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

Abstract

Positive results are derived concerning the long time dynamics of fixed step size numerical simulations of stochastic differential equation systems with Markovian switching. Euler-Maruyama and implicit theta-method discretisations are shown to capture exponential mean-square stability for all sufficiently small time-steps 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.

The moment exponential stability criterion of nonlinear hybrid stochastic differential equations and its discrete approximations

“…Theorem 5.5 shows that the EM approximation (4.2) can reproduce the mean-square exponential stability of the exact solution without the global Lipschitz condition. This improves the existing results about the numerical stability of nonlinear hybrid systems in the sense of moments (see [7,10,18]). …”

Almost sure and moment exponential stability of Euler–Maruyama discretizations for hybrid stochastic differential equations

“…The converse question is: if the numerical method is exponentially stable in moment or almost surely for small , can we infer that the underlying hybrid SDE is exponentially stable in moment or almost surely? It is only recently that Higham et al [8] have positively answered this question for mean-square exponential stability. It is not very difficult to develop their techniques to cope with the pth moment exponential stability for p 1 while it would be a challenge for p ∈ (0, 1) although it is highly possible given the techniques developed in [19].…”

Nonuniform mean-square exponential dichotomies and mean-square exponential stability

“…Recently, Zhu and Chu [51] presented the numerical methods for a mean-square exponential dichotomy (MS-ED) of a linear SDE and showed that the MS-ED is equivalent to the numerical results for sufficient small step sizes under natural conditions. We also refer to [18,23,24,52] for more related results and techniques about this topic.…”