Instability and stability of solutions of systems of nonlinear stochastic difference equations with diagonal noise

Abstract

We prove results about almost sure instability and stability of the equilibrium of asystem of nonlinear stochastic difference equation with a small parameter h. Thestructure of the system is motivated by the Euler–Maruyama discretization of an Itoˆstochastic differential equation where h is a discretization parameter. Our analysisrelies upon a variant of the discrete Itoˆ formula, which is derived for this system.