In this paper we shall establish a new theorem on the existence and uniqueness of the adapted solution to a backward stochastic differential equation under a weaker condition than the Lipschitz one.

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,...

The main aim of this paper is to discuss the almost surely asymptotic stability of the neutral stochastic differential delay equations (NSDDEs) with Markovian switching. Linear NSDDEs with Markovian switching and nonlinear examples will be discussed to illustrate the theory.

Stability of stochastic differential equations with Markovian switching has recently received a lot of attention. For example, stability of linear or semi-linear type of such equations has been studied by Basak et al. (1996, J. Math. Anal. Appl. 202, 604-622), Ji and Chizeck (1990, Automat....

Population systems are often subject to environmental noise, and our aim is to show that (surprisingly) the presence of even a tiny amount can suppress a potential population explosion. To prove this intrinsically interesting result, we stochastically perturb the multivariate deterministic...

Consider a stochastic differential equation with respect to semimartingales dX(t)=AX(t)d[mu](t)+G(X(t),t) dM(t) which might be regarded as a stochastic perturbed system of dX(t)=AX(t)d[mu](t). Suppose the second equation is exponentially stable almost surely. What we are interested in in this...

The aim of this paper is to investigate the almost surely polynomial stability of the stochastic differential equation with respect to semimartingales d[phi]t = F([phi]t, t) d[mu]t + G([phi]t) dMt + f([phi]t, t) d[mu]t + g([phi]t) dMt under the condition that its unperturbed equation d[psi]t =...

Although the Razumikhin-type theorems have been well developed for the stability of functional differential equations and they are very useful in applications, so far there is almost no result of Razumikhin type on the stability of stochastic functional differential equations. The main aim of...

This work is concerned with a class of semilinear stochastic functional parabolic differential equations of retarded type. We first establish conditions to ensure the existence of a unique non-negative solution of the stochastic delay partial differential equation under investigation....

A strong solutions approximation approach for mild solutions of stochastic functional differential equations with Markovian switching driven by Lévy martingales in Hilbert spaces is considered. The Razumikhin-Lyapunov type function methods and comparison principles are studied in pursuit of...