Chapter V* Stochastic Differential Equations with a Retarded Argument

Chapter V* Stochastic Differential Equations with a Retarded Argument 1.

Basic Concepts

Real systems with aftereffects, for the description of which areused differential equations with a deviating argument, of necessity are subjected to the influence of random effects. When random effects are small and their influence may be neglected, the mathematical apparatus for studying such kinds of systems is the theory of deterministic differential equations with a deviating argument, the basic problems of which were dealt with in the preceding chapters. Otherwise, it is necessary to make use of stochastic differential equations. This chapter is devoted to some aspects of the qualitative theory of stochastic differential equations with a retarded argument. We introduce some necessary concepts for the subsequent presentation. These concepts, in connection with stochastic equations without a deviating argument, are presented in detail in the monograph by 1.1. Gikman and A.V. Skorokod, "Stochastic Differential Equations", Kiev, Nauka, 1968, to which we refer the reader for the necessary refinements (in what follows this reference is denoted by SDE), We denote by S(t) the process of Brownian motion, i.e. a homogeneous, normal process with independent increments for which

*This chapter was written in collaboration with V.B. Kolmanovskii.

219

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

2

E ( 0 ) = 0, M<(t) = 0, ME (t) = t.

Here M is the sign of mathematical expectation. The term Brownian motion process has a physical origin. We consider the motion of small particles suspended in a liquid and subjected to collisions by the random thermal motion of the molecules of this liquid. Then the evolution in time of such particles will be exactly described by the process E(t). We will consider the simplest stochastic differential equation with a retarded argument dx(t) = fl(t,x(t) ,X(t-Tl))dt

+ (1)

where the continuous functions fi(t,x,y) (i=l,2) satisfy Lipschitz conditions with respect to the second and third arguments, the constant retardations ~ ~ 2 0 . The elementary physical representations connected with Brownian motion say that for its description it is natural to draw upon a stochastic process, each realization of which represents a continuous, nowhere differentiable function. Therefore, Eq. (1) with deterministic (for simplicity) initial conditions

where +(t) is a continuous initial function, is to be understood only as a written symbol of the integral equation t x(t) = 4(0)

+

1

fl(s,x(s) ,x(s-~~))ds+

0

(3)

220

V. STOCHASTIC DIFFERENTIAL EQUATIONS

x(s-T.) E + ( s - T . ) , if s-T.
M /tf2(s,x(s),x(~-r2))dc(s) = 0. 0

As shown in [165.1], under suitable assumptions, the initial value problem ( 1 ) , ( 2 ) has a unique solution. This means that there exists a continuous, with probability 1,process x(t), agreeing with 4 (t) on the initial set and satisfying with probability 1 the integral equation ( 3 ) ; in this connection, if x1 (t),x2 (t) are two continuous solutions, in the stated sense, of Eq. ( 3 ) with the fixed initial function +(t), then for any T>O the probability

Ix1(t)-x2(t)

P( sup

O
1

> 0 1 = 0.

Let the continuous functions f (t) and such that for any t and t2 1 f(t2)

-

f(tl) =

It2

J,

(t) be

J,(t)dt.

Then the differential of the function f(t) is defined by the formula df (t) = J,(t)dt. Analoqously, we will say that the expression (1) assigns a stochastic differential process x (t)the solution of the integral equation ( 3 ) . The most important property of the stochastic differential (1) is the Ito formula of differentiation of composite functions: Let the function u(t,z) be continuously differentiable in t and twice continuously differentiable in z and let the stochastic

221

DIFFERENTIAL EOUATIONS WITH DEVIATING ARGUMENTS

d i f f e r e n t i a l p r o c e s s x ( t ) b e g i v e n by formula ( 1 1 , t h e n t h e p r o c e s s q ( t ) = u ( t , x ( t ) ) a l s o h a s a stochastic d i f f e r e n t i a l , equal t o r

I

z=x ( t )

W e n o t e t h e scheme of t h e proof o f formula (5). For t h i s w e form t h e a r b i t r a r y p a r t i t i o n of t h e i n t e r v a l [tl,t2]by t h e p o i n t s t ( k : ) O < t l = t ( 0 ) <.< t (n)=

-

t 2 . Then

n-1 ( t 2-n ) (tl)' c [ u ( t(k+l) , x ( t(k+l)) -u ( t( k ) , x ( t( k ) ) ) 1. k= 8 (61 But, from T a y l o r ' s formula i t f o l l o w s t h a t u (t(k+l),x ( t( k + l )) ) -u ( t(k),x (t(k)) ) = Ut

( t(k)+Ok ( t( k + l )-t (k)) , x ( t( k ) ) ) ( t( k + l )-t ( k )

222

+

V. STOCHASTIC DIFFERENTIAL EQUATIONS

I

where all Ok,Ok ~(0,l). Substituting ( 7 ) into (6) and estimating the order of the quantities appearing here (the results of this calculation are simple, but cumbersome), one is convinced of the validity of formula (5) (for details, see [SDE]). The appearance f22dt, which in formula (5) of the term 1 a2u

2-1 z=x (t)

at first glance is unexpected, is connected with the fact that in the definition of the process c(t), 5(0)=0, M{(t)=O, and MS2(t) = t. 2.

Stability

By the stability of solutions of deterministic equations is understood the property of the solutions that, in some sense, small variations in the initial conditions or in the equations themselves result in small variations of the solutions. In an analogous manner, it is possible to introduce the concept of stability of solutions for stochastic differential equations, with only the difference that, for this case, it is necessary to speak about small variations of the solution with probability 1. Thus, for stochastic equations with a retarded argument, all definitions of stability introduced in Chapter I11 are carried over. However, there are possibilities presenting considerable interest of other definitions of stability, reflecting the specific peculiarities of stochastic differential equations: stability with respect to moments, with respect to probability, and 80 on. Restricting ourselves to the linear case, we consider the stochastic differential equation dx(t) = [alx(t)+a2x(t-Tl)]dt

+

with constant coefficients and constant retardations ~ ~ 2 with 0 , the initial condition ( 2 ) .

223

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

I.

Definition of Mean-Square Stability

(Stability relative to the second moment). The trivial solution of Eq. (8) is called mean-square stable if for each E > O , there exists a 6 ( ~ ) > 0such that from the inequality l#(t)1<6(e) on the initial 2 set, it follows that Mx (t)0, - where 4 $(t) is any continuous initial function. 11. Definition of Asymptotic Mean-Square Stability. Mean-square stability of the trivial solution of Eq. ( 8 ) - is called asymptotic mean-square stability if lim MxL (t) = 0, for any continuous ini4 t+m tial function O(t), satisfying the condition on the initial set for a sufficiently small I@(t) IS1>O. 111. Definition of Stability in Probability. The trivial solution of Eq. (8) is called stable in probability if for any c1>0, c2>0, there is a 6 > 0 such that any solution x (t) of Eq. (8) satisfies the inequality 4

if 1 4 (t)I < 6 on the initial set. For the stability of the solutions of stochastic differential equations with a retarded argument, it is possible to develop a generalization of Lyapunov's second method in terms of LyapunovKrasovskii functionals (see § 6 , Ch. 111) [170.1]. For concrete equations, these methods allow us to obtain stability conditions in terms of the coefficients. Thus, for the linear equation (8) we have the Theorem.

Let there be satisfied the conditions

224

V . STOCHASTIC DIFFERENTIAL EQUATIONS

Then the trivial solution of Eq. (8) is mean-square asymptotically stable. Proof.

We consider the functional

a32

lo

x2(t+s)ds

-

We calculate the stochastic differential of the By formula ( 5 ) of I t o and Eq. (8)I functional ( 9 ) . we obtain* 0 dV[x(s) ,tl = 2[x(t)+a2 x(t+s)ds] [x(t) (al+a2) dt

1

-‘I1

*For the calculation of the stochastic differential - change of variable of by formula (5), we make the integration t+s=S, t+sl = sl.

225

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

Keeping i n mind t h a t under t h e c o n d i t i o n s of t h e theorem al+a2<0, it i s e a s i l y shown t h a t 2a2 (al+a2) x ( t )

io

< - l a 2 1 (al+a2) x(t+s)ds -

x

Hence, from ( 1 0 ) f o l l o w s t h e estimate

1

0

dV[x ( s ) ,t l 1. 2 [x ( t )+a2

-.T

x (t + s )d s ] a3x ( t - T 2 ) dg ( t ) 1

2

+ qx ( t l d t .

(11)

For t h i s , j u s t as t h e d i f f e r e n t i a l r e l a t i o n (1) i s understood i n t h e s e n s e o f t h e c o r r e s p o n d i n g i n t e g r a l e q u a t i o n ( 3 ), so t h e i n e q u a l i t y (11) i s unders t o o d i n t h e s e n s e t h a t f o r any t > t > O w i t h proba2- 1b i l i t y 1, t h e i n t e g r a l on t h e l e f t s i d e o f (11) i s bounded from tl t o t 2 by t h e i n t e g r a l on t h e r i g h t s i d e of (11). I n t e g r a t e b o t h s i d e s of t h e i n e q u a l i t y (11) from 0 t o t > O ( s t i l l , w e r e c a l l t h a t t h e i n t e g r a l t e r m s c o n t a i n i n g d c ( t ) are u n d e r s t o o d i n t h e s e n s e of I t o ) and t a k e t h e mathematical e x p e c t a t i o n of both sides o f t h e r e s u l t i n g i n e q u a l i t y . Then, keepi n g p r o p e r t y ( 4 ) of t h e I t o i n t e g r a l i n mind, w e w i 11 have w [ X ( S ) ,t]-MV[X(S)

to] 2 q

I t h 2 ( T ) d T .

(12)

0

I n a d d i t i o n , by ( 9 ) t h e r e i s found a c o n s t a n t C>O, such t h a t 2 ~ [ X ( S ) , O l 5 C I 141 I I T

226

V. STOCHASTIC DIFFERENTIAL EQUATIONS

where

Hence, by (9) and (12), under the condition of the theorem, there occurs the inequalities r

m

and

On the other hand,

-1

1

0

la2[

x2 (t+s)ds.

-Tl From the inequalities (13),(14), and (15), it follows that for all t20

and mean-square stability of the trivial solution of Eq. ( 8 ) is demonstrated. For any continuous functions 4(t) on the initial set for which I I g l I + < rS1 (rS1>0 is a fixed constant), by (16) uniformly in tL-max{+1,T2)t we have

227

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

2

Mx (t) 5 K061

2

, KO

= maxIK,lI.

(17)

We shall now assume the Ito formula (5) for V[x(s),t]=x 2 (t). Analogously to (11) we obtain

For any tlft2~[0,-),integrating both sides of the inequality (18) from tl to t2 and taking the mathematical expectation o f both parts of the resulting inequality; by ( 4 ) and (1 ) we will have +21a21+a32~~~6~ 2 Itl-t21. 2 2 (t2)-Mx itl) I (21al

IMX

i

Hence, by (13) it follows

that lim Mx 2 (t)=O,

t+m i.e. the trivial solution of Eq. (8) is mean-square asymptotically stable. The mentioned proof of the theorem of meansquare asymptotic stability allows us to observe deep analogies between the classical second method of Lyapunov and its generalization to stochastic equations. The role of the derivative along a trajectory of the deterministic differential equation here is played by the Ito stochastic differential. Remark I. With the aid of the functional (91, it is possible to show that under the conditions of the theorem (p. 6 ) not only does there occur meansquare stability but also stability in probability (for the idea of the proof, see [SDE], p. 329). Remark 11.

We consider the equation

dx(t) = alx(t)dt

+

a3x(t)d5(t)

with constant coefficients, al>O. We assume 2 V[x(s) ,t]=x (t). Then by Ito's formula

228

(19)

V. STOCHASTIC D I F F E R E N T I A L EQUATIONS

dV[x(s) ,tl = (2al+a32)x2 (t)dt+2a3x2 (t)dC (t). 2 (t). It is not difficult to show that the function r(t) is differentiable and satisfies the ordinary differential equation 2 ?(t) = (2al+a3 )r(t) ,

W e set r(t) = Mx

the solution of which is unstable. However, from the proven theorem (p. 6 ) it follows that if in Eq. (19) a term with a retardation is added, then it may be made mean-square asymptotically stable. The latter means that the aftereffect in real systems may exhibit a stabilizing influence: a system goes from being unstable to asymptotically stable. 3.

Stationary Solutions of Equations with a Delay

With the question of stability of solutions considered in the previous paragraph, closely connected is the question of the existence of stationary solutions of stochastic equations of the form (1) or equations, containing on the right side, a perturbation in the form of an arbitrary stationary process. In this paragraph will be formulated some conditions for the existence of stationary solutions for elementary equations of the form A(t) = f(X(t),X(t-T))

c(t)r

(20)

where the function f(x,y) satisfies a Lipschitz condition in both arguments, and c(t) is stationary in the restricted sense of stochastic processes with continuous realizations. We recall that a stochastic process is called stationary if it is a finite dimensional distribution which is invariant relative to translation, i.e. for any n and any numbers tn,a we have the relation tl,

...,

229

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

...,

where A1, An are any intervals on the line. In this connection, a stationary solution of Eq. (20) is called stationary in the restricted sense of stochastic processes if it satisfies this equation with probability 1. We formulate, at first without proof, a general theorem about the existence of stationary solutions of Eq. (20). Theorem I. (V.B. Eolmanovskii r44.21). Let there exist a solution x(t) of Eq. ( 2 0 ) such that it satisfies the conditions:

uniformly in t>O; 2) we may find positive constants E~ and E~ such that €or some C>O, for any points s1 and s2 from the interval [ - h , m ) we have the inequality

-

M l x ( s l ) -x (s2)I

5 C I s1-s2 I

1+E2

(22)

Then Eq. (1) has a stationary solution. It turns out that the existence of a stationary solution of Eq. (20) depends on the character of the stability of the solutions of the deterministic equation with retardation *(t) = f (x(t),x(t-7)), t>O,

(23)

in which, without loss of generality, it is possible to assume that f (0,O)=O (if f (0,O) # 0, then this constant may be included in the process c(t)). A precise statement is

2 30

V . STOCHASTIC DIFFERENTIAL EQUATIONS

Theorem 11. Let the trivial solution of Eq. (23) be uniformly exponentially asymptotically stable. If, in addition, McL(t)<-, then there exists a stationary solution of Eq. (20). Proof. We denote by x(t) the solution of Eq. (20), defined by zero initial conditions and prove 2 that Mx (t) is bounded. From the first condition of Theorem 11, it follows that on the space of continuous functions on the interval - ~ l s i O there exists a functional V[+(s)], satisfying the estimates

lim+ sup A t+O

(26)

At

Here all Ci>O. Let there be given an arbitrary number h>O. We introduce into consideration the function z(t7 with the aid of the relations

2 (t) = f ( Z (t),Z (t-T)) , t>h,

,

z(t) = x(t)

(27) t2h.

Then by (24)-(27) lim+ sup At+O

Av[x(t+s)l < lim 1 [V[z(t+A+s)] At - a+()+*

-

-v[x(t+s)]]+ lim A+O

1 [V[x(t+A+s)l

-V[z(t+A+s)ll < -C5v[x(t+s)l+C31~(t)I =

c4.

I

It is easy to show that from the c2 preceding inequality it follows that

where C5

23 1

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

--

Consequently, by the second condition of Theorem

We now show that the process x(t) satisfies conditions (21) and (22) of Theorem I. By Chebyshev's inequality, for any t>Q we have t

On the other hand, for any t>O and arbitrary ALO, from ( 2 8 ) and the second codition of Theorem I1 it follows that

232

V . STOCHASTIC D I F F E R E N T I A L EQUATIONS

t where r is the Lipschitz constant of the function 2 2 f(x,y) I the constant C=2(r K+Mr; (0)). From (29) and (30),by virtue of Theorem I there follows the assertion of Theorem 11. Some properties of stationary solutions of stochastic differential equations with a delay are studied in the works [165.11 , [44.61. Other directions of investigations in the theory of stochastic differential equations with a retarded argument are pursued in the works [39.1], [61.1],[46.1], dealing with the still absolutely miniscule amount of study of questions about the properties of solutions of equations with a stochastic retardation.

233