Stability Analysis of Numerical Schemes for Stochastic Systems

Abstract

This paper presents a comprehensive analysis of stability properties for numerical schemes applied to stochastic differential equations (SDEs). We examine the mean-square stability, asymptotic stability, and numerical stability of commonly used methods including the Euler-Maruyama, Milstein, and implicit schemes. Through rigorous mathematical analysis and extensive numerical experiments, we establish stability criteria and convergence properties for these methods. Our investigation reveals that while explicit methods offer computational efficiency, implicit methods provide superior stability characteristics, particularly for stiff stochastic systems. We derive explicit conditions for step size selection that ensure stable numerical integration and validate our theoretical findings through Python implementations. The results demonstrate that proper understanding of stability properties is crucial for reliable numerical simulation of stochastic systems arising in finance, physics, biology, and engineering applications.