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Stability theorems of stochastic difference equations. (English) Zbl 0709.60068

Let {A n ; n0} be an independent sequence of d×d real-valued matrices and f(n,x) be an d valued mapping defined on × d . The author investigates various definitions of the stability of the solutions of the equation

x n+1 =A n x n +f(n,x n ),x 0 d ·

A typical result is the following: Let us suppose that

𝔼{A m-1 ···A n }Cform>n0,
𝔼{f(n,x)}b n 𝔼{x}with n=0 b n <·

Then for any positive ϵ there exists a positive δ (ϵ) such that for any n one has 𝔼{x n }ϵ provided that 𝔼{x 0 }δ(ϵ)·

An other kind of result is the following: Let us suppose that

𝔼{A m-1 ···A n }Cδ m-n+1 form>n0andδ<1,
𝔼{f(n,x)}b𝔼{x}withbsufficientlysmall·

Then lim n 𝔼{x n }=0·

Analogous results hold for the L p -norm and various conditions on the behavior of the products of the matrices A n and contraction properties of the function f.

Reviewer: J.Lacroix
MSC:
60H99Stochastic analysis
39A11Stability of difference equations (MSC2000)