Stability Analysis of Numerical Schemes for Stochastic Differential Equations

Stochastic differential equations (SDEs) represent physical phenomena dominated by stochastic processes. As for deterministic ordinary differential equations (ODEs), various numerical schemes are proposed for SDEs. In this paper we study the stability of numerical schemes for scalar SDEs with respect to the mean-square norm, which we call $MS$-stability. We will show some figures of the $MS$-stability domain or regions for some numerical schemes and present numerical results which confirm it. This notion is an extension of absolute stability in numerical methods for ODEs.