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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

60H99Stochastic analysis
39A11Stability of difference equations (MSC2000)
Full Text: DOI
[1] Bellman, R.; Soong, T. T.; Vasudevan, R.: On the moment behavior of a class of stochastic difference equations. J. math. Anal. appl. 40, 286-299 (1972) · Zbl 0244.60051
[2] Cesari, L.: Asymptotic behavior and stability problems in ordinary differential equations. (1973) · Zbl 0265.49002
[3] Friedman, A.: 4th ed. Stochastic differential equations and applications. Stochastic differential equations and applications 1 (1975) · Zbl 0323.60056
[4] Grace, S. R.; Lalli, B. S.; Yeh, C. C.: Comparison theorems for difference inequalities. J. math. Anal. appl. 113, 468-472 (1986) · Zbl 0595.39014
[5] Ladde, G. S.; Sambandham, M.: Random difference inequalities. Trends in the theory and practice of non-linear analysis, 231-240 (1985)
[6] Lakshmikantham, V.; Leela, S.: 4th ed. Differential and integral inequalities. Differential and integral inequalities 1 (1969)
[7] Lakshmikantham, V.; Trigiante, D.: Theory of difference equations. (1988) · Zbl 0683.39001
[8] Ma, F.; Caughey, T. K.: On the stability of stochastic difference systems. Internat J. Non-linear mech. 16, 139-153 (1981) · Zbl 0502.39002
[9] Ma, F.; Caughey, T. K.: On the stability of linear and nonlinear stochastic transformations. Internat. J. Control 34, 501-511 (1981) · Zbl 0473.93073
[10] Ma, F.; Caughey, T. K.: Mean stability of stochastic difference systems. Internat. J. Non-linear mech. 17, 69-84 (1982) · Zbl 0518.93060
[11] Ma, F.: Stability theory of stochastic difference systems. Probabilistic analysis and related topics 3, 127-160 (1983)
[12] Ortega, J. M.: Stability of difference equations and convergence of iterative processes. SIAM J. Numer. anal. 10, 268-282 (1973) · Zbl 0253.65054
[13] Pachpatte, B. B.: Finite difference inequalities and an extension of Liapunov method. Michigan math. J. 18, 385-391 (1971) · Zbl 0237.39001
[14] Redhefter, R.; Walter, W.: A comparison theorem for differential inequalities. J. differential equations 44, 111-117 (1982)
[15] Sugiyama, S.: Difference inequalities and their applications to stability problems. Lecture notes in mathematics 243, 1-15 (1971)
[16] Titchmarsh, E. C.: The theory of functions. (1939) · Zbl 0022.14602