Delay integro-differential equations are very important in biology, medicine and many other fields. If we take random noise into account, we can obtain many stochastic delay integro-differential equations (SDIDEs). As a special case of stochastic functional differential equations (SFDEs), the fundamental theory of existence and uniqueness of the solution of SDIDEs can be regarded similarly to t...

We consider an Ito stochastic differential equation with delay, driven by brownian motion, whose solution, by an appropriate reformulation, defines a Markov process X with values in a space of continuous functions C, with generator L. We then consider a backward stochastic differential equation depending on X , with unknown processes (Y, Z), and we study properties of the resulting system, in p...

A stochastic competition system with harvesting and distributed delay is investigated, which described by differential equations delay. The existence uniqueness of a global positive solution are proved via Lyapunov functions, an ergodic method used to obtain that the asymptotically stable in distribution. By using comparison theorem limit superior theory, sufficient conditions for persistence m...

This paper is devoted to the convergence analysis of stochastic θ-methods for nonlinear neutral stochastic differential delay equations (NSDDEs) in Itô sense. The basic idea is to reformulate the original problem eliminating the dependence on the differentiation of the solution in the past values, which leads to a stochastic differential algebraic system. Drift-implicit stochastic θ-methods are...

Abstract This article deals with approximations of center manifolds for delay stochastic differential equations additive noise. We first prove the existence and smoothness random these approximation equations. Then we show that C k {C}^{k} invariant system colored nois...