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About stability of nonlinear stochastic difference equations. (English) Zbl 0959.60056
Summary: Using the method of Lyapunov functionals construction, it is shown that investigation of stability in probability of nonlinear stochastic difference equation with order of nonlinearity more than one can be reduced to the investigation of asymptotic mean square stability of the linear part of this equation.

MSC:
60H25Random operators and equations
39A10Additive difference equations
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References:
[1] Kolmanovskii, V. B.; Shaikhet, L. E.: Control of systems with aftereffect. Translations of mathematical monographs 157 (1996)
[2] Kolmanovskii, V. B.; Nosov, V. R.: Stability of functional differential equations. (1986) · Zbl 0593.34070
[3] Kolmanovskii, V. B.; Myshkis, A. D.: Applied theory of functional differential equations. (1992) · Zbl 0917.34001
[4] Kolmanovskii, V. B.; Shaikhet, L. E.: New results in stability theory for stochastic functional differential equations (SFDEs) and their applications. Proceedings of dynamic systems and applications 1, 167-171 (1994) · Zbl 0811.34062
[5] Kolmanovskii, V. B.; Shaikhet, L. E.: General method of Lyapunov functionals construction for stability investigations of stochastic difference equations. Dynamical systems and applications 4, 397-439 (1995) · Zbl 0846.93083
[6] Shaikhet, L. E.: Stability in probability of nonlinear stochastic hereditary systems. Dynamic systems and applications 4, No. 2, 199-204 (1995) · Zbl 0831.60075
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[8] Kolmanovskii, V. B.; Shaikhet, L. E.: Matrix Riccati equations and stability of stochastic linear systems with nonincreasing delays. Functional differential equations 4, No. 3/4, 279-293 (1997) · Zbl 1148.93346
[9] Shaikhet, L. E.: Necessary and sufficient conditions of asymptotic mean square stability for stochastic linear difference equations. Appl. math lett. 10, No. 3, 111-115 (1997) · Zbl 0883.39005
[10] Shaikhet, L. E.: Some problems of stability for stochastic difference equations. 15th world congress on scientific computation, modelling, and applied mathematics (IMACS97, Berlin, August, 1997) computational mathematics 1, 257-262 (1997)
[11] V.B. Kolmanovskii and L.E. Shaikhet, Riccati equations and stability of stochastic linear systems with distributed delay, In Advances in Systems, Signals, Control, and Computers, (Edited by V. Bajic), pp. 97-100, IAAMSAD and SA branch of the Academy of Nonlinear Sciences, Durban, South Africa.
[12] Shaikhet, G.; Shaikhet, L.: Stability of stochastic linear difference equations with varying delay. Advances in systems, signals, control and computers, 101-104 (1998) · Zbl 0938.93059