About stability of nonlinear stochastic difference equations

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. Difference equations usually appear by investigation of systems with discrete time or by numerical solution of systems with continue time [1]. Lyapunov functionals are used for investigation of hereditary systems in problems of stability and optimal control [2,3]. One method of Lyapunov functionals construction has been proposed and developed for differential and difference equations in [4-12]. This method is used here to construct stability in probability conditions for nonlin-ear stochastic difference equations. 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. For stochastic nonlinear differential equations, a similar result was obtained in [6].