On Random Periodic Solution to a Neutral Stochastic Functional Differential Equation

Lili Gao,Litan Yan

doi:10.1155/2018/8353065

Abstract

In this paper, we consider the random periodic solution to a neutral stochastic functional differential equation driven by Brownian motion. We obtain the existence and uniqueness of the random periodic solution by Banach fixed point theorem. Moreover, we introduce two examples to illustrate our results.

Highlights

In the last two decades, the theory of stochastic dynamical systems has attracted much attention due to its wide applications in the fields such as financial market, insurance, biology, medical science, population dynamic, and control
The study of periodic solutions has occupied a central role in the theory of dynamical system since Poincare’s seminal work [12]
More recently researchers have given special interest to the study of equations in which the variable delay argument occurs in the derivative of the state variable, or so-called neutral stochastic functional differential equations (NSFDEs)

Summary

Introduction

In the last two decades, the theory of stochastic dynamical systems has attracted much attention due to its wide applications in the fields such as financial market, insurance, biology, medical science, population dynamic, and control (see [1,2,3,4,5,6,7,8]). The idea to regard stochastic differential equations (SDEs) as random dynamical systems can be traced back to late 1970s and early 1980s (see [9,10,11]). We consider the random periodic solutions of such stochastic equations. Later the definition of random periodic solutions and their existence for semi-flows generated by nonautonomous SDEs and SPDEs with additive noise were given by Feng and Zhao in [22, 23]. In [30], Feng et al study the existence of random periodic solutions to semilinear stochastic differential equations.

Preliminaries

Existence and Uniqueness of Random Periodic Solution

Examples

Conflicts of Interest