EXPONENTIAL STABILITY OF A KIND OF STOCHASTIC DELAY DIFFERENCE EQUATIONS

We present a Razumilchin-type theorem for stochastic delay difference equation, and use it to investigate the mean square exponential stability of a kind of nonautonomous stochastic difference equation which may also be viewed as an approximation of a nonautonomous stochastic delay integrodifferential equations (SDIDEs), and of a difference equation arises from some of the earliest mathematical models of the macroeconomic “trade cycle” with the environmental noise.


Introduction
The problem of stability of stochastic difference equation has been investigated in a number of papers.We refer the readers to [2,3,[13][14][15][16][17]. Some results on the asymptotic behavior of the moments were obtained in [18].But very few results on the Razumilchin-type theorem for stochastic delay difference equation have been published.In this paper, we present a Razumilchin-type theorem for stochastic delay difference equation, and use it to investigate the mean square exponential stability of a kind of nonautonomous stochastic difference equation.

Main result
Definition 2.1.The stochastic difference equation (1.1) is said to be pth moment exponentially stable if there are positive constants γ and N such that with initial data ξ n , n ∈ I, where ξ = max n∈I |ξ n |.

A numerical approximation to SDIDEs. We consider the stochastic delay difference equation
where h > 0 is a nonrandom parameter.For the functions a(•), b(•), and K(•,•), suppose that satisfy the conditions (H1) and (H2) as the following: From the conditions (H1) and (H2), we know that Equation (3.1) may also be viewed as an approximation of the stochastic delay integrodifferential equation where W(t) is a standard Brownian motion.Here, setting h = τ/m and approximating the differential part of (3.5) with the Euler-Maruyama method and the integral part with composite left-side rectangle rule [12], t n = nh, write X n for an approximate value to X(nh), and use X n−m to approximate the delayed argument X(t n − τ).When n ∈ I = {−m, −m + 1,...,−1,0}, we have X n = ξ(t n ).Moreover, the increments √ h μ n := W(t n+1 ) −W (t n ) are independent N(0,h)-distributed Gaussian random variables, so μ n are independent N(0,1)-distributed Gaussian random variables.We assume X n to be Ᏺ nmeasurable at the mesh-points t n .It is therefore to be hoped for h sufficiently small that solutions of (3.1) have similar asymptotic properties to those of (3.5).A statement of these asymptotic results for stochastic delay differential equations can be found in, for example, [1,[4][5][6][7][8][9][10][11].
Here, we use the above Razumilchin-type Theorem 2.2 to study the moment exponential stability of (3.1).

Models of macroeconomics.
Consider the following nonlinear delay difference equation: where c ∈ [0,1) and ε are constants, m is a positive integer.μ n are independent N(0,1)distributed Gaussian random variables.We assume that x(n) are Ᏺ n -measurable for all n ∈ N, and we have x(n) = ξ n when n ∈ I. f : R → R satisfies f (0) = 0, f (u) = 0 for u = 0, and there exists a constant α such that Such equation arises from some of the earliest mathematical models of the macroeconomic "trade cycle" with the environmental noise.
Theorem 3.2.Assume that the conditions (3.14) and are satisfied.Then there exists positive constants γ such that with initial data ξ n , n ∈ I, Xiaohua Ding 7 Proof.Also define a Lyapunov function V (n,x) = |x| 2 .Similar to the proof of Theorem 3.1, let Approximating the differential part of (3.5) with the Euler-Maruyama method and the integral part with composite left-side rectangle rule, we get the difference equation as follows: Here, h = τ/m and so that the (H2) is satisfied, and if then (H1) also is satisfied.By letting From Theorem 3.1 we know that the inequality (3.6) holds.
From the above analysis and (4.5), we can get the theorem as follows.