Stochastic averaging in discrete time

We investigate stochastic averaging theory for a class of discrete-time nonlinear systems with stochastic perturbation. Firstly, we introduce two average system: one is discrete-time average system, the other is continuous-time average system. Then we establish a general stochastic averaging principle between the continuous-time average system and the original system. With the help of the continuous-time average system, we establish stochastic averaging principle between the discrete-time average system and the original system. Finally, we establish some related stability theorems for a class of discrete-time nonlinear systems with stochastic perturbations. Our stochastic averaging results remove or weaken several significant restrictions present in existing results: (a) boundedness condition of the solution; (b) global Lipschitzness of the nonlinear vector field, and (c) too many limitations on stochastic factors.