The Almost Sure Asymptotic Stability and Boundedness of Stochastic Functional Differential Equations with Polynomial Growth Condition

Abstract

Stability and boundedness are two of the most important topics in the study of stochastic functional differential equations (SFDEs). This paper mainly discusses the almost sure asymptotic stability and the boundedness of nonlinear SFDEs satisfying the local Lipschitz condition but not the linear growth condition. Here we assume that the coefficients of SFDEs are polynomial or dominated by polynomial functions. We give sufficient criteria on the almost sure asymptotic stability and the boundedness for this kind of nonlinear SFDEs. Some nontrivial examples are provided to illustrate our results.

On the approximations of solutions to stochastic differential equations under polynomial condition

“…It should be stressed that, in the recent years, the existence and uniqueness of the exact solutions, development of the approximate methods, stability of the exact and approximate solutions and other qualitative and quantitative properties of the exact and approximate solutions, under highly nonlinear conditions on coefficients of the appropriate stochastic differential equations have attracted the attention of many researchers. We refer the reader, for example, to [4,8,17,20,21], among many other. So, the main aim of this paper is to provide a contribution to the analysis of stochastic differential equations with highly nonlinear drift coefficients, that is, with drifts which satisfy the polynomial condition.…”