On Global Asymptotic Stability of Solutions of Some in-Arithmetic-Mean-Sense Monotone Stochastic Difference Equations in $\rm{IR}^1$

Abstract

Global almost sure asymptotic stability of the trivial solution of some nonlinear stochastic difference equations with in-the-arithmetic-mean-sense monotone drift part and diffusive part driven by independent (but not necessarily identically distributed) random variables is proven under appropriate conditions in $\rm{IR}^1$. This result can be used to verify asymptotic stability of stochastic-numerical methods such as partially drift-implicit trapezoidal methods for nonlinear stochastic differential equations with variable step sizes.