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Stability theorems of stochastic difference equations. (English) Zbl 0709.60068
Let $\{A\sb n$; $n\ge 0\}$ be an independent sequence of $d\times d$ real-valued matrices and f(n,x) be an ${\bbfR}\sp d$ valued mapping defined on ${\bbfN}\times {\bbfR}\sp d$. The author investigates various definitions of the stability of the solutions of the equation $$ x\sb{n+1}=A\sb nx\sb n+f(n,x\sb n),\quad x\sb 0\in {\bbfR}\sp d. $$ A typical result is the following: Let us suppose that $$ {\bbfE}\{\Vert A\sb{m-1}...A\sb n\Vert \}\le C\text{ for } m>n\ge 0, $$ $$ {\bbfE}\{\Vert f(n,x)\Vert \}\le b\sb n{\bbfE}\{\Vert x\Vert \}\text{ with } \sum\sp{\infty}\sb{n=0}b\sb n<\infty. $$ Then for any positive $\epsilon$ there exists a positive $\delta$ ($\epsilon$) such that for any n one has ${\bbfE}\{\Vert x\sb n\Vert \}\le \epsilon$ provided that ${\bbfE}\{\Vert x\sb 0\Vert \}\le \delta (\epsilon).$ An other kind of result is the following: Let us suppose that $$ {\bbfE}\{\Vert A\sb{m-1}...A\sb n\Vert \}\le C\delta\sp{m-n+1}\text{ for } m>n\ge 0\text{ and } \delta <1, $$ $$ {\bbfE}\{\Vert f(n,x)\Vert \}\le b{\bbfE}\{\Vert x\Vert \}\text{ with } b\quad sufficiently\quad small. $$ Then $\lim\sb{n\to \infty}{\bbfE}\{\Vert x\sb n\Vert \}=0.$ Analogous results hold for the $L\sp p$-norm and various conditions on the behavior of the products of the matrices $A\sb n$ and contraction properties of the function f.
Reviewer: J.Lacroix

MSC:
60H99Stochastic analysis
39A11Stability of difference equations (MSC2000)
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Full Text: DOI
References:
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