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Int. J. Appl. Math. Comput. Sci., 2003, Vol. 13, No. 2, 151160

APPROXIMATION OF THE ZAKAI EQUATION IN A NONLINEAR

FILTERING PROBLEM WITH DELAY

KRYSTYNA TWARDOWSKA, TOMASZ MARNIK

MONIKA PASAWSKA-POUDNIAK

Faculty of Mathematics and Information ScienceWarsaw University of Technology

Plac Politechniki 1, 00661 Warsaw, Polande-mail: [email protected]

Department of MathematicsTechnical University of Rzeszw

ul. W. Pola 2, 35959 Rzeszw, Polande-mail: [email protected]

A nonlinear filtering problem with delays in the state and observation equations is considered. The unnormalized conditionalprobability density of the filtered diffusion process satisfies the so-called Zakai equation and solves the nonlinear filteringproblem. We examine the solution of the Zakai equation using an approximation result. Our theoretical deliberations areillustrated by a numerical example.

Keywords: nonlinear filtering, stochastic differential equations with delay, Zakais equation

1. Introduction

We study a nonlinear filtering problem with delay usingan approximation result of the Wong-Zakai type for thecorresponding Zakai equation with delay. The nonlin-ear filtering problem was considered in the literature, e.g.,by Bucy (1965), Kushner (1967), Zakai (1969), Liptserand Shiryayev (1977), Pardoux (1979; 1989), Kallianpur(1980), and others. Their studies concentrated mainly onfinding an equation for the conditional probability densityof an unobserved process given an observed path. It isknown that the conditional expectation gives the best es-timate in the mean square sense. The conditional densitycan be computed by two methods. The first method gives

the so-called Kushner equation (Kushner, 1967), which isa nonlinear stochastic partial differential equation. Thesecond method gives the so-called Zakai equation (Bucy,1965; Zakai, 1969), which is a linear stochastic partialdifferential equation for the unnormalized density. There-fore, the problem of constructing solutions of the Zakaiequation is more important for practical applications be-cause of the linearity.

In recent years, the Zakai equation has been exam-ined by many authors, e.g., by Bensoussan et al. (1990)using a splitting method, by Lototsky et al. (1997) us-

ing a spectral approach, by Crisan et al. (1998) usinga branching particle method, by Cohen de Lara (1998)

using invariance group techniques, by Elliot and Moore(1998) in Hilbert spaces, and by Atar et al. (1999) usingthe Feynman-Kac formula.

In our study we apply the approximation problemof the Wong-Zakai type for stochastic partial differentialequations. It was considered by Gyngy (1989), Gyngyand Prhle (1990), Brzezniak and Flandoli (1995), andTwardowska (1995). They showed that if in the Zakaiequation we replace the disturbance by its good approxi-mations, then the approximations converge to a limit equa-tion with the so-called It correction term. The aboveproblems were considered without delays.

The well-known result for the existence and unique-ness of a filtering problem with delays but in the lin-ear case belongs to Kolmanovsky (1973), see also (Kol-manovsky et al., 2002). The approximation result is notconsidered.

In this paper, the Zakai equation is a linear stochas-tic parabolic partial differential equation with delay. Itcorresponds to our nonlinear filtering problem with de-lay. We prove the existence and uniqueness theorem forthis equation. Also, we establish the approximation resultusing the correction term derived in (Twardowska, 1991;

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K. Twardowskaet al.152

1993; 1995) in the approximation theorems of the Wong-Zakai type.

An important part of the present paper contains a nu-merical example showing that a good stability result isachieved because in the approximation sequence of equa-

tions we have added the appropriate correction term forstochastic linear differential equations with delay. Us-ing the Galerkin technique and some numerical schemes(Kloeden and Platen, 1992; Sobczyk, 1991) we transformthe Zakai equation to a simpler finite-multidimensionalform. We solve this equation without any correction termand with a correction term in the approximation sequence.It is evident that the correction term has a crucial role andimproves our approximation results.

In the paper by Ahmed and Radaideh (1997), a nu-merical method for the approximation of a nonlinear fil-tering problem was developed. Using the Galerkin tech-

nique, the solution of Zakais equation was approximatedby a sequence of nonstandard basis functions given bya parameterized family of Gaussian densities. We takesome ideas from that paper. Other numerical techniquesfor the Zakai equation can be found in the papers by Bene(1981), Elliot and Gowinski (1989), Florchinger and LeGland (1995), and It (1996).

2. Definitions and Notation

We consider the probability space (,F

,Ft[0,), P)

such that it is the cannonical space of a process{(X(t), Y(t)), t [0, )} RM RN, where

= 1 2,1 = C(R+,R

M), 2 = C(R+,RN),

X(t, ) = 1(t), Y(t, ) = 2(t),

Ft = {(X(s), Y(s)), 0 s t} N,

F is a -algebra of Borel sets on N, where N is aclass of subsets with the P-measure equal to zero, P is

the probability law of the process (X, Y), C(R+,RM)is the class of continuous functions, and Cb(R+,RM) de-notes the class of bounded continuous functions.

For the stochastic process X(t, ) and for a fixedt [0, ) we define

Xt(, ) = X(t + , ), I = [r, 0].

Therefore Xt(, ) denotes the segment of the trajectoryX(, ) on [t r, t].

Let {(X(t), Y(t)), t [0, )} be the solution tothe following system of stochastic equations with delay:

X(t, ) = X0() +

t0

b(s, Ys(, ), Xs(, )) ds

+ t0

fs, Ys(, ), Xs(, ) dV(s)+

t0

g

s, Ys(, ), Xs(, )

dW(s), (1)

Y(t, ) = Y0() +

t0

h

s, Ys(, ), Xs(, )

ds

+ W(t), (2)

where X0() is an initial constant random vari-able independent of the standard Wiener processes

{(V(t), W(t)), t

[0,

)

}with values in RM

RN,

Y0() = 0. Moreover, b, f, g and h are measurable map-pings from R+ C(I, RN) C(I, RM) with valuesin RM, RM, RMN and RN, respectively. We assumethat they satisfy Lipschitz and growth conditions (see 4below). Then the system of equations (1)(2) has exactlyone solution. The uniqueness is understood in the senseof trajectories. We shall call X(t) the state and Y(t) theobservation process.

We define

a(t,y,x) = f f(t,y,x) + g g(t,y,x) (3)for t R+, y C(I, RN) and x C(I, RM), wheref

and g

are the transpose matrices of f and g, respec-tively. Moreover,

Z(t) = exp

t0

h(s, Ys(, ), Xs(, )

dY(s, )

12

t0

hs, Ys(, ), Xs(, )2 ds (4)for t [0, T].

We make the following assumptions:

(A1) For t > 0, n N and for a measurable function : 2 [0, 1] such that

(y) = 0 if sup0ts

| y(s) |> n,

we have

E

(Y)

t0

hs, Ys(, ), Xs(, )2 ds < .(A2) E[Z(t)1] = 1 for each t 0.(A3) The coefficients b, f, g and h are uniformly

bounded by a constant c.

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Approximation of the Zakai equation in a nonlinear filtering problem with delay 153

Having Assumption (A2), we define a new probabil-ity law P0 on (, F) by

dP0

dP

Ft

= Z(t)1, t 0. (5)

We know (Pardoux, 1989, p. 13) that for each t 0, L1(, Ft, P) we then have Z(t) L1(, Ft, P0)and

E( | Yt) = E0(Z(t) | Yt)

E0(Z(t) | Yt) ,

where Yt = {Y(s) : 0 s t}, E0 being the condi-tional expectation operator under P0.

Let M+(RM) denote the space of finite measureson RM. We define the processes {(t), t 0} and{(t), t 0} with values in M+(RM) by

(t)() = E0(X(t))Z(t) | Yt (6)and

(t)() = E

(X(t)) | Yt

(7)

for t 0, and Cb(R+,RM). The spaceCb(R+,RM) is endowed with the topology of the uni-form convergence.

Let us remark that (0) = (0) = law of X(0).We introduce some families of partial differential opera-tors indexed by (t, y) R+2 for C2b (R+,RM),y C(I, RN), x C(I, RM):

L(t,y)(x) =

1

2 a

ij

(t , y, x)

2

xixj (x)

+ bi(t , y, x)

xi(x), (8)

Aj(t,y)(x) = flj(t , y, x)

xl(x), (9)

Bi(t,y)(x) = gli(t , y, x)

xl(x) (10)

and

Li(t,y)(x) = hi(t,y,x)(x) + Bi(t,y)(x) (11)

for i = 1, . . . , N and j = 1, . . . , M . We have used herethe convention of repeated indices summation.

Now we are in a position to formulate the so-calledZakai equation in 3 (see Theorem 2.2.3 in (Pardoux,1989; Chaleyat-Maurel, 1990) for the case without delay):

(t)() = (0)() +

t0

(s)

L(s,Y)

ds

+

t0

(s)

Li(s,Y)

dYi(s) (12)

for every C2b (R+,RM) if all coefficients ofEqns. (1)(2) are bounded.

Note that this is a stochastic linear parabolic partialdifferential equation because of the form of the operatorL(t,y)(x).

Let us introduce the normalized law by

(t)() = E0(X(t))Z(1) | Yt. (13)The corresponding equation for the densities of the con-ditional probabilities cf. (7) can also be established.For the case without delay it is called the Kushner-Stratonovich equation (see, e.g., Pardoux, 1989).

3. Zakai Equation

Theorem 1. Let all coefficients in (1)(2) be bounded.Then for every C2b (R+,RM) the solution of (1)(2)satisfies the Zakai equation (12).

Proof. From (1) and (2) we have

dW(t) = dY(t) ht, Yt(), Xt()dt.From this we obtain the following relation:

t0

g

s, Ys(, ), Xs(, )

dW(s)

=t0

g

s, Ys(, ), Xs(, )

dY(s)

t0

g

s, Ys(, ), Xs(, )

hs, Ys(, ), Xs(, ) ds. (14)Using (14) we get

X(t) = X0 +

t0

b

s, Ys(, ), Xs(, )

gs, Ys(, ), Xs(, ) hs, Ys(, ), Xs(, ) ds

+

t0

f

s, Ys(, ), Xs(, )

dV(s)

+

t0

g

s, Ys(, ), Xs(, )

dY(s). (15)

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Using the It formula for the multidimensional case (seeLiptser and Shiryayev, 1977), we obtain

d(X(t))

= xbt, Yt(), Xt() gt, Yt(, ), Xt(, )

ht, Yt(, ), Xt(, )

+1

2

xx

f ft, Yt(), Xt()

+ g gt, Yt(), Xt()Big ] dt+ xf

t, Yt(), Xt()

dV(t)

+ xg

t, Yt(), Xt()

dY(t)

= L(t,Yt)X(t) dt hit, Yt(

, ), Xt(

, ) Bi(t,Yt)X(t) dt + Al(t,Yt)X(t) dVl(t)

+ Bi(t,Yt)

X(t)

dYi(t). (16)

Writing the above equation in an integral form we have

(X(t)) = (X0) +

t0

L(s,Ys)

X(s)

ds

t0

hi

s, Ys(), Xs()

Bi(s,Ys)

X(s)

dt

+ t

0

Al(s,Ys)X(s) dVl(s)

+

t0

Bi(s,Ys)

X(s)

dYi(s).

From the Girsanov theorem (see Liptser andShiryayev, 1977), we have

Z(t) = 1 +

t0

Z(s)hi

s, Ys(), Xs()

dYi(s).

Using once more the It formula for the multidimensionalcase for f(t, x1, x2) = x1 x2, we get

Z(t)(X(t)) = (X0) + t

0

Z(s)L(s,Ys)X(s) ds+

t0

Z(s)Al(s,Ys)

X(s)

dVl(s)

+

t0

Z(s)Li(s,Ys)

X(s)

dYi(s).

Taking the expected value E0( | Y) of both the sides andusing Lemma 2.2.4 from (Pardoux, 1989), we have

E0t

0

U(s) dYi(s) | Y

=

t0

E0

U(s) | Y dYi(s)

and

E0t

0

U(s) dYj(s) | Y

= 0

for t 0, i = 1, . . . , N , j = 1, . . . , M and for a pro-gressively measurable process

{U(t), t

0

}. From the

definition of (t)() we get

(t)() = (0)() +

t0

(s)(L(s,Y)) ds

+

t0

(s)(Li(s,Y)) dYi(s).

The existence and uniqueness of the solution of (12)follows, e.g., from the classical result of (Pardoux, 1979;Bensoussan et al., 1990).

4. Approximation Results of theWong-Zakai Type

We recall that for our numerical computations we shallneed the approximation result of the Wong-Zakai type(Wong and Zakai, 1965) of our filtering problem when thenoise in our Zakai equation is replaced by its polygonalapproximations. In practice we obtain the real observa-tions as a result of measurements of the process Y(t).But then, instead of the observations {Y(t) : s t}, weobtain the paths {Yn(t) : s t}, where the processesYn(t) have bounded variations and they are approxima-tions of Y(t). Using real Yn(t) instead of Y(t), we solvethe approximate equations with the operator (11), i.e., wesolve the equations

n(t)() = n(0)() +

t0

n(s)(L(s,Yn)) ds

+

t0

n(s)(Li(s,Yn)

) dYin(s). (17)

So we obtain n(t)() as the solutions and, consequently,we obtain the densities pn(t)() = dn(t)()/dx.

In our theorem we shall show that if Wn(t) W(t) and so Y

n(t)

Y(t), in a certain sense, as

n , then also n(t)() (t)() in an appro-priate sense.

We shall further see that applying the Galerkin tech-nique we shall obtain from (12) a finite multidimensionalsystem of stochastic ordinary differential equations withdelay (Ahmed and Radaideh, 1997).

So now we start from the investigation of a stochasticordinary differential equation with delay (in a more gen-eral form, i.e., the stochastic functional differential equa-tion when the delay is not constant with respect to time).

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Let us restrict our deliberations to t [0, T].For J = (, 0] we introduce some metric spacesC = C(J,R

d), C1 = C((, T],Rd) and C02 =C((, T],Rm) = of continuous functions. Thespace C is endowed with the metric

(f, g)C =n=1

2nf gn

1 + f gn

for f, g C, hn = maxnt0 h(t).For further consideration we also set I = [r, 0],

0 < r < , and we introduce the norm spacesC = C(I, Rd), C1 = C([r, T],Rd) and C02 =C([r, T],Rm) = of continuous functions with theusual norms of the uniform convergence.

Here d is the dimension of the state space and m isthe dimension of the Wiener process; in the space C02 all

functions are equal to zero at zero.Below we denote by X one of the above spaces. Let

F(X) denote the Borel -algebra of the space X. It isobvious that C02 is identical with the -algebra generatedby the family of all Borel cylinder sets in X (see Ikedaand Watanabe, 1991). So we construct the Wiener space(C02 ,B(C

02 ), P

w), where Pw is a Wiener measure. Thecoordinate process B(t, w) = w(t), w C02 , is an m-dimensional Wiener process.

The smallest Borel algebra that contains B1,B2, . . .is denoted by B1B2. . . ; Bu,v(X) denotes the small-est Borel

algebra for which a given stochastic process

X(t) is measurable for every t [u, v], and Bu,v(dB)denotes the smallest Borel algebra for which B(s)B(t)is measurable for every (t, s) with u t s v.

Let Bn(t, w) = wn(t) be the following piecewiselinear approximation of B(t, w) = w(t):

Bn,p(t, w) = wp

k

2n

+ 2n

t k

2n

wp

k + 1

2n

wp

k

2n

(18)

for each p = 1, . . . , m and kT /2n

t < (k + 1)T /2n

for k = 0, 1, . . . , 2n 1.Now we consider = C02 . Let X be a contin-

uous stochastic process X(t, w): [r, T] Rd,i.e., X: X=C1. We take some fixed initial con-stant stochastic processes for J for i = 1, . . . , d:Xi(0 + , w) = Xi0(w) = X

n,i0 (w) = Y

i0 (w).

We also consider operators b : C Rd, :C L(Rm,Rd) (where L(Rm,Rd) is the Banachspace of linear functions from Rm to Rd with the uni-form operator norm | |L).

In order to give a meaning to the stochastic integralsin (19) below, we introduce the following condition:

(A4) for every t (, T] the algebra B,t(X) B,t(dB) is independent ofBt,T(dB).

We consider the following stochastic functional dif-ferential equation:

Xi(t, w) = Xi0 +

t0

bi

Xs(, w)

ds

+m

p=1

t0

ip

Xs(, w)

dwp(s) (19)

for i = 1, . . . , d.

Replacing the Wiener process by Bn, we obtain thefollowing approximations of (19):

Xn,i(t, w) = Xn,i0 +t0

bi

Xns (, w)

ds

+m

p=1

t0

ip

Xns (, w)

Bn,p(s, w) ds. (20)

We also introduce another stochastic differential equation:

Yi(t, w)

= Yi0 (w) +

t0

bi

Ys(, w)

ds

+m

p=1

t0

ipYs(, w) dwp(s) (21)

+1

2

mp=1

dj=1

t0

DjipYs(, w)jpYs(, w)dsfor every i = 1, . . . , d , where the last term on the right-hand side of (21) is the so-called correction term that isdescribed as follows (Twardowska, 1991; 1993):

Let Dip denote the Frchet derivative from C toL(C,R) (the necessary assumptions are given below).From the Riesz theorem it follows that there exists a fam-

ily of measures = ipjg of bounded variation such that

Dip(g)() =d

j=1

0r

j(v)ipjg (dv)

is a directional derivative for any , g C. The mea-sure has the following decomposition:

(A) = (A (, 0)) + (A {0})= (A) + ({0})0(A),

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where 0 is the Dirac measure, A B((, 0)). Wedenote by Djip(g) the value ipjg ({0}), i.e.,

Djips() = ipjg {0}. (22)The second integral in (21) is the It integral.

Let us introduce the following conditions:

(A5) The initial stochastic process X0 is F0-measurable and P(|X0(w)| < ) = 1, where|X0(w)| =

dj=1 |Xi0(w)|, and B,0(X0) is indepen-

dent ofB0,T(B);

(A6) For any , C the following Lipschitzcondition is satisfied:b() b()2 + () ()2

L

L1

0

()

()

2

dK()

+L2(0) (0)2,

where K() is a certain bounded measure on J, and L1,L2 are some constants;

(A7) For every , C the following growthcondition is satisfied:

b()2 + ()2L

L10

1 + 2()

dK()

+L2

1 + 2(0)

,

where 2(0) = dj=1 2i (0);(A8) We have

PT

0

b(Xs) ds < = 1,PT

0

(Xs)2L ds < = 1;(A9) Let bi, ip be bounded functions and bi,

ip C1, for all i = 1, . . . , d , p = 1, . . . , m.We say that a ddimensional continuous stochastic

process X : (, T] C0

2 Rd

is a strong solutionof (19) for a given process w(t) if Conditions (A4), (A5)and (A8) are satisfied and (19) is valid with probability 1for all t (, T]. The uniqueness of strong solutionsis understood in the sense of the trajectories:

An absolutely continuous stochastic process Xn :(, T] C02 Rd is a solution of (20) if Conditions(A4) and (A5) are satisfied and (20) is valid with proba-bility 1 for all t (, T].

Notice that our conditions ensure the existenceand uniqueness of the strong solution Y of (21) since

Djip(Yt(, w)) is a real number (it is a value of a mea-sure). Moreover, for every n N, there exists exactlyone solution of the ordinary differential equation (20).

We have the following approximation theorem ofthe Wong-Zakai type for stochastic functional differential

equations (Twardowska, 1991; 1993):Theorem 2. Let Conditions (A4)(A7) be satisfied. Let

Bn(t, w) be an approximation of the type (18) of a Wienerprocess. We assume that Xn and Y are solutions of (20)and (21), respectively, with a constant initial stochastic

process. Then Conditions (A4) and (A8) are satisfied and

for every > 0 we have

limn

P

sup0tT

Xn(t, ) Y(t, )H

>

= 0. (23)

Remark 1. The proof in (Twardowska, 1991; 1993) isgiven for the interval J = (

, 0]. Instead of J =

(, 0], we can consider I = [r, 0], r > 0. Then,instead of considering Xi(tni + s) Xi(tni1 + s) on thewhole interval of the definition of time, we divide it intosome parts (see Twardowska, 1993) and we estimate eachpart separately by expressions converging to zero.

For example, consider the initial equation

dX(t) = b(Xt) dt + (Xt) dw(t),

X0(, ) = () for J, (24)where for some constants b0, b1, 0, 1 we defineb, : C

R as follows:

b() = b0(0) + b1(r),() = 0(0) + 1(r).

We note that (0) = Xt(0) = X(t), (r) =Xt(r) = X(t r) and

dX(t) =

b0X(t) + b1X(t r)

dt

+

0X(t) + 1X(t r)

dw(t), (25)

X0 = .

Then the limit equation (21) takes on the formdY(t) =

b0Y(t) + b1Y(t r)

dt

0Y(t) + 1Y(t r)

dw(t) (26)

+1

20

0Y(t) + 1Y(t r)

dt,

Y0 =

because 0X(t) is the only term for which the support ofthe measure contains zero. Therefore ({0}) = 0.

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Now we shall come back to our Zakai stochastic lin-ear parabolic partial differential equation (11). We havethe following approximation theorem of the Wong-Zakaitype (Twardowska and Pasawska-Poudniak, 2003).

Theorem 3. Let Conditions (A4)(A7) be satisfied. Let

Bn(t, w) be an approximation of the type (18) of a Wienerprocess. We assume that and n are solutions of (12)and (17), respectively, with a constant initial stochastic

process, and also

(t)() = (0)() + t0

(s)(L(s,Yn)) ds+

t0

(s)(Li(s,Yn)) dYi(s)+

1

2

t0

(s)(

DLi(s,Yn))(L

i(s,Yn)

) ds, (27)

where the last term is the so-called correction term of the

form (22). Then for every t 0 we have

limn

En(t, )() (t, )()2 = 0. (28)

Proof. For a proof of the Wong-Zakai type theorem forstochastic partial differential equations in Hilbert spaces,without delay, see (Twardowska, 1995). The convergenceis of the type (28). The case of the nonlinear filter-ing equation (12) without delay is covered by the theo-rem which we can be found in the paper (Pardoux, 1975,

pp. 130131). Now the technique of proving the Wong-Zakai theorem with delay can be copied from (Twar-dowska, 1991; 1993). We get the convergence of the typelimn E(supt |n(t, )() (t, )()|2) = 0 but theconvergence in (28) is weaker, so we prove (28) in our the-orem.

5. Approximation Result for the Zakai

Equation

From the numerical point of view, it is convenient to con-sider the Zakai equation (12) in the Stratonovich form

(Dawidowicz and Twardowska, 1995), i.e., subtracting thecorrection term appearing in (27). Then, after the Wong-Zakai approximation, we will obtain a limit equation with-out a correction term.

First, to obtain a system of stochastic ordinary dif-ferential equations from our Zakai equation, we applythe Galerkin method. We follow the idea of Ahmed andRadaideh (1997, 3.3). Therefore, using the Galerkinmethod based on the Fourier coefficients {Ni } and pro- jecting the Zakai equation onto the space spanned by{wi, 1 i N} (see Ahmed and Radaideh, 1997, 3.2,

Eqn. (8)) we can approximate the solution of (12) in theform

N(t, x) =

Ni=1

N(t)wi(t)

and then we obtain a system of stochastic ordinary dif-ferential equations in a matrix form. In our case it is asystem of linear stochastic ordinary differential equationswith delay, so we can use the theory from 4.

6. Numerical Experiments

We start with the following filtering problem:

dX(t) =

b0X2(t) + b1X

2(t 1) dt+

0X(t) + 1X

2(t 1)

dW(t), (29a)

dY(t) = a0X(t) + a1X2(t 1)dt + dW(t), (29b)where a0, a1 b0, b1, 0 and 1 are some constants,X(t) R, Y(t) R and W(t) is the one-dimensionalWiener process. We transform this problem to the fol-lowing stochastic partial differential equation of the Zakaitype (12):

t =1

220X

2(t) + 01X(t)X2(t 1)

+1

221X

4(t 1)

xx

+

b0X2(t) + b1X

2(t 1)x+

a20X2(t) + 2a0a1X(t)X

2(t 1)+ a21X

4(t 1)+

a0X(t) + a1X2(t 1) dW(t) (30)

and the correction term is of the form (cf. (22))

1

2a0

a0X(t) + a1X2(t 1) dt.

After discretization (see 5 and Ahmed and Radaideh,1997), we can restrict our analysis to the followingstochastic ordinary differential equation with delay on theinterval [0, 1]:

dX(t) =

aX(t) + bX2(t) + c

dt

+

a0X(t) + a1

dW(t), (31)

X0() = X(0 + ) = 1 for [1, 0],X(t 1) = 1 on [0, 1] as t 1 [1, 0],

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where

a = 01 + 2a0a1,

b =1

220 + b0 + a0,

c =1

221 + b1 + a

21.

We solve this equation with the following numerical meth-ods: Euler, Milshtein and Runge-Kutta schemes (Kloe-den and Platen, 1992; Sobczyk, 1991). But for our caseof stochastic differential equations with delay, we mod-ify the Milshtein scheme. It is well known that the Mil-shtein scheme can be obtained as the Euler scheme for theStratonovich version of (31) using the relation for the tran-sition between the It and Stratonovich integrals (Dawid-owicz and Twardowska, 1995).

Below we present some numerical computations toconfirm our theoretical result that the correction termplays a crucial role in numerical schemes, too.

Consider (31) with a0, a1, b0, b1, 0 and 1 givenby

a0 = 1, a1 =1

2, b0 = 2, b1 = 1

2,

0 =

2, 1 =

2

4.

Then b = 1220 + b0 + a0 = 0. Equation (31) without the

correction term has the following form:

dX(t) = 12 X(t) + 1316 dt+ (X(t) +

1

2) dW(t). (32)

Equation (31) with the correction term is

dX(t) =9

16dt +

X(t) +

1

2

dW(t). (33)

First, we obtain an exact analytical formula for t [0, 1] in the so-called step method (see 4). We use theform of the solution derived for the linear equation (4.9),pp. 119-120 in the book by Kloeden and Platen (1992),

i.e., for Eqn. (31) with b = 0. We have

X(t) = (t)

X(0) + (c a0a1)t0

(s)1 ds

+ a1

t0

(s)1 dW(s)

with the fundamental solution

(t) = exp

(a 1

2a20)t + a0W(t)

.

In our case

(t) = exp

W(t)

and X(0) = 1.

So we obtain the following solution to (31) for t [0, 1]:

X(t) = exp

W(t)

1 +5

16

t0

exp W(s)ds

+1

2

t0

exp W(s) ds. (34)

We recall that in the step method we set X(t 1) =1 for t [0, 1], so (t 1) [1, 0]. We have also usedthe following formula (Kloeden and Platen, 1992, p. 101):

t0

exp

W(s)

dW(s)

= U

W(t) UW(0) 1

2

t0

h(W(s)) ds,

where

h(x) = exp(x), U(x) = h(x).

This solution is used to test and compare numerical meth-ods in this paper. We solve the stochastic differentialequation numerically by the simulation of the approxima-tion of discrete trajectories in time. To construct a solutionfor a given discretization t0 = 0 < t1 < < tN = Twe used the Euler and Milshtein methods. We modifiedthe recursive formulae for the Milshtein method takinginto consideration the delayed argument.

The Euler approximation for (29) is generated recur-sively by

Yn+1 = Yn + (b0Yn + b1Ynk)n

+ (0Yn + 1Ynk)Wn (35)

for n = k + 1, k + 2, . . . , N 1 with initial valuesY0 = Y1 = = Yk = 1 and n = T /N (equidistantstep size), k = 1/n (an integer parameter related to thedelay),

Wn = Wtn+1 Wtn.

The random variables Wn are independentlyN(0, 1)-normally distributed random variables. We havegenerated such random variables in simulations from in-dependent and uniformly distributed random variables on[0,1] which are provided by a pseudorandom number gen-erator on a computer. The generation of the sample pathsof the process W(t) may be realized by W(0) = 0,W(t) =

n(1 + + t

n

), where i are indepen-

dent and identically N(0, 1)-normally distributed randomvariables.

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The Milhstein approximation scheme has the modi-fied form (see Kloeden and Platen, 1992) for the correc-tion term


+ (0Yn + 1Ynk)Wn

+1

20(0Yn + 1Ynk)(W

2n n). (36)

The Runge-Kutta approximation scheme (Kloeden andPlaten, 1992) is of the form


+ (0Yn + 1Ynk)Wn

+1

20(0n 0Yn + 1Ynk)

(W2n n)

1/2n , (37)

where n = Yn + b1/2n .We say that the approximating process Y converges

in the strong sense to the process X with the order (0, ] if there exist some finite constants K and 0 0such that

E|XT YN|K

for any time discretization with the maximum step size (0, 0).

In (Kloeden and Platen, 1992) it is proved that theEuler scheme has the strong order = 0.5 and the Mil-

shtein scheme converges with the strong order = 1(under some regularity conditions).

Our computations were performed using the MAT-LAB package. Figure 1 summarizes graphically the nu-merical experiment with Eqn. (33). It compares simulatedtrajectories of the examined Euler, Milhstein and Runge-Kutta schemes with the exact solution (34) of (32) forthe same sample path of the Wiener process. The solidline represents the exact solution, the dotted line the Eu-ler method, the dashed line the Milshtein method and thedotted-dashed line the Runge-Kutta method. In Fig. 2 wesolve Eqn. (32) without the corretion term and we com-pare it with the exact solution (34) of (32). We can ob-

serve that the simulated trajectories in Fig. 1 are close tothe exact solution because in (33) the correction term oc-curs. The results with the correction term in Fig. 1 arebetter.

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Received: 4 February 2002Revised: 16 January 2003

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