Chenggui Yuan

Robust stability and controllability of stochastic differential delay equations with Markovian switching

In this paper, we investigate the almost surely asymptotic stability for the nonlinear stochastic differential delay equations with Markovian switching. Some sufficient criteria on the controllability and robust stability are also established for linear stochastic differential delay equations with Markovian switching.

Stabilization and destabilization of hybrid systems of stochastic differential equations

This paper aims to determine whether or not a stochastic feedback control can stabilize or destabilize a given nonlinear hybrid system. New methods are developed and sufficient conditions on the stability and instability for hybrid stochastic differential equations are provided. These results are then used to examine stochastic stabilization and destabilization.

Asymptotic stability in distribution of stochastic differential equations with Markovian switching

Stability of stochastic differential equations with Markovian switching has recently been discussed by many authors, for example, Basak et al. (J. Math. Anal. Appl. 202 (1996) 604), Ji and Chizeck (IEEE Trans. Automat. Control 35 (1990) 777), Mariton (Jump Linear System in Automatic Control, Marcel Dekker, New York), Mao (Stochastic Process. Appl. 79 (1999) 45), Mao et al. (Bernoulli 6 (2000) 73) and Shaikhet (Theory Stochastic Process. 2 (1996) 180), to name a few. The aim of this paper is to study the asymptotic stability in distribution of nonlinear stochastic differential equations with Markovian switching.

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

Relatively little is known about the ability of numerical methods for stochastic differential equations (SDEs) to reproduce almost sure and small-moment stability. Here, we focus on these stability properties in the limit as the timestep tends to zero. Our analysis is motivated by an example of an exponentially almost surely stable nonlinear SDE for which the Euler-Maruyama (EM)method fails to reproduce this behavior for any nonzero timestep. We begin by showing that EM correctly reproduces almost sure and small-moment exponential stability for sufficiently small timesteps on scalar linear SDEs. We then generalize our results to multidimensional nonlinear SDEs. We show that when the SDE obeys a linear growth condition, EM recovers almost surely exponential stability very well. Under the less restrictive condition that the drift coefficient of the SDE obeys a one-sided Lipschitz condition, where EM may break down, we show that the backward Euler method maintains almost surely exponential stability.

Stabilization of a class of stochastic differential equations with Markovian switching

Stability of stochastic differential equations with Markovian switching has been studied quite extensively for a number of years, for example, by Basak et al. (J. Math. Anal. Appl. 202 (1996) 604–622), Ji and Chizeck (IEEE Trans. Automat. Control 35 (1990) 777–788), Mariton (Jump Linear Systems in Automatic Control, Marcel Dekker, New York, 1990), Mao et al. (Stochastic Process. Appl. 79 (1999) 45–67; Bernoulli 6 (2000) 73–90) and Yuan and Lygeros (in: R. Alur, G. Pappas (Eds.), Hybrid Systems: Computation and Control, Seventh International Workshop, HSCC 2004, Lecture Notes in Computer Science, vol. 2993, Springer, Berlin, 2004, pp. 646–659). By contrast, the problem of designing controllers to stabilize systems of this type has received relatively little attention. In this paper we study the problem of mean square exponential stabilization for a class of stochastic differential equations with Markovian switching.

Convergence of the Euler-Maruyama method for stochastic differential equations with Markovian switching

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Most of SDEwMSs do not have explicit solutions so it is important to have numerical solutions. It is surprising that there are not any numerical methods established for SDEwMSs yet, although the numerical methods for stochastic differential equations (SDEs) have been well studied. The main aim of this paper is to develop a numerical scheme for SDEwMSs and estimate the error between the numerical and exact solutions. This is the first paper in this direction and the emphasis lies on the error analysis.

Stability in distribution of stochastic differential delay equations with Markovian switching

In this paper we discuss stochastic differential delay equations with Markovian switching. Such an equation can be regarded as the result of several stochastic differential delay equations switching from one to another according to the movement of a Markov chain. The aim of this paper is to investigate the stability in distribution of the equations.

Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps

The existence, uniqueness and some sufficient conditions for stability in distribution of mild solutions to stochastic partial differential delay equations with jumps are presented. The principle technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to pass on stability of strong solutions to mild ones. As a consequence, stability results of Basak et al. (Basak et al. 1999 J. Math. Anal. Appl. 202, 604–622) and Yuan et al. (Yuan et al. 2003 Syst. Control Lett. 50, 195–207) are generalized to cover a class of much more general stochastic partial differential delay equations with jumps in infinite dimensions. In contrast to the almost sure exponential stability in Ichikawa (Ichikawa 1982 J. Math. Anal. Appl. 90, 12–44) and Luo & Liu (Luo & Liu 2008 Stoch. Proc. Appl. 118, 864–895) and the moment exponential stability in Luo & Liu, we present a new result on the stability in distribution of mild solutions. Finally, an example is given to demonstrate the applicability of our work.

On the exponential stability of switching diffusion processes

In this note, we investigate almost sure exponential stability for a class of switching diffusion processes. Lyapunov type sufficient conditions to ensure this type of stability for nonlinear switching diffusions are derived. The conditions are an improvement over related results in the literature, since they do not rely on the moment stability of the system. These conditions are then specialized to case of linear switching diffusion processes, to provide easily checkable sufficient criteria for exponential stabilization and robust stability.

Euler-Maruyama approximations in mean-reverting stochastic volatility model under regime-switching

Stochastic differential equations (SDEs) under regime-switching have recently been developed to model various financial quantities. In general, SDEs under regime-switching have no explicit solutions, so numerical methods for approximations have become one of the powerful techniques in the valuation of financial quantities. In this paper, we will concentrate on the Euler-Maruyama (EM) scheme for the typical hybrid mean-reverting θ -process. To overcome the mathematical difficulties arising from the regime-switching as well as the non-Lipschitz coefficients, several new techniques have been developed in this paper which should prove to be very useful in the numerical analysis of stochastic systems.