Almost Sure Exponential Stability in the Numerical Simulation of Stochastic Differential Equations

Abstract

Abstract. This paper is mainly concerned with whether the almost sure exponential stability of stochastic differential equations (SDEs) is shared with that of a numerical method. Under the global Lipschitz condition, we first show that the SDE is pth moment exponentially stable (for p ∈ (0, 1)) if and only if the stochastic theta method is pth moment exponentially stable for a sufficiently small step size. We then show that the pth moment exponential stability of the SDE or the stochastic theta method implies the almost sure exponential stability of the SDE or the stochastic theta method, respectively. Hence, our new theory enables us to study the almost sure exponential stability of the SDEs using the stochastic theta method, instead of the method of the Lyapunov functions. That is, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size Δt. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p ∈ (0, 1), we can then infer that the underlying SDE is almost surely exponentially stable. Our new theory also enables us to show the ability of the stochastic theta method to reproduce the almost sure exponential stability of the SDEs. In particular, we give positive answers to two open problems, (P1) and (P2) listed in section 1.

The partially truncated Euler–Maruyama method and its stability and boundedness

“…However, a nice numerical method for an SDE should also preserve some asymptotic properties of the underlying SDE, for example, stability and boundedness (see, e.g., [4,8,9,15,19,23]). …”

“…Although the stability of numerical methods for SDEs has been studied intensively (see, e.g., [4,8,9,19,23] asymptotic boundedness of numerical methods (see, e.g., [15]). …”

Convergence and stability of the one-leg θ method for stochastic differential equations with piecewise continuous arguments

“…It is also proved that the backward Euler method maintains the almost sure exponential stability under the one-sided Lipschitz condition. Instead of Lyapunov functions, Mao [9] used the discrete approximation to investigate whether the almost sure exponential stability of the SDEs is shared with that of a numerical method or not. There are also several works on the stability of stochastic delay differential equations(SDDEs).…”

Almost sure exponential stability of stochastic differential delay equations

“…But this is not the case for SDDEs as it has been pointed out in [5]. There is an extensive literature on stochastic stability, for example Mao( [9], [10]), Rodkinaa and Schurz [8].…”