Stochastic Differential Equations in Science And Engineering

) satisfies Z()d = 0.

(A.27)

The normalized radial distribution function G(r) is derived from (A.25) and (A.26). It has the form G(r) = 2vrA0r exp[-A(r 2 - 4R 2 ) 2 ].

(A.28)

We give in Figures 3.2 and 3.3 a comparison of the theoretical predictions of the radial and azimuth distribution functions with numerical simulations of the process (A.l). The corresponding numerical routines shall be covered in Chapter 5. Finally we present in Figure 3.4 a graph to get an impression in which way the noise destroys the limit circle curve of the deterministic oscillator and produces a banded diagram whose bandwidth increases with growing noise intensity T.

The Fokker-Planck

123

Equation

Fig. 3.2. A comparison of the numerically computed and the theoretically predicted radial distribution function (A.28); n = <5 = 0.001, T = 0.05.

6000

6000

4000

4000

2000

2000

-1,57 -1,047-0,5233

0

0,5233 1,047

1,57

if Fig. 3.3. A comparison of the numerically computed and the theoretically predicted azimuth distribution function (A.27); parameters as in Fig. 3.2.

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0,2

y

o

/!/) i If 11

-0,2

0 x

-0,2

0,2

Fig. 3.4. A single realization of the noisy oscillation phase-diagram; n = <5 = 0.001, T = 0.05; IC's: a;(0) = 0, j/(0) = - 0 . 2 .

Appendix B We discuss here two different concepts to study the stability of fixpoints (FP) of SDE's. Both strategies have their merits in the limit of deterministic problems and we give a brief introduction to the corresponding theories for ODE's. The stability of the FP's is investigated in the sense of Lyapunov, or we concentrated on asymptotic stability. B . l . The method of Lyapunov

functions

This method is well established in investigations of the stability of deterministic ODE's. We present here a brief survey of these deterministic problems. Given a system of n autonomous first order ODE's —— = ri\x)\ F i (0) = 0;

x t -ttn;

j = l,2,...,n;

x = [Xi, X2, • • •, xn)\ t>0.

(B.l)

A FP can always be replaced to the origin (x = 0) and this is expressed in (B.l) by Fj(0) = 0.

The Fokker-Planck

Equation

125

We introduce now a positive definite function, the Lyapunov function, L(x) that satisfies the following three conditions in (B.2) (i)

L(0) = 0;

(ii)

L(x)>0;

dL _ ^ ,^dL(x) — = Fp(x) —^ < 0,

(iii)

(it

OXr,

(B.2) where the conditions (ii) and (iii) hold in an open neighborhood (|x| > 0) of the FP. If we can find such a Lyapnov function then the FP is stable in the sense of Lyapunov [see (3.60)]. In the stochastic case we wish to investigate the stability of the FP's of the autonomous SDE's dxj = a,j(x)dt + 6jm(x)dB™;

j = l,2,...,n;

m = 1, 2 , . . . ,K. (B.3a)

A FP of (B.3) is defined by dxj = 0 => aj(x0) = 0; Vj = l , 2 , . . . , n ;

bjm(x0) = 0;

m = l,2,...,K.

(B.3b)

FP's of 1-D SDE's studied in Section 3.6.1 are analyzed in EX 3.14. Lyapunov functions can be used, however, in the case of higher order SDE's. To verify this we introduce a Lyapunov function that satisfies the conditions of (i) and (ii) of (B.2). Its differential [see (1.127)] is given by a m

^ ~

+

?bmrbkr?Sr

flr,

)

dt

+ bm

i^dBt-

(B

-4)

A stable system has now the property that L does not increase, i.e. dL < 0.

(B.5)

The latter equation means that the deterministic definition of the stability holds also for every single trajectory x(t,co). However, we cannot investigate every trajectory and this condition is thus impractical. Hence, we require only that the average of dL is negative definite

dL

+

+

dB s

< > = < (<-»£; s^slfe) > * <"-iS; ') °-

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We see t h a t the second term on the right hand side of the latter line vanishes. Thus we derive in the stochastic case as the third stability criterion U(L) = a m -

h -bmrbkr-

— < 0.

(B.6)

Example 1 We consider the 1-D population growth model (2.1) with the F P in the origin dx = rxdt + uxdBt;

r,u — const.

Its exact solution can be written in the form x(t) = x(0) exp((r - u2/2)t{l

+ uBt/[t(r

- u2/2)]}).

(B.7)

To study its stability, we use the relation — -»• 0,

t - • oo.

(B.8)

To prove (B.8) we can use the law of the iterated logarithm (1.63) or we can apply alternatively the way of EX 3.15 showing t h a t all moments of the right hand side vanish. (B.7) and (B.8) imply now t h a t the F P is stable for r - u2/2 < 0,

(B.9)

and unstable elsewhere. Now we apply the Lyapunov function approach and use as a trial function L(x) = \x\s;

s > 0.

(B.10)

It is easy to verify t h a t this choice of the Lyapunov function satisfies the first and second condition of (B.2). The substitution into (B.6) yields U(|a;| s ) = 0.5s\x\su2[{s

- 1) + 2r/u2}.

(B.ll)

If we choose 0 < s < 1 - 2r/u2,

(B.12)

we prove, with the choice of the Lyapunov function (B.10) and the compliance of (B.9), t h a t (B.6) is satisfied. Hence, we conclude t h a t the F P stable, provided t h a t Equation (B.9) is fulfilled.

The Fokker-Planck

Equation

127

Example 2 We consider a damped nonlinear pendulum where the damping as well as the frequency are stochastically perturbed x + S(l + a&)x + (1 + /?&) sin a; = 0,

(B.13)

where 5 is the damping and a > 0, (5 > 0 are intensity constants, respectively. The deterministic and the stochastic FP lie in the origin and the linearized theory of deterministic problems shows that this FP is stable for S > 0 and that at 5 = 0 a Hopf bifurcation appears. We study now the stability of the stochastic FP and rewrite (B.13) in the standard form dx = ydt;

dy =—(5y + sin x)dt — (a8y + B sin x)dBt.

(B.14)

With (B.6) we obtain the operator U = y(5y + sin x) - — I — (a6v + B sin x) r—^. J dx K y dy 2V y y ' dy2 We use now as trial Lyapunov function L = Ax 2 + Ba;y + y2 + 2Dsin 2 (x/2),

(B.15)

where the constants A, B and D are unknown as of now. (B.15) is a quadratic form plus a multiple of an integral of the nonlinear term / sin(z)d,2. Jo This leads to U = xy(2 A - BS) + y2(B -28 + a2S2) + y sin(ir)(D - 2 + 2aBS) , ^/„ ^osin(x) -xsin(x) I B-B2—— To comply with the condition U < 0 we use the relations between the constants A = BS/2;

D = 2(1 - aB8);

B - 25 - a252 < 0;

B//3 2 > 1. (B.16)

The Lyapunov function takes now the form

L = y+

( T) 2 + T i1 ~ I) x2 + 4(1 ~ a(3S)sin2^/2)-

^B-17)

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To achieve the condition L > 0 for (x, y) ^ (0,0) we require that 0 < B < 25;

a(38 < 1.

(B.18)

A recombination of condition (B.16) yields finally «

2

< ^ 2

with/3 = a 7 ,

7

>0;

B< - ^ .

(B.19)

With constants complying with (B.19) we can conclude now that the FP in the origin is stable. B.2. The method of

linearization

We begin with a linear nth order ODE with constant coefficients j/(")(x) + biy^-l\x) bj = const.;

+ ••• + bn.iy'{x)

+ bny(x) = 0;

Vj = 1, 2 , . . . ,n,

(B.20)

where primes denote derivations with respect to the variable x. (B.20) has the FP (y(0),y'(0),...,^-1)(0))=0.

(B.21)

The Routh-Hurwitz criterion (see [1.2]) states that this FP is asymptotically (for t —> oo) stable if and only if the following determinants satisfy the relations (see also Ex 3.16) Ai = 6i > 0;

A 2 = bib2 > 0;

A3 =

h 1 0

b3 b2 6i

0 0 > 0. b3

(B.22)

We generalize now the concept of the Routh-Hurwitz criterion for the following class of second order SDE's y" + [ai + Ci(t)]y' + [aa + Q2{t)y = 0; aj = const.;

Q(t)dt = const. dB];

j = 1,2,

[a.26)

with the two white noise processes [see (2.41)] (Cj(t)Ck(s)) = 6(t - s)Qik.

(B.24)

This stochastic extension was derived by Khasminskiy [3.17]. It states that the FP at the origin is asymptotically stable in the mean square IFF the following conditions are met a1 > 0;

a2> 0;

2axa2 > Qna2 + Q22.

(B.25)

The Fokkei—Planck

Equation

129

E x a m p l e (Stochastic Lorentz equations) We consider a Lorentz attractor under the influence of a stochastic perturbation. The SDE has the form dx = s(y — x)dt, dy = (rx — y — xz)dt + ay dBt, dz = (xy — bz)dt;

b:s,r

(B.26)

> 0.

This system of SDE's is a model for the development of fluid dynamical disturbances and their passage to chaos in the process of heat transfer. The dimensionless parameters in (B.26) are given by s (the P r a n d t l number) r (proportional to the Rayleigh number) and b (a geometric factor). (B.26) has the deterministic F P ' s (0,0,0);

(±y/b{r-l),

±y/b(r-l),

r - 1).

However, in the stochastic case (a ^ 0) only the F P in the origin survives. We use now the linearization routine to study the stability of the F P in the origin. The linearization of (B.26) leads to du = s(v — u)dt, dv = (ru - v)dt + avdBt,

(B.27)

dw = —bwdt. We infer from (B.27) t h a t the component w is decoupled from the rest of the system and because of w(t) = w(0) exp(-bt) this disturbance is stable for b > 0. Hence we will investigate only the stability of the system (u,v) taking advantage of the first two equations of (B.27). However, this system is not a member of the class (B.23). To achieve a transformation we eliminate the variable v by use of v = u + (du/dt)/s. This yields u + [(1 + s) - aC{t)}u + [s(l - r) - as((t)]u

= 0.

(B.28)

A comparison with (B.23) and (B.24) leads then to ai = l+s; Q

_ / a2 "Va2S

a2 = s(l-r); a2s \ a2s2)-

(i =-a((t);

(2 =

-as((t); (B.29)

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The Khasminskiy criterion applied to our example reads now l + s>0;

s(l-r)>0;

2(l + s)(l -r)

> a2(l -r + s).

(B.30)

Since we have by definition s > 0 the first part of (B.30) is automatically satisfied, while the second part gives r < 1. Finally, the third part of (B.30) yields for small intensity constants

r<1

-207)+0^-

(R31)

The origin is stable in the deterministic limit for r € [0,1). Hence, equation (B.31) tells us how the stability is reduced by a small intensity noise a £ [0,1). The study of Hopf bifurcations is beyond the scope of this book. The interested reader can find, however, pertinent literature in [3.4], Ebeling et al. [3.18], Arnold et al. [3.19], Schenk-Hoppe [3.20], [3.21], [3.22] and Keller and Ochs [3.23]. Exercises E X 3.1. (a) To find the solution to the random walk problem (3.13) use the initial condition Z n (0) = <5no and introduce the probability generating function (u is an auxiliary variable) oo

G(u,t)=

£

unZn{t).

n=—oo

Show that G(l,t) = 1; G'(l,t) = (n(t)>, G"(l,t) = (n2(t)) - ; ' = 3/3*. (b) Consider a random walk on a straight line with step size Ax and a time At between two steps. Equation (1.54) can be reduced to Pi(nAx, sAt) = 2_.Pi(mAx,

(s — l)At)

m

x p 1 | 1 (nAx,sAt|mAx, (s — I)At)). Assuming equal probability of the steps to the right and to the left Pi|i — 0.5(5m;Tl+i + 5mjTl-i) shown that we obtain pi(nAx, sAt) = 0.5[pi((n + l)Ay, (s - l)At) +

Pl((n-l)Ay,(s-l)At)].

The Fokker-Planck

Equation

131

Derive from the latter equation the quantity [pi(nAx, sAt) - pi(nAx, (s - l)Ai)]/Ai and apply the limit x = nAx;

t = sAt; Ax -> 0; At -»• 0; D = (Ax)2/(2At)

= 0(1)

to derive a diffusion PDE dpi dt

=r)9

2

Pi dx2 '

This is the FPE with A = 0; B = 2D. E X 3.2. Calculate the expression for d(y2)/dt to the derivation of (3.17), the relation

^(y 2 ) = jdyjdy'[(y

and use, in analogy

- y1)2 + 2y(y' - y)]W(y'\y)Z(y,t).

Verify that the evaluation of the last line leads to ^t(y2}

=

(a2(y))+2(ya1(y)).

The variance a satisfies a2' = (y2)' — 2(y)(y)'. Verify that this yields *2' =

(a2(y))+2((y-{y))al(y)).

E X 3.3. To calculate the transformation of the third term in (3.28) use the relation (B^T)" = (BT)'V + 2(BT)>' + ( B T ) < , to verify that f(BT)^"dz

= /"(BT)'Vdz.

E X 3.4. Derive the backward Chapman Kolmogorov equation. Use in (3.4) u = At and v = t, y% = z; y2 = w and y\ = x (x, w and z are backward, intermediate and forward spatial variables, respectively).

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Hint: This yields T(z|x; t + At)=

/ dwT(w\x; At)T(z\w;

t).

Multiply this equation by an arbitrary function ip{x, t) and note that x < w. Apply the other considerations that lead from (3.21) to (3.29). The result ^T{z\x;t)

= -^[A(x,t)T(z\x;t)]

E X 3.5. Consider [see (3.32)] 9P

the

-

three-term

^^[B(x,t)T(z\x;t)]. Kramers-Moyal

1 32

3

equation

Id3

Multiply this formula by Ay and (Ay) 2 (see (3.33a)). Show that the third term in this Kramers-Moyal equation does not affect the first and second order terms in (3.34). Hint: V

d

3

= {yv)

- Sv ;

y

d

3

= (yv)

- 6{yv) + 6^ .

E X 3.6. Continue the calculation of (3.33b) for k = 4 and 5 and find formulas in analogy to (3.33c) to (3.33e). E X 3.7. Consider the average S= / J

a(y(s),s)dsf

b(y(w),w)dBl

= ^2(a(ym, s m ) ( W i - tm)b{yn, s n )(B n+ i - B„)). n,m

The integrand is a non anticipative function. Thus, we obtain from

the independence of the increments of Brownian motion that the average written above vanishes: (S) = 0. E X 3.8. Use the third line of (3.38) to calculate dT 3 /di; T 3 = ((Ay) 3 ). Take advantage of (2.14') and compare the results with T3 calculated from (2.19a').

The Fokker-Planck

Equation

133

E X 3.9. Compare the Ornstein-Uhlenbeck-problem (2.14) with SDE that arises from the coefficients (3.49). Use a scaled Brownian movement (see (1.65)) to perform this juxtaposition. E X 3.10. Consider (3.54) and verify Bm =

1 N,m

6l(2(3/K)1/2z}exp(z2/2)Am(z)dz -1/2

A m (0)/N, The symmetry of the eigenfunctions (3.50) leads to B2m+i = 0. Eventually we obtain p(x,t) = y^

exjp(-2nt)B2nMn

n=0

We observe that (3.38a) tends for t —• oo to its asymptotic limit. This is the stationary solution l i m ^ o o p ^ i ) = Boexp(—(5x2/K). Note also that we can determine the normalization constant Bo in the form 1 = / p(x, oo)dx =$> B 0 =

\ZKTT/(3.

E X 3.11. To prove (3.70) we put the first term in (3.69) to zero. We obtain with the use of the first integral of the PD that leads to (3.63) A

1 f(2f/g2-g'/g)--gg" -2|(f/g): Psdx+ /

P s dx

fg'/g

P,dx.

The first part of the last line gives Equation (3.70). In the second integral we apply an integration by parts for the second term where substitute (3.63). This shows that the second integral vanishes. E X 3.12. (a) Show that (3.62) is equivalent to da; g(x)odB t . (b) Solve with the use of (2.35) the transcritical case dx = (ax — x2)dt +

axodBt-

f(x)dt +

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E X 3.13. The Hamilton function H is defined by (A.14). Using (A.10) show that H = const, along W = const. E X 3.14. Find possible FP's of the 1-D SDE's (3.72), (3.74) and (3.76) of Section 3.6.1. E X 3.15. Verify that all moments of Bt/i vanish for t —> oo. E X 3.16. The Routh-Hurwitz matrices An = (ay) are in (B.22). They are constructed in the following way. First we set the coefficient of ?/n) to bo = 1. Then we put together the first row as a sequential array of odd coefficients 6i,6 3 ,...,b2fc+i,0,..., Vl<2fe + l < n . Each subsequent row is then built according to C'ij = b2j-i Vi > 2, 0 < 2j — i < n

and

ay = 0 otherwise.

i) Construct in this way A2, A3 and A4. ii) Verify the stability criterion for the case of a deterministic linear damped and/or excited pendulum b\ ^ 0, 62 = A2. E X 3.17. Study the stability of the FP's of the pendulum (B.14) with a linearization routine.

CHAPTER 4 ADVANCED TOPICS In the first section we treat stochastic partial differential equations (SPDE). The second section is devoted to an example of the influence of stochastic initial conditions on the evolution of partial differential equation (SPDE). In the third section we give a brief introduction to stochastic eigenvalue problems. 4.1. Stochastic Partial Differential Equations We investigate here a SPDE that includes an additive stochastic term L $ = aW x t ;

$ = $(x,t),

(4.1)

with a linear operator containing one space (x) and one time variable (£) 2

~2

a

2

p.

fc=l

SK

= E ^^^ +E^+c;

L

Sft

k,m=l

S m

x0<x<xi;

to
(4.2)

(£i,6) = 0M); a*™ = afcm(£i>6); &* = &*(&, 6 ) ; c = c(^i,6)-

(4.3)

In this SPDE a stands for an intensity constant and Wxt denotes the additive stochastic term, a two-dimensional Wiener sheet [see (1.66) to (1.70)]. We have to supply boundary and initial values to the problem (4.1) such as *(aro,*) = «o(*);

$(x1,t)

= u1(t),

$ ( M o ) = T 0 (x).

(4.4)

We solve (4.1) with the method of Green's functions and we obtain the solution to (4.1) $(x,t)

= $o(x,t)+a

dy Jxo

dsG(x,t,y,s)Wy8. J t0

135

(4.5)

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The homogeneous solution satisfies L3>o = 0. The inverse operator of L is labeled with L _ 1 and we define the Green's function by G(x, t, y, s) = L-^y

- x)8(s - t),

(4.6)

and we note that the Green's function G is a deterministic function. Now we specify the stochastic term Wxt- We recall that for an ordinary SDE we use dB t = £tdt [see (1.77)] where £t is the white noise and Bj is the Brownian movement. In a two-dimensional (2-D) setting we have 32W

w

(47)

»- = w

'

Equation (4.7) characterizes now the most general 2D white noise. The space (x or y) and the time (t or s) variables are, however, independent and we can write [see (1.68)] Wys=W(y)W(s);

W(y) = ^ ;

W(s) = ^ ,

(4.8)

where B/- is the Brownian movement for the fcth coordinate. The calculation of the delta functions and the determination of the Green's functions is performed for a finite region with the initialboundary conditions (4.4) and the use of an eigenfunction expansion. To determine the statistical moments of the solution (4.5) where y and s are independent variables we use (W(y)W(s))

=

{W(y))(W(s))=0;

(W( y i )W(y 2 )W( S l )W( S 2 )) = <W(2/i)W(j/2))<W(si)W(s2)>

(4.9)

= (2/1 A y 2 ) ( s i A s 2 ) .

Using (4.9) we rewrite (4.5) in the form

$(x,t) = $0(a;,*) + a / dBy •Jxo

dsG(x,t,y,s)dBs.

(4.5')

Jto

We perform the average and we take advantage of (dB y dB s ) = (dB y )(dB s ) = 0. Thus, we obtain the mean value of the solution <$(x,t)) = $oOr,S/),

(4-10)

Advanced

Topics

137

and this is in analogy to linear ordinary SDE, the deterministic limit solution (a = 0) of the problem. Now we calculate the variance of the solution (4.5) and this yields in first place I

fX

pX

pt

J XQ

Jt0

Var($) = a 2 / / dB y / dB, / dB s \Jx0

x / J t0 I

PX

/ \JxIX00

dBuG(x,t,y,s)G(x,t,z,u) px

dBy /

pt

pt

dB, / dB s / dB u

J XX00

JtJt0 0

x G(x,t,y,s)G(x,t,

Jtn

z, u).

(4-11)

To simplify this integral we use the Dirac-function expansion of the quantity
+ a W(s)W(t);

0 < x < vr;

t>0; (4.13)

$ a: (0,t) = $ a: (7r ) t) = $(x,0) = 0. We use the deterministic eigenfunctions that are given by Vt = VXx ~ V;

V x (0, t) =

VX(TT,

t) = 0,

(4.14a)

and the Green's function takes the form oo

G(x,t,y,s)

= J2Vk(x,t)Vk(y,-s) fc=0

(4.14b)

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where Vk(x,t) and Vk{y,s) are the eigenfunctions satisfying (4.14a) for the pair of variables (x,t) and (y,s), respectively. The separation V(x,t) = exp(-Ai)U(a;) leads to U " - ( 1 - A ) U = 0 and the compliance with the boundary conditions yields Vfc = exp[-(l + k2)t]XJk(x);

k = 0,1,..., (4.15)

1 Tj0 = — ;

\jk

=

12 J-cos(kx)

\/k>l.

The eigenfunctions are orthonormalized and the orthogonality relation is given /

XJk(x)XJm(x) dx = 8km.

JO

Thus we obtain the Green's function oo

G(x, t, y, s) = ^

\Jk(x)Uk(y) exp[-A fc (i - s)];

\k = 1 + k2.

k=0

The application of (4.5) leads to />7r

rt

$ = a / dy Jo Jo

dsG(x,t,y,s)Wys

oo OO

„OT

„i

a ; V u f c ( x ) e x p ( - A f c t ) / dBj,Ufc(y) / dB s exp(A fc s). u-n JO JO fc=o

(4.16)

Note t h a t we set $o = 0 since (4.16) already satisfies the initial condition <&(x,i = 0) = 0. We can p u t (4.16) into a more elegant form if we define Ztfc = [ dBs f dBy\Jk(y) Jo Jo

= Btpk;

(3k = f dB y U*(y), (4.17a) Jo

Advanced Topics

139

and Ak(t) = f dBsexp[-\k(t-s)} Jo

I* &By\Jk{y) Jo

= (3k [ dB s exp[-A fc (i - s)]; Jo

Afc(0) = 0.

(4.17b)

The differential of (4.17a) is dZ^ = (3kdBt and the substitution of this line into (4.17b) yields Ak=

[ dZ s f e exp[-A f c (t- S )]. Jo Yet the differential of the latter line takes the form dAfc = dZ^ - Afc dt / dZ*exp[-A f c (£-s)], Jo and we obtain the ordinary SDE (we do not use a summation convention) dAfc = dZtfc - AfcAfcdt = -AfcAfcdt + (3kdBt.

(4.18)

Equation (4.18) represents a special case of the OrnsteinUhlenbeck SDE (2.14) and we obtain as solution Ak(t) = /3k [ exp[-A fc (t - s)]dB s . (4.19) Jo We use the variable Ak to simplify the formula the solution (4.16) and this yields oo


(4.20)

The mean value is (Ak(t)) — 0 and the autocorrelation function has the form C(Afc) = (Ak(t - r/2)Ak(t

+ r/2)) = <j32k) f exp[-2Afc(i - s)} ds, Jo

where the time 7 was defined in (2.47). Thus we obtain an autocorrelation function that is in an analogy to (2.48) given by C(Afc) = (/3fc2)[exp(-Afc|r|) - exp(-2Afct)]/(2Afc).

(4.21)

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The relation (4.20) is used in EX 4.1 calculate the variance of the solution $(x,t) and to compare this result with the prediction of the general variance formula (4.12). X We continue now to investigate SPDE in higher space dimensions. We do this for the stochastic Poisson equation. Thus we investigate the problem n-D problem A n $ = aU(x)W n ; 32 dx\ The function U(x) domain of (4.22) as The 2D and 3D Feshbach [4.1]) and

$(x); 32 9x2'

x = ( x i , . . . ,x„); (4.22) dB Xl dxi

n

dB x dxn

is a deterministic function and we specify the the entire R n . Green's functions are well known (see Morse & we obtain the 2D Green's function

G 2 (x, x', y, y') = ~ ln[(x - x'f + (y - y')\

(4.23a)

and the 3D Green's function has the form G3(x, x',y,y',z,z')

=

4TTR'

R2 - [(x - x')2 + (y- y'f + (z - z')2].

(4.23b)

Thus, we obtain the 2D solution $(x,y)

= ^

J dBu J dBv\J(u,v)ln{(x-u)2

+ (y-v)2},

(4.24a)

where we put the homogeneous solution, and with this the average of $, to zero. The variance is calculated with (4.12) and we obtain Var($) = (a/47r) 2 f du f dv\J2{u,v) ln 2 [(x - uf + (y -

vf\. (4.24b)

We recall the criterion of square-integrable Green's functions. We extend this criterion now to the square integrabihty of the product of

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the Green's function and the coefficient function U(x). As an example we set U(u,u) = y/6(u)5(v).

(4.25)

The substitution of (4.25) into (4.24) leads to the variance Var($) = (a/4vr) 2 In2(a;2 + y 2 ).

(4.26)

It is instructive to compare (4.26) with the result of the deterministic 2D Poisson equation /92

92\

U + Vy) ^ = aS{x)6^ with the solution <S>d = ^\n{x2

+ y2).

Thus, we see that the stochastic 2D Poisson equation (4.22) with the "charge" (4.25), leads to a zero mean solution and a variance that is the square of the deterministic solution (4.26). A similar result is obtained in EX 4.2 for the case of the 3D Poisson equation. In the case of SPDE in dimensions higher than three we cannot represent the solutions with stochastic functions but only in form of distributions. This subject is beyond the scope of this book and we refer the reader for investigations on this field to the article of Walsh [4.2] and the books of Holden et al. [4.3] and Krylob et al. [4.4] and Wojczinsky [1.10]. 4.2. Stochastic Boundary and Initial Conditions 4.2.1. A deterministic

one-dimensional

wave

equation

An interesting subject is the study of the evolution of deterministic PDE subjected to stochastic boundary or initial conditions. We investigate in this section the solution of a simple deterministic wave equation that is of importance in the study of nonlinear acoustics (see Whitham [4.5]) and we will introduce a stochastic initial condition in the next section.

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The problem is to find a solution of the PDE that governs the space (x) and time (t) evolution of the wave velocity \

ov

ov

.

.

,

,, ,__N

—- =—v—; v = v(x,t); T = t-x/c. (4.27) ox
(4.28)

In Equation (4.27) the parameter A is given by A = (1 + 7)/2 (7 is the ratio of the specific heats) and c is the sound speed at rest. To construct the solution to (4.27) we introduce the function $(x,i) = i>osin

LO[

t

(4.29)

c +A$

First we can simply find that $(0, t) complies with the initial condition (4.28). We substitute in (4.28) r and we obtain with the use of the Mach number MQ = VQ/C $(x,t)

= uosin

1

c 1 + \Mo$>/voJ\

y

'

Equation (4.30) defines the function $ only implicitly. We can, however, solve (4.30) explicitly for the convection time and we obtain ,

,

,

COX

wr = arc sin($v 0 )

AMn0

"

;

"

•

(4.31)

We differentiate (4.31) and this yields 9$ 9r

to G'

9$ ox

u> AMo$/wo c G l + AM 0 $/u 0 '

^'^

where G is a certain function. Without giving the details of G, that we determine in EX 4.3, we eliminate G from the two equations in (4.32) and this yields 9$ dx

=

1 \M0$/v0 8$ c 1 + AMo*/u 0 or'

[

'

}

Many applications of acoustics are, however, characterized by small values of the Mach number. Hence we obtain with an expansion

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from (4.33) 9$ __ A 3 * dx c2 dr' and this is the PDE (4.27). Since $(0, t) satisfies also the initial condition we can infer that the function that arises from an expansion of (4.30) $(x,t)

= fosin u>[ T + -AM 0 $/«o

(4.34)

is the solution of the PDE (4.26) and satisfies the initial condition. We replace now $ by v and use a Fourier series expansion to investigate the frequency components of its discrete spectrum. We write (4.34) in the form v(x, t) =

VQ

sin(u;r +

/3V/VQ);

(5 =

UJXMOX/C.

(4.35)

We obtain a Fourier series from oo

A

v/v0 = Y^

n(/?) sin(amr).

(4.36)

71=1

The coefficients of this series are given by

2 r An(/?) = - /

sin(iOT +(3v/vo)sm(ionT)d(u>r).

(4.37)

71" Jo

We substitute £ =

UIT+(3V/VO, V/VQ

= sin£ and we obtain from (4.37)

A„(/3) = - r s i n ( £ ) s i n [ n £ - n/3sin(0][l - /?cos(0]d£.

(4.38)

7T JO

To evaluate (4.38) we take advantage of the theorems of the harmonic functions and of the recurrence formulas of the Bessel function (see Abramowitz and Stegun [1.3]). We use the integral representation of the Bessel function

i r Jfc(n/3) = — / 7T Jo

cos[ka — n/3sin(a)]da,

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where 3n(z) denotes the Bessel function of order n. Thus, we obtain from (4.38) Ki{P) = Jn-l(-2) - Jn+l(z) - -[Jn-2(z) =

~ Jn+2^)]

2

^±z = n(5. z The substitution of (4.39) into (4.36) yields -} = f > ^ )nJ1 /

v0

^

2

sin(no;r); v

"

E K{ nJ

^

(4.39)

= H ^ M .

(4.40)

nf3

n=l

The relation (4.40), t h a t is sometimes called t h e Bessel-Fubini formula, represents now the explicit solution t o t h e P D E (4.27) for small values of the Mach number Mo- The latter condition limits the validity of (4.40) to a space variable in the order of 0(x) = 1. 4 . 2 . 2 . Stochastic

initial

conditions

Now we concentrate on a stochastic wave problem where the stochasticity is introduced by the application of a stochastic initial condition t h a t includes slowly varying amplitudes and phases. We follow here the ideas of Rudenko & Soluyan [4.6], Akhmanov & Chirkin [4.7] and Rudenko & Chirkin [1.9]. First, and we apply t h e initial condition v(x = 0,i) = vo(£lt)sm[u>t

+ ip(ttt)},

Q = ecj,

0 < £ « 1. (4.41)

The relation (4.41) represents a randomly modulated wave with a stochastic phase ip. In the following we will assume t h a t the parameter e is sufficiently small such t h a t we can use the solution v(x,

T) — VQ(CIT) sin LOT

+

-^UJV(X, T)X

(4.42)

+ ip(Qr)

where r is again the convection time defined in (4.27). (4.42) complies with the PDE (4.27) and with the initial condition (4.41). We rewrite (4.42) with the use of new variables V{z,9) = A(e)sm[e + zV(z,6) +
ip{6) = V'(fir);

z = Xcoax/c2;

a2 = (vg), (4.42')

where z is the nondimensional position of the wave.

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To find the explicit solution of (4.42) we take again advantage of the Bessel-Fubini formula and we put oo

V(z, 9) = A(0) sin[9 + zV + y?(0)] = ^

B

" 0 ) sin{n[0 + p(0)]}.

71 = 1

The inverse of this relation is given by Bn(z) = - A{9) ^ sin(0 sm[m{9 + ip)]d(6 + ip), Ti" Jo with £ = # + zV +
sin(0 sin[m(£ - zA sin(£)][l - zk cos(£)]d£.

The evaluation of this integral yields in analogy to (4.40) to the explicit solution with the series expansion V(z, 9) = 2 J2

3n[nzA{9)]

sm{n[9 + y(g)]}.

(4.43)

71=1

It is easy to verify that (4.43) tends in the limit z —> 0 to the intial condition (3.41). We use (4.43) to calculate the correlation function of the velocity V. We employ P/i(A, A', ip, ip') as the tetravariate distribution function of a stationary normal process. The amplitudes have zero mean and unit variance and they vary in the interval (—oo, oo) whereas the phases vary in [0, 2-7r]. We put 9' — 9 + LOT; A = A(0), A' = A(9'), ip = (p(9), ip' = ip{9') ((A, (p) and (A',
(V(z,9)V(z,9')) 0 0

A

= V ^

/-OO

5 1

nmz

/ J0

/"OO

dAAJ n (nzA) / J0

dA'A'J^mzA')

71,771=1 /•2ir

P2TT

x / dnpsm[n(ip + 9)] / dip's'm[m(,¥>')• The four-dimensional distribution was derived in [4.6]

( 4 - 44 )

P 4 (A, A',
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Note that the function b(j]) is the envelope of the input-signal C(z = 0,r/) = 6(T/) cos(77).

(4.46)

We begin the evaluation of (4.44) with the determination of the phase integrals. To achieve this goal we must use the expansion (ln(z) is the modified Bessel function of order n (see [1.3])) CO

exp[T cos(ip' — if v)} = ^2

SfcIfc T cos

( )

[Hf'

-
k=0

T = AA'bp(rj);

s0 = l, sk =

2Vk>l. (4.47)

The evaluation of the amplitude-integrals leads eventually to the correlation function (see [4.6]) C(z, rj) = 2 J2 ^ rc=l

UKv)(nz)2]

~V (nz) s

coS(nrj),

(4.48)

We compare now the growth (or decay) of the harmonic intensities for the deterministic (E^) and the stochastic case (E*). We have according to (4.40) and (4.48) 4 2 E^ = -jl(a) and E„ = - exp(-a)In[a\; a = (nz)2. (4.49) a a We obtain E^ from (4.48) for r = 0 and a = VQ. A graphical result of this comparison is given in Figure 4.1. 1,2-1 Ed's

[1,2 -1

1-

n

0,8-

0,8

0,6-

-0,6

0,4-

0,4

0,2-

0,2 — r

00

•—*'-'

r

' .

''

—•—^—•

•

1

1

1

0,5

1

1,5

- ••

-

0

Fig. 4.1. Variation of the harmonic (continuous lines) and stochastic (broken lines) intensities with the nondimensional position of the wave z. Upper pair of lines fundamental mode (n = 1), lower pair of curves: second harmonic modes (n = 2).

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4.3. Stochastic Eigenvalue Equations 4.3.1.

Introduction

To consider the nature of random eigenvalue equations we consider a boundary equation problem for an ODE that includes a parameter A Lu = \u;

u = u(x,\);

x\ < x < X2\ (4.50)

L=

gn

8^

+ an

gn-1 X A)

~^ " 9^=i + " '

g + ai(x A

' f e + a ° ^ A)'

where L is a linear nth order differential operator. The boundary conditions (BC's) are given by Zi(u) = 0;

i = l,...,n,

(4.51)

and they involve conditions at the endpoints x\ and x^. We assume that u\(x, A ) , . . . , un(x, A) is a fundamental set of (4.50). This leads to the general solution n

u(x, A) = y^KjUj(x,

A),

Kj = const.

(4.52)

3=1

The application of the BCs yields n

J]K j Z i (« j (C,A)) = 0;

i = Xlori

= x2-

i = 1,2,... ,n. (4.53)

i=i

Equation (4.53) forms a set of n linear homogeneous algebraic equation for the coefficients KJ; j = 1 , . . . , n. Hence, non-trivial solutions exist only if the determinant vanishes and this gives the eigenvalue relation $(A) = det[Z i (u j (^,A))]=0.

(4.54)

Solutions to (4.54) exist only for a parameter A, called the eigenvalue, that satisfies certain conditions. There exists in particular a discrete spectrum if the eigenvalue can be labeled with an integer number k and complies with Ao < Ai < • • • < An < • • •. By contrast we obtain a continuous spectrum if the eigenvalue is denned on a continuous interval a < A < b. There exists also the combination of an eigenvalue spectrum that contains both a discrete as well

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Stochastic Differential Equations in Science and Engineering

as a continuous part. Functions corresponding to the fcth eigenvalue Ufc = u(x,\k) are called eigenvectors or eigenfunctions. A stochastic eigenvalue problem arises when either the differential equations or the BC's contain a stochastic parameter 9. This means that we consider in Section 4.3 only ODE with random coefficient functions but not SDE. The main goal is the determination of the PD of the eigenvalues PA(A). The ODE (4.50) is linear and we can in principle calculate its solutions. However, we encounter in many applications a nonlinear problem with a nonlinear operator or nonlinear BC's or both the operator and the BC's are nonlinear. This means that the exact solution of (4.50) is inaccessible and exact solutions of the PD PA(A) are impossible. Furthermore, there is in various applications in chemistry and physics the problem that one must measure the coefficients of the operator L or the coefficients of the BC's. The latter measurements introduce random effects. Geometrical dimensions can be measured with rather high precision but other quantities such as the position or the velocity of a particle are less accessible to exact measurements. This means that the coefficients in the operator or in the BC's introduce a randomness. Thus, we must consider more modest goals and restrict the considerations to the calculation of the lowest moments of the PD PA (A). In practical cases one would be already satisfied with the calculation of the mean and the variance of the eigenvalue A. 4.3.2. Mathematical

methods

Every procedure to calculate the eigenvalues consists of two different parts: a deterministic and a stochastic part. Accordingly there are two categories of methods. In the first method we try to calculate or approximate the first eigenvalues and apply final stochastic routines to calculate the PD p(A) or at least the lowest order moments of this PD. Such an approach was called by Keller [4.8] and [4.9] an "honest" procedure. By contrast, we use in the second method statistical methods as the calculation of the moments of the ODE and apply approximate methods such as the assumption of stationary processes or weak correlations. A typical approach of this kind is used in the fluid dynamic

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theory of the turbulence (see e.g. Prisch [4.10]). This type of procedure was named by Keller as "dishonest". However, we have to keep in mind that an "honest" approach does not imply more accuracy for its solutions or approximations than the results of a "dishonest" routine. The methods that are used in both types of approaches are frequently one of the following routines or a combination thereof -

Variational principles, Perturbation expansions, Green's function and transformation to an integral equation, Asymptotic theory for high order eigenvalues, Asymptotic theory for singular differential operators, Iterations.

Furthermore, one introduces elements of statistical theory and statistical assumptions like independence of variables, stationary processes, weak correlations, etc. A survey of these methods is given by Haines [4.11] and Boyce [4.12]. Random boundary problems are studied in detail by Bass and Fuks [4.13]. Here we use an asymptotic theory to treat the problem of higherorder eigenvalues. Higher order eigenvalues We consider the Sturm—Liouville problem (see also (3.47)) and investigate the eigenvalue problem y" + P'V'/P - (r/p

y = y(x,X); ' = d/dx;

- M/p)y

p = p(x);

= 0;

q = q(x);

r = r{x);

(4.55)

0 < x < 1,

with the BC's y'(0,A) + Cy(0,A) = 0;

y'(l, A) + Dy(l, A) = 0;

C, D = const. (4.56)

We use a transformation to put (4.55) into a form amenable to the WKB routine (see also the appendix A of Chapter 3). We must

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(i) to eliminate the first derivative and (ii) obtain a constant as the coefficient of A. To this end we scale both the eigenfunction and the independent variable y(x,\)

= E(x)u(£,\);

£ = H(x)/K;

H(l) = K,

H(0) = 0;

0 < £ < 1,

(4.57)

where E(x) and H(x) are two unknown function to be determined such that the conditions (i) and (ii) are met. The derivatives y(x,X) and u(x, A) are calculated in EX 4.5 and the substitution into (4.55) leads to •/2 EH' E'H' + (EH')' + ^ E H ' H K 2 -«« + K P p'E' E + E" + A^ 0. (4.58) p PJ p Hence we can satisfy the conditions (i) and (ii) if we put

+

2

I+FF+P

= 0 ;

H(X)

=

[Vq(z)/P(z)dz,

(4.59)

where the first part of (4.59) leads to E = (w)~1/4.

(4.60)

Now we can write (4.55) in the form UK + [M2 - R(0]« = 0;

(4.61)

E" p'E'A K 2 E ~ ~pE ) W

V j)

R =

/x2 = K 2 A » l ;

We suppose that the eigenvalues increase with an order such that fj? is a large parameter. Thus we can take advantage of the WKB routine and we propose the asymptotic expansion u(£,/x) = e x p f ^ o (01

1 Go(£) + - ^ ( 0 + 0 ( / O

.

(4.62)

\r

This yields in leading order (/j?) to the eikonal equation x

0£

1

or

u0 = ±£,

(4.63)

l

and the first correction order {ji ) leads to G0 ^ _ Gn

l«o& 2 u0$

or

Go = const.

(4.64)

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Thus we find that the expansion (4.62) has the form u(£,/z) = A 1 cos(//0 + A 2 sin(/i0 + O(Ai"2).

(4.65)

We see that u is of 0(1) whereas the derivative u% is of order 0(/z). Using the results of EX 4.5 we find that the BC's (4.56) are in leading order independent of the constants C and D (4.56) reduces to ^(0,^=^(1,^=0.

(4.66)

The application of (4.56) to (4.56) leads to the eigenvalue and the eigenfunction (ik — irk;

/cGN;

k —> oo;

u(£, fi) = Ai cos(£;7r£),

(4.67)

where k is an integer number. The eigenvalue A of the original problem (4.55) is then given by A = (fc7r/K)2 + o(A;).

(4.68)

Now we make assumptions about the randomness of the problem. We introduce stochastic functions for the coefficients q(x) = 1 + au(x);

p=l

+ av(x),

a = const,

(4.69)

where u(x) and v(x) are random functions and a is an intensity parameter. To find the moments of eigenvalue A we must calculate the moments of K 2 . Hence, we obtain (K 2m ) = (fe7r)2m(A-m).

(4.70)

Example Here we use a Brownian motion for the coefficients (4.69) q(x) = 1+aB^;

p=l

+ a/3Bx;

0
0
(4.71)

where Bx is the Brownian motion with a spatial argument. The corresponding series expansion of K2 and K4 for a —> 0 is given in EX 4.6 and this yields (K2) = 1 + a2{(5 - 1)[(1 + 3/3)/2 + 2(J3 - l)/3]/8 + 0 ( a 4 ) } . (4.72)

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4.3.3. Examples

Equations in Science and

of exactly soluble

Engineering

problems

We consider here three examples of simple problems with exact solutions. In all of them we use the nomenclature that 9 and A represent a stochastic parameter and the eigenvalue, respectively. Example 1 We consider the linear problem u" + Xu = 0;

' = d/dx;

0 < x < 1, (4.73)

u(0,A)=0;

ti'(l,A) + 0it(l,A) = O,

with 9 6 [0, oo) appearing only in one of the BC's. We solve the ODE using u(x) = exp( 0 : a = ±ivA and u(x) = Csin[vAa;]; C = const, or alternatively (ii) A < 0 : a = ± v |A| and u(x) = Dsin/i[y|A|x]; C = const. The application of the second BC leads to (i) C[0sin(VA) + V/Acos(v/A)],

(4.74)

and (ii) B[6smh{^/\\~\) + V\cosh(y/\\~\)}.

(4.75)

Thus we obtain in case (i) the eigenvalue equation 9 = -VAcot(v / A) = G(A) > 0,

(4.76)

whereas there is no eigenvalue problem for the other alternative (ii) (see EX 4.7). The latter statement means that we have to comply with A > 0.

(4.77)

Equations (4.76) and (4.77) give for 9 > 0 the eigenvalue relation {2n + lf^

<

A n

<

( n + 1 ) 2 7 r 2,

n = 0

,i....

(4.78)

We reveal the first and the second eigenvalue in Figures 4.2(a) and 4.2(b). We see from these figures that the eigenvalues are monotonically increasing functions of 9 and we can uniquely calculate the

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Fig. 4.2. The zero and first order eigenvalues Ao and Ai of the problem (4.73) plotted versus the stochastic parameter 6.

inverse function of G(A). Thus, \ n = G~l{e);

n = 0,l...,

(4.79)

where the subscript of the function G _ 1 refers to the number of the eigenvalue. We continue with this problem in the next section. X Example 2 We consider a problem in polar coordinates r V + ru' + (Ar2 - 62)u = 0;

u = u(r, A);

' = d/dr,

(4.80)

with the BC's K0,A)|
u(l,A) = 0.

(4.81)

The solution to the ODE (4.80) takes the form u{r, A) = CJ e (rVA) + DYg(ry/\);

C, D = const.,

(4.82)

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where Jg(x)(Yg(x)) denote the Bessel functions of the first (second) kind of the order 6 with the argument x and 6 > 0 stands again for the random parameter. The function Yg(0) diverges, and we put in accordance with first BC D = 0. Thus, we obtain from the second BC the eigenvalue equation

j 0 (vT) - o.

(4.83)

We give in EX 4.8 hints how to solve (4.83) numerically and we reveal the results in Figure 4.3. These results show an approximate parabolic variation of the eigenvalues. However, we can also confirm this variation if we use for large values of \/A the asymptotic behavior of the Bessel functions. This yields with (4.83) (see [1.4]) Afcf« (vr/4)2(3 + 4A; + 20)2

(4.84) *

Example 3 A quantum-mechanical particle moves on the ti-axis in a potential that increases linearly with u. The separated Schrodinger equation 200

200

150

150

100

100

Fig. 4.3. The zero and the first order eigenvalues Ao (lower pair) and Ai (upper pair) of the problem (4.80) are plotted against the stochastic parameter 0. The continuous (broken) lines correspond to the numerical solutions of (4.83) (asymptotic approximations (4.84)).

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has in this case the form d2

-T^MU)

= ui/)(u);

ip{oo) = 0.

(4.85)

The solution to this equation is ip{u) = CAi(u);

C = const.;

Ai(oo) = 0,

(4.86)

where Ai(it) stands for the Airy function that oscillates for u < 0. After this physical motivation, we formulate a stochastic boundary value problem: y"(x)=en(x

+ \e-n/3)y(x);

' = d/dz;

y(-l) = y(oo) = 0. (4.87) We can transform (4.87) into a Schrodinger equation with a stochastic potential if we put u = X + x6n/3.

(4.88)

Indeed, using (4.88) we find that solution to (4.87) is y(u) = CAi(X + x6n/3),

(4.89)

and (4.89) satisfies already the BC at infinity. Complying with the second BC yields x = - 1 : Ai(A - 9n/3) = 0.

(4.90)

In EX 4.9 it is shown that the Ai-function in (4.90) exhibits the discrete spectrum Afc-0n/3 = - K | ;

A; = 0,1,2,...

with \uk\ = {2.33811,4.08795,5.52056,...}.

(4.91)

We may now solve (4.91) for the variable 9 and this yields in a special case n = 1 n = l : 0 = G(Afc) = (Afc + K | ) 3 ;

Xk e [-|ufc|, oo).

(4.92)

* We continue in the next section with the calculation of the moments of the eigenvalues and we will use the examples 1 through 3 to illustrate these ideas.

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4.3.4. Probability laws and moments eigenvalues

Engineering

of the

In Section 4.3.2 we could introduce random functions at will for the coefficient function p and q (see (4.69)). Then we calculate the moments of the eigenvalues, given the probability laws of the used random functions [like the Brownian motion in (4.71)]. Here we assume that the task of establishing the eigenvalue relation is already performed. We write the latter equation again in the form 6 = G(A). To determine the PD PA (A) we use the following theorem: Theorem 4.1. We recall the definition of the ID probability distribution density (PDF) of two continuous random variables X and Y F x (x) = Pr(X < x);FY(y) = Pr(Y < y) [see (1.1)]. We transform the variables X and Y into each other with the aid of a strictly increasing function Gy = G(x);x = G _ 1 (y), where the monotonic properties of G ensure the existence of the inverse function G _ 1 . It is now easy to find (see also EX 4.10) that X < x implies Y < y. This yields Pr(Y < G(x)) = Pr(X < x)

(4.93a)

and this is equivalent to the relation for the PDF's FX(x) = F Y (G(x))

or

FY(y) = Fx(G-1(y)).

(4.93b)

Equation (4.93b) also implies the rule for the PD's [see (1.3)] PY(y) = ^ = P x ( G - 1 ( y ) ) ^ ^ . (4.94a) Ay dy Note that if the function G(x) were strictly decreasing we would obtain on the right hand side of (4.94a) a minus sign (see EX 4.10). Hence we obtain in conclusion dG -l 1 (4.94b) P Y (y) = Px(G- ) dy Note also that (4.94b) is the ID version of the law of transforma8 5 tion of stochastic variables derived in Section 1.5. *

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The application of Theorem 4.1 to the characteristic equation 9 = G(A) with G a strictly increasing function, leads to pA(A)with \uk\ — pe(#)(d0/dA). Thus, we obtain for the nth moment of the kth eigenvalue rv

(Xk) = J

jn

PV

A^pA(Afc)dAfc = /

Ajjp fl (0)— dAfc;

u = ak,

v = bk,

(4.95) where we took the kth eigenvalue that is defined in the interval ak < Afc < bk. We apply now (4.95) for the Examples 1 to 3 of the Section 4.3.3. Example 1 We use G(Afc) given by (4.76) and we obtain the limits of the integral (4.89) for the first two eigenvalues ao = (7r/2) 2 ,ai = (37r/2)2;6o = 7r2,bi = 4-7T2. We introduce two different types of the PD's. (a) We suppose that the random variable 9 is uniformly distributed in the interval 0 < 9 < (3 and we put f 1//3 (J

for0e[O,/3] elsewhere

We obtain the after a substitution z = y \K the integrand A

d# ,, 1 zn9„ z — sin(z) cos(z) ^ ~ f c P,— e ^ ~ d\ dAKK = = -z2n -±£-—^ dz; a\K [3 smz z

for ^


zu,k. (4.97)

The upper limit of the integration is zUjk = y/rj, with n = —-v/A^cot(\/A^). Hence, we obtain the moments with (4.95) and (4.97) in the form /? 7 / ^

sin (z)

Note that as /3 —> 0 + we have zU;fc - • V ^ - The corresponding integral is an undetermined form and we obtain (see EX 4.11) (Afc)(/3 = 0 + ) = (2n + l ) V / 4 .

(4.99)

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Table 4.1. The moments of the zero and first order eigenvalues and the corresponding variances.

p

(Ao>

0.001 0.25

2.4687 2.7092 2.9354 3.3460 4.0290 4.5715 5.0118 6.1721 6.5456

0.5 1 2 3 4 8 10

°~o 0. 0.1373 0.2613 0.4750 0.7948 1.0135 1.1658 1.4456 1.4939

o"l



22.208 22.456 22.702 23.859 24.107 24.954 25.726 28.151 29.056

0. 0.1433 0.2846 0.5585 1.0628 1.5012 1.8742 2.8504 3.1293

We give the corresponding results now terms of Table 4.1, where we use ^i,2 = - (Ai,2)2, We calculate the integral (4.98) with the Mathematica program M41, that is given in the attached CD. Note that the moments for /3 = 0+ given by (4.99) coincide almost with the values given for /?-= 0.001 in Table 4.1. (b) Here we apply the normal PD with zero mean and we obtain PA(A) = p e ( 0 ) ^ = (27r
(4.100)

We use again (4.97) and we obtain moments in the form = ( 2 ^ ) - ^ r k V^k

z 2n*-Bin(*)coB(z) 2

x expf-z 2 cot2(z)/{2a)]dz.

sm (z)

(4.101)

The numerical results for the eigenvalue are calculated with the program M42 and they are displayed in Table 4.2. Jj» Example 2 We calculate here numerically the moments of the eigenvalues of the characteristic equation (4.83).

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Table 4.2. The moments of the zero order eigenvalue and the variances for the normal PD.


V^o) - <*o}2

0.2 1 5

2.38242 2.33368 2.58837

0.92072 1.64598 2.38830

Table 4.3. The moments of the eigenvalues with the characteristic equation (4.86). v



V ^ g ) - <*o>2

0.05 0.2 1

3.5919 4.3650 6.5718

3.6754 4.6818 7.9691

Note that Figure 4.3 indicate only minor variations of the eigenvalues with the stochastic parameter. Hence we can use a polynomial fit to approximate the eigenvalues with sufficient accuracy. We use again the POD (4.100) and we obtain the results disclosed in Table 4.3. The corresponding Mathematica program M43. £ Example 3 We apply (4.92) and this yields A£p*(G)—dAfc;

a=\uk\.

(4.102)

We employ the PD (4.96) and we obtain with (4.102)

(K) = ^J aK(h + a)2d\k = ^P~ [ * z\a-z)sdz; b = p1/3-a. (4.103) P Jo We can calculate analytically the integral (4.103) and we obtain (see also EX 4.12)

(K

! _ 3^1/3 + 0(^/3) 4a

(4.104)

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and this leads to (Afc) = - K | ;

a = 0(/3 4 / 3 ).

(4.105) *

4.4. Stochastic Economics 4.4.1.

Introduction

The London (FTSE) and the New York (NASDAQ) stock exchange data for about 13 years were fitted by Benth [4.14]. This leads to considerable differences with respect to the predictions of the theoretical approach by Black & Scholes [4.15] based on GPD. Particular unsatisfying is the skewness and tail heaviness of the empirical data that cannot be predicted by a GPD. Thus the normal inverted Gaussian distribution NIGD (see e.g. Barndorff-Nielsen [4.16] and Eberlein &; Keller [4.17]) was invented to fit the stock data. As opposed to the GPD with two parameters, the NIGD has the advantage of being equipped with four parameters: a: tail heavines, 0 < \f3\ < a, j3: skewness /3 > 0 (/3 < 0) positive (negative) skewness, /? = 0 symmetric NIGD, 5 > 0: scale parameter, fi: position on the real axis. With this parameters we can construct the NIGD I a x\ p{x;a,f3,n,5) =

kexp\P(x-[i)] 9

p = V<52 + (x -

M )2 ;

Ki(a/>); (4.106) k = — exp[5(a - 0 ) ],

where Ki is the modified Bessel function (see [1.3]) 1 f°° Ki(«) = x / exp{-uz/[2(l + z2)]}dz.

2

2 1/2

(4.107)

^ Jo

The NIGD governs the stochastic Levy process L(i) and its mean and variance have the form (L(i)) = (i + 05{a2 - / 3 2 r 1 / 2 ; Var(L(i)) = a25{a2 p2)'3'2;. (4.108)

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It was shown in [4.14] that approaches bases on Levy processes fit the stock data with respect to skewness and tail heaviness and simple price model can be written as S(t) = S(0)exp[L(t)].

(4.109)

However, Equation (4.108) is still of preliminary nature and in the next section we will discuss a more realistic price model. We complete now this section by explaining some elements of the financial terminology. Assuming that an investor makes two types of investments: a risky stock and a comparatively low risk bond. The investor's aim is to obtain a fair price when he sells his investments. To achieve this he makes an option contract, where the financial asset depends on another asset. Option means that the contract has specific choices or alternatives. Such contracts are also referred to as derivatives, because the value is derived from an underlying asset. There exists a great variety of derivatives, we mention only the call option and alternatively the put option. A comprehensive survey of options is given by Hull [4.18]. The problem we wish to solve is now: what profit should the seller accept at the time of the sale. The time of the sale (the strike) is called the exercise time and the price of the sale is denoted strike price. Therefore we have for a call option the price P (the payoff) P = (max(0, S(T) - K)),

(4.109a)

where K is the agreed strike price, and we perform an average since the stock price S(T) varies randomly. On the other hand we obtain for a put option P = (max(0, K - S(T))).

(4.109b)

Finally we give a heuristic explanation of self-financing portfolios. We assume again that an investor made two investments: a number of (3{t) stocks with the price S(i) and a number of j(t) of bonds of the price R(£). The value of its portfolio is therefore H(£) = P(t)S(t) + 7(t)R(<).

(4.110)

We assume that there is no information about future stock prices such that S(t) is adapted. A portfolio is self-financing if the investor

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does not withdraw funds or buy additional ones. He starts with an initial capital, and from this moment on all gains or losses result from increases or decreases of the stock and bond prices. This means that the differential of the portfolio takes the form dH(t) = P(t)dS(t) + 7 (t)dR(t).

(4.111)

We emphasize that (4.111) is valid only for self-financing portfolios. It contradicts the general formula (1.109) and its validity is restricted to semi-martingales (see [4.14]). Finally, we mention that a contingent claim gives a holder a random amount at the exercise time T. After these preliminaries we describe a stochastic theory for market prices. 4.4.2. The Black Scholes

market

A financial system (a market) consists of a stock with price S(t) and a bond with a price R(i). The price dynamics takes the form dS(i) = (adt + adBt)S(t);

a,a = const.,

(4.112)

r = const., R(0) = 1.

(4.113)

and dR(t) = rR(t)dt;

Equation (4.112) is an SDE of the population growth type (2.1), whereas (4.113) is an ODE with the solution R(i) = exp(rf). We also introduce the portfolio of the value H(i) given by (4.110). A claim that pays £ = f(S(T)) at the exercise time T has a price P(i) at time t, that is function of the corresponding stock price x = S(t). Hence, we can write P(t) = C(S(t), t) = C{x, t);

x = S(t).

(4.114)

We apply Ito's formula (1.127) to the function C(x,t) and we use the underlying SDE (4.112). This yields dP(t)

Ct + axCx +

1

^(ax)2Ca

dt + axCxdBt

= 0.

(4.115)

For non-arbitrage conditions (see [4.14]) we must set P(t) = H(t).

(4.116)

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Equation (4.110) combines with (4.116) to 7=[C(M)-/?S(i)]/R.

(4.117)

An insertion of (4.111) to (4.114) into (4.115) yields with a comparison of the coefficients of dBj to P(t) = Cx(x,t),

(4.118)

and the same procedure for the coefficient of dt gives rise to a PDE called the Black-Scholes equation Ct + rS(t)C+^[aS(t)]2Cxx

= rC.

(4.119)

Equation (4.118) is subjected to the terminal condition C(x,T) = f(a;).

(4.120)

The solution to (4.118) that satisfies (4.119) has the form C(x,T) = exp[-r(T - t)](f(^*(T))),

(4.121)

with zx't(s) = x e x p { ( r - s 2 / 2 ) ( s - i ) + [B(s)-B(i)]};

s>t.

(4.122)

The stochastic function (4.121) is the solution of a population growth SDE (2.1) that starts at time t from the position x: zx,t(t) = x. Note that the logarithm of (4.122) can be written as B(s) - B(t) = [In **•*(«) - In x - (r - s2/2)(s - t)]/a.

(4.123)

Since the right hand side of (4.123) is N(0, s — t) distributed we infer that k(s,x,t) — l n ^ ' ^ s ) is N(/j,,T1);fi = \nx + (r-a2/2)(s-t),

E = (s - t)a2

(4.124a)

distributed and this gives its distribution the form p K ( M ) = (27rS)- 1 / 2 exp[-(fc-//) 2 /(2S)].

(4.124b)

We focus now on the verification of the Black-Scholes solution (4.121) and the latter equation has with (1.124) the form C(x, t) = exp[-r(T - t)] f f(z(T))pz(z,

t)dz,

(4.125)

where we use the short cut z = zx,t(s). We see immediately from (4.125) that this solution satisfies the terminal condition (4.120).

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Now we perform the transformation z = exp(fc) and this gives (4.125) the form [see (1.44)] C ( M ) = e x p [ - r ( T - t ) ] /f(e fc )p K (fc,t)dk,

(4.126)

where the PD is pK.(k,t) is given by (4.124). Now we apply the operator of the PDE (4.119) on (4.126) and we obtain in the first place

0 = /dK(e*)(A+r^

^ )

+

.

P K ( M )

The integrand can be reformulated in terms of the equation (1-57) 3

^

+ rX

9

+

x2a2

92 \

9^ ^9^J

„

,

PK(fc i)=

(\

32

9 \

' Uv"^J

PK(

„

diffusion

''

,

) = 0n

'

and this verifies that (4.121) is the solution of the Black-Scholes PDE (4.119). It remains to say that Scholes and Merton (with his investigation [4.19], Black died in 1995) earned the Nobel award in economics in 1997. Exercises E X 4 . 1 . Use (4.20) to calculate the variance of the solution (4.16) and compare the result with the formula (4.5) where we take into account the orthogonality of the functions Ufc (x). E X 4.2. Calculate the variance of the 3D stochastic Poisson equation. The Green's function is defined by (4.23b). Use the "charge" U 2 (x) = 5(x) and compare the variance of the stochastic PDE with the corresponding deterministic solution <&d(x) = a/(47r|x|). E X 4.3. Determine the function G in (4.32). E X 4.4. Solve the stochastic heat transport equation 9T ^=

/ (a

dBt\92T ^r hx^

+ b

+

9T ^

;

m

«.M

= «H»t.;

T=

m.

N

T( M ),

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165

with the use of the separation T(x,t) = A1(t)cos(Lot) + A2(t)sm(cot). Show that the resultant ordinary SDE can be transformed into a deterministic ODE involving the two functions A i ^ i ) . E X 4.5. Calculate the derivatives of the functions y(x, A) and u(x, A) defined in (4.57). Hint: Verify EH'

Vx = E'u + — « f ;

pij'2

i

+ ^ [E'H' + ( E H ' ) > € +

yxx = ~^Hi

E u

"-

E X 4.6. Referring to (4.71), calculate the series expansions of the moments of (K2) and (K 4 ). Hint: Verify (K2) = [ dx [ dy{l + a2(P - 1)[(1 + 3/3)(x + y) Jo Jo +

2(P-l)(xAy)]/8

+

0(a4)}

= 1 + a 2 (/3 - 1)[(1 + 3/3)/2 + 2(/3 - l)/3] + 0 ( a 4 ) . E X 4.7. Explain why no eigenvalue problem arises for (4.76) in the case of A < 0. E X 4.8. To solve (4.80) plot numerically (with the aid of "Mathematica") J#(\/A) in the plane < 0 < # < 5 ; 0 < A < 150 and use the plotted zeros for as initial values for an iteration to compute the numerical values of An . Using "Mathematica" you can take advantage of the routine FindRoot. Explain why the asymptotic order relation (see [1.3])

J,(

^)a^3(^)"': ^°°; ^<0°'

cannot be used to compute the eigenvalues. E X 4.9. Apply the Mathematica routines AiryAi and FindRoot to verify the zeros of the Ai-functions given by (4.91).

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E X 4.10. Derived by inspection of a strictly decreasing function y = G{x) the relation for the PDF's Pr(X < x) = Pr(Y > G(x)) and use this to verify (4.94b). E X 4.11. Applying the rule of l'Hospital to verify (4.99). Hint: Show that the PD (4.96) gives rise to lim /" P e (0)H(0)d0 = H(O). E X 4.12. We consider the 4th order boundary value problem

yw - ^

y = o;

s/(°>A) = y"(°> A ) = y&A) = s/"(i>A) = °-

Verify that the eigenvalue equation has the form sm[y/9/(l + A)]. Use the PD (4.96) to find numerically the mean and the standard variance for Ai and A2.

CHAPTER 5 N U M E R I C A L SOLUTIONS OF O R D I N A R Y STOCHASTIC D I F F E R E N T I A L EQUATIONS In this chapter we will familiarize the reader with some numerical routines to solve ordinary SDE and we shall introduce these routines with the F O R T R A N programs t h a t are included on the attached CD. The latter programs are labeled with an " F " . So, for example, F 3 is the third F O R T R A N program.

5.1. R a n d o m N u m b e r G e n e r a t o r s a n d A p p l i c a t i o n s A prerequisite to solve SDE are r a n d o m n u m b e r s (RN's). There are various techniques to construct "random numbers". In fact, the numbers repeat after a period P. As a result, we should call these numbers pseudo-random numbers. To find an algorithm for a random number generator, we recall t h a t ever since the work of Feigenbaum [5.1] it has been known t h a t the iteration of numbers on the real axis (ID maps) can lead to irregular outcomes. It is convenient to apply a congruential iteration Nfc+i = (aN/- + u) mod v;

k = 0,1,...,

where a, u, i>,No are given positive integers and the symbol u m o d t ; (the modulo) denotes the remainder of u after dividing u by v. T h e choice of v is determined by the capacity of the computer and there are various ways to select a and u. The period P of this iteration was investigated by K n u t h [5.2] and it was found t h a t the condition t h a t u and v are relatively prime makes it important to increase the value of P . The IMSL (international mathematical scientific library) proposes for RN's with a uniform P D in the interval (0, 1) the iteration N f c + i = (16807 Nfc) mod (2 3 1 - 1); 167

k = 0,1,...,

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where 0 < No < 1 is an initial number and the factor 2 31 — 1 is a prime number. The numbers in the above formula are not RN's in (0, 1) but the sequence R fe+ i = N f c + 1 /(2 3 1 - 1), serves to produce a sequence of RN's, Rfc+i, in (0, 1). Other techniques are discussed by Press et al. [5.3] and we recommend in particular the routines called rani, ran2 and ran3, that are based on congruent iterations. The question of a uniform distribution and the independence of the RN's is investigated in the next section. RN's that vary in arbitrary intervals are obtained with transformations (see Section 1.5, where also Gaussian distributed numbers are introduced). 5.1.1. Testing of random

numbers

We will investigate here the property of uniformity and the independence of stochastic variables. To perform this task we introduce a few elements of the testing of stochastic (or probabilistic) models Ho on random variables X. The basic idea is that we perform a random experiment (RE) of interest and group the outcomes into individual categories k with K as the total number of categories. One particular outcome of the RE is given by Xi and this particular outcome is called a test statistic. We categorize the predictions of an application of the model Ho in the same way. If, under a given model Ho, values of the test statistic appear that are likely (unlikely), we are inclined to accept (reject) the model HoAn important statistical test is x 2 -statistics. The quantity X is a X2-random variable of m degrees of freedom (DOF) with the PD px(x,m) = [2sr(s)]-V-1exp(-:r/2);

s = m/2;

x > 0.

(5.1a)

The mean and variance of (5.1a) are calculated in EX 5.1. It can be shown that (5.1a) is the PD of a set of m independent identical N(a, a) distributed RV's, X i , . . . ,X m , with m

X = J](X p -a) 2 /^ 2 . P=i

(5.1b)

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We assume now that under a given hypothesis Ho (e.g. the uniformity of RN's) one particular simulation falls with the probability Pfc in the category k. We perform N simulations and the expected number of events under Ho in the category k has the value Npfc. This contrasts with the outcome of a RE where Z& is the number of events in the category A; of a particular simulation. Now we calculated the sum of the K squares of the deviation of Z^ from Npfc and divided each term by Npfc. This test statistics yields the RN K

DK-i = ^(Zfc-NPfc)2/(NPfc).

(5.2)

k=l

It can be shown that for N > 1 the RN in (5.2), X = D K _i approximately has the PD (5.1a) with the DOF m = K — 1. To verify this value of the DOF we note that we can sum RV's in (5.2) but we must consider the compatibility relation Ylk=i ^fc = NA close agreement between the observed (Z&) and the predicted values (under Ho) Npfc yields (Z^ — Npfe)2
Pr(X>6) = l - /

p x (:r,K-l)d:r = a;

b = *cK-i-

(5.3)

Jo

In Equation (5.3) we must substitute the PD (5.1a) and use a given and small level c t « l . Conveniently one chooses a = O(10~ 2 ) and solves (5.3) numerically to obtain %2 K - 1 . As a goodness criterion for Ho we employ now the condition that D K - I must not exceed the critical value. Thus, we accept (reject) the hypothesis for D K - I < Xc,K-l(DK-l

>Xc,K-l)-

Example (Uniformity test) As an application of the x 2 -test statistics we investigate now the hypothesis Ho of the uniformity of RN's. We categorize by dividing the interval (0, 1) into K subintervals of equal length 1/K. Then we generate N RN's and count the numbers that fall into the fcth

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subinterval. Now we assume that Zfc RN's in: ((k - 1)/K, Jfe/K);

k = 1 , . . . , K.

Under Ho, the probability of simulation RN's in each subinterval is pfc = 1/K. Thus, we infer from (5.2) K D

N

K-i = N ^ ( Z fc=l

f c _ N / K ) 2

(5 4)

-

-

We investigate in program F l the uniformity of RN's using 10, 30 and 50 subintervals and applying various RN generators. The results are given in Table 5.1. £ Now we discuss independence tests. Suppose that we have obtained a random sample of the size of n pairs of variables (xi, yj);j — 1,.. -, n. To investigate the independence of the two sets of variables X = (x\,..., xn), Y = (j/i,..., yn) and we calculate their covariance R = I>P-<*>)(%>-<3/>)/D; P=i

n

n

p—X

p=l

I

n

(5.5)

p=X

If the variables X and Y are uncorrelated we would find R = 0 and the variables are independent. We focus now on a test of the independence of N RN's x\,..., xn and we group them into consecutive pairs (x\,X2),{x2,x%),..., (xn-i,xn). We regard the first member of such a pair as the first Table 5.1. The evaluation of the uniformity test with the r a n i , ran3, the IMSL routine and a congruent method (CM) for a = 0.05. DK-I

D

DK-I

DK-I

K

XC,K-I

(ranl)

(ran3)

(IMSL)

(CM)

10 30 50

16.92 42.56 66.34

9.240 37.100 44.200

9.200 23.900 33.900

9.300 47.780 43.100

9.300 47.780 43.340

K

-I

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variable (say X) and the second member of the pair as another variable (say Y). The correlation coefficient Ri of these consecutive pairs is called the correlation coefficient with lag 1 n-l

Ri = E ( ^ - ^ ( 1 ) ) ) ( ^ + i - ^ ( 2 ) ) ) / D ; P

=i

n—l

2

D =5>

2

n—1

- <^))) J > p + 1 - <*(2)))2;

p

(5.6)

p=\

P=\ n—l

1

, ,

n — l P=I '-^

,-Ep+k—l-

When n is sufficiently large we can simplify (5.6) and this yields 1

n

( 5 - 7a )

(xW) « (x^)(x) = - J > p , n

P=I1

n-l

Ri = j >

p

- ^ » ( ^ P + I - ( * » / E ( X P - ^)) 2 -

(5-7b)

* p=i

P=I

It can be shown (see e.g. Chatfield [5.4]) that if independent then Ri, is for R » 1 , N(0,1/n) distributed. Under these conditions we expect |Rij < 1.96n -1 / 2 with a confidence parameter a = 0.95. However, there are cases when the pairs (x\,X2), (x2,xs), are independent but pairs such as (xi,xs), (^2,^4), are not independent. Hence, we compute the correlation coefficient with time lag k (see Tuckwell [5.5]) n—k

R

k = J2(XP P=I

,

x

x

x

~ ( )")( p+k - ( ))/ '

n

J2(XP p=i

~^)2'

(5-8)

where the mean (x) is defined as in (5.7a). We investigate the question of the independence of RN's in program F2. Finally we explain the term confidence interval. We know from the discussion of the central limit theorem (Section 1.4.1) that a sum of IID RV's converges to a normal distribution. Hence, it is of

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practical importance to find probabilistic limits of the normal distributions N(0,1) and N(/x,a). We start with a N(0,1) distributed variable and define the confidence parameter a by N(0,1) : Pr(Z > za/2) = a/2.

(5.9)

Equivalently we can rewrite (5.9) in the form 1 fb a/2 = 1 - —= \ exp(-u2/2)du] b = za/2. (5.10) V27T J-oo There is no analytic expression for the inverse function of the exponential integral in (5.10). Alternatively we take in EX 5.2 advantage of an asymptotic method to approximate (5.10) and we use there the program F3. However, it is easy to find numerically for every value of za/2 the confidence parameter a. Now we study a N(/i, a) distributed variable X with f x (x) = (27ra2)-1/2 e x p [ - 0 r -

^)2/(2a2)},

and we consider that it lies with the probability K = Pr(xi < X < X2)] x\,2 " i ^ i aza/2 m the indicated interval. Thus, we infer from (1.1a) K = (2ira2)-1/2

H exp[-(x -

n)2/(2a2)]dx

fb = (27T)-1/2 / exp(-y 2 /2)dy = 1 - a; J—b

b = za/2.

(5.11)

Equation (5.11) means that Pr(/x - aza/2

< X < \i + aza/2) = 1 - a,

(5.12)

and (5.12) defines a 100(1 — a)% confidence interval.

(5.13)

Example A 0.95% confidence interval is given by a = 0.05 and this implies with (5.10) zaj2 = 1.96. Numerically computed and approximate values of za/2 are calculated in EX 5.2 and in the program F3. •

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Detailed studies of stochastic model testing can be found in the books of Walpole and Myers [5.6] and Overbeck-Larisch and Dolejsky [5.7]. 5.2. The Convergence of Stochastic Sequences In Chapter 1 we introduced already the term almost certain, (AC). We focus in this section on the convergence of sequences of RV or stochastic processes. A classification of the convergence of the sequence of RV {Xn(co)};

n = N,

u> <E ft

is given by strong and weak modes of convergence . (i) Strong convergence There are two subclasses of convergence modes: (i-1) AC convergence (or equivalently called convergence with probability 1) if Prf lim |X(w) - Xn(cu)| = 0 J = 1.

(5.14)

(i-2) Convergence in the fcth mean. lim (\X(LU) n—>oo

- Xn(uj)\k) = 0,

fceN.

(5.15)

This means for k = 2 convergence in the mean square. By contrast, for k = 1 (5.15) reduces to the mean (or strong in the narrow sense) convergence. Note also that (5.15) is a straightforward extension of the deterministic convergence concept. We also note that the positiveness of the variance of a RV Y, with Y > 0, leads to the inequality

(Y) < v ^ > , and this means that the mean square convergence implies the mean convergence. (ii) For the case of weaker convergences we do not need the behavior of the RV, we focus only on the sequences of PD's {pn(x)} and the way they tend to their limit p(x).

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(ii-1) The convergence in distribution is defined by lim pn(x) = p(x),

(5.16)

n—>oo

where the PD's are supposed to be continuous. (ii-2) The weak convergence in the narrow sense is expressed by lim / g(x)pn(x)dx = / g(x)p(x)dx, (5-17) n^ooj J where we include in the integrals classes of test functions g(x) (e.g. polynomials). An important application of the concept of strong convergence is given in the following example: Strong law of large numbers (SLLN) We consider a sequence of RV xi,X2,--- with finite means fi^ ~ (xk), k = 1, 2 ... and we define a new RV 1

n

An = - V ] x k

n

1

with

M n = - V'/ifc.

fc=i

(5.18)

k=i

Then, provided that l i m S n = 0 with

n-^oo

Sn = ^ V a r ( y x f c ) n

\ *•—' \/c=l

(5.19)

/ /

the SLLN says that lim (|An - M n |) = 0.

(5.20)

n—too

Note that for IID RV we have /ife = /i = const. =>• M n = lim A n = /x.

(5.21)

n—>oo

Proof. First we note that (A„) = M,,,

Var(A n ) = S n .

Now we apply the Chebyshev inequality (see EX 1.2) and we obtain P r ( | A n - M n | > £ ) < ^ 4 ^ = %->0

forrwoo.

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This proof can be made more rigorous (see Bremaud [1.16]). The convergence in (5.20) is of mean square type. * An interesting application of the SLLN is given in the following: Example (Estimation of the probability distribution function, PDF) It is required to find (or numerically approximate) the PDF defined by (1.1) of the RV X. To this end we generate a sequence of independent samples of the RV X i , X 2 , . . . and we define the indicator function

{

1, n 0, Thus we obtain with the use of the

V Xi. < x, ^ otherwise. SLLN (5.20)

<&(a;)) = lim -(Ci + • • • + £„) = F x (x).

(5 22)

'

(5.23)

n—»oo n

We will use this method in program F4 to estimate a Gaussian PDF. 5.3. The Monte Carlo Integration Here we make a slight digression a from our path to the calculation of numerical solutions of SDE and we use RN's to perform a stochastic type of numerical integration. This method is widely used in integration and statistics. It is based on our ability to generate RN's and it can be done in principle by dice or a roulette wheel. For this reason the method is called the Monte Carlo method (MCM). As a typical application we might consider the MCM to calculate the value of the area A below a curve i(x) and the x-axis, see Fig. 5.1. The curve is located in a rectangular box of the area B = ab and we consider a pair of two independent RV (x, y) that are uniformly distributed in 0 < x < a, 0 < y < b. The probability that a random point (x, y) falls between the x-axis and the curve i(x) is given by Pr{(x,y) GA} = ^

f

f(x)dx.

(5.24)

We use a collection of n random points (xk,yk), k = 1,2,... ,n and define for the fcth member of this collection the indicator variable

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The function f(a;) whose integral is calculated.

[see also (5.22)] 1

for (xk,yk)

<E A,

0

otherwise.

(5.25)

Thus, we infer from the SLLN (5.20) that 1 n (Z) = - V z f c ^ P r { ( x , y ) G A }

for n —> oo,

(5.26)

fc=i

or equivalently ab

f(x)dx

y ^Zfc n fc=i

for n —> oo.

(5.27)

Example (Volume of spheres) We calculate the area F of a unit circle that is enclosed by a square of area B = 4. Clearly this example is a stochastic method of computing the value of TT. Here we infer from (5.25) 1

foTx2k +

0

otherwise.

yl
(5.28)

In (5.28) we use independent RN's that vary uniformly in the interval (0, 1). The area of the unit sphere is then by symmetry four times

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the one that would be obtained by the application of (5.28) to the first quadrant of the (x, y)-plane. Hence, we obtain from (5.27) n

A

F = -J2zk^TT.

(5.29)

k=l

However, it is easy to generalize this example to the calculation of the volume of the 3D sphere, and more interestingly, to the calculation of the volume m-D sphere with m > l . This is where the MCM is superior to traditional numerical calculations. Consider the computation of a volume in a 10D space. If we employ a minimum of 100 points in a single direction we would need for the traditional computation 1020 points. By contrast, the MCM would need according to (5.28) 10 points for one indicator variable and a collection of say 107 points would be sufficient for an accurate computation. Thus we generalize the present example and we calculate the volume of an m-D unit sphere. The indicator variable is now given by m

Zfc — 2_]X2A < 1; Zfc = 0 otherwise. i=i Thus, the MCM gives the volume V m of the m-D unit sphere the value (2 m is the number of sectors that make up the m-D unit sphere: 4 quadrants in case of a circle, 8 quadrants in case of a 3D sphere,...) tym

n

- V z ^ V n ^—'

m

.

(5.30)

k=i

Note that the exact value V ^ is given by (see Giinther and Kusmin [5.8])

v

" = w^wy

(5 31)

-

The numerical computation is made using program F5. We summarize the results of the computation of a 10-D sphere in Table 5.2. The error of the computation is defined by err = 100|(VfS-Vi 0 )/VfS|%.

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Table 5.2. The results of a MCM calculation of the volume of a 10-D unit sphere using ran3. n

Vio

err

60000 1100000 1150000

2.54123 2.54203 2.55029

0.35048 0.31884 0.00508

There are several improvements of the MCM for multiple integrals. In the case of the determination of the area between an irregular shaped curve i(x) and the x-axis we can write b

rb

i(x)dx =

dx p(x)i(x)/p(x),

(5.32a)

Ja

where p(x) is an arbitrary PD defined in x = [a, b]. If N RN's £&, k = 1 , . . . , n are generated from the PD p(x), then

One of the most widely used application is in statistical mechanics. The PD of a statistical mechanical state with energy E is given by the Gibbs distribution function lK

'

exp(-/?E) /exp(-/3E)dE'

where (3 is the inverse temperature. The computation proceeds by starting with a finite collection of molecules in a box. It can be made to imitate an infinite system by employing periodic BC's. Algorithms exist to generate molecular configurations that are consistent with p(E). The most popular one is the Metropolis algorithm. A presentation of this algorithm would take us too far afield. Details are given in Metropolis et al. [5.9]. Once a chain of configurations has been generated, the PD is determined and various thermodynamic functions can be obtained by suitable averages. The most simple example is (E) = V E f c p ( E f c ) .

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5.4. The Brownian Motion and Simple Algorithms for SDE's We recall that the Brownian motion is a non-stationary Markovian N(0, t) distributed process. We remind also the reader of the BoxMuller method (1.46) to (1.48) of generating a N(0, 1) distributed variable. The latter method contains the computer time-consuming use of trigonometric functions [see (1.47)]. Hence, it is more convenient to use the polar Marsaglia method. We start from a variable X that is uniformly distributed in (0, 1) and use the transformation V = 2x — 1 to obtain a variable V that is uniform in (—1,1). Then we apply two functions of the latter type Vi and V2 with W = V? + V | < 1 ,

(5.33)

that is distributed again in (0, 1) and the angle 8 is such that 6 = arc tan(V2/Vi) that varies in (0,2ir). The area-ratio between the inscribed circle (5.33) and the surrounding square has the value 7r/4. For this reason we see that a point (Vi, V2) has the probability 7r/4 of falling into this circle. We consider only these points and disregard the others. We put now cos 9 = V i / \ / W ; sin^ = "V^/VW and we obtain as in (1.47) yi = Vi V / -21n(W)/W;

y2 = V 2 y / -21n(W)/W.

(5.34)

This yields in analogy to (1.48) to the PD p(yi,y 2 ) = (27r)- 1 / 2 exp[-(y? + y22)/2}.

(5.35)

We calculate with this generation of the Brownian (or Wiener) processes in program F6 the numerical data pertaining to Figures 1.1 and 1.2 of Chapter 1. We present in Figure 5.2 the graph of the solution to population growth problem (2.4) and calculate the corresponding numerical data in F7. We give now a simple difference method to approximate solutions of ID deterministic ODE o\x — = a(x, t); x(t = 0) = XQ (5.36) is the Euler method. The latter discretizes the time into finite steps to = 0 < h < ••• < tn < tn+r,

Ak = tk+1-tk.

(5.37)

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Fig. 5.2. A graphical evaluation of the results of the population growth model (2.4), r = 1, u = 0.2. The figure reveals also one realization and the predicted mean (2.5) that coincides with the numerically calculated mean using 50 realizations.

This procedure transforms the ODE (5.36) into a difference equation 2-n+l — Xn -\- cl\Xn, tn)L\n]

Xn = X[tn).

(^O.OOJ

The difference equation must be solved successively in the following way xi = x0 + &(x0, t0)A0 = x0 + a(x 0 , 0)ii; x2 = xi + &(x1,t1)A1 = x0 + &(x0,0)ii + a(xi,tj)(t2-ti);

a(xi, h) = a(s 0 +

a(x0,0)ti,ti).

Equation (5.39) represents a recursion. Given the initial value XQ we obtain with the application of (5.39) the value of Xk for every following time £&. Now we propose heuristically the Euler method for the ID autonomous SDE dx — a(x)dt + b(x)dB 4 ;

x(t — 0) — XQ.

(5.40)

In analogy and as a generalization to (5.39) we propose the stochastic Eulerian difference equation xn+i =xn + a(xn)An + b(xn)ABn;

AB n = Bn+l

•Br

B, = B t (5.41)

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In the majority of the examples in this chapter we will use equidistant step widths, tn = to + n/S. = nA;

A n = A = const.

This yields (AB n ) = 0

and

((AB n ) 2 ) = A.

In the next section we shall derive (5.41) as the lowest order term of the Ito-Taylor expansion of the solutions to the SDE (5.40). 5.5. The Ito—Taylor Expansion of the Solution of a I D SDE We start here from the simplest case of an autonomous ID SDE (5.40). In the case of a non-autonomous N-dimensional SDE, that is covered in Section 5.7, we need only to generalize the ideas of the present derivations. We integrate (5.40) over one step width and we obtain /•*n+l

x(tn+i)

/"tn + 1

= x(tn) + / &(xs)ds+ Jtn Jt„ XQ = x(0); xs = x(s).

b(x s )dB s ; (5-42)

Now we apply Ito's formula (1.99.3) to the function f(x), where the variable x satisfies the ODE (5.40). This leads to df(s) = L°f(x)dt + L 1 f(x)dB i ; 1 L°f(x) = a(x)f'(x) + - b2(x)f"(x);

L : f(x) - b(x)f (x).

(5-43)

The integration of last line leads to f(x n + i) = f(x n ) + / " +1 [L°f(xs)ds + iMixJdB,].

(5.44)

Jtn

If we specify (5.44) for the case i(x) = x, this equation reduces again to (5.42). The next step consists in the application of (5.44) for f(xs): = &(xs) and i(xs): =b(xs) the substitution of the corresponding result

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into (5.42). Hence, we put g(x s ) = g(xn) + / [L°g(xu)du + L 1 g(x u )dB u ]; Jtn

g= a

or

g = b.

Proceeding along these lines we find xn+i =xn + a(x n )A n + b(x n )AB n + Ri; ftn + l

R1=

Jtn

fS

/ [L°&{xu)dsdu + L 1 a(a; u )dsdB u Jt„

(5.45)

+ L 0 b ( ^ ) d B s d u + L 1 b(x u )dB s dB n ]. Equation (5.45) is the simplest nontrivial Ito-Taylor expansion of the solutions of (5.40). Neglecting the (first order) remainder Ri we obtain the stochastic Euler formula (5.41). We repeat this procedure and this leads, with the application of the Ito formula the function i(xu) = L 1 b(x„) = b(xu)b'(xu), to x(tn+i)

= x(tn) + a(iEn)An + b(x n )AB n + b(xn)b'(xn)lijl

Ii.i = /

n+1

dB s f dB u = f " +1 dB s (B s - Btn)

+ R2; (5.46)

= i(AB2-An) where we have used the Ito integral (1.89). If we proceed to second order, the remainder R2 contains two second and one third order multiple Ito integrals that are based on the following integrals rtn+1

Ii,o= / Jtn ftn + l

Io,i= / Jtn

rs

ftn+i

/ dBsdu = / Jtn

(s-tn)dBs,

<Jtn

fS

(5.47)

ftn + i

/ dsdBu- / Jtn Jtn

(Bs-BtJds,

with Io,i + Ii,o = A„AB„,

(5.48)

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and rtn+i

Ii,i,i = / Jtn

rs

ru

ptn+i

/ / dBsdB„dB, = / Jt„ Jtn Jtn

fs

ru

d B s / dB„ / d B , Jtn Jtn

= f "+1 dB s J &BU{BU - B t n ) = ^[(AB n ) 2 - 3A n ]AB n . (5.49) We will verify (5.48) and (5.49) in EX 5.3. The application of (5.46) with R2 = scheme xn+1

0 yields the M i l s t e i n

= xn + M(xn,An,ABn),

(5.50a)

with the Milstein operator M(xn,An,ABn) — a(xn)An + b(xn)ABn + ^b(xn)b'(xn)[(ABn)2

- An}.

(5.50b)

Note t h a t the Milstein scheme (5.50) is again a recursive method and can be considered as a generalization of the Eulerian scheme (5.41). The numerical d a t a revealed in Figures 2.1 and 2.2 for the Ornstein-Uhlenbeck problem (2.14) are computed with the Milstein routine (5.50) and the corresponding program is F8. The latter is a general solver for first order autonomous SDE's. We calculate in particular the solution of the generalized population problem (2.35a). A sample p a t h of numerical solution of this equation is plotted in Figure 5.3. With the inclusion of more stochastic integrals, we obtain more information about sample paths of a solution. Hence we include more terms of the expansion of &(x) and b(x) into the remainder R2 of (5.46). We postpone this cumbersome algebra to treat the general case of non-autonomous N-D SDE's. We can, however, infer from (5.45) t h a t one term resulting from the expansion of Loa(x u ) must be given by pv

ps

/ ds / duL°&(xu) Jw Jw

1

= - &{w)a!(w) + -b2(-u;)a"(u;) A 2 ^ tn+l, W = tn.

(5.51)

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Fig. 5.3. Lower line: numerical average of the solution to (2.35a), middle (irregular) line: a SF of (2.35a). Upper line: graph of the exponential, exp(krt); r = 0.01, k = 100, u = 0.2, Ensem = 200, h = 0.001.

Proceeding along this lines we end up with the scheme x„+i = xn + M(x n , A n , AB n ) + H n (A n , AB n , AZ n ),

(5.52a)

with the operator R(xn,An,

AB n ,AZ n )

= a'bAZn + ~ LB! + ±b2A A2n + Lb' + \b2bA (ABnAn - AZn) + ^b(bb" + b /2 )[(AB n ) 2 - 3A n ]AB n ,

(5.52b)

b

where the drift and the diffusion coefficients depend on the argument xn. In Equation (5.52b) a new RV appears that is defined by the first of the integrals in (5.47) AZ n = I i , 0 = /

dBa(s-io),

Jw

where the integration boundaries are defined in (5.51).

(5.53)

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We easily find the moments (AZ n ) = 0 and Var(AZ n ) = M

dBs(s - w)

\Jw

dBr(r - w) Jw

I

= [ ds(s - wf = \A3U, Jw

(5.54a)

"J

and the covariance (AZ n AB n ) = / fV dBs(s - w) fV dB u = [ ds(s-w)

= lA2n.

Jw

(5.54b)

^

Note also that Equation (1.93) states that the variable AZ n is N(0,A£/3) distributed. In order to perform a numerical simulation of the RV AZ n we formulate the following theorem: Theorem 5.2 (Numerical simulation of the RV AZ n ) First we note that we can generate the Wiener variable AB n in form of N(0, A n ) distributed variable. We obtain, from the polar Marsaglia (or the Box-Miller) method (5.34), two independent N(0,1) distributed RV's, yi and y2- Now we put

ABn = v/A^il

AZn = ±A3J2 L + 4 p V

(5-55)

We obtain from (5.55) (AZ n ) = 0;

<(AZn)2> = ±A3n;

(AZ n AB n ) = ~A2n.

and this proves (5.54).

&

We verify (5.52) to (5.54) numerically in program F9A. There we use the traditional mean a of RV x by averaging over K realizations of this variable 1 K (x) = Jim — V xk,

(5.56a)

fc=l

where the RV's x^ are individual samples of x. We contrast this average in program F9B with the batch average. There we calculate

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the mean of x by computing K batches, each of which consists of M samples of x and has a different seed of the random generator. This yields 1

^

K

M

(5.56b)

= KMEE^ fc=l m = l

(k)

where x^ represents the mth realization of the variable x in the kth batch. Note that (5.56a) does not coincide with (5.56b), even not for identical seeds of the individual batches. We propose to use F9B with K = M = 25 and compare the results with the ones of F9A with K = 625. The numerical data for Figure 5.3 were calculated with the Milstein scheme (5.50) as well as with the higher order routine (5.52) and the SF's as well as the averages that coincide up to the first three digits. We tried also to compare the growth of the solution for generalized population growth (2.35a) with the solutions of the usual growth problem (2.1). To accomplish this we use in (2.35a) rk = 1 and compare the corresponding numerical results with to those of (2.1) with to those of (2.1) with r = 1. We also note that (2.35a) can be written as dX = rkX(l - X/k) + uXdBt, and the deterministic limit of this SDE has stationary point at X = k. The latter property prevents the solutions from tending asymptotically to each other. We can only expect an approximate coincidence in the first phase of the growth. Example The application of the routine (5.52) to the problem of OrnsteinUhlenbeck (2.14) with a — m — x; b = const, yields H(x n , A n , AB n , AZ n ) = -bAZn - - ( m xn+l

=xn + (m-x)

xn)A2n.

( 1 \ I An - - A n J + b{ABn - AZ n ).

(5 57)

-

* The schemes discussed in this section belong to the class of explicit schemes, where the advanced solution xn+\ is given by

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functions of the solution xn. Explicit routines have the disadvantage of including a great variety of functions and their derivatives. An additional disadvantage of these schemes is their instability. Hence, we replace these routines in the next section with modified routines. A particular modification is given by an implicit scheme that is known in deterministic problems to be less unstable. However, a general comment about the use of numerical routines to approximate the solutions of SDE'S must be given. To eliminate the possibility of spurious results, it is always advisable to confirm numerical results of a particular routine with the numerical data obtained from another method. We note also that we introduced at the beginning of this section the Ito-Taylor expansion that is based on the application of the Ito formula (1.99.3). In the case of Stratonovich SDE's there is also a Stratonovich-Taylor expansion that relies on the Stratonovich differential (1.100). The latter approach is not covered in this book and interested reader should consult the book of Kloeden et al. [2.5].

5.6. Modified I D Milstein Schemes We used the Milstein scheme in program F8 to solve the population growth problem (2.1) in the regime of the variable 0 < x < 10 and depicted the results in Figures 2.1 and 2.2. There we found that the numerically calculated moments agree very well with the theoretical ones. We did not also come across with any problems of a divergence of the numerical solution. However, the situation changes dramatically if we would like to extend the regime of this solution beyond x > 10. The Milstein routine is in this regime unstable and results in a function that diverges rapidly. The question of how to improve the stability of the Milstein scheme arises. Basically, there are two possibilities to modify given routine and we exemplify this for the Milstein routine. In the first case (i) we perturb the drift coefficient. In the second alternative (ii) we modify the coefficients that are based on the diffusion coefficient. We begin with the first alternative.

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(i) An implicit ID scheme Here, we replace the Milstein routine (5.50) by xn+i = xn + [aa(x n + i) + (1 -

a)&(xn)]An

+ b(x n )AB n + ^b(xn)b'(xn){(ABn)2

- An}.

(5.58)

Note that (5.58) is for a <E [0,1] a family of SDE's. We call (5.58) in the limit a = 1 (a = 0) a fully implicit (an explicit) SDE, and a is the degree of implicitness. Note also, that implicit effects refer exclusively to the drift coefficient, whereas the diffusion coefficient is always taken at xn. In an implicit scheme we must calculate the solution xn+\ as the numerical solution of the (generally nonlinear) Equation (5.58) with given xn. To solve this aspect of the problem one usually uses a Newton-iteration routine. However, convergence problems arise in this iteration, in particular if the stochastic influences are too strong. Thus, it is convenient to take advantage of this method, despite its popularity in deterministic problems, only for the case of SDE's with a linear drift coefficient, where no iterations are needed. In EX 5.4 we use (5.58) for the Ornstein-Uhlenbeck SDE (2.14). However, the employment of the corresponding program F l l shows that the scheme (5.58) again leads to unstable results in the regime x > 10. This contrasts with the result of the higher order explicit scheme (5.52) for the same problem and we must conclude that the implicit routine (5.58) is inconsistent with a ID autonomous Ito SDE. (ii) Modification of the diffusive terms Here we use for an SDE with a variable drift coefficient the scheme proposed by Kloeden (see [2.5]) xn+i = xn + a(x n )A n + b(x n )AB n + ^[(AB n ) 2 - A n ][b(S n ) - h{xn)]/y/E~n,

(5.59)

with the support value S n = xn + &(xn)An + b(xn)^/A^.

(5.60)

To motivate (5.69) we note that a Taylor expansion of b(S n ) for the usually small parameter An reproduces (5.50) to leading order.

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189

X, <X>

20

30

t

Fig. 5.4. A sample function and the mean value of the SDE (5.61); m = 3, 0 = 1, A„ = 0.005, Ensem = 200.

We test successfully (5.59) and (5.60) in program F12 for the SDE with variable drift coefficients dx = (m — x)dt + /3sin(x)dB i ;

m, f3 = const.

(5.61)

A particular test criterion for applicability is the agreement with respect to the average, since we obtain from (5.61) [in analogy to (2.17a)] (x) = m— (m — XQ)exp(—t). In Figure 5.4 we reveal a graph of a particular SF and depict the averages where the exact and t h e numerical one are almost coinciding. Thus, we may conclude t h a t the routine (5.59) represents a stable modification of the Milstein scheme.

5.7. T h e Ito—Taylor E x p a n s i o n for N - d i m e n s i o n a l S D E ' s We consider now the general case of non-autonomous N-dimensional SDE's with R > 1 independent Wiener processes. We introduce here the nomenclature t h a t Sk(tt,t)

=gk(xi,...,XN,t);

tit =

{xi(t),...,xN(t)),

is the fcth component of the vector function g at the space-time position (&,i).

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We write now the N-dimensional SDE (1.123) in the form dxk(t) = afc(£fc, t)dt + bkr(£t, t)dBrt; Jfc = l , . . . , N ; r = l , . . . , R .

(5.62)

For the differential of the vector function ffc(£t,i) we find now, with (1.127), the expression dffe(&,t) = [U&,t)ik{Zt,t)]dt+

[L^ t ,*)f fc (&,t)]dB[

(5.63)

with T _

9

9

dt

dx

k 2

U

^ dx dx '

(5.64)

9 Li — b m r OXr,

where the functions afc,bmr,ffc are taken at the space-time position (Ct,t) and B | , . . . , B ^ represent a set of R independent Brownian motions. We can integrate (5.63) formally and the result is ffc(6, t) = ffc(Co,«) + /

{Lo(6, s)ffe(6, «)ds,

+ L;(6,s)f f c (6,s)dBS}, *o = a;

€o =

(5 65)

"

(xi(a),...,xu(a)).

We apply now (5.65) to the functions xk to obtain Lo^^ = afe(a),L^Xfc = bkr and we infer from (5.65) Xk(t) = xk(a) + I {a fe (£ s ,s)ds + b fcr (£ s ,s)dB£}.

(5.66)

./a

However, we could obtain Equation (5.66) in a more simple way as the result of a formal integration of (5.62). We apply (5.65) to calculate the functions a&(£,£) and bkr{t],t) and substitute the results into (5.66). Proceeding in this way

Numerical Solutions of Ordinary Stochastic Differential

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we obtain afc(6>s) = afc(£o,a) + /

{L0(£u,u)a,k(£,u,u)di lit

Ja + Ll^u,u)ak^u,u)dBru},

(5.67a)

and bto(£ s , s ) = bfcu;(£o, a) + /

{M&*, «)bfcu,(^u, *t)du

./a

+ L^(^,B)bto(^,«)dB;}.

(5.67b)

The substitution of (5.67) into (5.66) yields + (Brt-Bra)bkr^0,a)

xk(t) = xk(a) + (t-a)a.k^0,a)

+ R^)

(5.68)

with R4 1} = f

f {{L0ak)dsdu + (L> f c )dsdB^ + (L 0 b fcr )dB^d«

Ja J a

+ (L^b fcr )dB^dB-}

(5.69)

where all functions and operators are taken at the space-time location (£u,u). Equation (5.68) with Kk = 0 represents now the non-autonomous N-dimensional Euler-scheme [see (5.45) for the ID autonomous case]. To obtain the N-dimensional Milstein routine we expand the 3rd order tensor L ^ b ^ in (5.69) into an Ito-Taylor series and truncate after the first term. This leads to TZ(£u,u)

= bmw(tiu,u)—-bkr(Zu,u)

<* TZ(to,a)

OXyyi

= bmw(€o,a)-— bkr(£0, a). oxm Hence, we obtain from the last term (5.69) <

}

= T£.(£o,«) f dB£ f dB™ = T&.(£0,a) f Ja

Ja

dB^(B™ - Bwa).

Ja

(5.70) The integral on the right hand side of (5.70) is again a new RV K?= fdB:(B™-B-). (5.71a) Ja

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We know only one of its diagonal elements, such as [see 5.46)] K\ = f d B j ( B j - B i ) = Ii, 1 .

(5.71b)

Ja

For the sake of simplicity, we reduce our goal to the investigation of an N-D non-autonomous SDE in the presence of only one Wiener process (this means (5.62) with R = 1, BJ = B t ) and we refer the reader for more details of the tensorial RV (5.71a) to the book of Kloeden et al. [2.5]. Now we can formulate the N-dimensional Milstein scheme for the case R = 1; this yields Sfc(tn+i) = Xfc(i„) + Mfc(£ tn ,A n ,AB n ),

(5.72)

with the N-D Milstein operator Mfe(&„, A n , AB„) = a fc A n + b fc AB n + I i , i b m - — b f c , (5.73) oxm where the coefficients a& and bfc refer to the space time position (&„,*n).

We apply the Milstein routine (5.73) now to in the following example E x a m p l e 1 (The linear pendulum) The linear pendulum with stochastic damping was analyzed in Section 2.3.2. We use for a 2D SDE the nomenclature xi{ts) = xs;

x2(ts) = ys.

(5.74)

We infer from (2.60) with a ^ 0, /? = 7 = 0 da; = ydt;

dy = — xdt — aydBt,

(5.75)

and this means «i = y,

h = 0;

a 2 = -x,

b2 = -ay.

(5.76)

With (5.72) to (5.75) we find %n+l — %n 1

yn^ni

Vn+i =Vn- xnAn

- aynABn

1 + - a 2 y n [ ( A B n ) 2 - A„],

where the summation convention is not used.

(5-77)

Numerical Solutions

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We solve the iteration problem (5.77) for an equidistant decomposition of the time-axis A n = A = const, in program F13. The corresponding initial condition reads XQ = x(0),yo = y(0). The results for the mean and variance of this process are already revealed in Figures 2.3 and 2.4 in the regime 0 < t < 10. If we wish extend this regime, we find that the scheme (5.77) is not stable and we will use an improved routine in the next section. Also we solve the nonlinear pendulum (2.105) with the Milstein scheme. This SDE is incorporated in program F13 also. £ 5.8. Higher Order Approximations A higher order scheme for the non-autonomous N-D SDE is derived in EX 5.5. We give here only the result Xn+i = xn + a fc A n + b fc AB n + -(L 0 a fc )A2 + (L0bfc)I0,i + (L ia fc )Ii, 0 + (L!bfc)Ii,i + (LiLib^Ii,!,!

(5.78a)

where we use the nomenclature [see (5.74)] xks=xk{ts),

fc

= l,2,...,N; s = 0,l,...,

(5.78b)

and where all coefficient functions, operators, afc,bfc,Lo,Ia, and all integrals (5.46)-(5.49) and (5.53) depend on the space-time variable (x^,... ,x%,tn). Equation (5.78) is the N-D generalization of (5.52) for a non-autonomous SDE. We employ also (5.48) and (5.53) to obtain the relation Io,i = A B n A n - AZ n .

(5.79)

We can easily write the compressed equation (5.78) in a clean-cut formulation if we use the definitions of the operators (5.64). Example (The linear and the nonlinear pendulum) Using (5.76) we infer from (5.64) L0 = ^ - s i n ( x ) - + - ( a y ) 2 g ? ;

U = -ay-.

(5.80)

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Thus, we obtain from (5.78) xn+i = xn + yn(An

- alifl) - - sin(x n )A^,

Vn+i = yn~ sin(:rn) A n - -yn cos(x n ) A 2 -

aynABn

+ 2 a V [ ( A B n ) 2 - A n ] + asin(a;n)Io,i - Ja3^[(ABn)2-3An]ABn.

(5.81)

6

Equation (5.81) applies to the nonlinear case. In the linear case we must replace in (5.81) by the linearization cos(x n ) := l,sin(x n ) := xn. The corresponding numerical work is done in program F13. Finally, we mention a modified Milstein regime for autonomous systems that generalizes the ID scheme (5.60). With the use of the definitions (5.78b) and the comments that follow, we write for N-dimensional systems 4 + i = xkn + a fc A n + b fc AB n + - [b f c (Y + ,..., Y^) -bfc(Yi,...,Y^)]l1,1/AV2 Y i = x£ + a f c A n ± b f c A y 2 , fc = l , 2 , . . . , N .

(5.82)

The motivation for this scheme comes again from a Taylor expansion of the last term in (5.82). We compare now the numerical data obtained for the variance V22(t) of the linearized pendulum [with the analytic formula (2.82)]. We apply (i) the Milstein regime (5.72), (ii) the higher order scheme (5.78), (iii) the modified Milstein scheme (5.82) and an implicit scheme. The latter is established in analogy to (5.58) and it is given for a general 2D SDE in the form xn+i = xn + [7ai(£ n+1 ) + (1 - 7)ai(£„)]A n + Zx, Vn+i = Vn + [7a2(£n+i) + (1 - 7)a 2 (£n)]A n + Zy;

(5.83)

where {Zx,Zy) are the components of the stochastic terms and 7 is the implicitness parameter. For the same reasons as discussed

Numerical Solutions of Ordinary Stochastic Differential

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195

in connection with (5.58), we use (5.83) only for SDE's with linear drift coefficients ai, a2. We specialize (5.83) to the case of the linear pendulum and this yields = (1 + 7 2 A2) _ 1 {x n + (1 - 7 ) y n A n + Zx

xn+l

+ jAn[yn yn+1 = (1 +

2 7

+ (7 - l ) A n x n + Zy}}, (5.84)

A2)

{yn + (7 - l)x„ A n + Zy

+ 7 A n [ - x n + (7 - l ) A n y n - Zx}}. The numerical data are produced, in the case of (i) to (iii), with the program F13 and the implicit scheme is executed with the aid of program F14. The comparison of the numerical data shows that — at least for the variance of the linear pendulum — the Milstein routine (5.72) and the modified Milstein routine (5.82) fail to give sufficient accuracy. By contrast, we see that the higher order scheme (5.78) and the implicit routine (5.84) reproduce satisfactorily the theoretical prediction (2.82) in the regime 30 < t < 50. The corresponding numerical solutions are almost identical for the routine (5.82) and the implicit scheme (5.84). Thus, in Figure 5.5 a comparison of variance produced by (5.82) with the analytic values (2.82) is shown. 0,25 0,2 0,15 V 22

0,1 0,05 0 30

35

40

,

45

50

Fig. 5.5. A comparison of the analytic variance (2.82) (continuous line) and the numerical data obtain by the routine (5.78); a = 0.1, A n = 0.005.

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5.9. Strong and Weak Approximations and the Order of the Approximation We begin with a definition. Definition 5.1 (Absolute, global and local error) We consider an n-D SDE for the variable X(i) e R n with the IC's X(*o) = Xo 6 R n in the interval t <£ [0,T}. A discrete approximation Y(t) G R n of the solution X(i) to this SDE has a local error that is defined by e^im^-Yttj)]);

0
(5.85)

t 0 = 0 < t i < - - - < t n = T,

(5.86)

The global error is given by n

e g = 5^(|X(i f c )-Y(t f c )|); fc=i

whereas the absolute error is defined by ea=(|X(T)-Y(T)|).

(5.87)

In (5.85) to (5.87) we used the symbol | | which is an abbreviation for the norm of a vector function. Thus, for instance, the absolute error is written in full length by

E^ro-WT)]2)

ea = \

(5.87')

fc=l

where the subscript k labels the vector components. The local error (5.85) measures the closeness of the solution and its discretization at just one internal point. The global and the absolute error measure the pathwise closeness of the exact solution and the approximation of the sum of K internal points and at the endpoint of the interval. • In the following we will investigate the dependence of the absolute error on the step width of a particular discretization routine. To realize this task with computer experiments, we choose a particular SDE with a known exact solution and approximate the solution with a given routine. Then we choose M batches of N simulations and we indicate the trajectories of the fcth simulation of the j t h batch at

Numerical Solutions of Ordinary Stochastic Differential Equations

197

the end of the interval by X j ^ T ) and Yj)/C(T). The average values (batch averages of the absolute errors) 1 N ^ = N^|Xj'fc(T)~Y^(T)l'

J' = 1 . 2 . - - - , M ,

(5.88)

fc=i

are for N » 1 independent Gaussian RV's. The batches are chosen to construct with the Student ^-distribution (a distribution that we will discuss in moment) confidence intervals for a sum of approximately Gaussian RV's with unknown variance. In this way we compute the batch averages 1

M

1= M 5 >

1

=

M N

MN £

i=l

£ |X*,*(T) " Yj>fc(T)|.

(5.89)

j=l k=l

We take advantage of the latter average to obtain the estimate of the variance of the batch averages 1

M

It is convenient to apply the Student distribution with M-l DOF to achieve a 100(1 — a)% confidence interval for r\ in the form ( T J - A T / . T J + ATJ),

A77 = K 1 _ a , M - i ^ / M ,

(5.91)

where the parameter Ki- a i M-i is calculated from the Student distribution. As an example we list the following data: For a = 0.05 and M = 20 we obtain Ko.95,19 = 2.09 and the absolute error will lie in the interval (5.91) with the probability 1 — a = 0.95. The Student or (£-) distribution governs an N(0,1) distributed RV X and x 2 is a RV following the chi-squared distribution (5.1a) with s DOF-parameter. Then we consider the ratio X / y x V s . The latter variable has the PDF [see (1.1)] F ( X ^ x V ^ ) = PrflX, ^fW*\ < t) •

(5.92)

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We give here the following represention of this PDF for even values of the DOF (see Abramowith & Stegun [1.3]) F(X/^Ts)

= sin(0) | l + \ cos2(0) + ^ | cos4(0)

+- ^ e f K H 9 = arctan(i/y / s),

(593>

(s even),

with an analogous formula for odd values of s. Definition 5.2 (Strong convergence and order of convergence) Strong convergence is a pathwise convergence characterized by the error criterion (5.87). A discrete approximation scheme with a maximum step size 5 converges with the order 7 > 0 at a time T if there is a constant C independent of S such that ea = < | X ( T ) - Y ( T ) | ) < G F

(5.94) D

In the framework of weak convergence we are not interested in pathwise approximation of the stochastic process. We focus there on the calculation of the PD, or its lowest moments, or functionals of the PD. Hence we do not compare the difference between exact solution and approximation as in (5.87), yet we concentrate on the errors of the moments. For the lowest moment, the mean, we introduce the mean error £m

= |(X(T))-(Y(T))|.

(5.95)

Definition 5.3 (Order of weak convergence) We use the weak convergence criterion (5.95) we define the order 7 > 0 by ||
(5.96)

where g can be a class of polynomials. Clearly, the simplest definition of the order comes with the application of the linear polynomial g(X) = X. •

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To perform numerical experiments of strong and weak convergence, we use in program F15 the population growth problem (2.1). To this end we take the logarithm of (5.94) and (5.96). The slope of the corresponding curve equals than the order of the convergence 7. The results are revealed in Figures 5.6 and 5.7, where we plot the logarithm of the errors versus the time steps. It is convenient to apply negative powers of 2 for the time steps and use the log 2 of the error.

-1.4-Log2(Err) -1.5-

-1.6-

-1.7--

-1.8-1

1

1

1

1

1

1

1

-6 -5.5 -5 -4.5 -4 -3.5 - 3 -2.5 -2 Log2(DT)

Fig. 5.6. Absolute error for the case of the polpulation growth problem (2.1). Upper curve: Euler scheme, Lower curve: scheme (5.52). Parameters: r = 1, u = 0.1, N(0) = 1, T = 1. O-i

Log2(Err)

-

. - - - •

- 6 -

-8-

.•'

-1 0-1

1

1

1

1

1

1

1

-6 -5.5 -5 -4.5 - 4 -3.5 - 3 -2.5

r

-2

Log2(DT)

Fig. 5.7. Mean error for the case of the population growth problem (2.1). Curves and parameters as in Figure 5.6.

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However, there is an ambiguity in the calculation of the strong order of convergence 7 in (5.94) related to the computation of the exact solution X(T). We discuss this problem only for a 1-D SDE and use this to calculate the numerical solution Y(T) alternatively by (5.41), (5.50) or (5.52) (Euler, Milstein or higher order scheme). Starting from the initial condition Yo we obtain after K equidistant steps Y K ( T ) , T = KA, where we need for each individual step the quantity ABfc = (z)fc-\/A; k — 1, 2 , . . . , K with (z)k as the kth simulation of a N(0,1) RV. At the end of the interval t = T we have for the WP K

K

B T = ] T Bk = \/A ^{z)k.

(5.97)

fc=i fc=i

To determine the strong order convergence we need next the exact solution X(T) that is in many cases a function of Br. There are now different ways to determine the latter quantity. We propose the following two alternatives (i) As in program F15 we calculate X(T) with the aid of BT =

VT(Z)K+1,

where we use the (K + l)-th simulation of the RVz. This is the simulation taken right after the computation of (5.97). This concept was employed in the determination of the data for Figures 5.6 and 5.7. (ii) Here we use (5.97) and we easily derive K


(B2T) = A £

K

<(Z)m(Z)fe) = A ^ ( ( Z ) | ) = AK = T.

k,m=l

k=l

This concept has the methodological advantage of assigning to the strong order of convergence for the exact 1-D SDE dx = adt + /3dBt;

a,0 = const,

(5.98)

the value 7 = 00. Furthermore, we obtain with concept trends of the order of convergence that apply to general class of 1-D SDE's. The Eulerian scheme leads for instance to 7 = 0.5 and 7 = 1 for strong and weak convergence, respectively.

Numerical Solutions of Ordinary Stochastic Differential

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We will give more hints to the determination of the order of convergence in EX 5.10. The important question is, however, what is the goal of the numerical simulation. Is a good pathwise approximation required or is an approximation of some functional of the PD, as the moments, sufficient? Finally, we wish to add some important references from the vast literature about strong and weak convergence schemes and implicit routines. We mention here only the following articles Milstein [5.10], [5.11], Drummond and Mortiner [5.12] Kloeden and Platen [15.13], [5.14], Saito and Mitsui [5.15] and Newton [5.16]. Exercises E X 5.1. Given the DOF m, calculate the mean value and the variance of the PD (5.1a). Verify (x) = m; Var(x) = 2m. E X 5.2. Use the asymptotic method of integration by parts to approximate (5.10). Using this method and applying subsequently Newton's iteration determine the parameter b(a). Hint: An integral such as the one in (5.10) can be transformed to poo

J(x) = /

exp(—x2 /X)dx

Ja

A f°° 1 d / da;-—-exp(—x2/X); 0 < a, A < oo. 2 Ja xdx Use integration by parts to find an asymptotic expansion of this integral. The subsequent Newton iteration is performed in program F3. E X 5.3. Verify (5.48) and (5.49). =

Hint: Use (1.89), (1.105b), and (1.110). E X 5.4. Find an implicit solver for the Ornstein-Uhlenbeck (2.14) problem based on the Milstein routine and compare your program with F l l . Hint: Use (5.58) to formulate the solver.

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E X 5.5. Derive the higher order scheme for a non-autonomous N-D SDE (5.62). Hint: Expand the error term (5.69). E X 5.6. Find a numerical solution of the SDE (A.l) with A := n (why?) of the appendix of Chapter 3 based on the routine (5.78). Here we obtain a\ = V,

& i = 0 ; a2 = -\g(y)y - f(x), b2 = V2AT = const, 9 9 32 , 9 L o = y ^ - + «2x- + A T —z ; Ll = V2XT— ox oy oy ay and hence (Lo&fc) = (Li&fc) = (LiLibfc) = 0, k = 1,2. Equation (5.78) leads to + ~a2A2n + V2ATAZ„

„ + ynAn

Vn+i = Vn + a2An + 72ATAB n + -(L 0 a 2 )A 2 +

(Ua2)AZn.

Find Loa2 and L\a2 and solve the numerical system using the parameters given in Figure 3.4 with step sizes An = 0.01 and 0.005. Plot the results and compare them with the phase diagram in Figure 3.4. E X 5.7. Apply the scheme (5.78) to solve the problem of the Brownian bridge (2.38). Use the step sizes An = 0.01 and 0.005 and the parameters r — 1 and x(0) = 0. Hint: In the case of a ID system we have 9

9

at

ox

! ,2

2

9 2

ox2

T

,

9

ox

E X 5.8. Use the Milstein (5.73) and the higher order scheme (5.78) to approximate the solution of the stochastic Brusselator problem da; = a(ic, y)dt +

ac(x)dBt,

dy = — [x + &(x, y)]dt — ac(x)dB t , a(x, y) = (0-

l)x + (3x2 + (x + l) 2 y;

c(x) = x(l + x);

a, /? = const.

Numerical Solutions of Ordinary Stochastic Differential

Equations

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The deterministic Brusselator (a = 0) equation was developed at the occasion of a scientific congress in Brussels, Belgium, to develop a simple model for bifurcations in chemical reactions. Solve this problem by developing the corresponding user-supplied subroutines for the program F13. Use step widths A n = 0.005, 0.01 and the IC's x(0) = —0.1, y(0) — 0, and the parameters a — 0.1, (3 = 2. Plot the results in form of a phase diagram (see Figure 3.4) and find traces of the bifurcations. E X 5.9. Solve numerically the stochastic Lorentz equations (B.26) of Chapter 3. To achieve this goal generalize F13 to the case of 3D SDE's. Use the step widths and IC's of EX 5.8 with z ( 0 ) = 0 and apply the parameters b = 1, r = 0.2, s = 0.1, a = 0.1. Compare your program with F17. E X 5.10. Modify the program F15 to emply (5.97) for the determination of the exact solution X(T). Use this concept after the computation of each individual batch in F15. Find with this modification a) Strong and weak convergence order that replace the data for Figures 5.6 and 5.7 in the case of the population growth that was originally considered in F15. b) Consider this alternative of F15 to find for the Euler routine (5.41) and calculate the strong order of convergence 7 for this case of the SDE (5.98). c) An other 1-D SDE with a simple exact solution is gtiven in EX 2.1 (iii). Determine again the strong and weak order of convergence.

REFERENCES

Introduction A. Einstein, Ann. Physik, 17, 549 (1905). M. von Smoluchowsky, Ann. Physik, 2 1 , 576 (1906). P. Langevin, Comptes Rendus Acad. Sci. (Paris), 146, 530 (1908). G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev., 34, 823 (1930). E. L. O'Neill, Introduction to Statistical Optics (Dover Publications, 1991). F. G. Stremler, Introduction to Communication Systems (Addison-Wesley Publishing Company, 1990). L. Hopf, Common Appl. Math., 1, 53 (1948). R. H. Kraichnan, Fluid. Mech., 67, 155 (1975). [i [1.9 O. V. Rudenko and A. S. Chirkin, Soviet Phys. Acoustics, 19, 64 (1974). [1.10; W. A. Wojczinsky, Burgers-KPZ Turbulence, Lecture Notes in Mathematics 1700 (Springer-Verlag, 1998). [1-1 [i-a: [1.3 [1-4 [1 [1

Chapter 1 [1.1] K. L. Chung, Elementary Probability Theory with Stochastic Process (Springer-Verlag, New York, 1979). [1.2] L. Arnold, Stochastic Differential Equations (J. Wiley & Sons, New York, 1974). [1.3] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964). [1.4] Y. S. Chow and H. Teichler, Probability Theory (Springer-Verlag, New York, 1978). [1.5] J. Doob, Stochastic Processes (John Wiley, 1953). [1.6] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North-Holland, Kodansha, 1981). [1.7] T. H. Rydberg, The normal inverse Gaussian Levy Process: Simulation and approximation, Commun. Statist.-Stoch. Models, 13, 887-910 (1997). [1.8] B. 0ksendahl, Stochastic Differential Equations. An Introduction with Applications (Springer-Verlag, 1998). [1.9] Z. Schuss, Theory and Applications of Stochastic Differential Equations (John Wiley, 1980). 205

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[1.10] K. L. Chung and F. Aitsahia, Elementary Probability (Springer-Verlag, 2003). [1.11] S. Ross, Probability Models (Academic Press, 2003). [1.12] P. Mallivan, Stochastic Analysis (Springer-Verlag, New York, 1997). [1.13] J. Pitman, Probability (Springer-Verlag, New York, 1997). [1.14] A. N. Shiryaev, Probability (Springer-Verlag, New York, 1996). [1.15] I. M. Ryshik and I. S. Gradstein, Tables of Sums, Products and Integrals (VEB Deutscher Verlag Wissenschaften, Berlin, 1963). [1.16] P. Bremaud, Markov Chains, Gibbs Fields, Monte Carlo Simulation and Queues (Springer-Verlag, New York, 1999). [1.17] A. R. Kerstein, Linear-eddy modelling of turbulent transport, Part 6. Microstructure of diffusive scalar mixing fields, J. Fluid Mech., 231, 361-394 (1991). [1.18] A. R. Kerstein, One-dimensional turbulence: Model formulation and application to homogeneous turbulence, shear flows and buoyant stratified flows, J. Fluid Mech., 392, 277-334 (1999).

Chapter 2 [2.1] H. Risken, The Fokker-Planck Equation (Springer-Verlag, 1984). [2.2] D. A. McQuarrie, Statistical Mechanics (Harper and Row, New York, 1973). [2.3] T. C. Gard, Introduction to Stochastic Differential Equations (Dekker, 1988). [2.4] R. L. Stratonovich, Topics in the Theory of Random Noise, Vol. 1 (Gordon and Breach, 1963). [2.5] P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of Stochastic Differential Equations (Springer-Verlag, 1994). [2.6] J. K. Hale and H. Ko§ak, Dynamics and Bifurcation (Springer-Verlag, 1991). [2.7] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus (Springer-Verlag, 1997).

Chapter 3 [3.1] N. G. V. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland Amsterdam, 1992). [3.2] S. R. Salinas, Introduction to Statistical Physics (Springer-Verlag, 2001). [3.3] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientist and Engineers (McGraw-Hill, New York, 1978). [3.4] L. Arnold, Random Dynamical Systems (Springer-Verlag, Berlin, 1998). [3.5] S. Wiggins, Introduction to Nonlinear Dynamical Systems and Chaos (Springer-Verlag, New York, 1990). [3.6] H. Crauel and F. Flandoli, Additive noise destructs a pitchfork bifurcation, J. Dynamics Differential Equations, 10, 259-274 (1998).

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[3.7] H. Rong, G. Meng, X. Wang, W. Xu and T. Fang, Invariant measures and Lyapunov exponents for stochastic Mathieu system, Nonlinear Dynamics, 30, 313-321 (2002). [3.8] A. H. Nayfeh, Perturbation Methods (John Wiley and Sons, New York, 1973). [3.9] W. V. Wedig, Invariant measures and Lyapunov exponents for stochastic generalized parameter fluctuations, Structural Safety, 8, 13-25 (1990). [3.10] S. Rajan and H. G. Davies, Multiple time scaling and the response of a duffing oscillator to narrow band excitations, J. Sound Vibrations, 123, 497-506 (1988). [3.11] A. H. Nayfeh and S. J. Serban, Response statistics to combined deterministic and random excitations, Int. J. Nonlinear Mechanics, 25, 493-509 (1990). [3.12] M. Van Dyke, Perturbation Methods in Fluid Mechanics (Parabolic Press, Stanford CA, 1975). [3.13] E. Zauderer, Partial Differential Equations (John Wiley and Sons, New York, 1989). [3.14] E. Ben-Jacob, D. J. Bergman, B. J. Matkowsky and Z. Schuss, Thermal shot effects and nonlinear oscillations, Annals of the New York Academy of Science, 410, 323-337 (1983). [3.15] P. Plaschko, Deterministic and stochastic oscillations of a flexible cylinder in arrays of static tubes, Nonlinear Dynamics, 30, 337-355 (2002) [3.16] P. Glendinning, Stability, Instability and Chaos (Cambridge University Press, Cambridge UK, 1994). [3.17] R. Z. Khasminskiy, Stability of Systems of Differential Equations in the Presence of Random Noise (Nauka Press, Moscow USSR, 1969). [3.18] W. Ebeling, H. Herzel, W. Richert and L. Schimansky-Geier, Influence of noise on Duffing-van der Pol oscillators, Zeitschrift f. Angew. Math. u. Mechanik, 66, 141-146 (1986). [3.19] L. Arnold, N. Sri Namachchivaya and K. R. Schenk-Hoppe, Towards an understanding of the stochastic Hopf bifurcation: A case study, / . J. Bifurcation Chaos, 6, 1947-1975 (1996). [3.20] K. R. Schenk-Hoppe, Stochastic Hopf bifurcation: An example, I. J. NonLinear Mechanics, 31, 685-692 (1996). [3.21] K. R. Schenk-Hoppe, Deterministic and stochastic Duffing-van der Pol Oscillators are non-explosive, ZAMP, 47, 740-759 (1996). [3.22] K. R. Schenk-Hoppe, The stochastic Duffing-van der Pol equation, Ph.D. Thesis, Univ. Bremen, Germany (1996). [3.23] H. Keller and G. Ochs, Numerical approximation of random attractors, Inst, of Dynamical Systems, Univ. Bremen, Germany, Rep. Nr. 431 (1998).

Chapter 4 [4.1] P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. I (McGraw-Hill, New York, 1963).

208

Stochastic Differential Equations in Science and Engineering J. B. Walsh, An Introduction to Stochastic Partial Differential Equations, Lecture Notes in Mathematics 1180 (Springer-Verlag, 1986), pp. 265-439. H. Holden, B. 0ksendal, J. Ub0e and T. Zhang, Stochastic Partial Differential Equations (Birkhauser Boston, 1996). N. V. J. Krylob, M. Rockner and N. D. J. Zabczyk, Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions, Lecture Notes in Mathematics 1715 (Springer-Verlag, 1999). G. B. Whitham, Linear and Nonlinear Waves (J. Wiley and Sons, New York, 1974). O. V. Rudenko and S. I. Soluyan, Theoretical Foundations of Nonlinear Acoustics (Consultants Bureau, New York, 1977) [Translated from Russian by R. T. Beyer]. S. A. Akhmanov and A. S. Chirkin, Statistical Phenomena in Nonlinear Physics (Izd. GMU, 1971). J. M. Keller, Wave propagation in random media, Proc. Symp. Appl. Math. Am. Math. Soc, Providence Rhode Island, 13, 227-246 (I960). J. B. Keller, Stochastic equations and wave propagation in random media, Proc. Symp. Appl. Math. Am. Math. Soc, Providence Rhode Island, 1963, 16, 145-170 (1963). U. Frisch, Turbulence (Cambridge University Press, 1996). C. W. Haines, An analysis of stochastic eigenvalue problems, Ph. D. Thesis, Renssellaer Polytechnic Institute, Troy, New York (1964). W. E. Boyce, in Probabilistic Methods in Applied Mathematics I, ed. A. T. Bharucha-Reid (Academic Press, New York, 1968). F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon Press, Oxford, 1979). F. E. Benth, Option Theory with Stochastic Analysis (Springer-Verlag, Berlin, 2004). F. Black and E. Scholes, The pricing of options and cooperate liabilities, J. Polit Economy, 8 1 , 637-654 (1973). 0 . E. Barndorff-Nielsen, Processes of normal inverse Gaussian type, Finance Stock, 2, 41-68 (1998). E. Eberlein and U. Keller, Hyperbolic distributions in finance, Bernoulli, 1, 281-299 (1995). J. C. Hull, Options, Futures and Other Derivative Securities (Prentice Hall, Englewood Cliffs, 1993). R. Merton, Theory of rational option pricing, Bell J. Econom. Manag. ScL, 4, 141-183 (1973).

Chapter 5 [5.1] M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statistical Physics, 19, 25-52 (1975). [5.2] D. E. Knuth, The Art of Computer Programming, Vol. 2 (Addison-Wesley, Reading, Massachusetts, 1981).

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[5.3] W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes (Cambridge Univ. Press, New York, 1992). [5.4] C. Chatfield, The Analysis of Time Series (Chapman and Hall, London, 1980). [5.5] H. C. Tuckwell, Elementary Applications of Probability Theory (Chapman and Hall, London, 1988). [5.6] R. E. Walpole and R. H. Myers, Probability and Statistics for Engineers and Scientists (Macmillan, New York, 1986). [5.7] M. Overbeck-Larisch and W. Dolejsky, Stochastic with Mathematica (Vieweg-Verlag, Braunschweig, Germany, 1998) (in German). [5.8] N. M. Giinther and R. O. Kusmin, A Collection of Examples of Higher Mathematics (VEB Deutscher Verlag d. Wissenschaften, Berlin, 1975) (in German). [5.9] N. Metropolis, M. N. Rosenbluth, A. W. Rosenbluth, A. H. Teller and E. Teller, Equations of state calculations by fast computing machines, J. Chem. Soc, 2 1 , 1087-1092 (1953). [5.10] G. N. Milstein, The Numerical Integration of Stochastic Differential Equations (Urals Univ. Press, Sverdlovsk USSR, 1988) (in Russian) [English Translation: Kluwer (1995)]. [5.11] G. N. Milstein, A theorem on the order of convergence of mean square approximations of systems of stochastic differential equations, Theor. Prob. Appi, 32, 738-741 (1988). [5.12] P. D. Drummond and I. K. Mortimer, Computer simulation of multiplicative stochastic differential equations, J. Comput. Phys., 93, 144-170 (1991). [5.13] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, Vol. 23 (Springer, 1992). [5.14] P. E. Kloeden and E. Platen, Higher-order implicit strong numerical schemes for stochastic differential equations, J. Statist. Physics, 68, 283-314 (1992). [5.15] Y. Saito and T. Mitsui, Discrete approximations for stochastic differential equations, Trans. Japan SIAM, 2, 1-16 (1992). [5.16] N. J. Newton, Variance reduction for simulated diffusion, SIAM J. Appl. Math., 54, 1780-1805 (1994).

FORTRAN PROGRAMS

Fl.f F2.f F3.f F4.f F5.f F6.f F7.f F8.f F9A.f F9B.f FlO.f Fll.f F12.f F13.f F14.f F15.f F16.f F17.f F18f F19f

Independence test: chi-squared criterion Independence test: Eqs. (5.5)-(5.8) The Newton Iteration of the integral Eq. (5.10) Estimation of PDF F(fc) according to Eq. (5.23) Volume of the D-dim unit sphere 1-D Brownian motion for Figs. 1.1 and 1.2 Population growth, solution (2.4), data for Fig. 5.2 Milstein scheme (5.50) for 1-D SDE's Numerical generation of the RV:DZ, Eq. (5.55) (no batches) N batches of M samples to verify Eq. (5.55) Higher order scheme (5.52) for a 1-st order SDE Implicit Milstein scheme (5.58) for a 1-st order autonomous SDE Modified Milstein scheme (5.59) for a 1-st order autonomous SDE 2-D schemes: Milstein routine (5.73), higher order routine (5.78) and modified routine (5.82) 2-D implicit scheme applied to the linear pendulum Weak and absolute error, confidence intervals Double precision version of F15.f Version of F15.f for the problem EX 5.11 Discretization for a 3-D SDE Numerical verification of the law of the iterated logarithm (1.40)

211

INDEX

adapted process, 13 additive noise, 70 algebra of events, 1 almost coincide, 75 asymptotic theory, 149 autocorrelation function, 11, 67, 139

compatibility, 9 conditional probability, 7 confidence interval, 171 convergence strong, 173, 198 weak, 173, 174, 198 convergence in the mean square, 173 correlation coefficient, 11 cross-correlation function, 11

batch average, 185 Bayes's rule, 7 Bernoulli differential equation, 80 Bernoulli distribution, 4 Bessel process, 86 Bessel-Fubini formula, 144 bifurcation, 81, 110 bivariate Gaussian, 15 bivariate GD, 15 bivariate PD, 19 Black-Scholes market, 162 Boltzmann transport equation, 62 Borel set, 2 Box-Miller method, 18 Brownian bridge, 66 Brownian motion, viii, xi, 21, 179 Burgers equation, xv

differences between the Ito and Stratonovich integrals, 37 Dirac delta function, 3 Dirac function, 67 eigenfunction, 136, 148 eigenvalue moment, 156 eigenvector, 148 eikonal equation, 118 ensemble average, 12 equivalent definitions of a Wiener process, 24 ergodic system, 12 error absolute, 196 global, 196 local, 196 Euler method, 179 existence theorem, 84 expectation value, 4

call option, 161 central limit theorem, 16 Chapman-Kolmogorov equation, 20, 50, 91 backward equation, 97 forward equation, 97 characteristic function, 6, 16 Chebyshev inequality, 13, 50, 174 coefficient with time lag, 171 colored noise, 67

flip-flop process, 49 fluidmechanical turbulence, xiv Fokker-Planck equation, 12, 21, 78, 91, 95 213

214 Gaussian (or normal), 6 Gaussian probability density, xiii, 6 Green's function, 136 growth bound condition, 85 Hamilton function, 82 Hermitian-chaos, 16 Hermite polynomial, 16 heteroclinic orbit, 82 homogeneous Poisson process, 46 Hopf bifurcation, 127 independence, 5, 25 independence test, 170 inhomogenous ordinary SDE, 59 investment, 161 Ito equation, 57 Ito formula, 38 the case of a multivariate process, 43 Ito integral, 30 properties, 37 is a martingale, 37 Ito-Taylor expansion, 181, 189 Jacobian, 18 Kamiltonian, 83 Khasminskiy criterion, 128 Kolmogorov's criterion, 10 Kramers-Moyal equation, 98 Langevin parameter, 62 law of the iterated logarithm, 17 Lebesque integration, 2 Levy process, 27, 160 linear equation homogeneous ordinary SDE, 56 nonlinear SDE, 56 Lipschitz condition, 85 Lyapunov function, 124 Lyapunov method, 108 Mach number, 142 marginal PD, 5

Index Markov or Markovian process, 19, 91, 179 stationary, 20 Markov property, 12 martingale, 12, 13 Ito integral is not a martingale, 37 Wiener process is a martingale, 24 master equation, 20, 91 Maxwell distribution, 61, 62 mean, 57 Metropolis algorithm, 178 Milstein method, 183 modified, 187 moments of a PD, 6 Monte Carlo integration, 175 multiplicative noise, 70 multivariate form of the Gaussian PD, 14 Navier-Stokes equations, xiv non-anticipative or adapted functions, 30 normal distributed variable, 6 normal inverted Gaussian distribution (NIGD), 27, 160 normalization, 5 numerical solution, 167 optical diffraction, xii ordinary differential equation (ODE), 55 Ornstein-Uhlenbeck, 26 problem, 59, 101, 186 process, 20, 27 SDE, 139 transition probability, 62 orthogonal, 138 pendulum, stochastic, xi, 70, 81 Poisson bracket, 83 Poisson distributed variables, 45 Poisson distribution, 4 polar Marsaglia method, 179

215

Index polynomials of the Brownian motion, 42 population growth, 56, 65 principle of invariant elementary probability measure states, 18 probability density, 3 probability distribution function, 3 probability of an event, 2 probability space, 2 put option, 161 quasi-chaos, viii random (or stochastic) variable, 2 random number, 167 random walk, 48 reduction method, 63 Reynolds number, xiv Riemann integral, 29 Routh-Hurwitz criterion, 128 sample function, 9 sample space, 1 sigma algebra, 2 stability of SDE's, 125 standard deviation, 6, 61 stationary normal distribution, 78 stationary point, 70 stationary process, 10 statistical mechanics, 11 stochastic effects, xi boundary condition, 141 boundary value problem, 155 cable equation, 137 damping, 73 differential equation (SDE), xi ordinary SDE, 55 partial SDE, 56 economics, 160 eigenvalue, 135 eigenvalue equation, 147 eigenvalue problem, 148 excitation, 72 growth, xii

hyperbola, 26 initial condition, 141 Lorentz equation, 129 Mathieu equation, 112 partial differential equation, 135 pendulum, xi Poisson equation, 140 process, 9 Stokes law, 68 stratonovich equation, 58 stratonovich integral, 30, 37 not a martingale, 37 properties, 37 strong law of large numbers, 174 student distribution, 197 Sturm-Liouville problem, 149 symmetry, 9 Taylor expansion, 7 transformation of stochastic variables, 17 transition probability distribution, 19 trivariate PD, 15 turbulence advection by a random map, 49 two independent Brownian motions, 45 uncorrelated function, 11 uniformity test, 169 uniqueness theorem, 84 variance, 6, 57 Volterra integral equation, 72 white noise, xi, xii, 21, 56, 57 W P or Wiener process, 14, 21, 25 is a martingale, 24 Wiener sheet, 25, 135 Wiener-Levy theorem, 54 WKB method, 117, 149

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Traditionally, non-quantum physics has been concerned with deterministic equations where the dynamics of the system are completely determined by initial conditions. A century ago the discovery of Brownian motion showed that nature need not be deterministic. However, it is only recently that there has been broad interest in nondeterministic and even chaotic systems, not only in physics but in ecology and economics. On a short term basis, the stock market is nondeterministic and often chaotic. Despite its significance, there are few books available that introduce the reader to modern ideas in stochastic systems. This book provides an introduction to this increasingly important field and includes a number of interesting applications.

ISBN 981-256-296-6

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