B0,...,X3+tm_1£Bm-1}PX(Xs+tm

G

Bm\Ts)

€ 5m|Xs+tm_1)|/'s).

(3.135)

By Lemma 2.5.14 there exists a measurable function g : R n —> [0,1] such that P*(X J + t r a G B

m

|X,

K

J = 5 ( X s + t m _ J P*-a.s.

Taking (p(x0,... , x m _ i ) = XB 0 (^O) • • • • • X f l m - i ^ m - O s K - i ) . w e m a Y a PP!y (3.134) to substitute in the last line of (3.135) J-a by <J(XS) which shows that (3.131) implies (3.132). The converse is trivial, compare (3.133). • Recall that for any filtration (Jri)t>o we have defined J 7 ^ = a{Tt,t > 0). Proposition 3.5.3. Let (n,A,Px,(Xt)t>o,(^t)t>o), x G K n , be a universal Markov process and fi an initial distribution. Moreover let Z be a J^-measurable bounded random variable. Then x i-> EX(Z) is measurable and

E>1{Z)= f Ex(Z)n{&x)

(3.136)

holds. Proof. This follows immediately from (3.51) and (3.54), resp., together with the required measurability properties. • Now we may reformulate the universal Markov property. Theorem 3.5.4. Let (fl,A,Px, (Xt)t>o, (^t)t>o) xeRn be a universal Markov process, (0s)s>o a family of shift operators with respect to (Xt)t>o, and Z any T^-measurable non-negative or bounded random variable. For every initial distribution /J, and all t > 0 E'l{Zo9t\J^) holds P^-a.s.

= EXt(Z)

(3.137)

3.5 The Shift Operator and the Strong Markov Property

99

Remark 3.5.5. If Z = X{xseB},B G B(n\ then (3.137) implies (3.131), i.e. (3.137) is equivalent to the universal Markov property. Proof of Theorem 3.5.4. By the definition of the conditional expectation we have to prove for every ^-measurable non-negative or bounded random variable the equality E" ((Z o 9t)Y) = E* (EXt (Z)Y).

(3.138)

Now, as argued several times before, it is sufficient to prove (3.138) for all random variables Y of the form Y = Ylj=i fj(Xtj), where fj is a positive Borel measurable function and tj < t, and Z is of the form Z = YliLi 9i(^u) with gi being positive and Borel measurable, £j < t. This however can be reduced as done repeatedly before to a statement for the transition functions • and the theorem is proved. Our next aim is to extend the Markov property to stopping times. For this we give Definition 3.5.6. Let (Q, A, PM) be a probability space with nitration (Tt)t>o and (Xt)t>o an adapted stochastic process with state space (M.n,Bn). Further let fi be a probability measure on (Rn,Bn). We call (Q, A, P*, {Xt)t>0, (Tt)t>o) a strong Markov process (with initial distribution /i) if i) for all B e B^ (3.139)

P»{X0 eB) = n(B), ii) for all (^)-stopping time a and alH > 0, B G S ( n ) P»{Xa+t e B\Fa+) = P»{Xa+t G B\XO)

(3.140)

holds P^-a.s. on {a < oo}. Recall, compare (2.135), that ?„+ := {A£A;AD{a
£ Tt+ for allt > 0}.

(3.141)

Next we generalize the concept of a strong Markov process to that of a universal strong Markov process: Definition 3.5.7. We (^,A,Px,(Xt)t>o,(J7t)t>o)xeRn strong Markov process if

call

a family of stochastic processes with state space (Rn,B(-n)) a universal

100

Chapter 3 Feller Processes

i) for all B G A the mapping x i-> PX{B) is universal measurable; ii) for all i 6 K " w e have PX{XO = x) = 1; iii) for all x G R n , i > 0, £ G B ( n ) and every ^-stopping time cr it follows P ^ + t G B|J>+) = P*(XCT+t G B\Xa) Px-&.s. on a < oo. (3.142) Remark 3.5.8. Since non-negative constants are stopping times, it is clear that every universal strong Markov process is also a universal Markov process. However the converse is not true. Remark 3.5.9. It is now an advanced exercise in measure theory to prove several equivalent conditions to (3.142). For example we have Px{Xa+t

G B\F.+) = (PtXB)(X
(3.143)

or Ex(u(Xa+t)\Ta+)

= (Ptu){Xa)

(3.144)

where these equalities hold P x -a.s. on {a < oo}. In addition, if a family of shift operators (#5)s>o exists on Q then we may define flff:{ff
Oa(u))(t) = ds(u)(t) on {cr = s}.

(3.145)

In particular, for cr(u>) < oo we find Xa{u)+t(u>)

(3.146)

= Xt(0a(u>)).

Now it is possible to prove as an extension of (3.137) E^{Zo9(r\^l7+)

= EX"{Z)

P M -a.s. on {a < oo}

(3.147)

for every ^"oo-measurable, non-negative or bounded random variable Z. For proofs of these statements we refer to K. L. Chung [75] or D. Revuz and M. Yor [299], some material is also contained in I. Karatzas and St. Shreve [216]. Let (fl, A, Px, (Xt)t>o, (Ft)t>o) be the cadlag modification of a given Feller process. For a probability measure [i on B^ we denote by T^ the completion of T^ with respect to P M , where P»{A) =

fPx{A)fi{dx).

3.5 The Shift Operator and the Strong Markov Property

101

Further, (^T)t>o is the filtration obtained by adding to each T® the PM-nullsets .A/"" of .F&, i.e. J* = a(J*>,M"). Finally we put

^i:=n^r

and

(3-148)

^oc-fi^

where the intersections are taken over all probability measures fx on B^n\ Elements of T^ &re called universally measurable sets. Theorem 3.5.10. The filtrations (Ft)t>o and (^t)t>o o,re complete and rightcontinuous, i.e. they satisfy the usual conditionsProof. Both nitrations are complete by construction and the right-continuity of (J~t)t>o will imply the right-continuity of (!Ft)t>o- Thus it is sufficient to prove for every J^-measurable, non-negative random variable Z that E^{Z\flt) = E»(Z\F?+)

P"-a.s.

(3.149)

holds. As before we may reduce this to random variables of the form z =Y17=ifj(xtj) where ft e C^QR") and h < t2 < . . . < tm,m G N. Moreover, for every t we have PM-a.s.

E»(Z\f?) = E»{Z\F°). Now, for t e K take /c e N such that tk-i < t < ife. Then for /i sufficiently large we find fc-i

£?"(^|^+fc) = I I fi(*tt)gh{Xt+h)

^-a-s.

where 9h(x) = I Ptk-t-h(x,dxk)fk(xk) •••

Ptk+1-tk(xk,dxk+i)fk+1(xk+1)

Ptm-tm-i(xm-i,dxm)fm{xm).

Here pt(a;, A) denotes of course the transition function of the Feller process (Xt)t>o under consideration. Since (Xt)t>0 is right-continuous, Xt+h tends to X t for h —> 0 and therefore we find by Proposition 2.6.12, JB"(Z|^+)=limJB"(Z|^;h)

fc-i

102

Chapter 3 Feller Processes

which proves the theorem.

•

Of course, we now want to consider (Xt)t>o as a stochastic process adapted to the nitration (Ft)t>o and (Jrt)t>o> respectively, and this causes some measurability problems. We follow in our presentation mainly the monograph [299] of D. Revuz and M. Yor. We assume that (Xt)t>o admits a family of shift operators (#s)s>oLemma 3.5.11. Let Z be a bounded T^-measurable random variable. Then x H-» EX{Z) is universal measurable and we have

E»(Z) = jE*(Z)n(dx). Proof. For every probability measure fi on B^ we find two .T^-measurable random variables Z\ and Z2 such that Z\ < Z < Z2 and E^(Z2 - Z\) = 0 and therefore EX{Z1) < EX{Z)

<EX{Z2)

for every x G K™. Since x H-> Ex{Zi),i = 1,2, is S(n)-6(^-measurable and

J(EX(Z2) - E^Z^^dx)

= E»{Z2 - Zx) = 0

it follows that x — i > EX(Z) is B^-B^the lemma is proved.

measurable and since [i was arbitrary •

Lemma 3.5.12. A. For every t > 0 the random variable Xt is Tt- B^measurable. B. For every t > 0 and h > 0 we have 0^l{Tt) C Ft+h, in particular 0s is measurable. Proof. Part A follows from [299], Proposition 0.3.2. Further we know that Oh is ^-^Y^-measurable and the assertion will follow from [299], Proposition 0.3.2. if for every initial distribution v there is an initial distribution /U such that 6h{Pv) = PM. For this define fx := Xh(Pv) and use the Markov property in the form of (3.137) to find for A £ F^ that

Pv{6h &A) = EV{EX*{XA)) = JpY(A)v(dy) which proves the lemma.

= P»(A) •

3.5 The Shift Operator and the Strong Markov Property

103

Now we can prove that (Xt)t>o is also a universal Markov process with respect to the new filtration (ft)t>oTheorem 3.5.13. Let (Xt)t>o be a Feller process with respect to (.Fto)t>o and state space (lRn, B^). Then for every Too -measurable non-negative or bounded random variable Z, every t > 0 and every initial distribution /z E"(Z o 9t\Tt) = EXt(Z)

P^-a.s.

(3.150)

on {Xt < oo}. In particular ((Xt)t>o, {F)t>o) is a universal Markov process. Proof. (Compare D. Revuz and M. Yor [299]/ Note that {Xt < oo} denotes the set { w e f i ^ i f w J ^ K ^ I 1 1 } , i.e. for a conservative Feller semigroup {Xt < oo} — £1 for t < oo. From Lemma 3.5.11 and Lemma 3.5.12 it follows that u> — i > EXt^(Z) is always .Ft-measurable and it remains to prove for all A&Tt = E»(XAEXt(Z)).

E»(XAZo6t)

(3.151)

Without loss of generality let Z be bounded. The definition of .Foo implies the existence of an J 7 ^- measurable random variable Z' such that and P ^ r ) = 0 with v = XtiPi1). It follows that {Z ^ Z1} C r , T G ^ 1 {Z o 6t + Z o 6t} C ^ - ^ r ) and P'1(9t1(T)) = PV(T) = 0. Now the Markov property with respect to {^t)t>o yields

P^r1?))

= E»{Ex>{xr)) = j PY{T)u{Ay) = P»(T),

and since E'1(EXt(-)) E"(EXt{\Z

= Ev{-) we find

- Z'\)) = EV{\Z - Z'\) = 0,

or EXt(Z) = EXt(Z'), P^-a.s. Thus we may substitute in (3.151) Z by Z' and then we may use the Markov property with respect to (Ff)t>o• Let r be a ^-stopping time, (Jrt)t>o as in (3.148). On {r = oo} we define XT = A e M^.Then XT is j'v-measurable. For the shift operator we set 0T{w) = et(u)

for

T(W) = t

(3.152)

T(W) = oo,

(3.153)

and eT{w) = A

for

where A G ft denotes the path t i-> A G M^. With these definitions we find Xto0T = XT+t and therefore 0 " 1 (.?£,) C o{XT+t ;t>0).

104

Chapter 3 Feller Processes

Theorem 3.5.14. Let ((Xt)t>o, (^rt)t>o) be the cadlag modification of a Feller process with state space (M n ,S ( n ) ) and the filtration is constructed as in (3.148). Then for every Too-measurable, non-negative or bounded random variable Z, every Tt- stopping time r and every initial distribution \x it holds = EX*(Z)

E»{ZoOT\fT)

P»-a.s. on {XT + oo},

(3.154)

i.e. since {Tt)t>o is right continuous, ((Xt)t>o, (^t)t>o) is a universal strong Markov process. Proof. Suppose r takes its values only in a countable set D. Then it follows

o eT\TT) = J2 x{r=d}X{xd^A}^(z

X{XT*A}E»(Z

o od\rd)

deD

Y,X{r=d}X{Xd¥:A}EX*{Z)

=

deD where we used Theorem 3.5.13, i.e. the fact that ((Xt)t>o, (Tt)t>o) is a universal Markov process. Hence, in this special case (3.154) holds. To prove the general case define by [2 n r] + 1 Tn-=V—^ , neN, a sequence of ^-stopping times having values in a countable set and which decreases to T. For fi G Coo(Mn), / , > 0, i = 1,..., k and t\ < ... < tk we put g{x) = j pti(z,dzi)/i(zi) / Pt2-ti(a:i,da;2)/(a;2) •••J

Ptk-t^{xk-i,dxk)f{xk).

Since (Xt)t>o is a Feller process we have g £ Coc(Mn) and g > 0, and due to the construction k

£"(n/iW°^i^)=^»)3= 1

The right-continuity of the paths and Proposition 2.6.12 yields now by a monotone class argument the assertion for all Z > 0, Z being .F^-measurable. It remains to show (3.154) for all Z > 0 which are .Foo-measurable. If we substitute P» by P»

~ ^

[A)

_ p»(An{xTn}) ~

P»(XT ^ A)

'

3.5 The Shift Operator and the Strong Markov Property

105

then the conditional expectations with respect to J-T are the same for P M and P» and we may therefore assume XT ^ A a.s. We now write once again P^ instead of P». Let v(A) = P»(XT e A), i.e. v — X T (P M ). By the definition of J7^ there are two T%^-measurable random variables Z' and Z" such that Z1 < Z < Z" and PV(Z" - Z' > 0) = 0. The first part of the proof yields P»{Z" oeT-Z'o9T>0)

= E^ (Ex- (Z" - Z' > 0)) = 0.

Since /x was arbitrary we derive that Z o 9T is ^"oo-measurable, hence E»(Z o 6T\FT) is well defined and it follows that E^iZ' o 9T\TT) = Ex- (Z) = E*(Z" o 6T\fT) proving the theorem.

n The main lines of the above proof had been once again borrowed from D. Revuz and M. Yor [299]. In Chapter 6 we will relate potential theoretical considerations to probabilistic ones. It is convenient to add already here some preparatory material. We start with Blumenthal's 0-1-law. Theorem 3.5.15. Let (Xt)t>o be the cadlag modification of a Feller process with respect to a right-continuous and complete filtration (Jrt)t>o o/nd state space ( R n , S ( n ) ) . For x G R™ and V € T%x it holds either PX(T) = 0 or PX{T) = 1. Proof. If T G cr(Xo), then PX(T) = 0 or PX{T) - 1 since PX{XO = x) = 1. But FQX — o-(Xo,Af€x) proving the theorem. • Corollary 3.5.16. / / r is an {T^*)-stopping PX(T = 0) = 1 or PX(T > 0) = 1. Now let i C I

n

time then it holds either

be a set. We define

aA •= inf{t > 0 ; Xt £ A}

(3.155)

aA := inf{t > 0 ; Xt e A}

(3.156)

and

106

Chapter 3 Feller Processes

where (Xt)t>o ls as in Theorem 3.5.15. We call DA the entry time of (Xt)t>o into A and TA is called the (first) hitting time of (X t ) t >o of the set A. As usual we use the convention inf 0 = +oo. For s > 0 we find s + aAo0a

= s + inf{t > 0 ; Xt+S e A} = inf{t > s ; Xt e A},

hence (3.157)

s + &Ao0s = &A on {&A > s}, and further we have \im(s + &Ao6s)

(3.158)

= <JA-

By analogous arguments we find t + aAo0t=aA

(3.159)

on {aA > t}.

It holds Theorem 3.5.17. Let A S B^n\

Then &A and a A are J-t-stopping times.

For a proof we refer to D. Revuz and M. Yor [299], p.90, but compare also Section 5.2, especially Theorem 5.2.15. Proposition 3.5.18. Let (Xt)t>o be as in Theorem 3.5.15 and for x £ K" set ox := inf{t > 0; Xt 7^ x}. Then there exists a = a{x) £ [0, oo] such that Px{ax>t)=e~at

(3.160)

Proof. By definition ax is the first hitting time of the Borel set {x}° hence it is an (.F^-stopping time. Further, by (3.158) we have on {ax > t} the equality ox — t + ax o 6t • Thus we find Px(ax =

>t + s)= PX{{(TX >t}D{ax>t

+ a})

x

E (X{ax>t}X{*x>s}°9t)

and the Markov property yields Px(ax

>t + s)=

Ex(X{ax>t}EXt(x{ax>s}),

where we used that Xt ^ A on {ax > t}. In fact, on {ax > i) we have Xt = x almost surely and therefore Px{ax

>t + s)= Px{o-X > t)Px(ax

> s).

The continuity of s t—> Px(crx > s) implies now the proposition.

•

3.5 The Shift Operator and the Strong Markov Property

107

With arguments similar to those used in the proof of Theorem 3.5.14 one shows, compare [299], p.98, Lemma 3.5.19. For every t > 0 and every Tt-stopping time r a Tt-stopping time is given by r + t and in addition ^~1(^rt) C TT+t holds. We use Lemma 3.5.19 in order to prove Lemma 3.5.20. For two Tt-stopping times a and r a further Tt-stopping time is given by a + r o Qa. Proof. Since (Tt)t>o is right-continuous it is sufficient to prove that for every * > 0 it holds {ex + T O Ba < t} £ Tt. Since {a + T o 6a < t} = (J {a < t - q} n {r o 6a < q} qSQ

and since by Lemma 3.5.19 we have {T o6a < q} £ F^+q the lemma is proved once we observe that

{a < t - q} n {r o 6a < q} = {a + q < t} ("1 {r o 9a < q}. D If for example r = TA = inf {i > 0; Xt £ A} and cr = TB = inf {t > 0; Xt G B} then TA + TB ° 8TA is the first time the process hits the set B after it has hit the set A. Let T be a ^-stopping time. We associate with T the kernel

Pr{x,A)=E?{XA°XT) and for a non-negative Borel function u we define PTu(x):=Ex(u(XT)).

(3.161)

Proposition 3.5.21. A. For two Tt-stopping times a and r we have PaoPT

= Pa+ro9a.

(3.162)

B. If T is an (Ft)-stopping time, then Yt := XT+t is again a universal Markov process with respect to the new filtration (!FT+t)t>o and it has the same transition function as (Xt)t>o-

108

Chapter 3 Feller Processes

Proof. A. Take u e B(Rn), u > 0, to find P
Ex(Ex'(u(XT)))

and the strong Markov property gives Pa(PTu)(x) = E*{X{XB^}E?(U{XT) =E*(x{x^A}u(XT)o6<7).

o ea\Ta))

However, u(XT) o 6a = u(Xa+TOga) and u(XT) o 9a = 0 for {Xa = A}, and part A is proved. B. Let u > 0 be a Borel function on Kn and let fi be any initial distribution. First we find on {XT+t ^ A} E"(u(X T+t+ ,)|:F T+t ) - E»{u{Xs) o eT+t\TT+t) =

Psu(XT+t),

and by our definition this equality extends to {XT+t = ^ } proving part B.

•

Remark 3.5.22. Note that if (Xt)t>o is a canonical process then (Yi)t>o of Proposition 3.5.21.B is in general not a canonical version of (Xt)t>o-

3.6

The Martingale Problem for Feller Processes: First Remarks

Given a (conservative) Feller semigroup (Tt)t>o o n Coo^") with generator (A,D(A)). In this section we will first associate certain martingales with (A,D(A)) using the Feller process (Xt)t>o corresponding to (Tt)t>o- Then we will take a converse point of view: We will try to construct the process (Xt)t>o associating a family of martingales to (^4, D(A)), i.e. by solving the martingale problem. We need some auxiliary results and for simplicity we restrict all of our considerations to processes with state space (R n ,B^ n '). Definition 3.6.1. Let (il,A,P,(Xt)t>o) be a stochastic process with state space (K n ,S' n ') and {Ft)t>o be a filtration in A. We call (Xt)t>o progressively •measurable with respect to (Pt)t>o if for all t > 0 the mapping X.(-) : [0,t] xQ^Rn (t,w)^X t (w) is B^([0,t]) ® Tt-B^cesses are adapted.

measurable. Clearly, progressively measurable pro-

3.6 The Martingale Problem for Feller Processes

109

Of importance is now Proposition 3.6.2. Let (Xt)t>o be a right-continuous stochastic process adapted to the filtration (J7t)t>o- Then (Xt)t>0 is progressively measurable with respect to (^t)t>oProof. For t > Q,v > l,k = 0 , 1 , . . . , 2 " - 1, and 0 < s < t we define for fc* M:=X-(»+i),M and

It follows that (S,LJ) •-> X{SV)(UJ),0 < s < t,w € ft, is BW([0,t]) ® Ft-B(n) -measurable. Since {Xt)t>o is right-continuous it follows for (s,w) 6 [0,i] x ft that lim X^u){w) = X s (w) implying the 5 (1 '([0,t]) ® J^-'B(™)-measurability of the limit mapping X(-).

D

Corollary 3.6.3. Let (Tt)t>o be a (conservative) Feller semigroup on Coo(Mn). Let cadlag modification of the associated Feller process is progressively measurable with respect to the canonical filtration {T^)t>o as well as to each of the filtration (T^)t>o and (^rt)f>o from (3.148). Corollary 3.6.4. Let (Xt)t>o be a stochastic process with state space (M", B^) which is progressively measurable with respect to the filtration (!Ft)t>o and let Then (u(Xt))t>0 is (!Ft)t>a progressively measurable and in u e Bb(Rn). addition (/ 0 u(Xs)ds)t>0 is {Tt)t>o-adapted. Proof. The first statement is trivial, the second one follows by approximating the integral. • A central observation is now Theorem 3.6.5. Let {Tt)t>o be a (conservative) Feller semigroup on Coo(Mn) with generator (A,D{A)). Further let ((Xt)t>o,(^t)t>o) be a cadlag modification of the associated Feller process. For every u G D(A) and every initial distribution p. Ml := u(Xt) - u(X0) - [ {Au)(Xs)ds, ./o

t > 0,

(3.163)

110

Chapter 3 Feller Processes

is an (J-f)t>o martingale with respect to PM. In particular, ifu is ^4-harmonic, i.e. Au = 0, then (u(Xt))t>o is an {J-^)t>o martingale with respect to P^. Proof. Since for u G D(A) we have Au € Coo(R™) and since (-Xt)t>0 is progressively measurable the integral in (3.163) is well denned. Moreover we find for u < s < t Ei*(M?\J*)

= E"(u(Xt) - u(X0) - y V«)(*r)dr|J?) = u(Xs) - u(X0) -

[S(Au)(Xr)dr Jo

+ E»(u(Xt) - u(Xs) - J

(Au)(Xr)dr\J*)

= Ms" + E» (u{Xt) - u(Xs) - J (^«)(X r )dr|^). Thus we have to prove that E»(u(Xt) - u{Xs) - j (Au)(Xr)dr|J?) = 0. The Markov property yields E»(u(Xt) - u(X.) - J

(Au)(Xr)dr\J*)

= Ex» (u(Xt_s) - u(X0) - J

(Au)(Xr)dr),

but for every y G R" and u G D(A) we have E» (u(Xt_s) - u(X0) - J

\Au)(Xr)dr)

pt — S

= Tt-,u(y) - u(y) - / Jo

TT{Au){x)dr - 0

by Lemma 1.4.1.14.C proving the theorem.

•

Remark 3.6.6. Note that nothing will change in the above proof if {!F^)t>o is substituted by any filtration (^t)t>o turning (Xt)t>o into a Markov process (with cadlag paths).

3.6 The Martingale Problem for Feller Processes

111

It is noteworth that we may use martingales in order to characterize elements in D(A). Proposition 3.6.7. Suppose that (Xt)t>o is as in Theorem 3.6.5 and suppose that toue Coo(Rn) there exists g G Coo(Mn) such that

(u(Xt) - u(X0) - f g(Xr)dr) j

\

/t>o

0

is a (Tt)t>o martingale with respect to every Px,x G W1. Then u G D(A) and Au = g. Proof. For x £ Rn we have Ttu(x) - u(x) = f Tsg(x)ds Jo and therefore sup -(Ttu(x) - u(x)) - g{x)

x€R n

l

1 /"*

= sup - / (Tsg(x) - g{x))ds x€R" I JO

< -1 f \\T,g - ffl^ ds —» 0 Jo

as t —* 0

proving the proposition.

•

Remark 3.6.8. Note that the last results allow us to extend the definition of a generator of a Feller semigroup: We may call u G B(Rn) to belong to De(A), the extended domain of A, if there exists g G B(Rn) such that almost surely /„* \g(Xr)\dr < oo for t > 0, and (u(Xt) - u(X0) - / \ Jo

g(Xs)ds) >t>0

is a right-continuous (or cadlag) (^rt)t>o-martingale with respect to every ,x t K . Our aim is to use the expression M" in (3.163) in order to construct "many" martingales, sufficiently many to recover the Feller process associated with the Feller semigroup generated by (A, D(A)). Given a generator

112

Chapter 3 Feller Processes

(A,D(A)) of a (conservative) Feller semigroup (Tt)t>o on CooQR"). The process which we want to construct out of the martingales (M")t>o should be a Hence it will have a cadlag Feller process (fl,A,Px,(Xt)t>o,(^t)t>o)xeRnmodification. Therefore we may try to reduce the problem by assuming that fi = £>n([0, oo)) equipped with the Skorohod topology and the associated Borel-cr-field A = Avn which makes the projections Xt : r>n([0,oo)) —> M™, LJ i-> w{t) measurable, compare Theorem 3.3.12. As filtration we may start with the canonical filtration. Now we face two problems: 1. For a given initial distribution \x on B^ find a probability measure P^ on (T>n([0, oo)), Avn) such that under P^ a martingale (M") t > 0 is given on (T>n([0, oo)), Az>n, PM) for every u e -D(^4) (or u out of a "nice" subset oiD(A)). 2. Use the family of martingales constructed in this manner, i.e. the measures PM, in order to obtain a Feller process, more precisely, try to prove that ((Xt)t>o, Px)xeRn is a Feller process. Let us give some precise definitions. Definition 3.6.9. Let A : D(A) -> Bb(E.n) , D(A) C Bb(Rn), be a linear operator. A probability measure P on 2?n([0,oo)) is called a solution to the T>n-martingale problem for the operator [A, D(A)) if for every u G D(A) (Mnt>o := (u(Xt) - ( (Au)(Xs)ds) \ Jo ' t>o

(3.164)

is a martingale on (P n ([0,oo)),^r» n ,P, (^)t>o)- We call the Pn-martingale problem well posed if for every initial distribution /x £ M\ (R™) there is a unique solution PM to the Pn-martingale problem for (A,D(A)) which satisfies (XO)PM

= M,

(3-165)

i.e. /x is the initial distribution of the process (Xt)t>o defined on (X» n ([0 l oo)) > ^ B ,P"). Let us make explicite that by (M^)t>o a martingale under P is given, i.e. we have for u € D(A) and s
P-a.s.

(3.166)

3.6 The Martingale Problem for Feller Processes

113

where E stands for the (conditional) expectation with respect to P. Clearly (3.166) is equivalent to m

m

E(M? H hv{X.v)) = E(M? I ] KiXsS)

(3-167)

or

0 = E((u(Xt) - u{Xs) - / (Au)(Xr)dr) JJ hv(X.u)) m

= E((u(Xt) - u(Xs)) J ] hv{X,S)

(3-168)

E((Au)(Xr)l[hu(XSL,))dr for all m G N, 0 < si < ... < sm < s o, P) solves the martingale problem is a statement on the finite-dimensional distributions of ((Xt)t>o,P). As a first result we prove Theorem 3.6.10. Suppose that the Vn-martingale problem for (A, D(A)) is well posed. Then for any distribution fi G Mi(M.n) a Markov process is given by((Xt),P»). Proof. (St. Ethier and Th. Kurtz [98]). Let ((Xt)t>o, (Jf)t>o,P M ) be the unique solution to the 2?n-martingale problem for the operator (A, D(A)) and initial distribution fi. For r > 0 and F e Tr, pi*(F) > 0, fixed we define F

i \B) —

pl^F)

'

(3.169)

•

(3 170)

and M

2 {

' "=

E(XFE(XB

\Xr))

PT(F)

-

With Yt := Xr+t it follows that

if (r0 e r) = P?(Y0 e r) = P"(xr G T\F).

(3.171)

114

Chapter 3 Feller Processes

Furthermore, for 0 < h < t2 < • • • < tm < tm+1,u G D(A) and hv £ Bb(Rn), v = 1 , . . . , m, we define

r,(Y.) := (u(Ytm+1) - u(YtJ - / By (3.168) we have E(rj(Xr+.)\J^)

MMy,) =

(Au)(yr)dr) ]\ K(YU).

(3.172)

= 0 implying

(3.I73)

EiM^ml,0

and (3.147) implying that (Yt)t>o is a solution for the Dn-martingale problem for [A, D(A)) on (X> n ([0,oo)),Ai,^,(J?)t>o) and on (2?n([0,oo)), An,P%, (J$)t>o). By the uniqueness assumption we find for all u € D(A) and t > 0 El(u{Yt))=E2(u{Yt)), or E{XFE{u{XT+t)\J*)) = E(xF^(«(^r+t)|X r )).

(3.175)

But F, P(^) > 0, was arbitrary, thus we arrive at E(u(Xr+t)\f?)

= E(u(Xr+t)\Xr)

P"-a.s.

(3.176)

which proves the theorem. • We will need some extensions of the results of Theorem 3.6.10, but these we will discuss a bit later. Let us indicate how to get existence results. Suppose that for a given kernel fi(x, dx) on K n x B^ a bounded operator is given by A : Coo(Rn) -> Coo(R") u i-> Au(x) — I (u(y) - u(x))fi(x, dy)

(3.177)

which extends onto Bb(Rn). As a bounded operator, A generates a strongly continuous semigroup (T t ) t > 0 on Coo(Kn) • Since for u(x0) = supw(a;) > 0 it follows that Au(x0) = /

(u(y)-u(x0))fi(xo,dy)<0,

3.7 Levy Processes and Translation Invariant Feller Semigroups

115

i.e. A satisfies the positive maximum principle, we conclude that (Tt)t>o is a Feller semigroup. Hence the £>n-martingale problem for A is solvable, compare Theorem 3.6.5. If now (A, D(A)) is any (densely defined) operator on Coo(Mn) satisfying the positive maximum principle, we may try, having in mind the structure result for such operators, i.e. Courrege's theorem, Theorem 1.4.5.21, to approximate in some sence A by a sequence of operators of type (3.177). Then we will get a sequence of corresponding probability measures on (Pn([0, oo)), Aj)n) and there might be hope that this sequence converges to a solution to the Pn-martingale problem for A. We will come back to all these problems, but now it is time to discuss classes of examples.

3.7

Levy Processes and Translation Invariant Feller Semigroups

To understand the importance of Levy processes for our theory let us recall some basic ideas from the theory of second order elliptic differential operators and the associated (elliptic) diffusion processes. The proto-type of all second order elliptic differential operators is the Laplacian — A n — — X^=i J^? with symbol a(-A)(x,£) — |£|2. A second order differential operator

L(x,D):=-Y,aki(x)——-+J2bj(x) ax

k,l=l

kOXi

+ c(x)

j = 1

is called elliptic if its principal symbol <rpr (L(x, D)) (x, £) = satisfies the ellipticity condition apr{x,0

J2l,i=iakl(x)^i

> Ao(z)|£|2 , Xo(x) > 0.

(3.178)

Let us restrict our considerations to uniformly elliptic operators with bounded coefficients. Then we find Ao|£|2 < apr(L(x,D))(x,0

< X1\^\2

(3.179)

for all l e i * and ^ £ l " with 0 < Ao, Ai € R. If we don't want to use the principal symbol, but the full symbol n

n

k,i=i

j=i


116

Chapter 3 Feller Processes

then we may consider instead of (3.179) Ao|C|2 < a(L(x,D))(x,£) < X^]2

(3.180)

for all x £ R™ and all £ € -Bg(O), where g > 0 is sufficiently large. Thus we compare the symbol of a second order elliptic differential operator with the symbol of the Laplacian, and typical results for L(x,D) are thought to be related to analogous results for the Laplacian. An example for this philosophy are the Aronson estimates, compare II.2.442. These estimates have of course the interpretation as estimates for the transition functions of the diffusion (Xt)t>o associated with L(x,D), i.e. they give information on the finite-dimensional distributions of (Xt)t>o by comparing them with those of Brownian motion. Hence they provide us with almost all what we really want to know about (Xt)t>oThe theory of pseudo-differential operators q(x,D) with negative definite symbols q(x, £) as developed in Volume II often relies on the estimates

W(0
(3-181)

which are supposed to hold for all x £ M",£ G Bg(Q),g > 0 large. Therefore the stochastic process associated with the operator —t})(D) should have for the process generated by —q(x, D) the same role as Brownian motion for elliptic diffusions, i.e. diffusion processes generated by second order elliptic differential operators. As we will see later in this paragraph the above philosophy is not working as smoothly as we have just indicated. This is best seen by discussing examples. Therefore, partly this section serves also to provide concrete examples for case studies. The processes associated with a continuous negative definite function are Levy processes which we are going to introduce in this section. Let ip : R™ - > C b e a continuous negative definite function and suppose for simplicity that V'(O) = 0, i.e. the constant in the Levy-Khinchin representation of ip vanishes, compare Theorem 1.3.7.7. By (nt)t>o we denote the corresponding convolution semigroup, i.e. we have

faiS) = (27r)-^e-t^).

(3.182)

According to Example 3.2.7 by pt(x,B):=fit(B-x)

(3.183)

3.7 Levy Processes and Translation Invariant Feller Semigroups

117

we obtain a semigroup of Markovian kernels and by Example 1.4.13 the family (Tt)t>0 define on Coo(»") by Ttu{x) = f u(x- y)iH(dy)

(3.184)

JUL"

is a (conservative) Feller semigroup. Thus we may associate with ip a (universal) Feller process (fi, A, Px, (Xt)t>o, (Ft)t>o)xeRn where the filtration is rightcontinuous and complete. Moreover almost surely all paths X.(tj) : [0, oo) —> M.n can be assumed to be cadlag. In addition for every initial distribution /i £ Mi(R n ), the Markov process ((Xt)t>o, -PM) is stochastically continuous, P" = fP*IM(dx). We want to explore some properties of the processes ((Xt)t>o, PM) and for this we need a few definitions. Definition 3.7.1. Let ((Xt)t>o,Pfl) be a stochastic process with state space (M n ,$("'). A. We say that (Xt)t>o has independent increments if for all 0 < s < t the random variable Xt — Xs is independent of Ts. B. The process (Xt)t>o is said to have stationary increments if for all 0 < s < t it follows that (3-185)

PZt-x.=P£t..-

Theorem 3.7.2. The process ({Xt)t>o,P^) constructed from the Feller process associated with the continuous negative definite junction ijj and the initial distribution /J, has stationary and independent increments. Proof. Denote by (/j,t)t>o the convolution semigroup of probability measures associated with ip by (3.182). First we prove p

xt-x3

= IH-, for 0 < s < t,

(3.186)

which will imply that the increments of ((X t ) t > 0 ,P M ) are stationary. Note that for t = s we have Xt-Xt ~ ^0 -

£

0,

i.e. in this case (3.186) holds. Thus suppose that s < t and put Y := Xs ® Xt, so Y is the product (mapping) of Xs and Xt, and further set q : W1 x W1 —> M™, (^1)^2) — • > %2 — x\. With these definitions we have pM

_ pM

118

Chapter 3 Feller Processes

For Q := q~x{B), B £ B^n\ we find P»{Xt -XseB)

= P"( g o Y e B) = P»(Y e Q)

and the construction of PM yields with (3.183) P^iXt - Xa G B) = / / /

XQ(xi,X2)pt-s{xi,dx2)ps(x,dx1)fj,(dx)

= ///

Xx1+B{x2)pt-s(xi,dx2)ps{x,dx1)fi(dx)

=

pt-s{xi,xi

+ B)ps(x,dxi)[i(dx)

=IH-.(B) JPs(x,Rn)fi(dx)

= fit-s(B),

where we used that Pt-.(x1,x1+B)=Pt-.(0,B)

= txt-s{B)

holds as well as the fact that ps(x,Wl) = 1 and fx is a probability measure. Thus (3.186) is shown for s < t and therefore the stationarity of the increments. Next we prove that the increments are independent. For this take to = 0
are independent. According to Theorem 3.1.4 we have to show that for Z = Yo <8> Y1 (2>... ® Ym m DM _ / O \ pM

3=0

holds, i.e. for Bo,...,

Bm e B^ we shall prove m

Pg(B0 x...xBm)

= H P% (Bj). j=0

(3.187)

3.7 Levy Processes and Translation Invariant Feller Semigroups

119

With Xt_1 = 0 we find using the transformation theorem

Pg(B0 x...xBm)

=J

XB0*...xBmdP$0^9Ym o (y 0 ® • • • ® Ym)dP"

= JxB0x...xBm . m J

j=o 771

/

j=0

From the construction of P M we find for P/J

t

,, the joint distribution of

and « e B t ( ( i " ) m + 1 ) that

Xt0,...,Xtm,

/ u{x0, x i , . . . , x m )P{ to> ... )tm }(d(x 0 , • • . , z m ) ) = / • • • / u ( i 0 , a;o + x i , . . . , x0 + ... + xm)fitm-tm-i

(dxm)...

x /xt2_tl(da;2)Mt1-to(da;i)At(da;o) implying with /xto = /J.O = e0

/

m

TT

f

t m }(d(a;o,...,a; m )) = / X B o ( ^ ) J I x B j f e -Zj-i)-P{t 0 = / / • • • XBoiX-x + X0)YlxBj(Xj)lJ,trn-tm-1{dxm) • • • Uti-toidX!) JJ x

J

j=1

^to(dx0)n{dx-1),

and further m

P^(Bo x ... x Bm) = /x(B0) I ]

IH.-U.ABJ).

Now, P^(5 (3.186) yields 0 x ... x^.^ABj) Bm) = M(5O)=I P^(Bj), ] Fv|, ( S J)-or m

(3-188)

120

Chapter 3 Feller Processes

Finally we have to observe that YQ = Xt0 = XQ and that

PP0(B0) = jPo(x,B0)Kdx)

= fi(B0)

which together with (3.188) implies (3.187) and therefore the theorem.

•

Having Theorem 3.7.2 in mind we give Definition 3.7.3. A stochastic process (Ci,A,P, (Xt)t>0, (Tt)t>o) with state space (R n ,S' n ') is called Levy process (or process with stationary and independent increments) if i) (Xt)t>o is .^-adapted; ii) (Xt)t>o has independent increments; iii) (Xt)t>o has stationary increments; iv) (Xt)t>o is stochastically continuous; Corollary 3.7.4. The process ((Xt)t>o,PM) constructed from the Feller process associated with the continuous negative definite function tp and the initial distribution fi is a Levy process. Let [(Xt)t>o, P) be a Levy process with state space (Mn, B^). By stochastic continuity it follows that Pxt —> Pxs vaguely as t —* s, i.e. for all u e C0(Mn) we have / u dPxt

—> / u dPXs

as t —> s.

Therefore Pxt-x0 converges vaguely to Px.-xQ as t —> s. Thus for a given Levy process with state space (M.n,B^) a family of probability measures (Mt)t>o o n &^ is denned by Hf-=Pxt-x0,

(3-189)

and this family is vaguely continuous, i.e. /xt —» ns vaguely as t —> s. In addition we have Mo = Pxo-Xo = ^o = £o,

(3-190)

3.7 Levy Processes and Translation Invariant Feller Semigroups

121

and lim^t = £0 (vague limit). (3.191) t-»o Further, since Xs+t — Xt and Xt — XQ are independent random variables we find, compare Theorem 2.4.11. Ma+t = Pxs+t-X0

= Pxs+t-Xt+Xt-XQ

= Pxa+t-Xt * Pxt-X0

and the stationarity of the increments yields Px3+t-xt

= Pxs-x0

= Ms,

and we arrive at (3.192)

Hs+t = ns*Ht for all t, s > 0. Hence we have proved

Corollary 3.7.5. Let (Xt)t>o be a Levy process with state space (M",S' n ^). Then the family (Pxt-xo)t>o is a convolution semigroup on Rn. The next theorem identifies a given Levy process with the (canonical) process generated by the convolution semigroup (/Ut)t>o,Att = Pxt-x0Theorem 3.7.6. Let (fi,«4, P, (Xt)t>o, {Ft)t>o) be a Levy process with state space (]Rn,Z?(n)) and denote by (/Ut)t>0)Mt = Pxt-xQ the corresponding convolution semigroup. Then the finite dimensional distributions of (Xt)t>o coincide with those of the canonical process (Xt)t>o associated with (/J.t)t>o o,nd the initial distribution /x := Px0, i.e. (Xt)t>o and (Xt)t>o are versions of each others. Proof. Let to = 0 < t\ < ... < tm and denote by Qto,...,tm the joint distribution of the random variables Xto,.. .,Xtm. Further let T : ( R " ) m + 1 -> (K n ) m + 1 be the mapping T(xo,..., xm) = (XQ, X0 - xi,..., xm - xm-i). For a Borel measurable function u : ( R " ) m + 1 —> [0, oo] it follows now

y udQto,...,*™ = Juo (Xt0 ® ... ® Xtm) dP

= fuoT-1oT{Xt0®...®Xtm)&P =

JuoT-\Xt0,Xtl-Xt0,...,Xtm-Xtm_1)dP

122

Chapter 3 Feller Processes

where r is the joint distribution of t h e increments Xto, Xtl —Xto,..., Since r = /j, /x tl ® . . . (g> /x t m _t m _ 1 we find

Xtm

-tm-i-

/ udQt O i ..., t m = / • • • / u(a; 0) a;o + a ; i , . . . , x o + a;i + • • • + z m ) x

lHm-tm-i

( d a ; m ) . . . /z tl (dzi)/i(d:ro).

On t h e other h a n d we have

/ u d P £ o ] t m } = / • • • / u ( a ; _ i + x 0 , . . . , a;_i + a;0 + ... + x m ) x

^^-tm-iidxm)...

fito(dx0)fJ.(dx-1)

= / • • • / u(a;_i,a;_i + x i , . . . , a ; _ i + x i + . . . + z m ) x

fJ'tm-tm-i (daj m )... / i t l (da;i)/i(da;_i)

where Pf^ t , is defined accordingly to (3.44), compare also Corollary 3.2.12, and the last line follows since fio = /xto = eo- Thus we have proved for all u as above

/«d&o tm =

JudP{totm}

implying Pxt0®...®xtm = ?{t0

(3.193)

tmY

In order to establish the theorem we still have to consider the case For this take B € (B^)m and t0 = 0. By (3.193) we 0
... x

XR"XB(XO,XO

tm} (R"

x B)

+ xi,... ,x0 + xi + ... + xm)

(J>tm-tm-i(dXm)..-HtAdxi)lJ-(dxo)

= / • • • / XB(XO,XO + xi,... ,x0 + xi... + xm) x IMm-tm-i(dxm)

•••

fJ,tl(^xi)n(dx0)

=K t^w proving the theorem.

•

3.7 Levy Processes and Translation Invariant Feller Semigroups

123

In the following, when discussing Levy processes we also assume that they have cadlag paths (almost surely) and if necessary we assume that the underlying nitration is right-continuous and complete. This is of course best justified by our previous results on Feller processes and Theorem 3.7.2 and Theorem 3.7.6. Let (Xt)t>o be a Levy process with state space (M.n,B^) and associated convolution semigroup (nt)t>o,^t '•= Pxt-Xo- We know that (/J.t)t>o is completely characterized by a continuous negative definite function ip : M" —• C , ip(0) = 0, since we assume the process to be conservative. We are longing for a probabilistic interpretation of I/J and start with Definition 3.7.7. Let Y be a n-dimensional random variable on the probability space (Q,A,P). Its characteristic function Ay : Rn —> C is denned by Ay(O := E(eiY<).

(3.194)

Denoting as usual by Py the distribution of Y under P, the transformation theorem yields

M 0 = fe^xPY(dx),

(3.195)

i.e. XY(O = (27r)*(iV)A(O =

(2TT^F-1(PY)(O,

(3.196)

and therefore Ay shares the properties of Fourier transforms of probability measures. In particular, Ay is a positive definite function on R" bounded by 1. Remark 3.7.8. Of course it would be more straightforward to define Ay as the Fourier transform of PY but we would run into some trouble with different definitions of the Fourier transform in probability theory and analysis. Now, let (Xt)t>o be a Levy process with state space (M n ,S (n) ) and denote by (Mt)t>o the convolution semigroup \it := Pxt-x0- Each random variable Xt — XQ> has a characteristic function and we find using (1.3.112), i.e.

MZ) = (2*)-*e-*«\

124

Chapter 3 Feller Processes

that

= f eix^t(dx)

=

(2TT)-S/I(-0

= e- t t / ) ( - ? ) =

e'*^.

The continuous negative definite function i[i is called the characteristic exponent of the Levy process (Xt)t>o- On the other hand, the continuous negative definite function ip is the negative symbol of the pseudo differential operator generating the Feller semigroup associated with (/xt)t>o> i-e-

-i/>(D)u(x) := -(27r)t f extends from

CQ°(R")

e te -«V(O^)d^

to the generator of the Feller semigroup

Ttu(x)= f u(x-y)lH(dy) = (2w)-i f ete*e-^«>u(£)d£Definition 3.7.9. Let (Xt)t>0 be Levy process with state space (M™,S'™') and corresponding convolution semigroup {/J.t)t>o,^t = Pxt-x0, We call the continuous negative definite function ip ]ut(O = (2n)~^e~t^'^. the symbol of the Levy process (Xt)t>oRemark 3.7.10. Due to the tradition of different definitions of the Fourier transformation we have to distinguish for a given Levy process - the symbol of its generator which is —ip; - the characteristic exponent which is ip. - the symbol which is ip. In order to avoid confusions we will use in the following the notion symbol of the process and almost never that of the characteristic exponent. Every continuous negative definite function ij) has a Levy-Khintchin representation, compare Theorem 1.3.7.7 and (1.3.184),

V(0 =c + i(d-Q + q(0 + /

( l - e-te>« - 7 ^ 4 ) «/(<**),

^R"\{0} V

l

+ \x\ J

where the Levy measure v satisfies / ( | x | 2 A l)^(d:r) < oo. We call (c,d,q,v) the Levy quadruple characterising ip, and if V'(O) — 0> i-e- c = 0, then we

3.7 Levy Processes and Translation Invariant Feller Semigroups

125

call (d, q, v) the Levy triple characterising ip. Let us remind the reader on Remark 1.3.7.10: in the Levy triple (or quadruple) of ip the items d and u are not independent of each other and they depend on the choice of the cut-off function in the Levy-Khinchin formula. If (Xt)t>o is a Levy process associated with the negative definite function tj) the term Levy quadruple or triple is also attached to {Xt)t>oUsing the calculation done in discussing Example 1.4.1.12. we find a probabilistic expression for ifi V(0 = - l i m ^ -1(3.197) t-»o t Let (Xt)t>o be a Levy process with state space (IRn, £?("') and symbol tp. Further let / be a Bernstein function. We know that / o ip is once again a continuous negative definite function. Hence it is the symbol of a Levy We call (X/) t > 0 the Levy process process (X{)t>o with state space (E.n,B^). subordinate to {Xt)t>o by the Bernstein function / , or equivalently by the corresponding convolution semigroup (r]t)t>o, compare Theorem 1.3.9.7. We are longing for a probabilistic interpretation of (X/) t > 0 . For this we need to discuss more carefully paths properties and path-wise decompositions for Levy processes, see Section 3.8, which requires some knowledge of certain special classes of Levy processes. The remaining part of this section is therefore (and because of some reasons already mentioned above) devoted to examples of Levy processes. We start our investigation of special Levy processes with the Poisson process. Clearly, there are a plenty of ways to introduce the Poisson process and there is a universe of literature about the Poisson process. We have chosen a way which gives emphasis to the point important for our purpose namely to understand the role of the symbol when studying the process. The Poisson semigroup on M with jumps of size A > 0 and intensity v > 0 is given, compare Table 1.3.9.19, by M* = Z ^ e fc=0

jfei

6xk

'

( 3 - 198 )

and the corresponding negative definite function ip is given by V>(0 = 1/(1 - e " a ? ) -

(3.199)

By our general theory we know that there exists a Feller process associated with (/Zi)t>o and we may work with its cadlag version as well as with a complete

126

Chapter 3 Feller Processes

and right-continuous nitration. This process we call the Poisson process with intensity v and jumps of size A. Thus the Poisson process has stationary and independent increments and is stochastically continuous. We want to give a different characterization of the Poisson process and need some preparations. Let (fi, A, P) be a probability space and (7t)t>o be a filtration in A which is complete and right-continuous. In addition let (Tk)k>o be a strictly increasing sequence of non-negative random times, Tfc : fi —> [0, oo] with To = 0 almost surely, i.e. we have Tfc(w) < Tfc+i(a>) almost surely. Definition 3.7.11. We call the stochastic process (Nt)t>o denned on (fi, A, P) by Nt(u) := J2 AX{t>rfc(w)}, A > 0,

(3.200)

fc>i

the counting process associated with the sequence (Tk)ken and A. With T(w) := sup Tfc (u) < oo being the explosion time of (Nt)t>o we define [TfcH.oo) := {{t,u);Nt(u>) > k}, [Tk(u),Tk+1(u;)) := {(t,u);Nt(w) = k}

(3.201) (3.202)

and [T(w), oo) := {(t,w); Nt(u>) = oo}.

(3.203)

If T = oo almost surely we call (Nt)t>o a counting process without explosion. The increments of (iVt)t>o are given by Nt-Ns

= ^2\X{30

and Nt~^Na counts the numbers of random times falling into (s,i\. Thus v — E{ *~^ t - 1 ) gives the expected number of jumps in a time interval of length 1. It is easy to see that a counting process (Nt)t>o is adapted to a filtration (J7t)t>o if and only if the defining sequence of random times (Tk)ken0 is adapted. Now we can prove T h e o r e m 3.7.12. The Poisson process (Xt)t>o with symbol given by (3.199) (and therefore the characteristic exponent is ijj(£) = v(l — e%x^)) and with initial

3.7 Levy Processes and Translation Invariant Feller Semigroups

127

distribution e0 is a counting process without explosion and jumps of size A. In addition it holds E{Xt) = Xvt, V(Xt) = X2vt.

(3.204)

Proof. Since (3.204) follows from (3.198) we need to establish the path properties of (Xt)t>o which we already assume to be a cadlag process. Prom (3.198) we find that P(Xt = Xk) = ^ e " " * and therefore for every countable set Q C [0, oo) it follows that P(Xt e AN0; t G Q) = 1. Thus, if Q is dense in [0, oo) the cadlag property implies that almost surely (Xt)t>o takes only values in ANo- In addition, since Pxt+s-xt = Pxs and since Xs > 0 a.s., it follows that the paths of (Xt)t>o are almost surely nondecreasing functions. Hence we have established that {Xt)t>o has (a version with) values in ANo and only positive jumps. It remains to prove that almost surely these jumps have the size A > 0 and that there are only finitely many jumps in finite time. As usual we denote by Xt- the left limit of Xt. We are going to show that P ( sup (Xt - Xt-) < A) = 1. 0
For the cadlag version of any Levy process we have always P(\Xt - Xt.\ > 0) = 0 for all t > 0, i.e. there are almost surely no fixed jumps. Thus for t G Q+ we first observe that P(Xt - Xt- > 0) = 0 and it follows that sup (Xt — Xt-) = lim

0
max (Xk_ — Xk^x) a.s., m m

irnooKKm

and further P( max ( I , - Xk-i)

< A) = P(Xx. < \)m = e~» (l + -)m -» 1

as m —» oo which proves the theorem.

•

128

Chapter 3 Feller Processes For (Xt)t>o as in Theorem 3.7.12 the following times are stopping times To

= 0 , n := inf{i > 0 ; Xt - Xt- = A},

Tfc+i := inf{£ > Tfe ; Xt - Xt- = A} and we have

fc>l For smooth functions, for example u € <S(R), we may calculate the generator and find of the Feller semigroup (Tt)t>0 , Ttu{x) = Ex(u(Xt)), Au(x) =

-(2TT)-5

I eixt{v{l

= v(u(x — A) -

- e - iAf ))i2(C)dC

(3.205)

u(x)),

i.e. A is a difference operator. Our next aim is to find an analog to a Poisson process when the state space is R n . It turns out that the first idea to consider instead of (3.199) the continuous negative definite function

1>(£) = v(l - e- iA '«), A e (R+)n,Z e Rn, is not very successful. Instead we suggest to consider a convolution semigroup (nt)t>o on R™ satisfying MO = (27r)-t e -"'( 1 -( 2 -) t v(«))

(3.206)

where v > 0 and

(£)>£ ^ ^™> a n < i m addition by i > 1 — (2n)%o is indeed a convolution semigroup of probability measures on R". Obviously we have (3.207) fc=0

3.7 Levy Processes and Translation Invariant Feller Semigroups

129

Recall Theorem 2.4.11 which together with the convolution theorem yields that for independent random variables Y j , . . . , Ym we have

(PYl+...+Ym)A =

((2^)mPYl.....Pym.

Thus we may interpret ((2?r) ?<£>(£)) as the Fourier transform of the distribution of the sum Sk = Y\ 4- ... + Yk of independent random variables with the same distribution, namely a. Now, let (Yi)ieN be independent random variables on some probability space (fl,A,P) with the same distribution a and let us suppose that c({0}) = 0. We set Sk := Ylt=o ^ where YQ = 0. Further let {Nt)t>o be a Poisson process with intensity v and jumps of size 1 defined on (fi, A, P) and independent of ODieN- Consider the process (3.208)

Xt = SNt. For B € B^

the independence of (Yj)/eN and (Nt)t>o yields oo

P(Xt €B) = P(SNt € B) = Y; P(Nt = k)P(Sk G B) k=0

which gives

k=0

Thus we arrive at Theorem 3.7.13. Let (Yi)iG^ be a sequence of independent random variables with values in 1™ defined on the probability space (Cl, A, P) with identical distribution a,o be a Poisson process defined on (fl, A, P) with intensity v and size of the jumps equal to 1 independent of (yi)/gN- Then a Levy process is given by (Xt)t>o where Xt is given by Xt = 5jvt, with SN = X ) j l o ^ > ^o = 0. In addition Nt(w) gives the number of jumps of s H-» Xa(u>) for s € (0,i\, and the size of the kth jump is Sk(u)-Sk-1(w) = Yk(u>). Definition 3.7.14. The Levy process denned according to Theorem 3.7.13 is called the compound Poisson process associated with the intensity parameter v and the distribution a.

130

^

Chapter 3 Feller Processes

It is easy to determine the Levy-Khinchine formula for the continuous negative definite function

£_>,,(l-(27r)MO). In fact, since we have 1/(1 - ( 2 T T ) M O ) = v(v(Rn) - (27r)*(Fa)(O) = / (I - e-™*)v
(l-e-ix<)ua{dx),

=[

n

JU \{0}

where we used for the last line that cr({0}) = 0, it follows with 1 + \x 2 l 2 /x(da:) := va(dx), \x compare (1.3.183), that ^(l-(27r)^(O) =i(d • 0 + f kn\{0}\

=**•*)+[ JK"\{O}

(l- e~ix< -

lX

''*•' W ( d a : ) l + \x\2) K '

(3.209)

l

V

(l-e-<-^) -±^^x), l + | a ; / \x\ 2

2

where i/ij . , . f Xj 1 + |x| 2 ^—7:(T(Ax) = I f-pr—;—p3—H(dx). 2 7H-\{0} i + M A"\{o> i + M 2 l^l2

, /" d,- = /

When we denote by (X\3 )t>o the j t h coordinate process and by yfe component of 1^, a straightforward calculation yields E(X^) and

=

(2n)ntu^=0

the j t h

3.7 Levy Processes and Translation Invariant Feller Semigroups provided &

£=0

131

exists. In this case we find

E{X{ts)) = tuE{Y^)

= t f xjt/
note that ua is the Levy measure associated with the continuous negative definite function v(l - (2w)%ip(-). The process (X[ )t>o changes only by jumps X\3 — X\_ which are distributed by prj (a), i.e. the image of a under jrrj, and the expected number of jumps in a unit interval is v. Thus we have here a first indication that the Levy measure contains information on the structure, more precisely, on the jumps, of the paths of a Levy process. We will return to this point in Section 3.8. Let (Tt)t>o be the operator semigroup, say on Coo(]Rn), associated with the compound Poisson process discussed in Theorem 3.7.13. For its generator A we find on S(Rn) -Au(x) = (2TT)-? / eixiv(l = (2TT)-2

/

eixt f

r

JM"\{O}

or Au(x)


(1 - e-iv*)vo(dy)u(Z) d£

= (27r)-*/" f. JK"\{0} 7R"

= /

(2TT)I

elxS{l-e-iyS)u{t)dZv(T{dy)

{u{x) - u(x - y)) v{dy),

= / (u(x -y)JR"\{0}

u(x)) i/a(dy)

(3.210)

which for a(dy) = £\{dy) reduces to (3.205). The compound Poisson process has the interpretation of a time-changed process: From (Sk)k>i we pass to (Xt)t>o = (<5Vt)t>o- We will now turn to a new interpretation of subordination in the sense of Bochner, compare 1.3.9 and 1.4.3, in the context of Levy processes and time-changed processes. For this let (rjt)t>o be a convolution semigroup on R with supp r\t C [0, oo) for all t > 0. We denote by / the corresponding Bernstein function, see Theorem 1.3.9.7, i.e. f}t(y) = (27r)-ie-*^i»>

132

Chapter 3 Feller Processes

or

where C denotes the Laplace transform. Denote the cadlag version of the Levy process associated with (rjt)t>o and which is starting at 0 G M by (5t)t>o- It follows for t > s > 0 that P(St -SseA)

= P(St-t eA) = TH-a(A)

implying that (St)t>o is an almost surely increasing process obtaining its values only in [0, oo). Conversely, if (St)t>o is a Levy process with almost surely increasing paths starting at 0 G K, then we find for the corresponding convolution semigroup (rjt)t>o on K that Tfc((-oo,0)) = P(St G (-oo,0)) = 0, i.e. supp rjt C [0,oo). Definition 3.7.15. A Levy process with state space M and almost surely increasing paths starting at 0 G K is called a subordinator. Next let {Xt)t>o be a Levy process with state space R" associated with the convolution semigroup (/it)t>o a n d the continuous negative definite function V> : K n -» C, i.e.

ft(fl = (2W)-*c-**tt). In addition let us assume that (St)t>o and (X t )t>o are defined on the same probability space and that they are independent. Definition 3.7.16. Let (Xt)t>o and (<St)t>o t"e a s above. The stochastic process Yt{w) := XSt{u){v)

(3-211)

is said to be the process obtained from (Xt)t>o by subordination with respect to (St)t>o, or shortly the subordinate process. L e m m a 3.7.17. Let (Yt)t>o be the process subordinate to (Xt)t>o by (St)t>oThen it holds P(Yt G B) = [ Hs(B)rit(ds). Jo+

(3.212)

3.7 Levy Processes and Translation Invariant Feller Semigroups

133

Proof. The independence of (X t ) t >o and (St)t>o implies P(Yt GB)= P(XSt G B) /•oo

= /

Jo

P(XS e B\St = s)ds

= I™ P(XS £ B)r)t(ds) Jo = [ Jo

n.(B)TH(d3).

n

From Proposition 1.3.9.10 or more easily from Remark 1.3.9.11 we find from (3.212) that

(3.213)

P(Yt€B)=rf(B), i.e. we have

Theorem 3.7.18. Let (Xt)t>o and (St)t>o be as in Definition 3.7.16. The subordinate process (Yt)t>o is a Levy process corresponding to the convolution semigroup (fJ-{)t>o- The generator of the corresponding Feller or Lp-subMarkovian semigroup restricted to S(R") is given by Au(x) = - ( 2 T T ) - * /

e te «/(tf(O)«(Od£.

(3.214)

compare 1.4-3 or 11.2.9. Remark 3.7.19. Let {St)t>o be a subordinator with associated Bernstein function / , ,oo

(1 - e-xs)v{ds)

(3.215) ./o and let (Xt)t>o be a Levy process with state space R" associated with the convolution semigroup (fit)t>o , Mt(£) = (2ir)~^e~t^\ The Levy measure of the continuous negative definite function / o ip is given by f(x)=a + bx+

/•OO

af =b(i+ / ntv{dt) (3.216) Jo where /J, is the Levy measure associated with tjj and the integral is understood in a vague sense (compare Remark 1.3.9.11). For a proof of this result we refer to Chr. Berg and G. Forst [35], p.175, or K. Sato [310], p.l98ff.

134

Chapter 3

Feller Processes

We want to discuss one class of subordinate Levy processes in more detail, namely certain subordinate Brownian motions. By definition Brownian motion is the Markov process associated with the Brownian semigroup, i.e. the convolution semigroup associated with the continuous negative definite function £ l~^ l£|2- If wefixthe initial distribution we obtain a Levy process. The corresponding continuous negative definite function is of course again the function £ H~> l£|2> £ € ^™- The subordinate processes we are interested in are those subordinate to Brownian motion which do not have a killing or a diffusion part, i.e. according to Theorem 1.3.9.26 the continuous negative definite function associated with such a process is of the form 1>(t)= [

n

JR \{0}

(I - cos(y • O)m(\y\2)dy

(3.217)

where m(r) = [ e" r V(ds) Jo+ and v is a measure on (0,oo) such that J^s~^u(ds) < oo and 1 fx s~%~ v(ds) < oo. The corresponding Bernstein function / is then given by /(r) = /"°°(1 - e- rs )(47r S )t$(^)(d s ),

(3.218)

where ${v) is the image measure of v with respect to the mapping SH->$(S) = -L.

If (Tt)t>o denotes the operator semigroup on CooQR") or Lp(Mn) associated with such a Levy process we find on 5(R") for its generator A -Au(x) = (2TT)-? /

eix«V(O«(Od£

= (2TT)-* /

efa« /

= (2TT)-» /

(f

f = /

u(x + y) + u(x-y)\

VR"\{0}

( \u{x) V

(1 - cos(y • 0)m(\y\2) dy u{£) d£ efa«(l - cos(y • O)u(O^) m(\y\2)dy 0

I

L

. 2> )m(\yY)dy,

J

3.7 Levy Processes and Translation Invariant Feller Semigroups

135

or, using the symmetry of m(|-| 2 ),

Au(x) = /

n

(u(x + y) - u(x))m(\y\2)dy.

JR \{0}

In case where {r]f)t>o is the one-sided stable semigroup of order a € (0,1) we find

(3.219) i.e. ( j

~

Trfr(l-a)

r «+f

-^^a+f-

Thus on 5(M") we find

The corresponding stochastic processes are called symmetric stable processes (of order a). For a = \, i.e. £ H-> |£|, the corresponding process is of course the Cauchy process (Xt)t>o- Most important for us is the observation that the Cauchy process has no expectation since S(l*,l)=r(^)/i>w'W2)!ifldI

=

oo.

(3.220,

The Cauchy process is the Levy process with symbol £ i-> |£|. We are going to discuss now in detail a one-dimensional Levy process, the Meixner process, < Ai holds for some having a symbol £ i-> V ' M ( 0 such that A0 < 1+ |^jS| 0 < Ao < Ai, but which has many different properties than the Cauchy process, for example it has finite expectation. Thus already in the realm of Levy processes partly our philosophy that processes with comparable symbols have comparable properties break down, while of course other parts of this philosophy work very well, compare for example the following Proposition 3.7.21. In the following discussion of the Meixner process we rely mostly on the paper [52] by B. Bottcher and our joint paper [54]. However we emphasize that the original sources are B. Grigelionis' paper [131], W. Schoutens [325] and W. Schoutens and J. Teugels [326].

136

Chapter 3 Feller Processes

Definition 3.7.20. A real-valued Levy process with symbol

Vv*,a,b(0 = - i m £ + 2 6 ( l o g c o s h ( ° * ~ l b J - l o g c o s f - J J , £ e R, (3.221) where m e t , 5 > 0,a > 0 and — IT < b < •n is called a Meixner process. A Meixner process (X™1 'a' ) t > 0 has a density with respect to the Lebesgue measure A ^ given by (3.222) and for its finite expectation we find E(Xm,5,a,bj

= aSt t a n

I

+ mt)

(3.223)

and its variance is given by Var(X--) . ^

^

.

(3.224)

Further we find ReVw,a,&(0 = <^log(cosh(^) - sin 2 (^)) - 25 log cos (^) and

Im ipm,s,a,b(O = -m^ + 25arctan^-tanf-Jtanh^-r-JJ. Moreover it follows that

ReW<W>(£)>yl£l

(3-225)

for £ G R, |C| > ^ , as well as for some c0 > 0 |Im VmAa,6(01 < CO (1 + R e VmAa,b(0) , £ G »•

(3-226)

In addition it follows that |V- m Aa,6(0l
(3.227)

3.7 Levy Processes and Translation Invariant Feller Semigroups

137

implying with suitable constants 0 < Ao < Ai that Ao

<

l

+ -
(3228)

holds. Hence (X(m' '"' )t>o has a symbol comparable with the symbol of the Cauchy process, but (Xtm'(5'a'b)t>o has a finite expectation contrary to the Cauchy process. Therefore it does not make sense to try to get a pointwise comparison of corresponding transition functions: we will lose the integrability in case of the Meixner process! As a direct consequence of (3.226) and (3.227) we find by an application of Example 1.3.7.32 Proposition 3.7.21. The Meixner process (X™'S'a'b)t>o is associated with the non-symmetric Dirichlet form given on S(M.n) by

(3-229)

£(u,v) = [ 1>m,s,a,b(£)v(£)W)#-

Its domain D{£) is the classical Sobolev space H?(M.), hence (£,D(£)) is regular, and on //^(R) we have

£(u, v) =d f ^P-v(x)dx + \f JR

$ (u(x) - u(y)) (v(x) - v(y))

&X

I JR JR

Seb^)

, ,

•; r7-IMEVI (x — y)smh K a '

ydx

(3.230)

Remark 3.7.22. A. Prom (3.228) we derived that D{£) = H*(R) and in fact we have that £\(u, v) = £(u, v) + (u, v)0 is on H* (R) equivalent to the Dirichlet form (£C,H*(R)) associated with the Cauchy process in the sense that Xo£i(u,u) < £i{u,u) < \i£i(u,u) holds for all u G H* (W). Hence here our philosophy works very fine. B. We refer to 1.4.7, especially pp 417-18, where we discussed a representation of the drift part in (3.230) on ifi(R) with the help of fractional derivative, compare also [198].

138

Chapter 3 Feller Processes

Remark 3.7.23. A further Levy process of considerable interest, see 0 . Barndorff-Nielsen [22] and 0 . Barndorff-Nielsen and S. Levendorskii [25], is the normal inverse Gaussian process which has the symbol "0NIG(O

= -«m£ + 5(^a? ~(b + iO2 - Va2 - 62),£ e R,

(3.231)

where 0 < \b) < a, 5 > 0 and m £ R. Again as in the case of ipm,6,a,b the symbol V'NIG is comparable with the symbol of the Cauchy process but it has a finite expectation. Moreover its transition function has the density Pt{x)

=

£ ew^+Kx-mt)

V

V

^ t >)t

(3 232)

where

which differs much from (3.222). Thus we face the situation that |£|~V>NiG(£)~tfw,«,6(0

(3.233)

but one should not long for a pointwise comparison of the corresponding transition densities. Our final example of a Levy process is the most important Markov process of all — Brownian motion. As already mentioned the n-dimensional Brownian motion is the Levy process with symbol |£|2,£ £ M.n. Instead of listening some of the many known properties of Brownian motion or to give credit to its importance by quoting some results, we refer to A. N. Borodin's and P. Salminen's "Handbook of Brownian Motion" [51]. Within our considerations Brownian motion is an outsider: it has continuous paths and its analysis is related to an elliptic differential operator, i.e. to a local operator, namely the Laplacian. Again, we do not intend to make a lot of remarks to the interplay of second order elliptic differential operators and diffusion processes, here are just three references: R. Bass [27], L. C. G. Rogers and D. Williams [301]-[302], and D. Stroock and S. Varadhan [351]. However we want to supply a nice short proof due to R. Schilling [312] of the fact that Brownian motion has almost surely continuous paths using as assumption that it is already known that Brownian motion is a cadlag process. The latter fact follows indeed from our previous considerations.

3.7 Levy Processes and Translation Invariant Feller Semigroups

139

In the following let (Xt)t>o be a stochastic process on some probability space (fi, A, P) with state space (E™, B^). In addition we assume that it has almost surely cadlag paths, i.e. there exists a set M € A such that P(Af) = 0 and for all w £ Afc the mapping t *—> Xt(oj) is a cadlag function. For a partition II = {to,ti,... / C [0, oo) and (3 > 0 we define

,tk},to

< £i < . . . < £ & , of an interval

k

var^((X t ) t >o,/,n)( W ) = ^T\Xtj(uj) - Xti_x{u>)f.

(3.234)

Further we set J{t,u>) := Xt{uj) - Xt-{u)

:= Xt{uj) - limX s (w),

(3.235)

i.e. J(t,u) gives the size of a jump of the path s H-> XS(UJ) at time t. Let ( I I ^ g N , 11^ = {*o J*I '•••>*j(i/)}i ^ e a sequence of partitions of / . If II,, C Uu+i and if the meshes, mesh(IIy) := max (t^ — tj2\), of the par\<3
titions tend to zero as v —> oo, then we say that the sequence of partitions (IIJ / )^ € N becomes dense in I. Lemma 3.7.24. Let {Xt)t>o be a cadlag process with state space (M",S^"^) as above. For allT > 0 and for any sequence of partitions of[0,T] which becomes dense in [0, T], we have for almost all w 6 Q. Y \J{t,w)f t
< liminf var /3 ((X t )t>o, [0,T\,Hv){u). "-*°°

(3.236)

Proof. A cadlag function is bounded on bounded intervals and the number of jumps on [0, T] with size larger than e > 0 is finite, compare Section 3.3. Thus for w € Q,\Af,P(Af) = 0, there exists only a finite number N = N(u) = NT,e{u>) of jumps of 11-> Xt{u>) larger than e, i.e. \J(t,u/)\ > e. These jumps occur at (random) times T\{UJ) < ... < TJV(W). Fix w e fi\A/" and choose i/ such large that Tj{u>) is contained in the interval (im]_ 1 ,im]] or

140

Chapter 3 Feller Processes

[ O , ^ ] . Now it follows that

V

| J(t, uj)f = V liminf X&) (LJ) - X&) (UJ) < lim V XM - XW) 1/-.00 r—f

Xm

i

tm

< liminf var / j((Jf t ) t >o,

i-i

[0,T\,Uv).

Since the right-hand side is independent of e, the lemma follows as e —> 0.

•

Theorem 3.7.25. Let (Xt)t>0 be a cddldg process with state space (M.n,B^) as above. In addition assume that E(\Xt - Xsf)

< co|t - s\1+a

(3.237)

holds for all s,t > 0 and some constants a,/3,co > 0. Then almost all sample paths 11—> Xt(u>) are continuous. Proof. Let T > 0 be fixed an choose a sequence of partitions (Tl^)uS^, !!„ = {t^,... ,t|rj)}, of [0,T] that becomes dense. Applying Lemma 3.7.24 and Fatou's lemma we find

E(y2\J(t,-f) t
< sfliminfvar^((Xt)t>o,[O,r],nv)) "-00

< liminf £(vai7j(pr t )t>o, [0,T],n y ))

l{v)

^coliminfVl^-^

1+

< coliminf^eshCn,))" VftJ^ - &\) < coTliminf^esh^))11 = 0. V—»OO

3.8 A Summary of Some Path Properties of Levy Processes

141

Thus for T > 0 exists AfT e A, P{MT) = 0, such that for w e U\AfT the path t >-> Xt(w) has no jumps in [0,T]. Therefore, if {Tk)ke® is a sequence, Tfc > Tfe-i > 0, such that lim Tk — oo, we find for the set No = WkLi-^n k—»oo

that P(A/o) = 0 and for w € N§ the path 11-> X4(w) is continuous.

D

Now we can prove Corollary 3.7.26. An n-dimensional Brownian motion (Xt)t>o has almost surely continuous paths. Proof. The result will follow from Theorem 3.7.25 once we have proved £(|Xt-Xs|4)
(3.238)

But for 0 < s < t we have

[ |x|4Pxt-xs(dx)

E(\Xt~Xs\4)=

- / \x\4PXt_a(dx)

= f MVt-s(dz) =

1

f

(47r(i-s)) J^

=,

(t

" s)2 .(t-^k-^d,

(47T(t - S))

=

C l

2

•/

2

| t - S| ,

and the corollary is proved.

3.8

| a ,(2

w I n \x\Ae~'^r^dx 2

•

A Summary of Some Path Properties of Levy Processes

The analysis of the sample paths of a stochastic process is one of the core subjects in the field, and a lot of results are known for Levy processes. Our main interest in this monograph is to deduce from a certain comparison of a negative definite symbol q(x,£) with a fixed continuous negative definite function ip a comparison of properties of the process (-X't)t>o having q(x, £) as

142

Chapter 3 Feller Processes

its symbol (see (1.03)) with properties of the Levy process (Yt)t>o associated with ip. Since there are several good monographs and surveys dealing with paths properties of Levy processes we decided not to rework these results in detail, but to give a summarizing discussion, pointing out the concepts and results but not providing all proofs. Certainly we will give very precise references. Our topic is the Levy-Ito decomposition of the sample paths of a Levy process. This will give us also a deeper probabilistic understanding of the Levy-Khinchin formula, compare 1.3.7, especially Theorem 1.3.7.7. We follow in our presentation the monograph of K. Sato [310], but refer also to the notes. We need a few conventions to start with: We set No := No U {oo} = N U {0} U {co}, and we say that a random variable X is Poisson distributed with mean 0 (with mean +oo) if X = 0 almost surely (if X = +oo almost surely). Definition 3.8.1. Let (£l°,A°,P°) be a fixed probability space and (S,5, N o , LJ .-> N(B)(u>) be given. We call the family {N(B); B € S} a Poisson random measure on S with intensity measure a if i) N(B)(-),B a(B);

G S, is a Poisson-distributed random variable with mean

ii) if Bi,... ,Bk S <S are mutually disjoint, then the random variables iV(Bi)(.),..., N{Bk){-) are independent; iii) for every w g 0° a measure on S is given by N(-)(w) : 5 ^ I

+

,BM

N(B)(LJ).

If {N(B); B 6 S} is a Poisson random measure we use often the notation N(B,w) instead of N(B)(u), and consequently we will write N(dx,uj) for N(dx)(u)). The existence of a Poisson random measure is granted by the following proposition, compare K. Sato [310], Proposition 19.4, for a proof. Proposition 3.8.2. For any cr-finite measure space (S,5,cr) there exists on some probability space (fi°,.4 0 ,P 0 ) a Poisson random measure on £ with intensity measure a.

3.8 A Summary of Some Path Properties of Levy Processes

143

Further we have, following K. Sato [310], Proposition 19.5, Proposition 3.8.3. Let (E, <S, a) be a finite measure space and {N{B); B £ <S} be a Poisson random measure with intensity measure a and underlying probability space (Q,°,A°,P0). For a measurable function u : £ —» R" we define the mapping Y : ft0 -> Rn by (3.239)

Y(w) := [ u(x)N(dx,u). JT.

Then Y is a random variable, i.e. it is measurable, and we have i) the random variable Y is compound Poisson distributed with characteristic function Ay : K™ —> C given by A y (O=exp(/ 1 ( e i «^-l)a(da ; ) N ) )f

{

= exp (j yt-x _ l)au(dx)\ ,

(3-240)

where au is the image measure of a under u; ii) ifue

L2(E,a) then E(\Y\2) < oo and E{Y)=

(3.241)

f u(x)a{dx)

as well as Var(r) = E(\Y - E{Y)\2) = f \u(x)\2a(dx); Hi) for mutually disjoint sets B\,..., Yi{w):=

f u(x)N(dx,u;),l

B\. £ S the random variables =

l,...,k,

JBi

are independent. For 0 < f l < f c < o o w e write AO)b := A(a, b] := {x € Kn; a < \x\ < b} and AaiOO := A(a,oo) := {x e l ° ; a < |a:| < oo},

(3.242)

144

Chapter 3 Feller Processes

which yields A0,oo = K n \{0}. In addition we set H := (0,oo) x R"\{0} = (0,oo) x A0]OO. The Borel cr-field of H is denoted by B(H); a typical point h G H has components s and x, i.e. h = (s,x). If Q is a mesasure on B{H) we write for the corresponding integrals

f f(h)g(dh)= f

J(0,oo)xA 0 ] 0 0

JH

f(s,x)g(d(s,x)).

Following K. Sato [310], p.120, we state now Theorem 3.8.4 (Levy-Ito-Decomposition). Let (Xi)t>o be a Levy process on the probability space (fl, A, P) with state space M n and Levy triple (Q, d, v). On B(H) we define the measure v by v((0, t]xB) = tu(B) , B G B(n).

(3.243)

Further let QQ C fl,P(flo) = 1, be a set such that for all a; G f2o the paths t H-» Xt(uj) are cddlag functions. For C G B(H) we define (3.244) Then it holds i) by {J{C); C G B(H)} a Poisson random measure with intensity measure v is given, ii) for some set fii C £l,P{Vl{) — 1, we may define for all LJ G fii and t G [0, oo)

X}(u>) := lim / {xJ(d(s,x),w) -oy ( o,*]x A ( e ,i) + /

J(0,t]xA(l,oo)

-

xv(d(s,x))} (3>245)

xJ(d(s,x),Lj)

and the limit is uniform with respect to t on bounded intervals. Furthermore (Xt)t>Q is a Levy process with state space M.n and Levy triple (0,0,1/)/

3.8 A Summary of Some Path Properties of Levy Processes

145

Hi) for u> € Qi we define X?{LJ)

:= Xt(w) -

X](LO)

(3.246)

and there exists a set 0.2 C fi, P{0.2) = I, such that for all w £ f22 the path t i-> X?(UJ) are continuous. The process (X?)t>o is a Levy process with state space Rn and Levy triple (Q, d, 0); iv) the two processes (Xf)t>o and (X^)t>o are independent. Thus we have decomposed the Levy process (Xt)t>o into the sum of two independent processes (X})t>o and (Xf)t>o '•

(3.247)

Xt = Xl + X},

where one process is a pure jump process and the other has continuous paths. Remark 3.8.5. A. A proof of Theorem 3.8.4 is given in K. Sato [310], §20. B. Since the Levy triple (Q,,d,v) is in general not unique, compare Remark 3.7.10, the decomposition (3.247) is not unique either. But of course, once (Q, d, v) is fixed, then it is unique with respect to this Levy triple. Definition 3.8.6. A. Let (Xt)t>o be a Levy process with state space R™ and decomposition (3.247). We call (X^)t>o its jump part (with respect to the given Levy triple (Q,d, v)) and (X?)t>o its continuous part (with respect to the given Levy triple (Q, d,v)). B. The limit lim /

e-»0 7(0,t]xA(e,l]

{xJ(d(s,x),w) - xV(d(s,x))}

is called the compensated sum of jumps. If the Levy measure v of the process (Xt)t>o satisfies the additional condition r .<1|a;|j/(da;) < 00, then we may decompose (Xt)t>o a bit differently and will have a unique decomposition into jump and continuous part. Again we quote K. Sato [310], p.121. Theorem 3.8.7. Suppose that {Xt)t>o is a Levy process with state space R n and for its Levy measure v we assume

/ J\x\
IzKdz) < 00.

(3.248)

146

Chapter 3 Feller Processes

Further let ^ i and J{B,w) be defined as in Theorem 3.8.4. Then there exists a set fl3 c fii,P(fi3) = 1 such that X?{u)=

I

xJ(d(s,x),uj)

(3.249)

is defined for all t > 0 and {Xf)t>o is a Levy process with state space Mn and characteristic exponent f (e-ix^

(3.250)

- l)u(dx).

In addition for ui £ $72 n O3, JI2 as in Theorem 3.8.4, the process

Xfa) := Xt(w) - X?(u) has continuous paths, i.e. t *—» X^(w) is continuous for to E fiinf)3. Moreover the two processes (Xf)t>o and (Xt4)t>o are independent.

3.9

The Symbol of a Feller Process

When we combine Example 3.2.8 with the considerations preceeding Corollary 3.2.13, see p. 59-60, we find that every Feller semigroup (T"t)t>o on Coo(]R") gives rise to a Feller process {{Xt)t>o, Px)xemri• (In c a s e of non-conservative semigroups we have first to work in the one-point compactification of M.n.) The relation between (T t ) t > 0 and ((Xt)t>o, Px)xem. is of course given by Ex{u{Xt))=Ttu{x).

(3.251)

In 1.4.4.5 we studied the structure of generators of Feller semigroups and in 1.4.4.8 we had a closer look to extended Feller semigroups. We were led to the fact that generators of Feller semigroups are under some reasonable assumptions on their domains and their mapping properties extensions of pseudodifferential operators with negative definite symbols, i.e. the generators are completely characterized by their symbols. More precisely, if (Tt)t>o has an extension (f t ) t > 0 to 5 6 (M n ;E) mapping Bb(Rn;R) into C 6 (R";R); i.e. if (Tt)t>o is a strong Feller semigroup, compare Definition 1.4.8.6, then we have proved in Volume I, p. 444, that ftu(x) = (2TT)-* /

efa«At(a;,Ou(Od£

( 3 - 252 )

3.9 The Symbol of a Feller Process

147

holds on CQ 0 , where \t(x,Z) = e-ixtTt(ei(-'V)(x),

(3.253)

i.e. Tt has a representation as a pseudo-differential operator. Now, having in mind that Tt is obtained from Tt by using the kernel representation we first note that (3.251) extends to Ex(u{Xt)) = Ttu{x)

(3.254)

for all u S C&(Rn;R) or even £ b (R";R). Combining now (3.253) with (3.254) it follows that Xt(x,O=Ex(e^Xt-^<).

(3.255)

Thus in the case under consideration the symbol of Tt or Tt can be expressed in pure probabilistic terms. If ((Xt)t>o,Px)xGRn is a Levy process (lt)t>o associated with the convolution semigroup (/it)t>o, then it follows from the fact that the increments are stationary that \t(x,£) is independent of a;: \t(x,€) = Ex{ei{Yt~x^<) = E°(eiYtS) = (27r)-*/I(-O=e~ t * ( " € ) where tp is the continuous negative definite function associated with (/i*)t>o, i.e. Mt>o(0 = (27r)-*e-**«>. Thus the symbol of Tt is expressed by the characteristic exponent of the corresponding Levy process. (The reader may have noticed that in this discussion our presentation suffers from the fact that the Fourier transform is defined in the theory of stochastic processes usually by Ex(etXt'*) whereas we used always the definition u(£) = (2?r)~? JRn e~lx^u(x)dx for functions and its extension to measures and distributions.) ^Let us return to the general case. Following further 1.4.4.8 we find for (A,D(A)), the Cb-extension of (A,D(A)), where (A,D(A)) is the generator of (T t ) t > 0 , that on C£°(Rn) it holds

Au(x) = -(27r)-t / e^qfaduffidt,

(3.256)

148

Chapter 3 Feller Processes

where q(x,O = -e-ix^A(e^^)(x),

(3.257)

and, compare Volume I, p.445, e-^A(e<-'^)(x)

= lim t-*o

)[X)

- ^

.

t

Taking once again (3.254) into account we finally arrive at q{x, 0 = - lim — ^ t-*o

t

(3.258)

'—=•

which is (0.3). Hence q(x, £) is also completely expressed in probabilistic terms. Clearly, for a Levy process (Yt)t>0 we have q ( x , 0 = VKOi i-e- w e obtain its symbol in the sense of Definition 3.7.9, or equivalently, ip(~0 1S * n e characteristic exponent of the Levy process. This leads to a nice interpretation for q(x,£): If we fixed x G M™ then £ t-> q(x, £) is the symbol of a Levy process, thus at each point x G R™ we may associate with ((-^t)t>o,-P ;c ) xe ] Rn a Levy process (Y^)t>o, and "moving" from xi to X2 does mean to "visit" at each point on the way connecting xi and X2 a Levy process. Most interesting is of course what happens when we "move" along a path of (Xt)t>o- In [347] and [348] D. Stroock has followed a similar idea which he refers to K. Ito. We want to emphasize a different aspect and start with a definition which is now well justified: Definition 3.9.1. Let ((Xt)t>o, Px)xeRn q(x, 0 := - lim ^ t->o

be a Markov process. If the limit (3.259)

> t

exists, it is called the symbol of ((Xt)t>o,

Px)x£Rn•

Remark 3.9.2. A. In case of larger classes of strong Feller processes as well as for all Levy processes we have seen that the symbol of the process equals the symbol of —A, A being the generator of the corresponding operator semigroup. B. Suppose for a moment that (Xt)t>o is conservative. Then is a positive definite function with -{Ex{ei{-Xt~x^) - 1)

3.9 The Symbol of a Feller Process

149

is negative definite. Hence the pointwise limit (3.259) must be a negative definite function for x fixed. In case that (X t ) t >o is not conservative we note that ip(x, 0) < 1 and it follows that

-fEx(ei{Xt~xH)

- l) = -(Ex(e«Xt-xH)

-
is again a negative definite function with respect to £ for x being fixed. Now let ((Xt)t>o, Px)xe^n be a Feller process (or more generally a Markov process) with symbol q(x,£). Suppose we want to study (Xt)t>o after it has started in xo € K™ for a short time only. If (Ytx°)t>o is the Levy process with symbol V>(£) = o,.P Xo ) and (Ytx°)t>0 near the starting point XQ € Mn have quite similar behavior. If in addition there exists a fixed continuous negative definite function ip such that for all x € R n the function £ — i > q(x,£) behaves "essentially" as the function £ i-» ip(£), then nas properties analogous to (Yt)t>o we should expect that ((Xt)t>o, Px)x€Rn the Levy process with symbol I/J. In order to get some flavour for results of such type we will briefly discuss two of them following our survey [199] with R. Schilling. The following results are due to him and we refer to his papers [311], [316] and [318], and to his forthcoming monograph [323]. Let tjj : R" —> C be a continuous negative definite function and define the Blumenthal-Getoor upper index by P := /3(V») := inf (A > 0 ; lim sup ^ ^ - = o\.

(3.260)

A result which is classical by now and due to R. Blumenthal and R. Getoor [45] states for the Levy process (Yi)t>o with symbol ip that dimH{Yt{u>); t G E] < 0dimHE

a.s.

(3.261)

where dim^f(C) is the Hausdorff dimension of a set C and E C [0,oo) is an analytic set, compare Definition II.3.1.1. Now let (A,D(A)) be a generator of a Feller process ((X t ) t >o, Px) R n such that C^°(K n ;K) C D{A) and ^|co~(K";K) = -q(x,D) is a pseudo-differential operator such that sup |q(a:,£)| < c(l + |£| 2 ) and that the sector condition xeMn

|Im q(x, 01 < K Re q(x, 0 holds for all x . ^ e l " . Define

{

sup A > 0; lim sup l«l-oo

sup|g(j/,|£|j7)| BL

-r7rx I£|A

> = 0 S. I

(3.262)

150

Chapter 3 Feller Processes

Then for every analytic set E c [0, oo) din^{X t (u); t € E} < (sup (/%,))dim H £

(3.263)

ySffi"

holds Px-almost surely for all x € Kn. In particular, if sup|g(z,OI
(3.264)

holds, then it follows that sup (/^) < /?(V)

(3.265)

y£Rn

and therefore we have dinif^XtH; t € E} < P(ip)dimHE.

(3.266)

Although (3.263) is the more sharp and general result, in estimate (3.266) we encounter in explicit form our general philosophy: (3.263) compares the symbols-(3.266) compares properties of the processes. Remark 3.9.3. We have taken the formulation (3.266) from [199], but it was originally proved in [315], the version given here follows from [315] in combination with results in [316]. The second result we want to discuss identifies paths of ((Xt)t>o, as elements of certain Besov spaces. For this we also need the index

{

sup^ sup |g(?/, |f|7?)| A > 0; lim sup > e R " l > ' 1 ^ , .

Px)xeRn

"j = 0\ .

(3.267)

We remind the reader to the definition of the Besov spaces S* g(M") as given in 1.3.11. We need now two extensions: We say that a function u belongs to B*;^oc(]Rn) if for every

\ \L + \X\ > aX I ' then we arrive at the weighted Besov space B* q (W1 ;(1 + |-| 2 )^). The following theorem is due to R. Schilling [316] and [320].

3.9 The Symbol of a Feller Process

151

Theorem 3.9.4. Let ((Xt)t>o,Px)x€Un be a Feller process (or more generally a Markov process) and for the generator (A, D(A)) of the corresponding semigroup assume that CQ°(M.";R) C D(A) and /l|c~(K";K) = ~q(x,D). Further assume that \q{x,£)\ < c(l -f VKO) and Vm0 define a® := sup {p, q, /?£,} if q < oo and

a^:=

sup{p,/?^}.

Then we have Px -almost surely (t ~ Xtvo(w)) G ^ , , ( R ; (1 + * 2 )- f ) /or so^ ^-

+ ^; (3.268)

0C

(i ^ X tvo (o0) G ^;!, (K) /or s a ^ < 1;

(3.269)

(t ^ Xty0(uj)) G s|,i'° c (R) /or p > sup {/3^}.

(3.270)

Moreover, if sp > 1 , p G (0, oo] or sp = 1,^ < 1, i/ien we /tawe P x almost surely that (t ^ Jftv0(w)) ^ B^,{OC(K). Remark 3.9.5. We stated this result in its most precise form. When we have however upper bounds for sup {/%>}, for example sup {Z?^} < /?(V)) for a fixed y€R"

yeWL"

continuous negative definite ip, then we obtain once again results of the type: comparison of symbols leads to comparable properties of the processes. In [191] we gave a lot of constructions of Feller semigroups generated by pseudo-differential operators. In most cases we posed conditions on a negative definite symbol q(x, £) by using a fixed continuous negative definite function tp. — Only when considering in II.2.10 operators of variable order of differentiability the condition were different. Thus the two results discussed above, many of those described in [199], and most of the results in R. Schilling's monograph [323] will apply to the (Feller) processes associated with the Feller semigroups constructed in [191].

152

3.10

Chapter 3 Feller Processes

Notes to Chapter 3

The construction of the canonical process is of course quite standard and our main source is H. Bauer [30] which we use in some sense indirectly since Section 3.1 and later sections are partly translations of our lecture notes [180]. The same remark applies to Section 3.2 and in both sections the reader will not encounter some unexpected material with exception of our discussion of the continuity of the transition function, Theorem 3.2.17-Lemma 3.2.21, compare also the references given in that part of Section 3.2. Section 3.3 starts with a discussion of versions and modifications of a given process and soon turns to cadlag functions. In our presentation we mixed several sources (as we did in [180], partly our template for this section). The basic properties of cadlag functions are proved by following essentially St. Ethier and Th. Kurtz [98], and this is also the main source for our discussion of the Skorohod space. Clearly, one has to mention here A. V. Skorohod's original paper [334], his monograph [335], the monographs of I. I. Gihman and A. V. Skorohod [121]-[123], as well as P. Billingsley [38], J. Jacod and A. N. Shiryaev [206], D. Revuz and M. Yor [299], and the classics of K. R. Parthasarathy [284]. We also benefit much from W. Hoh [155]. The Markov property is discussed in Section 3.4 and once again H. Bauer [30] is the source with K. L. Chung [74] and [75] being helpful. The proof that Feller semigroups lead to canonical processes admitting a cadlag modification is borrowed from D. Revuz and M. Yor [299], results on stochastic continuity are taken from K. L. Chung [75]. The discussion of the shift operator and the strong Markov property is once again influenced by H. Bauer [30], K. L. Chung [75] and D. Revuz and M. Yor [299] with some additional input of I. Karatzas and S. Shreve [216]. In some sense Section 3.1-3.5 is a type of advanced crash-course on stochastic processes and should be well-known to probabilists. Therefore we may also refer to the monographs mentioned above and in the beginning of Section 2.6 and 2.7 for further references. The martingale problem is discussed in much more detail in Chapter 4 and we add remarks to Section 3.6 in the Notes to Chapter 4. It is possible to read Section 3.7 as a paragraph providing examples to the theory discussed before. But for us this section is more central. The whole approach to Markov processes by studying pseudo-differential operators relies on the model given by Levy processes and we present the subject mainly under this point of view. As modern standard references we mention the monograph J. Bertoin [37] and K. Sato [310], and of course the classics of S. Bochner

3.10 Notes to Chapter 3

153

[47] and P. Levy [244]-[245] and the more hidden work of K. ltd [174]-[176]. Further let us mention the surveys of S. J. Taylor [357], B. Fristedt [107], and N. Bingham [40], as well as the most recent monograph of D. Applebaum [21]. The Levy-Khinchin formula which we derived in Volume I by considering convolution semigroups was proved first by P. Levy [243] and A. Khinchin [219] by probabilistic methods. P. Levy and especially S. Bochner emphasized the role of Fourier analysis in the study of Levy processes and made already much use of the characteristic exponent of a Levy process as did many of their followers. The characteristic exponent has an interpretation in the frame of pseudo-differential operators: it is essentially the symbol of the generator of the corresponding Feller semigroup. "Essentially" refers to the fact that traditionally different normalizations of the Fourier transform are used in probability theory and analysis. Since in our general theory we start with a symbol or equivalently with a pseudo-differential operator, our approach to Levy processes is to start with a characteristic exponent and then to explore the corresponding process. First we discuss the Poisson process and we borrowed much from Ph. Protter [292], as we did later on when investigating the compound Poisson process. An additional source was K. Sato [310]. K. Sato [310] and Chr. Berg and G. Forst [35] were used in our discussion of subordination, but partly we used of course Section 1.3.9. The Cauchy process is treated as an example of a subordinate process. We spent some time in discussing the Meixner process using results from B. Grigelonis [131], W. Schoutens and J. Teugels [326] as well as B. Bottcher [52]. There are three reasons for this: recent interest in financial mathematics, see W. Schoutens [325] (and [326] again), recent results on developing a Malliavin-type calculus for the Meixner and related processes, compare E. Lytvynov [248] and [249], and further it serves as a (counter-)example to the idea that q(x, £) ~ 4>(0 will imply that the corresponding processes have comparable properties. Since diffusions are not so much in our interest, we mention Brownian motion only briefly. However we give a nice proof due to R. Schilling [312] that Brownian motion has almost surely continuous paths using the a priori knowledge that the paths are almost surely cadlag. Our rather "analytic" approach to Levy processes is in some sense not very suitable for pathwise considerations or an analysis on path spaces. In Section 3.8 we give following once again K. Sato [310] the Levy-Ito decomposition of the paths of a Levy process and we refer to [310] for comments on this central and important topic. So far there are three types of results which I would quote to justify an approach to jump-type processes by using pseudo-differential op-

154

Chapter 3 Feller Processes

erators: 1. The fact that we can construct processes, starting by giving a symbol. This was outlined in [178] and [179] with large classes of non-trivial examples in [181] and [182] as well as in [185], and this part of the theory owes most to W. Hoh's work on the martingale problem [150]-[153] as well as to his symbolic calculus [154] and [156], see also his habilitation thesis [155] and Chapter II.2 as well as Chapter 4. 2. Establishing path properties of processes by using the symbol with lasting contributions due to R. Schilling [311], [313]-[316], [320] and [322]. A reminiscene: When I suggested to use pseudo-differential operators to study Markov processes, for the first time in [178], some probabilists reacted a bit hostile (as some of them often do when a bit more analysis is involved) and asked what is gained for probability theory by this approach. Bearing in mind the work of R. M. Blumenthal and R. K. Getoor [43]-[44], J. Hawkes [140]-[141], P. W. Millar [271], W. E. Pruitt [293] and [294], and S. J. Taylor [357] on Levy processes, I suggested to R. Schilling for his thesis [311] to try to extend some of these results to Markov processes generated by a pseudo-differential operator by substituting the characteristic exponent of a Levy process by the symbol of the pseudo-differential operator, hence the process, under consideration. 3. It is possible to recover the symbol of the generator in pure probabilistic terms. This was first outlined in [189] and considerably extended by R. Schilling in [318] and [321], see also E. Popescu [288]. It is of great interest to relate this fact to D. Stroock's [347] and [348] latest approach to K. Ito's ideas [174]-[176]. In Section 3.9 we deal with the third item and first we discuss the central formula

q(x,0 = - lim —^ 1 . t-»o t Then we discuss some path properties of processes which could be characterized by their symbol. These results are mainly due to R. Schilling (see item 2), in special cases Z. Ciesielski, G. Kerkyacharian and B. Roynette [77], B. Roynette [304] and [305], as well as V. Herren [145] gave related contributions.

3.10 Notes to Chapter 3

155

It is of some historical interest that (to my best knowledge) the results of R. Schilling [320] are the first non-trivial example outside pure analysis for the need to consider weighted Besov spaces with p < 1, compare D. Haroske and H. Triebel [137]-[138] and H. Triebel [359]-[362]. Since R. Schilling is preparing a monograph [323] dealing with these and related results, we only give a brief discussion of some of his results. A more recent survey on these and related topics is Y. Xiao [372].

Chapter 4

The Martingale Problem In this chapter we discuss the X>n-martingale problem for pseudo-differential operators with negative definite symbols. For this we need a deeper understanding of probability measures on the Skorohod space and consequently Section 4.1 is devoted to this issue. We discuss first probability measures on general metric spaces. After having studied the Prohorov metric we prove Prohorov's theorem on the tightness of families of probability measures. This result is then applied to the Skorohod space. In particular the importance of the compact containment condition is investigated. In Section 4.2 we derive existence results for the Pn-martingale problem for some classes of pseudo-differential operators. The main result states that if q(x, £) is uniformly bounded with respect to x and satisfies q(x, 0) = 0 for all x, then we get always a solution to the •Dn-martingale problem for —q(x,D). However, in order to obtain a Markov process we need well-posedness and in Section 4.3 we discuss a uniqueness criterion for the martingale problem. We proceed in two steps. We first give a general result and then reduce this to a criterion for a given pseudodifferential operator. Essentially we must solve a backward "heat equation" to get uniqueness. After having investigated a localization procedure in Section 4.4, in Section 4.5 we prove the well-posedness of the Dn-martingale problem for a large class of pseudo-differential operators. This result, Theorem 4.5.12, is due to W. Hoh and a cornerstone of the whole theory. In the final section of this chapter, Section 4.6, we give sufficient conditions in order that the Markov process associated with a pseudo-differential operator by the martingale problem is a Feller process.

158

4.1

Chapter 4 The Martingale Problem

Probability Measures on P n ([0, oo))

As pointed out in the introduction to this chapter and in Section 3.6 in order to handle the martingale problem for pseudo-differential operators we need to study families of probability measures on the complete, separable metric space Vn ([0, oo)). Therefore, in this section we will first study families of probability measures on complete separable metric spaces and then we will have a closer look at probability measures on 2? n ([0,oo)). In our presentation we follow the monograph [98] of St. Ethier and Th. Kurtz, sometimes we use the remarks in W. Hoh [150] and [155]. In the following let (S,d) be a metric space and denote by M\(S) the probability measures on 5. As usual in case of a metric (or topological) space the underlying cr-field is always the Borel- / fdfj. for all

0 define the open set F £ : = ( x e 5 ; inf d{x,y) < e\ , FeC. <•

y€F

(4.1)

J

We introduce the Prohorov metric p on M^S)

by

p(P, Q) := inf{e > 0 ; P(F) < Q(FE) for all F e C}. Proposition 4.1.1. The Prohorov metric p is a metric.

(4.2)

4.1 Probability Measures on Vn ([0, oo)) Proof. We prove first that for P,Q £ Ml(S) P(F) < Q(Fa) + 0

159 and a,0 > 0 the inequality

for all F € C

(4.3)

for all F&C.

(4.4)

implies Q{F) < P{Fa) + 0

For this take first Fi <E C and let F 2 := S\F?. Fi C 5\F 2 a . Therefore by (4.3) with F = F2

It follows that F2 G C and

P(F?) = 1 - P(F2) > 1 - Q(F?) > 1 - Q{F?) -0>

Q(Fi) - 0,

(4.5)

and we get (4.4) with F — i"\. Now we can easily check that p is a metric: The symmetry of p is now trivial, i.e. p(P,Q) = p(Q,P) holds for all P,Q € M j ( 5 ) . Further, p(P,Q) = 0 implies P(F) = Q(F) for all F e C, hence for all F € a(C) = B(S), i.e. P = Q. Since P = Q implies trivially that p(P, Q) = 0 we have proved that p(P, Q) = 0 if and only if P = Q. In order to see the triangle inequality take P,Q,R£ Ml{S) such that p(P,Q) < 6 and p(Q,R) < e. Then it follows for all F G C that P(F) < Q{FS) + 6< Q(F*) + 5 < R{W)£) + 6 + e s+£

< R(F

(4.6)

) + 6 + e,

i.e. p(P, R) < 6 + e implying the triangle inequality for p.

•

Without proof we state Theorem 4.1.2. If the metric space (S,d) is separable then (A4l(S),p) is separable, and if in addition (S,d) is complete, then (Ml(S),p) is complete. For a proof we refer to St. Ethier and Th. Kurtz [98], p. 101. We want to investigate the relationship between convergence in the Prohorov metric with weak convergence of measures. Theorem 4.1.3. Let (S,d) be a metric space and let (Mfc)fceN, Mfc € Further let fj, <=Ml(S). A. Convergence o/(/Ufc)feeN to fi in p implies weak convergence, i.e. lim /o(/Xfc,/z) = 0

k—*oo

Ml(S).

implies for all

B. The following statements are equivalent

k—*oo J

J

160

Chapter 4 The Martingale Problem

1- (Atfc)fc6N converges weakly to fi; 2.

lim j ¥>d/ifc = / (pd/j, for all ip G

k—>oo

CU(S);

3. lim sup Hk(C) < /z(C)

/or aH closed sets C C 5 ;

^. liminf/ifc(f^) > M^O

/or a// open seis U C -S;

k—»oo

fe—>oo

5. lim /ifc(A) = /x(A) fc—>oo

for A € 5 ( 5 ) witt P(dA) = 0.

C. If (S, d) is separable then weak convergence implies convergence in the Prohorov metric p. Proof. A complete proof of Theorem 4.1.3 is given by St. Ethier and Th. Kurtz in [98], p.108-110. We will prove here only part A and part C using part B. A. For k € N let ek := p(fJ-k,v) + p With ip € Cb(S),

0, it follows that for every k £N /•llvlloo

/ /•llvlloo

< /

/x({9>0 £fc )d* + ^IMIoo,

where we used the notation (4.1). Thus we find for all

0, /• /-llvlloo lim sup / yd^fc < lim / ^(f k—>oo •/

fc—>oo

=y

7o

/•llvlloo

> tek)dt /•

M({V > t})dt = j tpdfi,

implying also limsup [(\\ip\\oo + oo 7

J

(4-7)

as well as limsup / (|M|oo - oo V

^

(4.8)

4.1 Probability Measures on T>n ([0, oo))

161

From (4.7) we derive for tp > 0 0 < limsup / (pdfik < / yd/it fc->oo J

J

and (4.8) gives liminf / / fdjj,, 7 fc-»oo i hence liminf / (pdfik = / pd/j, and further 0 < limsup / oo J

it—»oo J

i.e. lim yd/Zfc = / ^ d / x for

0. The general case follows now fc—>oo

by decomposing

0. C. We will prove that iv) implies convergence in the Prohorov metric provided that 5 is separable. For e > 0 let (Ek)ken be a partition of 5 into Borel sets Ek with diam(jBfe) < | . Further let N be the smallest n G No such that A*(U™=i Ej) > 1 — | , and let 5 be the (finite) collection of all open sets of type (Ujej -^j) 2 1 where / C { 1 , . . . , N}. Since the collection Q is finite we can find &o such that G £ Q and all A: > Aro it holds K G ) < fik(G) + | .

For every closed subset C c 5 w e put Co := \JiEj ;l<j
0},

and note then Co5 e Q, and further we find for all k > k0 MC)<MCOS) + | ko proving part C.

D

162

Chapter 4 The Martingale Problem Now we introduce the concept of tightness.

Definition 4.1.4. Let (S, d) be a metric space. A. A measure P G Ml(S) is said to be tight if for each e > 0 there exists K C S, K compact, such that P(K) >l-e holds. B. A family V C Ai\{S) is called tight if for each e > 0 there exists a compact set K c S such that inf P{K) > 1 - e Pev

(4.9)

holds. As a first result we note L e m m a 4.1.5. For a complete, separable metric space (S, d) every P G M.\{S) is tight. Proof. Let (xk)ken be a dense sequence in S and take P G .M£(5). For e > 0 given choose Nv £ N, v G N, such that for v G N P(\jBh{xk))>\-^

(4.10)

fc=i

holds. The closure K of fl^N Ufc=i B\(xk) is compact and further

P(jir)>i-f;i7 = i- e .

• In the proof of Lemma 4.1.5 we used already the fact in a complete metric space a closed subset K is compact if and only if it is totally bounded in the sense that for every e > 0 there are finitely many open balls with radius e whose union contains K, compare W. Rudin [307], p.369. With these preparations we can now prove Prohorov's theorem. Theorem 4.1.6. For a complete and separable metric space (5, d) and a subset V C M-l(S) the following are equivalent: 1. V is tight;

4.1 Probability Measures on Vn([0, oo))

163

2. for every e > 0 there exists a compact set K C S such that

inf P(Ke) > 1 - e Per where Ke is defined according to (4.1); 3. V is relatively compact in

(4.11)

(Ml(S),p).

Proof. Obviously i) implies ii). Next we prove that ii) implies iii). The closure of V is complete since by Theorem 4.1.2 the space {M\{S),p) is complete. Thus it is sufficient to prove that V is totally bounded. For this, given 6 > 0 we have to construct a finite set Q C Ail(S) such that P C{Q; p(P, Q) <5 for some P £ Q}. Let 0 < e < § and K C S be a compact set such that (4.11) holds. Since K is compact there exist finitely many points xi,.. •, xn € K such that Ke C \J"=1

P>2£{XJ).

Fix

XQ

€ S and m > j ,

and denote by Q the family of all P G Ail(S) of the form (4.12) 3=0

where 0 < kj < m and 5^?=o ^j kj := [mQiEj)} for j = l,...,m,

=

Now, for a given Q £ V set

m-

with E, = B2e{xj)\[Jizl

B2£(xi)

and set

further k0 := m - £)" = ifc^.With P as in (4.12) it follows that

U

Q(F)
y^

B2e(Xj))+e

K(^fe))l |

n

|c

(4-13)

Fns 2 e (x 3 )^0

< P ( F 2 e ) +2e for all closed sets F C S such that p(P, Q) < 2e < 5. Finally we prove that iii) implies i). The set V is relatively compact, hence totally bounded and therefore given e > 0 there exists for n e N a finite subset Qn C Ml(S) such for some P G Qn}. By Lemma 4.1.5 for n € N that V C {Q; p(P, Q) < ^ we may choose a compact set Kn C S such that P(Kn) > 1 — ^hr for all P G Qn- Now, for Q G P given, we find for n G N a measure P n G Q n such that

g ^ ^ * 1 ) > Pn(Kn) - JL- > i _ JL.

(4.14)

164

Chapter 4 The Martingale Problem

Taking K to be the closure of Hn>i ^ » " + 1 ** follows that K is compact and oo

Q(A-)>l-52|- = l - e

(4.15)

n=l

proving i). D Without proof, compare [98], p.105, we state Corollary 4.1.7. Let (S,d) be an arbitrary metric space and let V C IfV is tight, then V is relatively compact.

M\{S).

In order to characterise the relatively compact subsets of Vn ([0, oo)) let us recall some notations from Section 3.3. For r\ : [0, oo) —> [0, oo) belonging to the class L, compare p. 76, let | M | L = sup l o g ^ " ^

(4.16)

and for w\,u)2 € 2?n([0, oo)) we introduced p(u>i, u>2,r],u) := sup|a>i(i Aw) — u>2(r](t) A u ) | A 1, u > 0, t>o

(4-17)

as well as dD(ui,L02) := inf (max(||»7||i , / r,£L^

e~up(ui,uJ2,ri,u)du)).

JO

(4.18) '

By Theorem 3.3.12 we know that the Skorohod space (P n ([0,co)),d£)) is a complete separable metric space, compare also [98], Theorem 3.5.6. Thus the Skorohod space is covered by Theorem 4.1.6. Let w G P n ([0, oo)) be a step function and define so(w) - 0,

(4.19)

sk{u) = inf{t > s fc _i(w); u(t) ^ w ( t - ) }

(4.20)

if Sfc_i(a>) < oo, and Sfc(w) = oo if Sfc_i(w) = oo. Lemma 4.1.8. Lei <5 > 0 and for F C R" compact define A(T,6) to consist of all step functions u> £ 23n([0, oo)) such that u(t) £ F for t > 0 and Sfc(w) - Sfc_i(w) > ^ /or eac/i k £ N /or w/jic/i Sfc_i(u>) < oo. Tfte closure of A(T,S) is compact in P n ([0,oo)).

4.1 Probability Measures on X>n([0,oo))

165

Proof. We need to prove that every sequence in A(T, 6) has a convergent subsequence. Given a sequence (wn)neN> vn € A(T,S), by a diagonalization argument we may obtain a subsequence (wm)TOgN such that for k € No we have either &) Sk(uJm) < °° for each m,

lim Sfe(w) = tk £ [0, oo] exists, and 771—>0O

lim w m (s fc (w m )) = : ak exists, m—•oo

or b) Sk{wm) = oo for all m. Now, since Sfc(u>m) — Sfc_i(o3m) > <5 for each /c > 1 and m for which s t - i ( w m ) < oo it follows that (wm)TOGN converges tow £ D n ([0, oo)) proving the lemma.

•

The classical Arzela-Ascoli theorem and its generalization to L p -spaces by A.N. Kolmogorov requires a type of uniform equi-continuity condition. In order to formulate a compactness result for £>n([0,oo)) we introduce a generalized modulus of continuity for u 6 P n ([0,oo)). Fix UJ <E X>n([0,oo)), 6 > 0 and T > 0. By Z(S,T) we denote all finite partitions 0 = t0 < ti < ... < tm < T < tm+i with m G N and inf (tj+1 - tj) > S. j=0,...,m

We define w'(w,5,T):=

inf

Z(6,T)

sup

s,t€[tjttj + l) j=Oy...,m (t0 tm+1)eZ(s,T)

|w(s) - w{t)\.

(4.21)

It is clear that <5 — i > W'{UJ,8,T) and T \-^> w'(ui,5,T) are non-decreasing functions, and an application of the triangle inequality yields v/(w1,S,T)
sup

0
U(t) - w2(*)|.

L e m m a 4.1.9. A. The function 6 — i > w'(w,6,T) holds limti/(w,<J,T) = O.

(4.22)

is right continuous and it

(4.23)

166

Chapter 4 The Martingale Problem

B. Let (u)n)neN be a sequence in 2? n ([0,oo)) and u> £ X>n([0,oo)) such that lim do(wn,Lj) — 0. Then for every 5 > 0, T > 0 and e > 0 we have

n—*oo

limsupu/(w n ,£,T) < w'(u>,6,T + e).

(4.24)

n—>oo

C. The mapping w — t > w'(u>,8,T) is Borel measurable. Proof. A. If Z(S, T) is admissible in (4.21) then for suitable 5' > 5 the partition Z{8', T) is admissible too which yields the right continuity of 8 t—> W'(LV, 5,T). Further, define T^ = 0 and for k <= N

T? = T?(«) := inf{i > r^-a ; |u;(t) - 0,(^)1 > 1 } if

T£!_1

< oo, and

T^

= oo if r ^ j ^ = oo. If now

0<J<min{rf+1-rf ; r f
(»?n)n€N, »7n € X, Such t h a t lim

sup

\rjn(t)

—t\=0,

n—too 0
compare also (3.81), and lim

sup

n-»oo 0
\wn(t)-u;(Tjn(t))\=O.

Now, with ujn(t) := w(rjn(t)) and <5n := sup (rjn(t + S) - rj n (t)) we find by o
,S,T)

n—»oo

< lim sup w' (w, 5 n , r]n (T)) "-*00 < lim sup w'{u, 8n\/8,T + e)

(4.25)

n—*oo

= w'(w,8,T + e) for all e > 0. C. First we claim that w ^

w'(u,8,T+)

is upper semicontinuous, hence

4.1 Probability Measures on Vn ([0, oo))

167

measurable. TH->

Indeed we have from part B and by the monotonicity of W'{W,6,T) that w'(w,6,T+) = luaw'((jJ,6,T + e).

But for every w £ T>n([0, oo)) and e > 0 sufficiently small it holds w'(u, 6, T) = lim w' (w, 6, (T - e)+) proving part C.

n

Theorem 4.1.10. For A C P n ([0,oo)) to be relatively compact it is necessary and sufficient that the following two conditions hold: 1. For every t £ + there exists a compact set Tt C W such that u)(t) £ Tt for all LJ £ A. 2. For every T > 0 limsupw'(w,(5,r) = 0

(5-+0u/SA

(4.26)

holds. Remark 4.1.11. A. It is actually necessary that condition i) holds for every T > 0, i.e. for T > 0 must exist a compact set TT C K™ such that w(t) £ TT for 0 < t < T and all u £ A, compare [98], Problem 16 on p. 152. B. Since we are mainly interested in the sufficieny of condition i) and ii) we do not provide a proof that these conditions are necessary. For this we refer to St. Ethier and Th. Kurtz [98], p.123-124. Proof (of the sufficiency of i) and ii)). Let A satisfy i) and ii) and let / £ N. We may choose <5; £ (0,1) such that supw'(u>,Shl)<-,

(4.27)

and we may further choose mi £ N, m; > 2, such that ^ - < Si. For T (() := U £ o I ) m ' r ^ L we denote by At = A(T^,6i)

as in Lemma 4.1.8

the family of all step functions u> £ T>n([0,oo)) such that uj(t) £ F^ and

168

Chapter 4 The Martingale Problem

Sfc(w) — Sfc_i(w) > 5 for each k £ N for which Sk-i(w) < oo, Sfc being defined by (4.19), (4.20) and Sfc(w) = oo if Sfc_i(o;) = oo. For w e i w e may find a partition 0 = i 0 < h < • • • < <»-i < Z < U < I + 1 < ti+i = oo with min (tj — tj-i) > Si such that max

sup

i<j<«*,te[tj_i,t3)

2 |w(s) - w(<)| < y '

(4.28)

holds. Next we define u>' £ Ai by W'W = W

(M±1)

,ti
It follows that sup \w'(t) - w(t)| < I and further 0
r°°

dD(Lj',uj)< /

Jo 2 ,

e~usup(\u;'(tAu)-uj(tAu)\Al)du

t>o 3

( 4 2 9 )

For every / it follows that A C Aj, hence A C D/eN A' • ^ o w t n e compactness of each A/, compare Lemma 4.1.8, entails that A is totally bounded, hence it is relatively compact.

•

We want apply Theorem 4.1.10 to certain solutions of the P n -martingale problem, compare Definition 3.6.9. More precisely, for a certain (infinite) parameter set 7, for each a € I a probability measure Pa on £>n([0, oo)) will be given and we want to know whether the family (Pa)aei is tight or relatively compact. Note that each probability measure Pa determines a stochastic process with state space R n by considering the projection Xt : X>n([0, oo)) —> R", u> >-> u(t), as random variables. Theorem 4.1.12. Let (Pa)a£i, Pa £ Mf(Vn([0,oo))) be a family of probability measures. This family is tight if and only if the following two conditions hold: 1. (Compact containment condition) For every TJ > 0 and T > 0 there exists a compact set KT,V C Mn such that inf Pa({LJ

e Vn([0, oo)); w(i) £ KTit, for allO 1 - rj. (4.30)

4.1 Probability Measures on Vn ([0, oo))

169

2. For every r) > 0 and T > 0 there exists 5 > 0 such that supPa({veVn([0,o6));w\tJ,6,T)>ri}) v

aei


(4.31)

'

Remark 4.1.13. Note that (4.30) is equivalent to sup Pa ({u e Vn ([0, oo)); w(t) $ KT,V for some 0 < t < T}) < rj. (4.32) Proof. The necessity follows from Theorem 3.3.12, Theorem 4.1.6 and Theorem 4.1.10. Now, let £ > 0 and let T & N such that e~T < §. We choose 5 > 0 such that (4.30) holds with rj = f. For m > j put if := U ^ ^ e 2 - i - 2 , J - anc ^ note that inf P Q ( { w ( - ) £Ki;j

= 0,.. .,mT})

>1- J .

(4.33)

With the notation of Lemma 4.1.8 put vl := yl(r,<5) and observe that A has a compact closure. For UJ € T>n([0, oo)) such that W'(UJ,S,T) < f and ^ ) 6 J f f , j = 0 , . . . , mT, we choose 0 = to < *i < • • • < *m-i < ^ < tm such that min (tj — tj-i) > 5 and l<j<m

max

sup

l<J
|w(s) — w(t)| < —.

(4.34)

4

Now select y^ £ if such that |w(^) — yj| < | , j = 0 , . . . , m T , and define S ( t ) =

W i H i

.*i-i<*<*i.J = l

m-1

(435)

We find LJ e A and sup |w(t)—5(£)| < f implying that do(u;,a;) < f + e~ T < 0
£

e, i.e. u) G A . Therefore we get inf Pa({u(t)

G A£}) > 1-e and an application

of Theorem 3.3.12 and Theorem 4.1.6 yields the result.

•

We will use later on a further criterion for relatively compact sets in Vn([0,oo)) which again is taken from [98]. Let us first fix some notations. For x,y £Rn we p u t q(x,y)

:= \x-y\Al

a n d for (3 > 0 we write q&(x,y) for

(| a; - y\ A I)' 3 . Since in the following results conditional expectations will be used we will now use the projections Xt on P n ([0,oo)), Xt(uj) = u>(t), which

170

Chapter 4 The Martingale Problem

we will interpret as random variables. Further, if Pa is a probability measure on D n ([0, oo)) the expectation with respect to Pa is denoted by Ea. In the following on P n ([0,oo)) we will consider the canonical nitration {J^)t>o and by S0(T) we denote the family of all discrete (^"°)-stopping times bounded by T. Theorem 4.1.14. Let (Pa)aei be a family of probability measures on P n ([0,oo)). Equivalent to the relative compactness of (Pa)a€l ? s each of the following conditions: 1. For every T > 0 there exists p > 0 and a family of measurable mappings IS • £>n([0, oo)) —> [0, oo), 0 < 8 < 1, such that limsup£; a (7 ( 5) = 0,

(4.36)

i5-*0ae/

as well as Ea(q0(Xt+u,Xt)q0{Xt,Xt-v)\f?) Pa-a.s.,

< Ea{ls\T°t)

(4.37)

and further

lim sup£ a (/(X t ,X 0 )) = 0.

(4.38)

t—>ooa€l

2. For every T > 0 there exists S > 0 such that Ca{8):=

sup

sup £ " ( sup g^(X T + U ) X T )^(X T ,X T _,,))

rGSo(T) 0<«<<5

(4.39) (4.39)

0
is defined for 0 < S < 1 and all a £ I and satisfies lim sup CQ (£) = 0 . c5-»0 ael

(4.40)

In addition (4.38) is supposed to hold. The proof of this result requires some technical preparations which are worked out in detail in St. Ethier and Th. Kurtz [98], Sections 3.6-3.9, and for this reason we do not repeat the proof here. The next two results, the compactness results we will finally use, are in principle again from [98], namely Lemma 3.9.1 and Theorem 3.9.4. But we will take them in the formulation of W. Hoh [155], Lemma 3.9 and Theorem 3.10.

4.1 Probability Measures on Vn ([0, oo))

171

Theorem 4.1.15. Let {Pa)aei, Pa G M^(T>n([0,oo))), fulfill the compact containment condition, see Theorem 4.1.12, part i). Further let D C Cb(M.n) be a subspace dense with respect to uniform convergence on compact sets. For f G C(,(Mn) consider the real-valued cadlag process (/(^t)) t > 0 and denote by Pf,a G A^i"(£>i([0, oo))) the image measure of Pa induced by f. If for every f G D the family (Pf,a)aei is tight in Mf(Vn([0,oo))) then (Pa)aei is tight. Proof. We first show that if (P/ jQ ) a6 j is for all / 6 D tight, then (P/, Q ) Qe j is for all / G C&(IRn) tight. Given / G Ci(lRn) choose a sequence (fn)n€M, fn G D, such that (/n)neN converges uniformly on compact sets to / . For T > 0 and e > 0 by the compact containment condition there exist compact sets KT,e C Kn such that M r , £ := {u> G X»n([0, oo)) ; w{t) G # ? > for all 0 < t < T + 1} satisfies for all a G / Pa(Mr,e)>l-|.

(4.41)

For u sufficiently large it holds sup \f{x) — fn(x)\

< | and therefore, by

x£KT,c

(4.22), we get for these n and 0 < S < 1 sufficiently small Pf,a{{w'(;6,T) > e}) = Pa{{w'(f(X.),6,T) > e}) < Pa({w'(f(X.),8,T) > E} n MT,e) + Pa{Mle) sup \f(Xt)-fn(Xt)\>}nMTie)

<Pa{{w'(fn(X.),6,T) + 2

0
£

<Pa{{W'(fn(X.),S,T)> -})

+ £6

+~

= -P/n,a(M-,«,T)>|}) + | < e , where we used in the last step the tightness of (P/ n , Q ) a e / and Theorem 4.1.12 part ii). It follows that (Pf,a)aei also satisfies condition ii) of Theorem 4.1.12 and of course it fulfills the compact containment condition. Thus by Theorem 4.1.12 the family (P/, Q ) a € /, / G Cb(Rn) is tight. We apply the latter result now to the bounded and continuous function x H-» \x — z\ A 1 where z G Mn is fixed, and we will prove the tightness of (Pa)aei by using Theorem 4.1.14 with j3 = 2. For this let e > 0, T > 0 and KT,E , MT,E be as above. By compactness there are points z\,..., z^c G KT,E

172

Chapter 4 The Martingale Problem

such that the open balls BE{zi) cover Kx,e- Therefore, for y G KT,E there exists i e { 1 , . . . , NE} such that q(x, V) < q{x, zj + q(y, Zi) < \q{x, zt) - q(y, Zi)\ + 2e holds for all x e R™. Thus for 0 < t < T, 0 < <5 < 1 and 0 < u < S, 0 < v < 6At we find q2(Xt+u,Xt)q2(Xt,Xt_v) <

max

<

q(Xt+u,Xt)q(Xt,Xt-v)

{\q{Xt+v.,Zi)-q(Xt,Zi)\\q(Xt,Zi)-q(Xt-v,Zi)\}

i=l,...,Nc

+ 4(e + e2) + XM^C <

(4.42)

max w'{q(X.,Zi),26,T+l)Al

t=l,...,JV e

+ 4(e + e2)+XMi

= ls, where we used the fact that w'(-,25,T + 1) is calculated by taking only into account the oscillation of the path on intervals of length not smaller than 25. Note that at least one of the intervals [t — v, t) or [t, t + u) is completely contained in such intervals and therefore (4.37) holds with 7,5 as above. Given r\ > 0 we fix e > 0 and therefore iVe such that

holds. By Theorem 4.1.14 and 6 > 0 sufficiently small it follows that

supSQ(W'(g(X,^).2^r+l)Al) < -Jr for i = 1 , . . . , Ne. Now (4.41) and (4.42) yields supEa(l6)
(4.43)

a€l

implying (4.36) since r\ > 0 was arbitrary. By the definition of w' for 0 < t < T and e > 0 we find that q2(Xt, XQ) < q(Xt, XQ) < max \q{Xu i=l,...,Nc

<

Zi)

max w'(q(X.,Zi),t,T)+2£ i=l,...,Ns

- q(X0, zt)\ + 2e + XM$.

+ XM%c-

4.1 Probability Measures on Vn ([0, oo))

173

Arguing as above with Theorem 4.1.14 and (4.41) we finally arrive at

lira sup Ea{q2{XuX0))

t->0a£7

thus (4.38) holds implying that (Pa)aei is tight.

=0, •

In Section 3.6 we discussed already some idesa related to the X>n-martingale problem. In particular we have shown that if the X>n-martingale problem for (A,D(A)), (A,D(A)) being an operator in Coo(Kn), is well-posed, then we may associate a Markov process ((Xt)t>o,PM) with (A,D(A)) and a given initial distribution fi. In order to prove that for certain operators the T>nmartingale problem is indeed well-posed we will need the following tightness result, compare St. Ethier and Th. Kurtz [98], Theorem 3.9.4, which we take once again from W. Hoh [155], Theorem 3.10. Theorem 4.1.16. Let D C Cb(M.n) be a subalgebra being dense with respect to uniform convergence on compact sets. Further let (^(Aa,D)s)aeI be a family of linear operators in C^IR") such that sup||A Q /|| 0 0
(4.44)

holds for all f G D. If for each a £ I the measure Pa £ M^(T>n([0,oo))) is a solution to the Vn-martingale problem for Aa and if (Pa)aei satisfies the compact containment condition, Theorem 4.1.12 i), then (Pa)aei is tight. Proof. By Lemma 4.1.15 it is sufficient to check that for all f £ D the process {f (Xt)) t>0 S^ve r ^ se to a tight family of distributions (P/ !Q ,) ag / in Ml(T>i([0,oo))). We introduce the notation Xf : £>i([0,oo)) -> R for the canonical projection Xt : 2?n([0,oo)) —> Kn when n = 1, and further we put Jf := a(Xf ; s < t) and TM = a(Xf ; t > 0). For / £ Cb(Rn) we consider the induced mapping / : Vn ([0, oo)) —> V\ ([0, co)) , W H / O W , which is T- J * as well as Ji-^f-measurable and P/, a is the image of Pa under / . Now we will use Theorem 4.1.14 to prove the tightness of (P/ >Q ) a£ /. Since (Pa)a&i satisfies the compact containment condition and / is bounded it follows that (Pf,a)aei also satisfies this condition. We want to verify (4.36)-(4.38) in Theorem 4.1.14. For this take T > 0 and 0 < 5 < 1. For every A £ Ff, 0
174

Chapter 4 The Martingale Problem

and 0 < u < 6 we have

/ (Xf+U - X?)2dP/>Q = / JA

(Xf+U o / - Xf o f)dPa

Jf~1(A)

= / (/(**+„) - /(X t )) 2 dP Q Jf'HA)

= f

Ep°{(f(Xt+u) -

(4.45)

f{Xt)f\Tt)APa.

Jf-HA) Using the fact that Pa is a solution for the Pn-martingale problem for Aa and that f,f2 S D we find further PQ-a.s. that

Ep°((f(Xt+n)-f(Xt))2\rt) = Ep" {f(Xt+u) - f{Xt)\Tt) - 2f{Xt)Ep° (f(Xt+u) - f(Xt)\Ft)

a

t+U

pt+U

Aa{f){Xs)ds\Ft) - 2f(Xt)Ep" [J^

Aaf(Xs)ds\Ft)

< S\\Aa(f)U + 2||/||oo SWA.fU < CfS, (4.46) where Cf is independent of a. Thus by (4.45) we arrive at EP'-» ((Xf+U - Xf)2|J?) < CfS which yields (4.36)-(4.38) for (Xf)t>0 with (3 = 2 and 7,5 = cfS. Thus by O Theorem 4.1.14 it follows that (P/, Q ) a 6 / is tight for all / G D. Remark 4.1.17. There is no problem to state and prove Theorem 4.1.16 when M.n is substituted by a suitable metric space. In particular W& will do, compare [98].

4.2

Existence Results for the P n -Martingale Problem for some Pseudo-Differential Operators

The purpose of this section is to establish the existence of a solution to the Pn-martingale problem when A is a certain pseudo-differential operator. For this let us recall Definition II.2.3.1: A function q : Rn x Kn —> C is called a continuous negative definite symbol if q is a continuous function and for each x G W1 the function q(x, •) : W1 —> C is negative definite. As usual

4.2 Existence Results for the ©n-Martingale Problem

175

we associate with a continuous negative definite symbol the pseudo-differential operator q(x,D) defined on C£°(R") by q{x,D)u{x) = {2ir)-i [

efa-««(a;,O«(O^.

(4-47)

The main result of this section is Theorem 4.2.1 (W. Hoh). Let q : Rn x M" -> C be a continuous negative definite symbol such that q(x, 0) = 0 for all i £ R " and \q(x,Z)\ < co(l + lei2),

x G Kn and £ € Rn,

(4.48)

where CQ is independent of x and £. Tftera for all initial distributions y. e ,M£(Rn) f/iene exists a solution to the Vn-martingale problem for —q(x, D). Remark 4.2.2. Note that (4.48) has the interpretation that the "coefficients" of q(x, D) are uniformly bounded. This condition is also sufficient to guarantee that q(x,D) maps Cfi°(Rn) into C6(K"). Indeed, the continuity of x — t > q(x, D)u{x) is clear since x H-» q(a;, ^) is assumed to be continuous and u G 5(M"). The boundedness of q(x, D)u follows from \q(x,D)u(x)\ = (27r)-*| /

ete-«g(i,$)G(^)de

x(£) •= q(x,£) the constant term is zero. The proof of Theorem 4.2.1 needs quite a lot of analytic preparations. The theorem as well as its proof is due to W. Hoh and we follow his work [155] closely. The final construction of the probability measure on Vn ([0, oo)) solving the Pn-martingale problem for —q(x, D) uses an approximation procedure and an application of Theorem 4.1.16. But instead of working on R™ we have first to work on the one-point compactification M^ and then to prove that A is never reached in finite time. Furthermore, for doing this we need to decompose q(x, D) into an operator mapping functions with compact support into functions with compact support, and an operator which can be considered as a bounded pertubation. On a more technical level we will often switch

176

Chapter 4 The Martingale Problem

from the pseudo-differential operator representation of — q(x, D) to its LevyKhinchin representation. We start our preparations by discussing the solvability of the martingale problem for operators of type Ku{x) = X f

(u{y) - u{x))n{x, dy), A > 0,

(4.49)

where fi(x, Ay) is a kernel on R n x B^ such that /x(x,M") < 1, i.e. each measure n(x,-) is a sub-probability measure. If necessary passing from K n to R^ and substituting JU(X, dy) by fi(x, dy) + (1 - [i(x, {A})e{dy) we may assume for simplicity that n(x, •) is always a probability measure. The operator if is a bounded oeprator on Bf,(Mn) which satisfies the positive maximum principle. Further it generates on Bb(M.n) a strongly continuous semigroup by oo

Ttu:=etK = J2^(tKyU.

(4.50)

Introducing the operator

Tu(x) = I u(y)[i(x,dy) and noting that u{x) = f u(x)fi(x, dy) we find that K

= A(r - id),

and therefore (4.50) becomes

TtU = f y A t ^ - P .

(4.51)

From (4.51) it is also clear that Tt is positivity preserving. Comparing K with the operator (3.212) it should be possible to associate with K or (Tt)t>o a Markov process which has a certain similarity to a compound Poisson process, see Definition 3.7.14. A detailed existence proof of such a process is given in St. Ethier and Th. Kurtz [98], p. 163-164, we just describe briefly the construction. Let (Nt)t>o be a standard Poission process with parameter A > 0 and let (ll)ieNo be an Rn-valued independent Markov chain with initial distribution /xo and transition probability /j,(x, dy), i.e. P(YQ 6 A) = Ho{A) and

P(Yk+1£A\Y0,...,Yk)

= fi(Yk,A).

4.2 Existence Results for the Vn-Martingale Problem

177

We define the process {Zt)t>o by (4.52)

Zt:=YNt.

As proved in [98], {Zt)t>o is a Markov process with initial distribution /i 0 , and it holds for all u G Bb(Rn) and s,t>0 a.s.

E(u{Zt+s)\F?)=Tsu{Zt)

(4.53)

where {^Ff)t>o is the canonical filtration for (Zt)t>o- In particular, (Zt)t>o is associated with (Tt)t>o in the usual way, i.e. Ttu(x) = E{u(Zt)) for all u G Bb(Rn).

Now we claim

Proposition 4.2.3. Let K with domain Bb(M.n) be defined by (4.49). Then for every initial distribution JJLQ £ Ail(M.n) there is a solution to the martingale problem for K. Remark 4.2.4. As indicated before, instead of W1 we may work on R^, in fact the whole argument works for every suitable metric space. Proof of Proposition 4.2.3. (compare W.Hoh [155]J. By construction the process (Zt)t>o has cadlag paths and Pz0 = (J.Q- We prove that for u £ BbQ&n) an (Jrtz)-martingale is given by

(u(Zt) - [ (Ku)(Zs)ds) v

./o

(4.54)

.

/ t>a

This will imply that the distribution of (Zt)t>o on the path space, i.e. the distribution of Z : Q —> D n ([0, oo)), W H ^ H Zt(u)), is a solution to the martingale problem. Using properties of the strongly continuous semigroup (Tt)t>o, compare especially Lemma 1.4.1.14, we find for 0 < ii < t2 and u G Bb(Rn)

E(u{Zt2) - J

= Tt2-tMZtl)-

I* Ts-ti{Ku){Ztl)ds ~

[\Ku)(Zs)ds

^ Jo

' Ts(Ku)(Ztl)ds

[\Ku)(Zs)ds

[\Ku)(Zs)ds

a.s.

= Tt2-tlu(Ztl)= u(Ztl)-

*(Ku)(Zs)ds\ftf)

Jo

Jti

Jo

-

Jo

178

Chapter 4 The Martingale Problem

and the proposition is proved.

•

We do need also a pertubation result which we take from St. Ethier and Th. Kurtz [98], Proposition 4.10.2, p.256, in the formulation of W. Hoh [155]. For the more involved proof we refer to [74]. As usual we formulate the result for K n but we will use it later on in more general situations. Proposition 4.2.5. Let A be a linear operator on Bt.(M.n) such that the martingale problem for A is for every initial distribution solvable. Further let K be given by (4.49). Then the martingale problem for A + K is also for all initial distributions solvable. Proposition 4.2.6. Let •& £ 5(M"), 0 < d < 1,tf(0)= 1, be an even function and let q : K™ x R" —> C be a continuous negative definite symbol with LevyKhinchin representation n

n

fc,J=i

i=i

+/

(4.55)

(l-e-**- T ^| 5 W,dy),

VR"\{O}V

i + lJ/r^ an

where a,ki = aik '• ^™ ~* K d bj : M.n —> M are continuous functions and £fei=i aki(%)€k£i > 0. Further, for x £ R™\{0} fixed the Levy kernel fj,(x,dy) satisfies

I

TXTi2^ a: ' dj ') <00 -

7K"\{O} i + \y\

We define q?(x,0 = (2*)-* [ {q{x,S + TJ)-q(x,rj))d(Ti)dri

(4.56)

and qt(x,t)

= (2n)-1

[ {q(x,Q-q(x,Z

+ Ti) + q(x,Ti))d(Ti)dri.

(4.57)

Then qf, q^ : W1 ~x M.n ^> C are continuous negative definite symbols and it holds

q(x,t) = qt(x,O + qt(x,Z)

(4.58)

4.2 Existence Results for the Z?n-Martingale Problem

179

as well as n

n

fc,i=i

3=1

VR\{O} V

(4.59) -1- + 12/1

y

and

«2 (*, 0 = /

(1 - e" i y 4 ) (1 -
(4.60)

where bj(x) = - / K n V{0} (l - ^(yJJiqfjj^M^.dy).

Proo/. First note that for a; € K, K C Mn compact, we have |g(a;,f)| < C^r(1 + |^|2) implying that the integrals in (4.56) and (4.57) exist and define continuous functions leading to the decomposition (4.58) of q(x, £) — note that

(2TT)-*

/

6{r,)dr, = F~1{d){0) = -d(0) = l.

Once we have proved (4.59) and (4.60) it is obvious that qf and q$ are continuous negative definite symbols. Using a Taylor expansion, compare also the beginning of the proof of Theorem 1.3.7.7, we find

1 _ e-iy((+v) _ iV(^ + V) _(-,__ e-iyn _ iyi 1 + M2 V l + \y\2)

^ l ^ i i C i + lc + ^ + l ^ ^ T ^ d + lvl8)-

\

180

Chapter 4 The Martingale Problem

An application of Fubini's theorem yields now (2*)"*/ /

(l-e-^)-^+^

JR"\{O}JR^X

1 + 12/1'

>/R-\{O}

i + \y\

(4.61) Since 1 — •&(—y) vanishing at 0 of order 2 the second integral exists and we define

W =~ I

(4.62)

(l-&(-y))^^^x,dy).

V/K™\{0}

! + 12/1

Further observe for f = ( ^ , . . . , £ n ) £ K" and ?? = (771,..., r]n) £ K" that (27r)-t f ((Cfc + Tjfc)(6 + »7l)-»)fc»?l)^(rJ)d»J VR" = &&0(O) - t€fc(ft
= Cfc6

where we used that grad$(0) = 0. In addition we find (2TT)-* / (fe- + r?,) - v^i^dr] = Zj0(O) = fr Thus it holds

JRn

f / " " (27r)"f / ( J2 aki(x)(tkti + VkVi) +iYJbj{x){£i •/Kn

fc,/=i n

n

- ^ akl{x)r]krji - iY^bjixjrij^ViriJdr) n

n

fc,/=i

j=i

+ rjj)

j=i

(4.63)

4.2 Existence Results for the Dn-Martingale Problem

181

Taking into account that tf(-ry) = 19(77), from (4.61), (4.62) and (4.63) we deduce finally (4.59) and (4.60).

•

Theorem 4.2.7. Let q : M™ x M™ —> C be a continuous negative definite symbol such that q(x, 0) = 0 and the Levy-Khinchin representation (4.55) holds. Further let d G C£°(R") be an even function such that 0 < i9 < 1 and i?(0) = 1. Decompose q{x,£) according to Proposition 4.2.6 as

(4.64)

q(x,Z) = qt(x,Z)+qi(x,O. Then -qf(x,D) maps C^°(M") into C0(Rn) operator of form -qt{x,D)u{x)

= f

(u(x -y)-

and -q$(x,D)

u(x))Jl(x,dy),

JE"\{0}

is a Levy-type

u

G CO°°(R"),

(4.65) where the Levy-kernel Jl(x, dy) satisfies ji(i,R"\{0})
(BeqNL(x,S))v(d£)
(4.66)

Here qNi,{x,€) denotes the non-polynomial part of q(x,£) and v is the finite measure defined by

compare the proof of Lemma 1.3.7.2. Proof. We know already that qf (x,£) is a continuous negative definite symbol and therefore qf(x,D)(p is for each cp € Co°(K") a continuous function. It remains to show that qf(x,D)

-qf(x,DMX) = f^Jv{x

- y) - „(*) + ±

-M^^.)miiM)

182

Chapter 4 The Martingale Problem

= / (p(x-y)d(y)[i(x,dy). JRn\{o] We use that supp $ is compact, which implies that supp
we find with Jl(x,dy) — (l — $(?/))/i(:r,ch/) the representation (4.65). Since 1 — •& is non-negative,bounded by 1, and vanishes of order 2 at the origin we get

and therefore we derive

Jl(x, R"\{0}) < a, f

7 X T ^ ( z , dy)

VR"\{O>

= c^ /

JR"

+ \y\

/

JRI\{O}


X

(1 - cos y • €)v{d£)(j.(x, dy)

JW-

ReqNL(x,£)i>(d£)

< oo,

proving the theorem.

•

We now extend the operator -qf(x,D), an operator acting on functions n defined on M. , to an operator Ad acting on functions defined on the onepoint-compactification R £ of E". Recall that u e Coo(Rn) can be extended to Cb{RX) = C ( R A ) b y defining u(A) = 0. Thus we may consider Coo(K n ), hence C£°(R n ) and 5(R"), as subspaces of C(R^). With q{x,$,) and & as in Theorem 4.2.7 we define on the Banach space (C(R^, ||.||oo) the operator A# by D(M)

•= W e C(Rl);

(ip(-) - ip(A)) e C0°°(Rn)}

(4.67)

and

A.rtX):=H<X>D)«*-«A»MM [0

>XGRn ,x = A.

(4.68)

4.2 Existence Results for the £>n-Martingale Problem

183

According to Theorem 4.2.7 the operator A$ maps D(A$) into Co(R"), hence into C0R2O which implies that we may consider (Ag,D(A#)) as an operator on C(R£). Note that D(A&) is dense in C(R£). Further the function XM£, i.e. the function x — i > 1 for all x £ M^, belongs to C(IR^). In fact this function belongs to D(A#) since XRJC 1 ) ~ X K ^ ( ^ ) = 0 for all a; £ K^. Hence ^4^XR^ is defined and gives for i g l " (AdXRl)(x) as well as (j4,9X]Rn)(A)

=

AoXKi = A»l = 0.

= -q?(x,D)(Q) = Q,

0, i.e. we have (4.69)

In addition, the operator (Atf,D(A$)^ satisfies the positive maximum principle. Indeed, since K^ is compact, ip obtains its supremum. Let tp(x0) = ma.xip(x) > 0. If x0 = A, then (A#(p)(x0) = 0, but for x0 £ Rn zeR£ it follows that max(<^(a;) - 0 and by the positive maximum principle applied to —qf(x,D) and 0. The clue in the proof of Theorem 4.2.1 is now to find a good approximation of A$ by Levy-type operators on R^ with bounded Levy kernels and then apply Propositions 4.2.3 and 4.2.5. For this a result on measurable selections is required which is essentially due to K. Kuratowski and C. Ryll-Nardzenski [240] which we state here (without proof) in a convenient form taken from St. Ethier and Th. Kurtz [98], Appendix 10, see also C. J. Himmelberg [147], Theorem 3.5. Definition 4.2.8. Let (f2, A) be a measurable space and (5, d) be a complete, separable metric space. For x £ Cl let r x c S. A measurable selection of (r x )zeft i s a measurable function / : Q, —> S such that f(x) € Tx for every x £ fl. (As usual, on 5 we take the Borel cr-field.) Theorem 4.2.9. Let (Cl,A) be a measurable space and (S,d) be a complete, separable metric space. Suppose that for x £ Q, a closed subset Tx c S is given and that for every closed set C C S the set {x £ £1; Tx D C ^ 0 } belongs to A, i. e. is measurable. Then for k 6 N there exists fk'.H^S S such that fk is measurable, fk {x) £ Tx for every x £ f2 and Tx is the closure of the sequence (fk(x))kEN.

184

Chapter 4 The Martingale Problem

In our application we will take as fi the set K^ with the Borel cr-field and as metric space Ml(M.%). Proposition 4.2.10. Let (A9,D(A^)) be as defined (4.67) and (4.68). There exists a sequence of kernels fik on R^ x S(M^),/ifc(a;,K^) = 1, such that for all tp G D(A#) and uniformly on R ^ it holds k /

(
> A#tp(x).

(4.70)

Proof, (compare W. Hoh [155],). Before going into details recall that we know already an approximation procedure for the generator of a strongly continuous semigroup by bounded operators: the Yosida approximation, see Theorem 1.4.1.29. Thus we would like to use the operators k(R$)kAtf, where (R$)k denotes the resolvent of A$ at k, and try to find an appropriate kernel representation for them. Since A$ satisfies the positive maximum principle on C(R^) it is dissipative, compare Lemma 1.4.5.2, hence k — A$, k 6 N, has a continuous inverse which is an operator in C(R^) but which need not necessarily be densely denned. On the range R(k — A$) of k - A# we define for x G R^ and k G N fixed the linear functional 1% by

Zg(v>) : = * ( * - 4 > ) - V ( * ) Since A$ is dissipative we find

Ufc(v)l < l|fe(*= - A*)-Vll«x, < HvllooFurther we have 1 = (k — A#)(^) G R(k — Ad), recall that A$l = 0, and therefore l%(l) = 1. In addition for

0 w e have

ZfcM = ^(IMIoo) + /£(¥>-IMIoo) >IM|oo-||(V-|M|oo)||oo>0.

Thus we find that 1% is a positive linear functional of norm 1 defined on R(k Atf) C C(W&). A variant of the Hahn-Banach theorem, compare Theorem 2.3.1, gives the existence of a (not necessarily unique) positive linear functional of norm 1 on C(R^) extending If.. According to the Riesz representation theorem there exists a probability measure /ijF £ AfJ(R^) such that for all


A*)

lUv) - /

^(yK(dy)

(4.7i)

4.2 Existence Results for the Z>n-Martingale Problem

185

holds. Clearly /ijF depends on the extension of 1%. Now, let

Ml := U G MlQBLl) ; l%(
G R(k - A*)} ? 0 .

The aim is to get for fixed A; G N a kernel ^(x, dy) on (R£ x S(R^) such that for each a; e M^ it holds /ife(a;, •) G M£. We want to apply the measurable selection theorem, Theorem 4.2.9, where fi = R^ equipped with the Borel
lim

m—>oo

lim k(k - Ao)~ V(z m )

m—»oo

'fcm(v) = lim /

m—>cx> 7 K "

p(y)Hm(dy) = /

JK

(p(y)noo{dy),

i.e. Hoo G Ml, proving that { i € l ^ ; M£ n C ^ 0 } is closed. Finally, for k G N we fix a kernel fik(x, dy) as above and for cp G D(A$) it follows that / = T. \ =-ll((k


K

= V(T) + T /

(A#ip)(y)nk{x,dy)

(At(p)(y)iik(x,dy)

K ,/Rn

(i4*v»)(i/)/ifc(a:, dy).

186

Chapter 4 The Martingale Problem

Since A$(p is bounded we find by our previous considerations that - / K

{A^ip)(y)/j,k(x,dy)

-> 0 uniformly on R £ ,

JR\

hence / v{y)^k{xAv) JRI

-*
uniformly on R^.

(4.72)

Since D(A$) is dense in C(R^) with respect to uniform convergence, we conclude that (4.72) holds for all ip G C(R£). Thus we get k

/ {f(y)-tP(x))fik(x,dy) JRI

=k

ip(y)nk{x,dy)-k(p(x) JR%

r

= /

A4(p(y)nk(x,dy)

which by (4.72) tends to A$ip as k —> CXD and the proposition is proved.

D

For the bounded operator A$ we can easily solve the martingale problem: Proposition 4.2.11. Let (Atf,D(A$)) be as in Proposition 4.2.10. Then for every initial distribution v £ . M ^ R ^ ) the Vn-martingale problem for A$ admits a solution P £ M.\^D^) Proof.(W. Hoh [155];. For k € N let Ak be denned by Ak
(4.73)

with the kernels /j,k(x, dy) as in Proposition 4.2.10. From Proposition 4.2.3 we deduce for every k €N and every initial distribution v that there is a solution to the martingale problem for Ak- Since Akip —> Pk € Ml(V^([0,oo))) A$ip as k —> oo uniformly for ip S £>(A^), it follows that sup ||j4fey||oo < ° ° feeN for all

4.2 Existence Results for the X>n-Martingale Problem

187

denote this subsequence once again by (Pk)keN- We want to prove that P solves the martingale problem for A$ and initial distribution v. First note that Xo : P K n ([0, oo)) —> W& is continuous implying that Pk o XQ- 1

k—»oo

> P o XQ- 1

in

Ml(K£)

which yields P o X " 1 = i/. It remains to check that


(Aw)(Xs)ds,t>0

Jo

is for all ip € D{A$) a martingale under P. This follows once we have proved that

/ ft2 ¥L \ Ep({v(Xt2) - 0 ; Ppf t _ = X t ) = 1}. This is a dense subset in [0, oo), compare Lemma 4.2.12 below, and due to the right continuity of paths we may assume that s j , . . . ,SM,^i,*2 G The distributions (-X"Sl,... ,XSM,Xtl,Xt2)(Pk) converge weakly to Tp. and since Pk solves the martingale problem for (XSl,.-.,XSM,Xtx,Xt2)(P), Ak we find

Ep« (Up(Xt2) -
Akip(Xs)ds) J ] hm(XSm))

= 0.

m=l

Jtl

The proposition is proved once we have shown that

a

t2

.

M

AW{Xs)ds J ] hm(XSm)) i

i

(4.74)

'

M

Atf^X.Jds JJ/i ro (X. m )J. 1

m=l

We know that (Afe<^)fcepj is uniformly bounded and converges uniformly to Atfip. Therefore we may interchange the order of integration in (4.74), i.e.

188

Chapter 4 The Martingale Problem

"E ft* = ft* E", and it remains to prove that M

M

m=l

m=l

lim Ep*{Akip{Xs) [ ] hm{Xs)) = Ep(AMXs) J ] M * O ) (4-75) fc

^°°

holds for all s £Tp. But the weak convergence of (Pfc)fceN yields P

M

M

P

lim £ *(A^(^) I ] M * O ) =E {AMXS) I ] M * O ) (4.76) and the uniform convergence of (Ak
lim Ep« ((At
fc

^°°

implying (4.75), hence the proposition.

(4.77)

m=l

•

Lemma 4.2.12. Let (Pk)k€N be a sequence of probability measures in Ail (2?n([0, oo))) which converges weakly to a probability measure P. Further put P(Xt- = Xt) = 1}. Tp:={t>0; Then T£ C [0,oo) is at most countable and for all t\,... ,tm £ Tp the finite-dimensional distributions (Xtl,... ,Xtrn)(Pk) converge weakly to (Xtl,...,Xtm)(P). Remark t o the proof. The convergence result is a bit more involved, a detailed argument is given in St. Ethier and Th. Kurtz [98], Theorem 3.7.8 on page 131. The statement that T£ is countable uses essentially Lemma 3.3.8, details are given in [98], Lemma 3.7.7. As final preparation for the proof of Theorem 4.2.1 we give Lemma 4.2.13. Let q : R" x R™ —» C be a continuous negative definite symbol satisfying (4.48) and q(x,0) — 0 for all x 6 W1. Then there exists a sequence (tpk)k€N, Vk G Co°(K"), such that both sequences {
4.2 Existence Results for the Pn-Martingale Problem

189

Proof. Take

C £ i ( 0 ) and <^|B = 1. Now define

(l + \t\2)kn\
< c sup (1 + |£|2) f
(1 + |£|2)fc"|£(fc0|d£

kn\
|€|2fcn|fce|-(n+«d$

+ c" j

|^|-(n+1)d$,


where we used the fact that (p £ S(M.n) to get the estimate

l£(*OI<«-(n+3)

,l^l>i-

For k —» oo we arrive at sup |g(a;,D)pfe(a;)| < C, i.e. (g(a:,D)(/?fc)fceN is uniformly bounded. Now, for xo S K" fixed we find \q(x,D)
sup |g(a:o,OI /

<

sup \q(xo,O\ [ +c/

\q(xo,€)\kn\v(kQ\dS

f

kn\v(k£)\dti + c [ mtm

+ cf

(1 +|$| 2 )fc n |^(^)|dC kn\k$\~(n+Vd$

|^| 2 A; n |^r(" +3) d^

J|CI>i

< c ( sup 19(3:0,01 +-L(fc*-1) + _L), (4.78)

190

Chapter 4 The Martingale Problem

i.e. \q(x,D)tpk(x0)\

< c( sup \q(xo,S)\ + ^(k?

\(\<js

- 1) + - ^ ) .

k

k

(4.79)

'

Since q(xo,£) —•> 0 as £ —> 0 we deduce from (4.79) that q(x,D) 0 as k —> oo and the lemma is proved. • Proof of Theorem 4.2.1. (W. Hoh [155],). Let q : Rn x M" -> C be a continuous negative definite symbol satisfying q(x,0) = 0 and (4.48). We decompose

q(x,$)=qf(x,0+qi(x,0 according to Theorem 4.2.7 and denote by A$ the extension of —qf(x,D) : C0°°(R") -» C 0 (M n ) as given by (4.67), (4.68). By P e M\{V^([0, ([0, oo))) we denote a solution to the PR«-martingale problem for ^4^ with initial distribution ii e Ml(M.n) C Ml(W%.)- According to Proposition 4.2.11 such a solution exists. We prove next that almost surely the paths of this solution are in X>n([0,oo)), i.e. do not reach A in finite time. For this we define the (Ji+)-stopping times

rm := inf{t > 0 ; d(A,Xt) < ^ } ,m£Ff, where d is the metric on R^. For T > 0 define Zt := lim XTfnw,

Xs,

m—*oo

s > 0, denoting the projections on D^n ([0,oo)). It is sufficient to prove that for all T > 0 we have P{Zt e R n ) = 1. Take (ipk)keN as in Lemma 4.2.13 with the usual extension (p(A) = 0. We know that the sequences (pk)keN and (A^ipk)keN as functions on R^ are uniformly bounded and tend pointwise to XK" and 0, respectively, as k —> oo. Since the paths are right continuous we find that 0, Jo is an (Ji + )-martingale. Now, optimal sampling, compare Section 2.6, gives for k, m G N that EP (fk(XrmAT) For m —> oo we find

~ Jo

a

A^k(Xs)ds)

sup (r m AT)

=

Ep(
(Awk)(Xs)ds)=Ep(
4.3 A Uniqueness Criterion for the Solvability of the Martingale Problem

191^

and as k —> oo we get

Ep{X^{Zt))

= Ep(XRn(X0))

= M(K") = 1,

thus we may assume that the martingale problem for A$ and /J, has a solution with paths in £>„([(), oo)), i.e. the Z>n-martingale problem for —qf(x,D) has a solution. According to (4.65) we have for ip € Co°(Rn) -qt(x,D)ip(x)= /

(ip(x + y)-(p(x))ji(x,dy)

= /

(
JI"\{0} JMn\{0}

and the Levy kernel /x consists of uniformly bounded measure since by (4.66) sup ||/Z(:r,-)|| < ci? SUP / zSR"

and further, by (4.48) we find

x€R JR"

sup f ReqNL(x,Qv(dt)

x€R" JRn

RegjvL(z,fMdf)>

n

< c [ (1 + |£|2)i/(d£) < oo. 7R"

Thus —qt{x, D) is a perturbation of —qf(x, D) in the sense of Proposition 4.2.5 implying that the T>n-martingale problem for —q(x, D) = —qf(x, D)—q$(x, D) is for every initial distribution solvable. • As pointed out in Theorem 3.6.10, we need to prove the well-posedness of the T>n-martingale problem for —q(x, D) in order to associate a unique Markov process with —q(x,D). Thus we have to establish the uniqueness for a solution of the X>n-martingale problem for —q(x, D). This is the topic of the next sections.

4.3

A Uniqueness Criterion for the Solvability of the Martingale Problem

In this section we discuss a general uniqueness criterion for the Pn-martingale problem, Proposition 4.3.1, and then we transform it into a criterion applicable to pseudo-differential operators, Theorem 4.3.2. As usual we work on £>n([0, oo)), but all arguments hold when R™ is substituted by a suitable metric space, in particular M^ is allowed. If P e

192

Chapter 4 The Martingale Problem

Ml(Z?n([0,oo))) and Xt : D n ([0,oo)) -> W1 is the projection Xt(u) = u(t), then P is determined by its finite dimensional distributions, i.e. by the family , for t\,..., tk S [0, oo). (Note that it is now more conveP o (Xt!,...,Xtk) nient to write (Xtx ,••••>Xtk) for the product mapping instead of (^) = 1 Xtj.) The following result is taken from W. Hoh [155] whose presentation relies on St. Ethier and Th. Kurtz [98], Theorem 4.42. Proposition 4.3.1. Let A : D(A) —> Bb(M.n) be a linear operator with D(A) C Cb(Rn). Suppose further that for all fi £ Ml(Rn) and all solutions Pi,P2 G A4l(T>n([0,oo))) to the martingale for A and n it holds P x o X^1 = P2 o Xf1

for all t > 0.

(4.80)

Then for every initial distribution there is at most one solution to the martingale problem for A. Proof. Let P, P' e M\ (•£>«([(), oo))) be solutions to the martingale problem for A both with the initial distribution v. In order to show P = P' it is sufficient £ Bf,(M.n) to prove that for all m £ N , all strictly positive functions fi,...,fm and all 0 < t\ < ti < ... < tm we have m

m

fc=l

k=l

Ep(j[fk(Xtk))=Ep'(j[fk(Xtk)).

(4.81)

We prove (4.81) by induction with respect to m. Note that the case m = 1 is just assumption (4.80). Assume that (4.81) holds for m e N and define the probability measures Pi,P2 £ A4l(T>n([0,oo))) by m

i

Pi(S) = Ep(xBoBtm) J ] fk(Xtk))/Ep(j[ fc=i

'

m

fk{Xtk))

(4.82)

fe=i

and m

I

P2(B) = Ep> ({XB oetm)Y[ fk(Xtk)) fc=i

m

Ep' (J[ fk{Xtk)\.

'

(4.83)

fe=i

Here B is a measurable subset of 2?n([0, oo)) and 9tm : P n ([0, oo)) —> Vn([0, oo)) is the measurable shift operator 0tm(oj)(t) — u(t + tm). Note that by our assumptions the denominators in (4.82) and (4.83) are equal. Since for

4.3 A Uniqueness Criterion for the Solvability of the Martingale Problem

193

C G S ( n ) the function xc = (1 + xc) - 1 is the difference of strictly positive functions belonging to B(,(Rn) we may apply (4.81) to find m

,

m

P1(X0 e C) = £p(xc(*tm) I I A ( * t j ) / £ P ( I I /*(x**)) fe=l

'

fc=l

771

i

171

- sp/ (xc^u) n / * M A p/ (n A(^*j) = P2(X0eC).

k=i

'

(4.84)

k=i

Moreover, for r G N, 0 < Sx < . . . < sr < sT+\ < sr+2, functions hi G Bb{M.n), I = 1 , . . . , r, and y? £ D(A) it follows that

^ P l ((
flhtist))

r+2+tm(A
(4.85)

Jsr+i+tm r

m

I

• n w x « + t j n /*(^*))

Ep{fk{xtk))=o

since 0 < t± < . . . < tm < si + tm < ... < sr + tm < sr+1 + tm < sT+2 + tm and P is a solution to the martingale problem for A. By the same reasoning we find

W M * S r + 2 ) - f(XSr+1) - r+\AV)(Xs)ds)f[hl(XSl))

= 0.

(4.86)

It follows that Pi and P^ are solutions to the martingale problem for A which by (4.84) have the same initial distribution .Using assumption (4.80) and (4.81) we get for all tm+1 = tm + t', t' > 0, and all fm+1 G Bb{Rn) m+l

EP(U

m

MXtk))=Epi(fm+1(Xt,))Ep(j[fk(Xtk))

fc=i fc=i

=

m

Ep>(fm+1(Xt,))Ep'(jlfk(Xtk)) k=i m+l

-Ep>(U

Mxtk))

fc=l

which is (4.81) for m replaced by m + 1 and the proposition is proved.

•

194

Chapter 4 The Martingale Problem

In order to formulate a uniqueness criterion to the martingale problem for a pseudo-differential operator q(x, D) let us recall some notions on function spaces. Let ip : Rn —> R be a continuous negative definite function and set as usual

\Hl,.=

I (l + iKO)*|S(£)|2d£

(4-87)

H^s(Rn)

= {ne S'(Rn);

(4.88)

and ||u||^, < co}.

We know that VKO > co|C|ro

(4.89)

for some c0 > 0 and r0 > 0 and all £ € R™, |£| > -R, will imply tfiM(Rn)

^ Coo(R") C Cb{Rn)

(4.90)

for all s > ^ , compare Theorem 1.3.10.12. For T > 0 fixed we denote by C([0, T]; ^ - " ( R " ) ) the space of all strongly continuous mappings u : [0, T) -> ff^' 5 (K") which is equipped with the norm sup ||u(i)||^ s a Banach space. 0
Further we need the space C 1 ([0, T]; H^'s(M.n)) which consists of all mappings it: [0, T] —> H^'s(W.n) which are strongly continuous and strongly continuously differentiable. The norm sup ||u||^,s + sup ||u'(t)||^, iS

0
0
turns C1 ([0, T]; # ^ s ( R n ) ) into a Banach space. Note that if (4.90) holds then we have a continuous embedding C([0,T];

tf^(R"))

-» C b ([0,r] x R").

(4.91)

Finally since C^°(R") C H^'s(Rn) we may consider C£°([0,T] x R") as a subspace of ^ ( [ O . T ] ; i/^> s (R n )), s is chosen such that (4.90) holds. Our aim is to prove Theorem 4.3.2 (W. Hoh [155]). Let q : R n x R " -> R 6e a continuous negative definite symbol and q(x, D) the corresponding pseudo-differential operator

4.3 A Uniqueness Criterion for the Solvability of the Martingale Problem

195

defined on C£°(R n ). Fix s0 such that (4.90) holds and assume that q{x,D) extends to a bounded operator q(x,D) : H^S0+2(Rn) If for allT>0 problem

-> H^'s°(Rn).

(4.92)

and for all ip € C 0 °°([0,r] x R") C ^ ( [ O . T ] ; #*••<> (R n )) the

u' - q(x, D)u = -ip

on [0,T),

u(T) = 0

(4.93) (4.94)

then has a unique solution u € C([0, T]; iJ^ s °+ 2 (R n )) n C 1 ([0, T]; H^'s°(Rn)) n for any initial distribution v £ Ail(M. ) there is at most one solution to the martingale problem for —q(x,D). Remark 4.3.3. In volume II we gave a lot of examples of pseudo-differential operators satisfying the assumption (4.92). The proof of Theorem 4.3.2 requires the following Lemma 4.3.4. Let q(x,D) and so be as in Theorem 4.3.2 and let P e A4l[T>n([0,oo))^ be a solution to the martingale problem for the operator Fix T > 0 and choose u e C([0,T] ; H*''0+2(Rn)) n (-q(x,D),CS°(Rn)). s ^ ( [ O . T ] ; tf^ °(IR™)). Then the process

Mt = M?:=u(t,Xt)-J

((^-s-q(x,D))u)(s,Xs)ds,0
(4.95)

is a P-martingale up to time T. Proof. Given

0. Jo

(4.96)

Next we want to show that it suffices to prove the lemma under the asFor u & C([0,T] ; #^ s ° +2 (]R™)) n sumption u G ^ ( [ O . T ] ; H^>So+2{Rn)). ^ ( [ O j T ] ; H^'So(Rn)) we construct an extension, again denoted by u, belongs DC1 {I; H^<s°(Rn)). (For example we may extend u by to C(I; H^'S0+2(Rn))

196

Chapter 4 The Martingale Problem

reflecting at the boundary points.) With tp £ CQ°(M),

0 and J R 0 sufficiently small the vector-valued Priedrichs mollification of u by u£(t) := / -ip(t-—^)u(s)ds, 0
(4.97)

where the integral is a Bochner integral. It is straight forward to see that for £->Owe have ue -> u in C([0, T] ; JT^°+ 2 (R n )) and ue^u

inC^ICT]; H^s°(Rn)).

Now (4.91) yields that for e -> 0 it holds ue - > « , « ; - > u', 9(1, £>)«e -» q(x, D)u

(4.98)

where the convergence is bounded and uniform and all functions are considered to be elements in C&([0,T] x R n ). Thus the martingale property in (4.96) is preserved under such an approximation. Now take u G C^fO.T]; #^ s ° + 2 (R n )) and note that v! = §-tu when u is considered as a function defined on [0,T] x R". For all 0 < tx < t2 < T and A £ T% we get Ep((u(t2,Xt2)-U(i1,Xtl))X^) = £; p ( ( « ( * 2 , ^ 2 ) - « ( * ! , X t 2 ) ) x ^ ) + ^ P ( ( ^ i , X t 2 ) - n ( i 1 , X t l ) ) X A ) . Since u{t{) e iJ^' So+2 (R n ), and taking into account that (4.96) is a martingale we find further Ep([u{h,Xt2)-u{h,XH))xA)

= EP(f2 As, Xt2 )dsXA) + Ep U \-q(x, D)u(t!, Xr)ArXA) = Ep(|*2 (A - q(x, D))u(s, Xs)dsxA) +

Ep([t2(u'(s,Xt2)-u'(s,Xs))dsXA

+ f *{{-q{x,D)u{tuXr) - (-q{x,D)u){r,Xr))drXA).

4.3 A Uniqueness Criterion for the Solvability of the Martingale Problem

197

Clearly, the lemma is proved when we can show that the second expectation vanishes. Using once again (4.96) we observe that EpU\u\s,Xt2)

- u'(s,Xs))dsXA) (4.99)

= Ep(£2(J2(-q(x,D)u')(s,Xr)dr)dsXA), while on the other hand Ep(f\(-q(x,D)u))(t,Xr)

- (-q(x,D)u)(r,Xr)drxA) (4.100)

= - Ep ( jf 2 (^

^(-q(x, D)u)(s, Xr)ds)drXA)

holds. Under our assumptions on u we have

q(x,D)u' = j-t{q(x,D)u), and since the iterated integrals in (4.99) and (4.100) are over the same domain • they are equal implying the lemma. Now we can give Proof of Theorem 4.3.2. For
Mt = u{t,Xt)- f Jo

{^--q(x,D))u(s,Xs)ds a s

is a P-martingale up to time T for any solution P to the martingale problem for — q(x, D) and initial distribution v. Therefore it follows that

J V1(s)Ep(
Ep(u(0,Xo))=E"(u(0,.)),

and the last expectation is uniquely determined by v. Since
198

Chapter 4 The Martingale Problem

s < T, is uniquely determined by v. But also ifi2 and T have been arbitrary and therefore we may conclude that the one-dimensional distributions PoXj1 are uniquely determined by v for s > 0 and do not depend on a particular choice of a solution to the martingale problem. Now the theorem follows from Proposition 4.3.1.

•

Theorem 4.3.2 tells us that whenever we can solve in a suitable function space a "backward heat equation" for a pseudo-differential operator with a continuous negative definite symbol then the martingale problem for this operator has at most one solution. Theorem 4.2.1 gives an easy criterion for the existence of a solution. Clearly the results in II.2.6 give examples for pseudodifferential operators where we can apply both theorems. However we do not really get new insights since in this situation we can already construct Feller semigroups, but, using a localization procedure to be discussed in the next section we can considerably improve the results of Theorem II.2.6.4.

4.4

A Localization Procedure

In Theorem II.2.6.4 it was proved that a pseudo-differential operator — q(x, D) = —qi(D) — q^{x, D) with a continuous negative definite symbol q(x, £) generates a Feller semigroup provided that the "coefficients" of q2(x, £) have only a small oscillation where smallness relates to the "ellipticity constant" of <7i(£)- But also smallness of the derivatives d"q2(x,£) upto a certain order \a\ < m was required which can be interpreted that for |£| large the coefficients of q(x, £) must become constant. This is quite a restrictive condition, but the advantage of Theorem II.2.6.4 compared with Theorem II.2.6.9 is that no smoothness of q{x,£) with respect to £ is required. It turns out that combining the results of Theorem II.2.6.4 with some martingale problem techniques yields a quite general existence result for Markov processes generated by a pseudo-differential operator. We will prove this result in the next section. In this section we will summarize some preparatory material. The main result in Section 4.5 will depend on a localization procedure which is discussed in detail by St. Ethier and Th. Kurtz [98], p. 216-221, and due to D. Stroock and S. Varadhan [350] who introduced it in case of diffusion operators. In our presentation we follow closely W. Hoh [155], see also [150], since this is most appropriate for our purposes. However we do not give proofs for the general results, for this we refer to [98].

4.4 A Localization Procedure

199

Let A : D(A) -> Bb(Rn) be a linear operator with domain D(A) C C b (R"). Further let P G M\(T>n([0,oo)) be a solution to the martingale problem for A and some initial distribution v. In particular

(4.101)

U{Xt) - [ A
is for all tp G D(A) a martingale with respect to (Ff)t>o under P. For an open set U C R™ we consider the (.Ft0)t>o-stopping time rt/ := inf | t > 0 ; (Xt $ U) or (t > 0 and Xf_ g 17)}.

(4.102)

The stopping time T\J is the /irsi contact time of the closed set Uc, i.e. the first time t such that the closure of the path LJ up to time t hits Uc. The proof that T[/ is indeed a stopping time is similar to the proof of Theorem 3.5.17 and uses the cadlag-structure of paths, for details see [98], Proposition 2.1.5. We may now apply optional stopping, see Section 2.6, to find that for all

(4.103)

(A^iXM /t>o

is an {T^)-martingale under P. Definition 4.4.1. A. In the situation described above we say that P £ A4l(pn([0,oo))) is a solution to the stopped martingale problem for A and U if (4.103) is for all ip G D(A) an (.^-martingale under P and P(Xt = XtAT for all t > 0) = 1.

(4.104)

B. If in addition for every initial distribution v there is a unique solution to the stopped martingale problem such that POXQ1=V,

(4.105)

then the stopped martingale is called well-posed. If we compare the original martingale problem for A and the stopped martingale problem for A and U we find that upto the stopping time TV there is no difference, but for t > Ty, in case of the stopped martingale problem the process is prescribed by (4.104). According to [98], Theorem 4.6.1, it holds. Theorem 4.4.2. The well-posedness of the martingale problem for A entails the well-posedness of the stopped martingale problem for A and U.

200

Chapter 4 The Martingale Problem

Most important for us is that a converse result holds: If the stopped martingale problem is well-posed for a family of open sets forming an open covering of M™, then the martingale problem for K n is well-posed provided it has a solution. More precisely we have T h e o r e m 4.4.3. Let (t/fc)fceN be an open covering o/R". Suppose that for all initial distributions v there exists a solution to the martingale problem for A (and M.n). If for all k G N the stopped martingale problem for A and Uk is well-posed then the martingale problem for A and M.n is well-posed too. Once again we refer to [98], Theorem 4.6.2, for a proof. Let us make explicit the procedure suggested by Theorem 4.4.3. Existence of a solution to the martingale problem for a pseudo-differential operator —q(x, D) with a continuous negative definite symbol is granted for a large class of symbols by Theorem 4.2.1. Using Theorem 4.4.3 we decompose this solution into solutions of stopped martingale problems. More precisely we use stopping times Tuk of type (4.102) where (Uk)ke.N is an open covering of M™. Next we (try to) prove that for each fceN the stopped martingale problem for —q(x, D) and Uk is well-posed, where qk(x,0 coincides with q(x,£) on Uk x K n . This we will achieve by applying Theorem 4.4.2. As a consequence the martingale problem for —q(x,D) is well-posed and gives rise to a Markov process. The exact formulation of this procedure is given in Theorem 4.4.4 (W. Hoh). For k G N let q,qk : Mn x E " -» 1 be continuous negative definite symbols, and suppose that the corresponding pseudodifferential operators q(x,D) and qk{x,D) mapC0Xi(M.n) into Cb(R"). Assume that for every initial distribution v the martingale problem for —q(x, D) has a solution. Moreover we assume that that there is an open covering (Uk)k£H of W1 such that for allkGN , x e Uk and £ e Mn

q(x, 0 = Qk(x, 0

(4.106)

holds. If for all k G N the martingale problem for —qk{x, D) is well-posed, then the martingale problem for -q(x, D) is well-posed too. Proof. For t < T\jk we have Xt G Uk and therefore
=
rtATuk

Jo

rt/\ruk

Jo

(-q(x,D)
{-qk(x,D)
4.5 On the Well-Posedness of the Martingale Problem

2(H

for all t > 0 and all

4.5

On the Well-Posedness of the Martingale Problem for a Class of Pseudo-Differential Operators

Let us recollect the ideas behind Theorem II.2.6.4 which gives the existence of a Feller semigroup generated by a pseudo-differential operator —q(x, D) with a continuous negative definite symbol q : R n x R" —> R which splits according to q(x,t)=Qi(O+12^,0-

(4-107)

The crucial assumption was that ^2^<m sup |<9f 0 and sufficiently many functions / . This was achieved by combining Hilbert space methods to obtain first a variational solution of this equation and regularity results. The smallness assumption mentioned above was needed to prove certain lower bounds for q(x,D). We want to combine similar ideas with the localization result of last section to prove the well-posedness of the martingale problem for -q(x,D). The construction of the symbols %(£,£) is a bit different than the decomposition (4.107) and therefore we have to modify some technical results used in the proof of Theorem II.2.6.9. We follow once again W. Hoh [155], but refer also to Section II.2.6, and the papers [182] and [185]. Let us fix a continuous negative definite function ip : K" —> R and assume

V(O>coie°

for all ICI > 1

(4.108)

202

Chapter 4 The Martingale Problem

with some ro > 0 and CQ > 0. Further let 5:R"xKn-4i

(4.109)

be a continuous negative definite symbol satisfying q(x, 0) = 0 for all x G R.

(4.110)

For M G N being the smallest integer such that (4-111)

M > (— V2) +n

suppose that x <—> q(x,£) is a C 2 M + 1 ~"-function satisfying for all /? G NQ, \/3\<2M+l-n

\d£q(x,f)\
(4.112)

In addition we assume the existence of a strictly positive function 7 : R n —> R with the property that q(x,t)>

7(1) (1 +V>(0)

(4-113)

holds for all a; € R™ and |^| > 1. Now we want to construct out of q(x, £) symbols which behave almost as those considered in Theorem II.2.6.4 but which will allow us to use the localization procedure of Section 4.4. For this fix xo 6 K n and take p G C§°(Rn) such that 0 < p < 1, p\Bl (0) = 1, suppp C fii(0) and for R > 0 define

( 4 - 114 )

P«(I)=P(£X2)We consider now QR(X,0

=pR(x)q(x,£) +

(l-pR(x))q(x0,Z)

= gfao, 0 + /»fl(a;) (g(a;, £) - q(xo,£)).

(4.115)

Clearly,
(4.116)

4.5 On the Well-Posedness of the Martingale Problem

203

which we will consider as a perturbation of q(x0, £). But unfortunately 92(x, 0 will in general not satisfy the assumptions of Theorem II.2.6.4 and for this reason we need to rework some of the auxiliary results involved in the existence proof for a Feller semigroup generated by —Qk{x, D). Let TV G N be the smallest integer such that (4.117)

N>(—V2), or M = N + n. By Taylor's formula we get

g(z,0=9(zo,0+

iX o)0

£ 0<|/3|<JV

^ (d^q)(xo^)

+ RN(x,O

(4.118)

with remainder RN(X,£)

= (N + 1) f\l-t)N

Yl

Jo

\0\ = N+l

{X

~*0)\d2q)(tx P

'

+ (l-t)xo,Odt. (4.119)

Combined with (4.115) this yields QR(X,0

0<|/3|
=«(a:o,o+ E 6/3(x)9/3(o+ 53 fofoo 0<|/3|
|/3|<JV+1

(4.120) where h(x)

= PR(x)^X

~p°^

9/9(0 = (^«) (io,O

,O<\0\
(4.121)

,0<| / 8|
(4.122)

and for |/3| = N + 1 9/3(x,O = ( ^ + l ) y ' ( l - i ) ; V P f l ( x ) ^ p ^ ( 5 ^ ) ( t e + ( l - i ) a ; o , O d i .

(4.123)

Note that bp and ^3 are depending on R and this dependence we have to take into account in the following.

204

Chapter 4 The Martingale Problem

L e m m a 4.5.1. For 0 < s < M — n there is a constant Ki(R) with lim Ki(R) = 0 and such that for all (3 € NJJ, |/?| = N + 1, and u € C£°(R71) ii R-+0

holds ||((l + ^(I>)) 4 ,? / s(a;,D)u|| o = l f 1 ( / J ) | H U i , + 1 .

(4.124)

Proo/. Since 2 + n < M, see (4.111), it follows for 0 < s < M - n that s — l | + l + n < M and an application of Theorem II.2.3.9 implies (4.124) provided we have \d2qp(x,Z)\ < $a(x)(l + V(0)

(4-125)

for a G NJ, |a| < M, and functions $ a G L ^ R " ) . For Ki{R) we will find

Ki(R) = cM,Bli> J2 ll*«|Ui

(4-126)

|a|<M

where CM,S,V i s determined similar to Kn,miS,v> i n (H.2.148). Using Leibniz' rule and (4.112) we get

< (7v + I) l ^ i - t)Nd;(pR(x)(x~*o)\dgq)(tx

+ (i - t)io,O) |dt

= (iv + i ) / ' 1 ( i - ^ E (a)dr"(pR^){jLZ^-)thl(d^Q)(tx

+ (1 - t)x0,o|d*

<{N + 1) jf'(l - t)» 53 Q l ^ - ^ ^ W ^ ^ J I ^ ' c t l + *«))«

Since |/? + 7| < N +1 + M = 2M + 1 - n, it follows that all derivatives needed in the above calculation exist. Therefore ,M$tt(x):=c^Mar(pfiW^)

(4.127)

4.5 On the Well-Posedness of the Martingale Problem

205

belongs to L^R") and (4.125) follows with (4.126). Next we will prove

lim sup /

<%-> (PR(x)

R->0 y
SU

P(E(T)/

h\<M^f^\dJ

~°;O)0) p \

which will imply the lemma by (4.126). Leibniz' rule yields

±

(X

V

n

\dx = 0

'\

A straightforward application of

BTW\&P)C^.) to K

I

JBR(X0)

x sup l ^ ^ ^ l ) x£BR(x0)

P-

W

^ SUP ( E ftV"l+fl / |^P(»)|d» |7|<M V J ^ V d /

x c

M

V Bl (0)

sup K x - i o ) ^ 7 - ^ ! ) ,

and it is sufficient to sum only over 7 — 8 < /? since otherwise d2~s{x — XQ)^ will vanish. Hence |/3 — (7 — S)\ = \j3\ — \~f\ + \S\ and therefore

sup/

ar>K0r)^^)dz

7
^

P-

SU

P ( E (lV*r.rf-"l+B /

|7|<M

V

J ^ \<5/

|^W|cte^ | -^ + l "

7Bi(0)

< c sup ir»+i^i-iTi - • 0 as i? -» 0 because n + \/3\ - \^\ > n + N + 1 - M = 1 by the choice of TV.

•

Next we need a commutator estimate involving 6/3, bp = PR(X)^X~Q°' , 0 < \P\ < N. Since bp is a smooth function with compact support we find for

all a 6 NJJ

|^(6p(x)^(0)| < l ^ ( z ) | c ( l + V(0),

(4-128)

206

Chapter 4 The Martingale Problem

where qp is as in (4.122). Further we have II^MOIILI

< C[R)

(4.129)

and therefore we find analogously to Theorem II.2.3.9 Lemma 4.5.2. Let s > 0 and 0 < |/?| < N. Then it holds \\[(l+i;(D)y,b0(-)q0(D)]u\\o
(4.130)

for a M u e C £ ° ( R n ) . Finally, we can restate the results for commutators involving the Friedrichs molhfier, compare Theorem II.2.3.16. There are no substantial changes in the proofs required. Lemma 4.5.3. A. Let 0 < s < M — n and e > 0 be given. Then there is a constant C(R) independent of e such that for all |/?| = TV + 1 and all u G C£° (R n ) \\[Js,qp(x,D)u\\^s

< C(R)\\uU,s+1

(4.131)

holds. B. Let s > 0 and let e > 0 be given. Then there is a constant C(R) independent of e such that for all 0 < \/3\ < N and for all u € Cg°(M.n) \\lJe,b0(-)qp(D)]u\\^s

< C(R)\\uU,s+1

(4.132)

holds. Our next aim is to solve the equation qR(x, D)u + Xu = f

(4.133)

in order to apply then the Hille-Yosida theorem and to get a semigroup generated by —qR.(x,D). However now we want to use this semigroup to solve the backward heat equation associated with —qn{x,D), see Theorem 4.3.2, and it turns out that it is more convenient to construct this semigroup as a semigroup on the Hilbert space H^'^W1), s suitable, and not on L2(Rn). The basic ideas are much the same as in the proof of Theorem II.2.6.4 but we have to use the newly derived commutator estimates and we have to take into account that the Hilbert space under consideration is H^'s(M.n). We start with

4.5 On the Well-Posedness of the Martingale Problem Lemma 4.5.4. A. Let s > 0 and 0 < |/3| < N. u € C£°(Rn)

207 Then it holds for all

||g(a:o,^)«IU,.
(4-134)

\\qp{D)u\\^s < c||u||^ + 2

(4-135)

and (4.136)

||&,j(-)
Then there is a constantK2(R) such that lim ^(-R) =

0 and for all |/?| = iV + 1 and a« w € C^°(ffi") ^ AoWs

H^fe D)u\\^s < /ir2(JR)||u||^i,+2.

(4.137)

Prw/. Clearly (4.134) and (4.135) follow as (II.2.145) in Proposition II.2.3.6. In order to see (4.136) we combine Lemma 4.5.2 with (4.135) to find \\be(.)q0(D)uU,s

< \\{l +

i;(D)y{b0(.)q0(D)u)\\o

f

< \\b0(.)q0(D)(l + ^(£>)) u||o + ||[(1 + iP(D))i,b0(-)q0(D)]u\\o

(4-138)

< c||&/9||OD||u||¥l,s+2 + C(R)\\u\\^,t+1 which proves (4.136). To prove part B, note that for \/3\ = N + 1 we find (as in the proof of Proposition II.2.3.11) for all u G C0°°(Mn) the estimate

Qe(x,D)u\\0{D))f q0(x, D)u\\0 <\\q0(x,D)(l

+ i}(D))iu\\o

< KR\\ (1 + i,(D))

f

u|U,2 +

+ \\[(l +

iP(D))i,q0(x,D)}u\\o

tf1(Ji)||u||,M+1

< (A-R + /JfiC^Z)) ||«||^,.+2 proving the lemma.

•

208

Chapter 4 The Martingale Problem

Having in mind the decomposition (4.115) we deduce from Lemma 4.5.4 that qR(x,D) maps H^-s+2(Rn) continuously into #^' S (R") for all 0 < s < M — n. As a next step we investigate the bilinear form associated with qR(x,D). For thisfixs0 > 2 such that (4.139) M -n > s0 > — 2r 0 holds. Note that this choice of so guarantees the embedding <^ Coo(]Rn). We are longing for a type of variational solutions to ff*'s"(ln) qk{x,D)u + Xu = f and for this we introduce for u, v £ Co°(R") the bilinear form BR(u,v)

(4.140)

= (qR(x,D)u,v)^sg.

This bilinear form is decomposed according to

BR(u,v) = B0R(u,v) + J2 B0^v)+ O<\0\
J2 B$(u,v)

(4.141)

\0\=N+1

where B*(u,v) := (qixo.Dfav)^, B${u,v)

:= {b0(-)qi3(D)u,v)^so

(4.142) ,0<\P\
(4.143)

,\(3\ = N+1.

(4.144)

and B>,«):=(^,i))v)^

Proposition 4.5.5. The bilinear form BR extends from CQ° (Rn) continuously Cg°(Mn) (or H^'So+1 {«"•)) it holds to H * ' ' » + 1 ( I n ) andforu,v£ \BR{u,v)\

< C r (fl)||«||^ i i o + i||«||^, s o + 1 .

(4.145) R

(The extended bilinear form will be denoted once again by B .) Proof. We estimate each term in (4.141) separately. Using the Cauchy-Schwarz inequality and Lemma 4.5.4 we get for u, v € Co°(K n ) \BR(u,v)\ = \((l+ij(D))^q(x0,D)u,(l

+ TP(D))S?v)o

= | ((1 + i>(D))e~^q(x0, D)u, (1 + V(£)) ^ ) J <

c\\q(x0,D)u\\^:So-x\\v\\$:So+i

< c\W\\ip,so+i\\v\y,so+i-

(4 . 146)

4.5 On the Well-Posedness of the Martingale Problem

209

Analogously, by Lemma 4.5.2 and Lemma 4.5.4 we obtain for 0 < \/3\ < N \B*(u, v)| = | ((1 + 1>{D)) ^bp(-)q0(D)u,

(l +

jj(D))^V)Q

<\(b0(.)q0(D)(l+i;(D))£Slflu,(l+ij(D))^v)o + |([(l + V( J D)) £ f l ^,6/ 3 (-)^P)K(l + V ' ( J D ) ) ^ ) o < cllb^Hoolliell^.'o+l ll«IU,.o+l

+C(R)\\uU,so\\vU,so+l

( < c||u||^,so+i||u||v-.so+i + C{R)||u|U,S0+iNk*,+i)

• (4.147)

Finally, for \(3\ = N + 1 we apply a further time Lemma 4.5.4 to find \B*(u,v)\ = |((1 + ^(D))^q0(x,D)u,

(l +

^{D))S^V)Q

< 115^(1, I > ) U | U , . 0 - I | | « I U , . 0 + I <

(4J48)

K2{R)\\uU,,o+i\\vU,ao+1.

But together (4.146), (4.147) and (4.148) prove the proposition.

•

Now using the function 7 from (4.113) we are going to prove a lower bound for BR, i.e. a Garding inequality. Theorem 4.5.6. For R > 0 sufficiently small there is a constant A = X(R) depending on R such that for all u € # ^ S o + 1 ( R n ) it holds BR(u,u)

> 2 ^ | | u | | 2 s o + 1 _ X(R)\\u\\lS0.

(4.149)

Proof. By (4.113) we can find c > 0 such that <7(zo,O > 7(*o)(1+(£))-c holds. Therefore, as in the proof of (II.2.146) in Proposition II.2.3.6 we find B*(u, u) > 7(*o)||u|ft,.o+i - c\\u\\l>so. For e > 0 we get for 0 < |/3| < N using (4.147) \B*(u,u)\

< c||ft / j|| oo ||u||^, 0+1 +

c(R)\\u\\^S0+1\\vU,S0+1

^(cllb/jHoo + ^ I M I ^ o + i + c ^ e ) ! ^ , . , , ,

(4.150)

210

Chapter 4 The Martingale Problem

where we used the estimate ab < ea2 + c(e)b2 to handle the second term. Since Halloo =

{x PR(x)

sup

~*o)0

xEBR(x0)

->0

astf^O

(4.151)

P-

we can choose R and e sufficiently small such that

Y

\BR{u,u)\ < ^ | 2 ) | | u | | 2 s o + 1 +c(fl)||u||^ 0

(4.152)

0<\P\
holds. Further, with (4.148) we obtain

Y

\B${u,u)\<

\0\=N+1

Y

K

< ^ I M f t , . 0 + i (4-153)

2(R)\Hl,S0+1

\0\=N+1

provided -R is sufficiently small,recall lim K^^R) = 0 by Lemma 4.5.4.B. From (4.150), (4.152) and (4.153) we derive

BR(u,u)>B*(u,u)- Y

\B*(U'U)\~

O<|/3|<JV

E \p\=N+i

and the theorem follows.

KMI (4.154)

D

Theorem 4.5.7. Suppose that (4.108)-(4.113) hold, decompose q(x,£) according to (4.115) and take A > X(R), A(i?) and R as in Theorem 4.5.6. Then for every f G H^So(Rn) there is a unique u £ H^'S0+1(Rn) such that BR(u, v) + X(u, v)^, so = (/, v)^so

for

allveH^'

S0+l

(4.155)

n

(R ).

Proof. We need only apply the Lax-Milgram theorem, Theorem 1.2.7.41, to • the bilinear form (BR, H^'So+1(Rn). We want to prove that the element u G H^'So+1{Rn) belongs already to H^'So+2{Rn). For this we need

satisfying (4.155)

Proposition 4.5.8. Let q(x,£) be as in Theorem 4.5.6 and suppose that R > 0 is sufficiently small. Then there is a constant C{R) such that N k . 0 + 2 < C(R) (\\qR(x, D)uU,S0 + \\uU,S0) holds for all u G

H^'So+2(Rn).

(4.156)

4.5 On the Well-Posedness of the Martingale Problem

211

Proof. We proceed, in principle, as in the proof of Theorem II.2.3.13. By (4.113) we have \\q(x0,D)u\\lrSO > 7 ( z 0 ) 2 N k S o + 2 ~ c\\u\\liao.

(4.157)

Further, for 0 < \/3\ < N, by Lemma 4.5.4 and its proof, especially (4.138), and using the estimate \\u\\^tS2 < e||u||v>,s3 + c(e)||u|| v ,, si for u e H^'S3(M.n) and S\ < S2 < S3, we get \)b0(-)qp(D)uU,So = \\{l +

^(D))^(bp(.)q0(D)u)\\o

< c|]6^||oo||w|U,So+2 + C(-R)||u||^,so+ 1 <(c||6 i 9 ||oo+e)Nk.o+2 + Cr(/i)e)||u|U,.0) which yields for R and e sufficiently small, compare (4.151), that

£

||^(-)5M^HI^o< 2 ^lkll^o+2 + C(JR)||U|U,S0.

(4.158)

0<|/3|<JV

Further, Lemma 4.5.4.B gives for R > 0 sufficiently small

]T

\\q0(x,D)uh,so<

\0\=N+1

E

K2(R)\\uU,S0+2<^\\uU,So+2.

\P\=N+1

(4.159) Hence, from (4.157), (4.158) and (4.159) we deduce

\\qR(x,D)u\yiS0 >

WqixQ^^y^

0<|/3|
\0\=N+1

>2^ll«IU..o+2-C(iZ)||u|U,,o implying (4.156).

D

Finally we can prove Theorem 4.5.9. Let q(x,£) be a continuous negative definite symbol and assume (4.108)-(4.113). Further let R > 0 be sufficiently small and let X(R) be as in Theorem 4.5.6. Then for all f e tf^s°(]Rn) and all X > \{R) there is a unique solution u G H^'S0+2(Rn) of the equation qR{x,D)u

+ Xu = f.

(4.160)

212

Chapter 4 The Martingale Problem

Proof. We know by Theorem 4.5.7 that for R sufficiently small there is a unique solution u G H^'S0+1(Rn) to the problem BR(u,v) + X(u,v)^S0 = {g,v)^S0

iora\lveH^S0+1(Rn),

(4.161)

where g G H^'s° (W1). We take in (4.161) g = / and choose a sequence (uk)keN, uk G C£°(M n ), which converges in H^'s°+1(Rn) to u. For v G Cfi°(Rn) it follows that BR(uk,v) = (qnfaD^v)^

= (qR(x,D)uk,(l + iP(D))s°v)Q. (4.162)

By Lemma 4.5.4, as k —> oo, we find qR(x,D)uk implying that

—> qji(x,D)u

( 9 / e (x I £>)u,(l + V(I?))' o u) 0 = B f l (u,v) = ( / - A « , t ; ) ^ 0

= (/-A U ,(i+^(£')r o ) 0 .

in L 2 (M"),

(4.163)

Since (l + V(D))'°(Cg°(R n )) C L 2 (K") is dense, (4.163) yields inH^s°(Rn).

(qR(x,D) + X)u = f

We claim u G i7 i/l ' So+2 (]R"). Denoting as usual by Je, 0 < e < 1, the Friedrichs mollifier, compare Proposition II.2.3.15, we find that Je(u) G H^>So+2(Rn) and as e —> 0 we have J e (w) —• u in H^'3°(M.n). Hence (Je(u))0<6<1 is bounded in H^'So(Rn). Further, by Lemma 4.5.3 we obtain \\qR{x,D)Je{u)\\i>tS0 <\\qR(x,D)u\\il,S0+

< Je(qR{x,D)u)\\TptS0 Y,

+

\\[JE,qR(x,D)]u\\^So

We,b0{')Qp{D)]uU,to

O<\0\
< WfU,s0 + |A|||u||^ 0 + C(U)||u||, Mo+1 < oo, and the right hand side is independent of e, 0 < e < 1. Thus by Propowith a sition 4.5.8 it follows that (^e(«)) 0 < e < 1 is bounded in H^'So+2(Rn) bound independent of e which implies, compare Proposition II.2.3.15, that D u G H^'S0+2(Rn). Now we are in position to prove the well-posedness of the martingale problem for q(x,D). First we show

4.5 On the Well-Posedness of the Martingale Problem

213

Theorem 4.5.10. Suppose that (4.108)-(4.113) hold and let qR(x,£) be as in (4.115), XQ € M.n being fixed. Then there exists R > 0 such that the T>n(j0, oo))martingale problem for —qR(x,D) with domain C^(Rn) is well-posed. Proof. Let R > 0 be sufficiently small such that Theorem 4.5.6 and Proposition 4.5.8 hold. Then -qR(x,D) - X(R) with domain H^<So+2(Rn) is a densely denned operator on H^tS°(Rn) which by (4.149) is dissipative. Further, by Theorem 4.5.9, for all A > 0 the range of - (qR(x, D) + X(R)) - A is H^s° (Rn). Therefore by the Hille-Yosida theorem, Theorem 1.4.1.33, —qR(x,D) is the generator of a strongly continuous semigroup (St)t>o on H^'s°(Rn). Now we apply Lemma 4.5.11 below to deduce that for ip e ^ ( [ O ^ ] , ^ ' 8 0 ^ " ) ) u(t) := / Jt

(4.164)

S3-ttl>(s)ds

gives a solution u : [0,T] - • H^'s°+2(Rn),u backward heat equation u' -qk{x,D)u

= -ip

on[0,r)

G ^([O.T] ; ff^- So (R n )), of the (4.165)

and u{T) = 0.

(4.166)

Moreover, using that u is a solution to (4.165), (4.166) we find qR(x,D)u£C([0,T]; H^'^iR71)) and by (4.156) we deduce further that An application of Theorem 4.3.2 gives now the u G C([0,T] ; H^'so+2(Rn)). result.

•

Lemma 4.5.11. Let (A,D(A)) be a generator of a strongly continuous semigroup (St)t>o on a Banach space X. For f £ C([0,T]; X) define u{t) := f 5 s _ t /(s)ds. Jt

(4.167)

Then we have u' = Au + f and u(T) = 0. Proof. Clearly we have u(T) = 0. Differentiating (4.167) with respect to t and using that ^Stu = AStu for u € D(A) we obtain the lemma, compare also Lemma 1.4.1.14. •

214

Chapter 4 The Martingale Problem

Finally we have Theorem 4.5.12 (W. Hoh [155]). Let tp : R™ -> R be a continuous negative definite function satisfying (4.108) and take M as in (4.111). Further let q : R" x Mn —> R be a continuous negative definite symbol such that (4.110), (4.112) and (4.113) hold. Then the T>n-martingale problem for the operator —q(x,D) with domain Co°(R™) is well-posed. Proof. Taking in (4.112) as multiindex /? = 0 € NQ , we see that condition (4.48) in Theorem 4.2.1 is fulfilled, hence there is for every initial distribution always a solutin to the martingale problem for —q(x,D). Next, for every XQ £ Rn we define qii(x,£) according to (4.115). From (4.112) we may deduce further that q(x,D) as well as qii{x,D) map C£°(Rn) into C b (R n ). Now, we know that if R — R(xo) is sufficiently small the martingale problem for —qjt(x,D) is well-posed by Theorem 4.5.10. Next we choose a countable set of points (£ofc)fceN such that the ball B R ^ J ^ O O cover the whole space R n . Since on 2

Bn(Xek) (xok) we have q(x,£) — qn(x,£) we may combine Theorem 4.5.10 with • Theorem 4.4.4 and the theorem is proved. By the general theory, compare Section 3.6, we know that under the assumptions of Theorem 4.5.12 we can associate a Markov process with —q(x, D). In the next section we will investigate when this process is already a Feller process.

4.6

The Martingale Problem for Pseudo-Differential Operators and the Feller Property

Under the assumptions of Theorem 4.5.12 we know by Theorem 3.6.10 that —q(x,D) generates a Markov process. In fact, compare St. Ethier and Th. Kurtz [98], p. 184, we know that this process is in fact a strong Markov process. A precise formulation of this statement adapted to our situation is the following theorem, see W. Hoh [155]. Theorem 4.6.1. Let q(x,^) be a continuous negative definite symbol and suppose that the Vn-martingale problem for —q(x,D) is well-posed. Consider the process {(Xt)t>o,Px) with Px being the unique solution to the Vn-martingale problem for —q(x,D) with initial distribution ex, x G R™. Then the universal

4.6 The Martingale Problem and the Feller Property

215

process ((^t)t>Oi Px) mn 2S a universal strong Markov process with respect to the filtration (ff)t>o, i-e- for every Tt+-stopping time r and all t > 0 and u G Bb(M.n) we have Px-almost surely Ex{u(Xt+T)\FT+)

= Ex^ (u(Xt)) on {r < oo}.

(4.168)

Our aim is to find sufficient conditions which turn ((Xt)t>o,Px)xeRn m t o a Feller process. Of course according to our philosophy such conditions should be expressed in terms of the symbol q(x,£). In this section we will always assume besides q(x, 0) = 0 for all x G Kn,

(4.169)

| 9 ( x , 0 l < c 0 ( l + |e| 2 ),

(4.170)

and

the additional condition sup \q(x,£)\ - + 0 a s ^ 0 .

xGKn

(4-171)

Note that according to Theorem 4.2.1 condition (4.169) and (4.170) imply already the existence of a solution to the martingale problem for —q(x,D). The assumption (4.171) is an equicontinuity condition (with respect to x) at £ = 0. We prepare the proofs of our main results, Theorem 4.6.6 and Theorem 4.6.7, which are due to W. Hoh [155] with some auxiliary considerations. Lemma 4.6.2. Suppose that the continuous negative definite symbol q(x,£) satisfies (4.169)-(4.171). Further let (p € C£°(1R™) and put yR{x) := y?(f), R > 0. Then it holds lim sup \q(x,D)(pR(x)\=0.

R->oo zeR n

(4.172)

Proof. We may argue as in the proof of Lemma 4.2.13 with R replacing k and then we should note that by (4.171) it follows that sup \q{x,£)\ tends uniformly (with respect to x) to 0 as R tends to infinity. A major step to our final result is

D

216

Chapter 4 The Martingale Problem

Theorem 4.6.3 (W. Hoh). Let q(x,£) be a continuous negative definite symbol satisfying (4.169)-(4.171) and let (nk)k€N be a tight set of probability measures on M.n. Denote by Pk G M^(T)n([0, oo))) a solution to the martingale problem for —q(x, D) with initial distribution fik. Then the set (-Pfc)fceN is tight inMl(Vn([0,oo))). Proof. First note that by Theorem 4.2.1 we know that at least one solution P^ to the X>n-martingale problem for —q(x, D) and Hk exists. We want to apply Theorem 4.1.16 with I = N, (Aa,D) to be {-q{x,D),C^(M.n)) for all k € N, and of course (Pa)aei to be (Pk)ken- For this we have to prove the compact containment condition to hold for (Pfc)fceN, i.e. we have to show for all T > 0 and £ > 0 the existence of a compact set K c Mn such that snpPk(Xt £ K for some 0 < t < T) < e, fc£N

compare Theorem 4.1.12. We choose tp G C^°(M") such that 0 <