Delay-dependent stability analysis of numerical methods for stochastic delay differential equations

This paper is concerned with the numerical solution of stochastic delay differential equations. The focus is on the delay-dependent stability of numerical methods for a linear scalar test equation with real coefficients. By using the so-called root locus technique, the full asymptotic stability region in mean square of stochastic theta methods is obtained, which is characterized by a sufficient and necessary condition in terms of the drift and diffusion coefficients as well as time stepsize and method parameter theta. Then, this condition is compared with the analytical stability condition. It is proved that the Backward Euler method completely preserves the asymptotic mean square stability of the underlying system and the Euler–Maruyama method preserves the instability of the system. Our investigation also shows that not all theta methods with θ≥12 preserve this delay-dependent stability. Some numerical examples are presented to confirm the theoretical results.