Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations

Positive results are derived concerning the long time dynamics of fixed stepsize numerical simulations of stochastic differential equation systems with Markovian switching. Euler–Maruyama and implicit theta-method discretizations are shown to capture exponential mean-square stability for all sufficiently small timesteps under appropriate conditions. Moreover, the decay rate, as measured by the second moment Lyapunov exponent, can be reproduced arbitrarily accurately. New finite-time convergence results are derived as an intermediate step in this analysis. We also show, however, that the mean-square A-stability of the theta method does not carry through to this switching scenario. The proof techniques are quite general and hence have the potential to be applied to other numerical methods.