Mean stability of stochastic difference systems

Inf. 1. Non.Lincor Printed in Char

Mrchsnicr. Brilain.

Vol.

17. No.

2. pp. 69-34.

002~7462/82/020069-16~3.00/0 Pergnmon Press Ltd.

1982.

MEAN STABILITY OF STOCHASTIC DIFFERENCE SYSTEMS FAI MA and T. K. CAUGHEY* California Institute of Technology, Pasadena, CA 91125, U.S.A. (Received

6 March

1981)

Abstract-In this paper the mean stability of linear and non-linear stochastic difference systems is considered. For linear systems the relationship between mean stability and other stability definitions is explored. For the non-linear system explicit criteria for mean stability are derived when the non-linear term satisfies a certain realistic condition. I. INTRODUCTION

An accurate mathematical model of a dynamic system in electrical, mechanical, or control engineering often requires the consideration of stochastic elements. Physical systems with random parameters are often modeled by stochastic differential equations, and an extensive study of such equations was made in the sixties. However, the advent of many modern-day sampled data control systems has necessitated a study of stochastic difference systems, for they invariably include some stochastic elements that can only change at discrete instants of time. Examples of sampled data systems are digital computers, pulsed radar units, and coding units in most communication systems. One of the most important qualitative properties of stochastic difference systems is the stability of such systems. In [l], we considered, among other things, the moment stability of linear stochastic difference equations. The purpose of this paper is to study the mean stability of stochastic difference systems. We shall establish conditions for mean stability of linear and non-linear systems and extend the results in [l]. We shall also attempt to initiate a systematic study of the stability of stochastic difference systems. 2. MATHEMATICAL

PRELIMINARIES

A stochastic difference system is one in which one or more variables can change stochastically at discrete instants of time. A linear stochastic difference system is described by the equation

(1)

xk+i = &(w)xt

which is statistically independent of the A,(w). In [I] a theory of stability of the matrix. By this we mean that the elements of An(w) are stochastic variables, which can be continuous or discrete, with w E Q where fi is the sample description space of the stochastic variables. We assume that A,(w) are independent stochastic matrices, and have a distribution which may depend on the state k. For generality we assume that the initial state x,, is excited by a certain noise process,

P(w),

w E

a,

(2)

which is statistically independent of the A,(w). In [I] a theory of stability of the system (1) was developed. Explicit stability criteria were also derived in many cases. Since then there have been questions on whether the definitions that we adopted constitute a consistent theory. Indeed, one of the difficulties involved in the stability analysis of stochastic differential equations is the proper choice of stability *Address for correspondence: Professor T. K. Caughey. Thomas Laboratory of Technology. Pasadena. CA 91125. U.S.A.

NLM

Vol.

17. No. 2--A

(1044%).California lnsGWe

F. MA and T. K. CAUGHEY

70

definitions. There are at the present time a large number of different stability definitions, even for the same mode of stochastic convergence [2]. In general, different sets of definitions lead to different relationships among the various stability properties of a stochastic differential equation. We shall show that the definitions that we adopted do constitute a logical development of stability theory by demonstrating that they lead to a class of well structured properties. Let us first review the stability definitions that we adopted. Given the stochastic difference system XP+I =

Ak(@)Xk+ fbk, k)

(3)

where xk E R” for all k, and Ak(w) are stochastic matrices as described in (1). The non-linear term f(xk, k) satisfies f(0, k) = 0 for all k. For simplicity we shall only consider the stability of (3) about the equilibrium solution x = 0. The following stability definitions are discussed in the present paper. Definition 1. The equilibrium solution of the stochastic difference system is said to be stable in pth moments if for every E > 0, there is a S > 0 such that

IEM’WXPW . . . x$(k)]] for all k and for any set of non-negative pn = p, where

The system (3) is asymptotically

< E whenever

llx,J < 6

integers pl, p2,. . . , p. such that p, + p2 + - - - +

stable in pth moments if moreover

vz E[xfl(k)xf*(k)

. . . x?(k)] = 0.

Definition 2. The equilibrium solution of (3) is stable in probability there exists 6 > 0 such that /xJ < 6 implies

if given e, E’ > 0,

for all k. The equilibrium solution of (3) is asymptotically stable in probability stable in probability and if there exists S’> 0 such that ]lxJ < 6’ implies

vz &llxkll > E) =

if it is

0.

Definition 3. The stochastic system (3) is pth mean stable for p > 0 if for every E > 0 there is a S > 0 such that E(llx#‘) < E whenever

llxoll> S

for all k. The system (3) is pth mean asymptotically fi

E(jXkIIP)

=

0.

stable if moreover

Mean

stability

of stochastic

difference

systems

71

It is clear from the definitions that pth moment asymptotic stability expresses a convergence to zero property of the state vector only if p is an even integer. Notice that while we investigate the pth mean stability of a system for any positive real p, we generally consider moment stability of integral orders only. The following two lemmas yield some qualitative information on the relationship between mean and moment stability. Lemma 1. The pth moment (asymptotic) moment (asymptotic) Proof. For every p

mean (asymptotic) stability is for any p at least as strong as pth stability. For even integers p, mean (asymptotic) stability and stability are equivalent. > 0 and any norm

. . . xf)(k)] =sE[xpqk)xp(k).

lmiw)xPw

. . x$-(k)/]

(4)

+ pn = p, proving the first part of Lemma 1. Let p = 2r be an even for p,+pz+.... integer and 11~11~ be the Euclidean norm of the vector XI X2 x=

*

(5)

i:i X”

defined by

J/XII* =
(6)

Then

E = E{b:W + x:(k)+ . . . + x2,(k)]‘} = ,,,,;+

,“=,

r,,r2,r! . . . ..r.. ,EWWX:‘W.

. . d'W1

(7)

by using the multinominal formula. The second part of Lemma 1 is thus established for Euclidean norm. But any two norms in a finite dimensional linear space are equivalent [3], hence (6) holds for any norm and the proof is complete. Lemma 2. The pth absolute moment (asymptotic) stability of the absolute moments E11xP1(k)x2pG) . . . x?(k)Jl, pI + p2 + . . . + pn = p is equivalent to pth mean (asymptotic) Proof. Let

stability for any integer p.

(8) be the absolute sum norm of the vector (5). Then . . . x?(k)ll. p,lP2r ! p EII-G’l(~)~P~(~) E(llxdl~) = E [ (2I’il)‘] =p,+p2+:+p ”=p . . ...“. I

(9)

Lemma 2 is thus true for the absolute sum norm. The general case follows from the fact that any two norms in a finite dimensional linear space are equivalent.

72

F. MA

and

T. K. CAUGHEY

The following lemma relates mean stability of different orders. Lemma 3. If a stochastic difference system is p,th mean (asymptotically) then it is prth mean (asymptotically) stable for any p2 satisfying 0 < p2 C pI. Proof. We need only consider 0 < p2 < pI. Let t= k= P2

Applying Holder’s inequality

stable,

1.

[3]

~(~~x#z) < E(llxlillq’)(‘l’))(“‘)E(1(‘fr-‘))‘-(llr) = E(jX#h)(Ph).

(10)

Hence p,th mean (asymptotic) stability implies p2th mean (asymptotic) stability. The results that we have established should, with an appropriate choice of definitions and modifications in the proofs, apply in the continuous case to stochastic differential systems. 3. LINEAR

STOCHASTIC

SYSTEMS

The moment stability and the stability in probability of linear stochastic difference systems were considered in great detail in [l]. In this section we shall consider mean stability of the linear stochastic difference system (1). The homogeneous system (1) may be regarded as the linearized version of the non-linear stochastic difference system (3). A solution of (1) satisfying Xn+, = A,X, (11) x, = I is called the fundamental satisfies

solution.

i

It is clear that the fundamental

X, = A._,A._2..

solution

of (1)

. A2A,Ao

(12) X,,,=A,_,...A,X,,m>n

i .

From (12), we immediately have the following result. Lemma 4. The linear stochastic difference system (1) is pth mean stable if and only if there exists a constant C such that E(IIA,A,-, . . . &%Aoll”) for all n. It is pth mean asymptotically

(13)

d C

stable if and only if

lim E((IA,A,_, . . . A2A,A,#‘) = 0. n-rThe system (1) is said to be strongly pth mean stable [l, 41 if there exists a constant C such that E(ilA,,,_, . . . A$)

s C

for all m, n such that m > n. It is strongly pth mean asymptotically

(14) stable if

lim E(j(A,_, . . . A./(P) = 0 Ina for all m, n such that

m > n. Observe

that since AL are independent

(15) stochastic

73

Mean stability of stochastic difference systems

matrices, it follows from (12) that

E(IIX#) c E(lI&-I - . . A#‘)E(llXd”),m ’ n.

(16)

As we shall see, strong stability is a very useful concept in non-linear analysis. The following theorem shows that mean stability is surprisingly stronger than stability in probability. Theorem 1. If the stochastic difference system (1) is pth mean (asymptotically) stable for any p > 0, then it is (asymptotically) stable in probability. Proof. We first establish the following inequality.

P(IlX”II 3 c)+ for all p, E > 0. Let f(x) be the probability

E

(17)

density of j/x,,IIp.Using the fact that Ilx.Ijb 0,

E(b,#) = Lxf(x)

dx

m

2

I

LI

x f(x) dx

for each a > 0. Choose a = EP.

It follows that

P(IIX”ll34 = P(IlX.IIp 3 EP) s f NX”llP) and (17) is established.

Suppose E, E’ > 0 are given. Choose r > 0 such that 2p 7E r-c E.

0

(1%

Since (1) is pth mean stable, there exists S > 0 such that E(llx#‘) < t whenever

llx,J < 6.

(20)

Hence

II%11 <6 implies

PCllXkll > E’)s Pcllxkll’; 6’) G

0f ’ E


for all k. Thus the stochastic

system

(1) is stable in probability.

From (17), it is

14

F. MA and T. K. CAUGHEY

obvious that lim P(]lxJ > E) = 0 “if lim E(I]x#) II-

= 0.

Hence asymptotic pth mean stability for any p > 0 implies asymptotic stability in probability. Notice that the above proof does not require linearity. Hence Theorem 1 also applies to the non-linear stochastic system (3). By our previous analysis, we have the following relationship between moment stability and stability in probability. Corollary to Theorem 1. If the stochastic difference system (1) is (asymptotically) stable in even order moments, then it is (asymptotically) stable in probability. In particular, stability in second moments implies stability in probability. A discussion of the stability of some specific linear systems is contained in [l]. 4. STOCHASTIC

STABILITY

OF NON-LINEAR

In this section we shall consider the stability non-linear stochastic difference system X ,,+I

=

AA + fb., n) J%or>

SYSTEMS

of the non-linear

< O”

I

system

(3). The

(21)

can, with an appropriate choice of the non-linear correction term f(x, n), serve as a realistic model for many real life sampled data control systems. We assume that the linearized system X ,,+I=

(22)

Ax,

corresponding to (21) is stable, and we would like to know how the non-linear term f(x, n) can affect the stability of the system. We have considered the above problem in [l] for the case of first mean stability. The following theorem is a direct generalization of a previous result. Theorem 2. Given the stochastic difference system (21) and any 0 <$ < m. If (i) the corresponding linearized system (22) is strongly pth mean asymptotically stable and that the solutions converge in pth mean to zero sufficiently fast. (ii) f(x, n) satisfies the non-linearity condition: there exists an integer N and a sufficiently small constant L such that

E(IJf(x, n)l/q < m, 0 c n s N

1

(23)

then the system (21) is pth mean asymptotically stable. Let us first explain the conditions of Theorem 2. In condition (i), we require that (15) be satisfied. It remains to specify a rate of convergence to zero. We know that when there is a geometric decay

E&L-r . . . A,(I) =s C6”-“, then Theorem

m> n

(24)

2 holds for p = 1 [l]. Now by Lemma 3 we should expect to have more

Mean stability of stochastic

75

difference systems

restrictive assumptions when p increases. This is reflected in the rate of convergence to zero of the solutions of (22). We now assume, for example, that the solutions of the strongly pth mean asymptolically stable system (22) decay at least as fast as specified in the following E&4,-,

. . , A,/(P)<

m>n

cd~)6"'-~,

1

(25)

where 0 < S < 1. Since ffij need only be defined for i > j 2 0, one easy way to check (25) in practical calculations is to demonstrate that aij can be dominated in the following way

where pi = 0(1/P) and C is a constant. The rates as specified in (25), (26) do not represent a measure of the optimal rate, and for this type of analysis the exact minimum rate is not significant [4]. In most real life sampled data systems, the inherent non-linearity is such that within the normal operating range, the non-linear term f(x, n) generally decreases in magnitude as x decreases in magnitude. This fact is reflected in condition (ii), which is a typical cone condition. It means that the pth mean of the non-linear term must lie within a half cone whose apex angle depends on L. We shall need the following auxiliary result. Lemma 5. Let e(i) and @l(i) be two non-negative sequences and p(i) a positive sequence satisfying

e(n)6

p(n)[ C+ 2 +(i) e(i)]

for some C > 0 and for all non-negative

integers n. Then with the customary

that l? 0(i) = 1 for m > n, the following inequalities i-m

(27) notation

hold

(4 i-0

for all n if p(i) are bounded above by 1

09

e(n)s

C[i-0ll p(i)]{ a[l + 4Ni,l)

(29)

for all n if p(i) are bounded below by 1. Proof. If 0 < p(i) S 1, then from (27)

(30) Hence

1+

G,(n)ff$ . s 1+4(n)

C+YZi $(i)g

F. MA and T. K. CAUGHEY

76

giving

c+

2

i-0

e(i) + 44n)l p(i)<11

+(i)

[c+Z i-0 $(i)~].

(31)

By iteration

c+zo *(i)p

Rrl+w)l[c+~(0,fg]. i=l

But from (27)

(32) yielding

If p(i) P 1, then from (27)

e(n) P(n) [C+ g e(i) B(i)]

c’

giving

C+

so W) Nil 1

4%) e(n)

cl+

p(n) [C + z: $(i) O(i)]

0)

[C + 2 Hi) e(i)]

Hence

C+

2W

i-0

e(i) d

[l + W)lp(n)

[C+ 2 i+b(i) e(i)]. i-0

By iteration

C+$Hi) WF i-0

[

fr(I+ WI]

i-l

[C+

+(o)e(o)l [fi

p(i)].

But from (27) e(0) c CP(~) giving C + 440) e(o) c

C[l + P(o)w)]

s c~(o)[l +

w)i.

Thus

ems p(n) [C+ The proof is therefore

complete.

2 WNW] s #I_dO][ )Yj [l + WI. i-0

Mean stability of stochastic difference systems

II

We consider Lemma 5 to be a generalization of the discrete form of the classical Bellman-Gronwall lemma [5]. Proof of theorem 2. First assume that N = 0 in (23). The general solution of (21) can be written as & = MO

for all n z 0, and Bi = fi

i=i

+ “gi Bi+lftXi,

(33)

3

A, Hence

1IX.P s [ lI%oll+ 2 IIBi+lf(xi* iN]p

C(n+ l)‘( OCiS#l-I max [ll&~lI,IIBi+, fh i>lll’ Q (n + 1)’

1ll~oll”lbol1”

for all n 3 0. Since P(w) is independent

of

= E(

+ n$i IIBi+dlpllfCxi~

WI

(34)

Ao,A,,A*,...,A,-,.

IiIYJ Ail>

E
= WW’>%#‘).

(35)

EWi+~II’Ilf<~i~ i>ll”l= Ell’l

(36)

Similarly

Ai- P(o), on which f(X, i) depends, are independent of Ai+,, Ao,A1,A~,..., Ai+ ***9A,+ Using (23), (25), (26), (35) and (36)

because

E(llxnlP)c (n +

1y(an.o~“E(ll~llP) + “2 La,i+lS”-i-lEQIxiIl’))

s C(n +

lyB[S”E(ll~#)+ “$1 LG"-im'E((IXillp].

(37)

Now p.sM$ Without loss of generality,

n3l.

we can choose the constant MC3

M such that

1.

It follows that E@d”)

s (I+

;r[

MCG"E(~~X#)+~MCL~"-'-'E(]IX~~~)] (38) for n 3 1.

78

F. MA and T. K. CAUGHEY

Define a sequence p(i) z 1 by P(i)=

I(

1p 1+7 ,ial

l,i=O.

>

Using (38), we have (39)

Multiplying both sides of (39) by 6-” and applying Lemma 5

EUlxnllP) s MC fi p(i)E(J(xdlp)(S+ MCL)” f LOall n 3 0.

If L is sufficiently small, then S+MCL=y
W)

Thus

~
(41)

l? (1 + (l/j))’ is a divergent infinite product, the product l? _ _ [( 1 + ( l/j))pr] convergel to zero (or diverges to zero. For terminology, cf. [a].) Her&

for all n. Although

proving pth mean stability for n = 0. If N 3 1, then by the assumption

EUlfll’> < O3

(43)

for 0 S n G N, we can shift indices and take xN as the initial vector by using the transformation Yk = xk+N WI Bk = &+N

I

or we may apply the previous analysis directly to n+

Xa+N

=

1-N

This completes the proof.

-I

II

+N

+ng’(,:nI,A,)fh,

9, n 3 0.

(45)

Mean stability of stochastic difference systems

79

Theorem 2 is a direct extension of a corresponding result in [l]. Naturally, we would like to extend the same result in other possible ways. We shall establish that if, instead of a decay as described by (29, the following inequalities are satisfied: E(jA,_,

. . . A$)

s a,,,.(p), m > n 3 0

(46) (47)

where pi and fi are positive sequences x

satisfying

tj<”

j

(48)

then the apex angle of the cone as determined by L in condition (ii) can be arbitrarily enlarged and the vertex of the same cone can be freely translated along the horizontal axis. If E((IA,l)“) =0(1/P) and p > 1, then (46)-(48) obviously hold. The following property of sequences will be useful. Lemma 6. If Ui3 0 for all i, then the product

I7(1 +

(49)

6)

i=l

and the series

converge or diverge together. Proof. See [6]. Theorem 3. Given the stochastic difference system (21) and any 0 < p < Q).If: (i) the corresponding homogeneous system (22) is strongly pth mean asymptotically stable and the solutions converge in pth mean to zero with a rate, for example, exceeding that in (46)-(48); and (ii) f(x, n) satisfies the non-linearity condition that there exist an integer N and arbitrary constants L,, L2 such that E(llf(x, n)IP) s L, + MHllxl1”), n ’ N (51) (jjf(x, n)Ip) < CQ,0 s n s N

I

then the system (21) is pth mean asymptotically stable. Proof. We can assume without loss of generality that N = 0 in (51). It is easily seen that (34) holds in the present case Il%IIp s (n + l)‘[

Il&011P

+ 2 llBi+l(lPllf(Xi* i=O for all n 2 0.

i)lP)

Using (46), (47) and (51)

E(IIXnllP)s tn + lIp[ Let

PntOECllxoll”>

+ 2

i=O

LIPnti+l

+ 2 i=O

L2~~l,+,~(l~~illp)]-

(52)

80

F. Ma and T. K. CAUGHEY

By (48), we can choose a constant K < 1 satisfying 0 < K(n +

I)‘& d 1, n 2 1

K-cc,. It follows from (52) that

Thus (54) for all n 2 0, where t = (v/K). Applying Lemma 5,

Since

the infinite product

is bounded by Lemma 6. But lim p(n) = 0. “-

(56)

Hence

proving pth mean asymptotic stability. It is surprising to find that (46)-(48) are strong enough to allow for a ‘projection vector’ type or a ‘combination cone’ non-linearity condition in f(x, n). The following generalization of Theorem 3 is useful in practical calculations. Theorem 4. The assertion of Theorem 3 holds if condition (ii) is replaced by (ii) f(x, n) satisfies the following non-linearity condition: there exist an integer N, r real numbers 0 < pI, p2, . . . , p, G p, and r + 1 constants L, L,,L2,...,L, such that

E[lIf(x, n)IPld L + 2 L&+(llXl~)v n> N (57) E< m, 0 s n c N.

81

Mean stability of stochastic difference systems Proof.

As before, assume N = 0 in (57). For 0
IlxllP’ s 1+ llxllp (58)

i= 1, 2,. . . , r. Hence

ri:

E[IJf(x9 n)IPlc L + i-l

LiE(JlXll+>

s

(L+2 4) +(2 Li)E(jlXlp) i-1 i-l

(59)

(ii) of Theorem 3. The result follows. Now we have shown that under a certain realistic assumption on the non-linear term f(x, n), the asymptotic stability of (21) can be deduced from the strongly asymptotic stability of the corresponding linearized system (22). Suppose the homogeneous system (22) is only strongly pth mean stable (13), we would like to know the kind of assumption that we can impose on f(x, n) to make the non-linear systems (21) pth mean stable. As in [l], we find that a variable cone condition is sufficient. Theorem 5. Given the stochastic difference system (21) and any 0 < p < Q),If: (i) the corresponding linearized system (22) is strongly pth mean stable; and (ii) f(x, n) satisfies the following non-linearity condition: for every n there exists a non-negative number B(n) such that and this is condition

(60)

Hllf(x, dIPI s mvmlJP) and B(n) can be dominated

by

B(n) c PA

(61) n-l

n-l

in such a way that the sequence

s(n) = np I: pi is bounded

and the series

i-0

X ti i-0

converges, then the system (21) is pth mean stable. Proof. The general solution of system (21) is given by (33) n-l X, = B&+ 2 Bi+lf(Xi, i) i=O

n-l

for all n, with Bi = ll A, Hence j=l

Ilx,lIpd ll&#

n-l

+ np E IIBi+tfh W’

(62)

for all n CO. Using (13), (60) and (61) n-l

W~.ll”) s CHllx~ll”) + Cnp z. BCWSIlx#‘>

s CE+ Cnp 6 Since the monotonic

‘$: PitiE(IIxiIl”)

CEhJ”>+ Cn’nz:Pi‘12tiE*

sequence s(n) = np “$ pi i=O

63)

a2

F. MA and T. K. CAUGHEY

is bounded, there exists a constant

K such that

for all n. Applying Lemma 5, with p(i) = 1 for all i

E(IIXnllP) s cECll%llp) “fl (1 + CKfi) i=O for all n. Since 5 ti
stable. The proof is complete. The stability analysis of the general non-linear implicit stochastic

difference

system

%+I = f(& xlr+1,k) 64) XtER”

I

remains an open problem. We believe that the development in this paper can contribute to the stability analysis of stochastic differential systems with colored noise parameters. The question of how the analysis in this paper can be applied to the continuous case will be taken up in a future paper. 5. ILLUSTRATIONS Example 1. Consider system (1) with sample space n = {w,, w2}. Suppose that PM

1 = P{WJ = -2

(65) (66) (67)

for all k. It can be readily checked that

grows without bound for any 0

E = O(k)

(68)

ECllxl~ll’> = W*).

(69)

and

Example 2. Consider system (21) with R = {w,, 02, wg}. Suppose that lz=p{w,}=az=O lLp{w*}=p30 1~/3{0,}=1--cr-p”0 and (Y,/3 are otherwise

arbitrary.

(70)

Mean

stability

of stochastic

difference

systems

83

Let

(71)

AL(w,) = 6

(72)

1 5 (73) 1 z

1

for all k

(74)

where A, B, C are constants, x=

0

XI . x2

Let (75) be the simple absolute

value norm. Then for the matrix (76)

the natural norm induced by (75) is the maximum absolute column sum

llAll= my (la,kl+Ia& k = 1,~

(77)

It can be readily checked that the linearized system in the present case is strongly second mean asymptotically stable and that (46)-(48) are satisfied. Moreover E[ljf(x, n)Ir] s E[((AI + lB1 11x/1+ICI Il~ll”“>~l = A’ + B%Cjlxll’>+ C’EC~~x~~“‘, (78) + $4 BIE(((xl()+ 21BCl~(llx11”‘6)

84

F. MA and T. K. CAUGHEY

where we have made use of the fact that for a non-negative

real number x

1 2

e’>-x2.

Hence by Theorem 4 the non-linear system in this example is second mean asymptotically stable, as can be checked by direct calculations. It follows that the system is also pth mean asymptotically stable for p c 2. 6. CONCLUSION

In this paper we have considered the mean stability of linear and non-linear stochastic difference systems. For linear systems the relationship between mean stability and other stability definitions has been explored. For the non-linear system explicit criteria for mean stability have been derived when the non-linear term satisfies a certain realistic condition. We have also presented two examples to illustrate the results. Clearly all the results established in this paper can be extended to complex stochastic difference systems at the expense of increased mathematical complexity. REFERENCES 1. F. Ma and T. K. Caughey. On the stability of stochastic difference systems, Ink 1. Non-lineor Mechanics 16. 139-153 (1981). 2. F. Kozin. A survey of stability of stochastic systems, Aufomalica 5, 95-112 (1969). 3. E. Hewitt and K. Stromberg, Real and Abstracr Analysis. Springer, Berlin (1965). 4. L. &sari, Asymplotic Behavior and Skzbiliry Problems in Ordinary Differential Equations, 3rd ed. Springer (197 1). 5. T. K. Caughey. Nonlinear Ordinary Diference Equations. pp. 50-79. JPL Publications (1979). 6. E. C. Titchmarsh. The Theory of Functions 32nd ed. Oxford University Press (1939).

R&urn@: Dans cet article on considke la stabilit6 moyenne de syst&es diff&entiels s:ochastiques li&aires et non lin& aires. Pour les sJstemes lineaires on esplore la relation entre la stabilite moyenne et d'autres definitions de la stabiliti. Pour le syst%nes non liniaire on 6tablit des crit&es explicites de stabiliti moyenne lorsque le terme non liniaire satisfait i une certaine condition &aliste.

Zusamnenfassunq: In dieser Arbeit wird die mittlere StabilitSt linearer und nichtlinearer stochastischer Differenzensystemeuntersucht. FPr'lineare Systeme wird die Beziehung zwischen der mittleren Stabilittt und anderen Stabilitztsdefinitionenuntersucht. Flirnichtlineare Systeme werden explizite Kennzeithen fiirmittlere Stabilita'thergeleitet wenn das nichtlineare Glied eine bestimnte realistische Bedingung erftillt.