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

Abstract

Based on the martingale theory and large deviation techniques, we investigate the pth moment exponential stability criterion of the exact and numerical solutions to hybrid stochastic differential equations (SDEs) under the local Lipschitz condition. This new stability criterion shows that Markovian switching can serve as a stochastic stabilizing factor by its logarithmic moment-generating function. We also investigate the pth moment exponential stability of Euler-Maruyama (EM), backward EM (BEM) and split-step backward EM (SSBEM) approximations for hybrid SDEs and show that, under the additional linear growth condition, the EM method can share the mean-square exponential stability of the exact solution for sufficiently small step size. However, the BEM method can work without the linear growth condition. We further investigate the SSBEM method under a coupled condition.

The asymptotic stability of hybrid stochastic systems with pantograph delay and non-Gaussian Lévy noise

“…In many engineering and control problems, many systems are in operation for very long time, it is very important to determine whether these systems are stable. There is an intensive literature on the stability of stochastic hybrid system and we mention, for example, Mao et al [21,22,23,24,27,28], Xi and Yin [25,32], You et al [30], Zhu and Cao [35,36,37], Zong et al [40]. It is worth noting that most existing works of research on the stability of stochastic hybrid system require that their coefficients are either linear or nonlinear but bounded by linear functions, which are somewhat restrictive for non-linear stochastic systems, such as stochastic Lotka-Volterra equation, stochastic interest rate models.…”

Euler-Maruyama Method for Regime Switching Stochastic Differential Equations with Hölder Coefficients

“…Numerical schemes for regime switching SDEs therefore have become an active area since the pioneer work by Yuan and Mao [31] with numerous results on various aspects [11,12,15,16,19,22,23,24,25,27,28,32,33]. See, for instance, [31] for Euler-Maruyama method, [24] for weak Euler-Maruyama method, [22] for tamed-Euler method, [23] for Milstein-type algorithm, [11,32] for stability of numerical approximations, [33] for stabilization of numerical solutions, [15] for approximation of invariant measures, [25,27] for numerical scheme for state-dependent switching systems, [28] for scheme for hybrid systems with jumps, [16] for approximation of delayed hybrid systems (see also [12]). However, most of the aforementioned works (except [19], to the best of our knowledge, which focuses on somewhat specific models) require the global or local Lipschitz conditions for the drift and diffusion coefficients despite of a vital fact that many models in reality violate these conditions.…”

The Numerical Invariant Measure of Stochastic Differential Equations With Markovian Switching

“…Motivated by the papers above, this paper focuses on using the backward Euler-Maruyama (BEM) method to approximate the invariant measure of nonlinear SDEs with Markovian switching that the drift coefficients need not to satisfy the global Lipschitz condition. The BEM scheme, which is implicit in the drift term, has been implemented for SDEs with Markovian switching to investigate the strong convergence and the approximation of the almost sure stability as well as the moment stability (see, e.g., [15,34,35] and the references therein). The main aim of this paper is to study the existence and uniqueness of the numerical invariant measure of the BEM method and the convergence in the Wasserstein metric to the invariant measure of the corresponding exact solution colorblue as well as the convergence rate.…”

“…However, solving the SDEs with Markovian switching is still a challenging task that requires using numerical methods or approximation techniques, see, e.g., the monographs [10,13,14,32]. Some long-time behaviors of the SDEs with Markovian switching, for instance, the almost sure stability and the moment stability, have been preserved by the numerical solutions, see, e.g., [7,13,15,23,32,35] and the references therein. For deterministic systems, the stability of equilibrium point is among of the interesting topics.…”