Periodic solutions of stochastic delay differential equations and applications to logistic equation and neural networks

TLDR: In this paper, the authors considered a class of periodic Ito-delay differential equations by using the properties of periodic Markov processes, and some sufficient conditions for the existence of periodic so-lution of the delay equations are given.

Abstract: . In this paper, we consider a class of periodic Itoˆ stochasticdelay differential equations by using the properties of periodic Markovprocesses, and some sufficient conditions for the existence of periodic so-lution of the delay equations are given. These existence theorems improvethe results obtained by Itoˆ et al. [6], Bainov et al. [1] and Xu et al. [15].As applications, we study the existence of periodic solution of periodicstochastic logistic equation and periodic stochastic neural networks withinfinite delays, respectively. The theorem for the existence of periodic so-lution of periodic stochastic logistic equation improve the result obtainedby Jiang et al. [7]. 1. IntroductionSince Itoˆ introduced his stochastic calculus about 50 years ago, the theory ofstochastic differential equations has been developed very quickly [1–15,17]. Itis now being recognized to be not only richer than the corresponding theory ofdifferential equations without stochastic perturbation but also represent a morenatural framework for mathematical modeling of many real-world phenomena.Now there also exists a well-developed qualitative theory of stochastic differ-ential equations [6,10,12]. However, not so much has been developed in thedirection of the periodically stochastic differential equations. Till now only afew papers have been published on this topic [1,3,4,15,17]. In papers [3,6], theauthors got the conditions for the existence of periodic solution of differentialequations with random right sides. Hasminskii in [4] gave some basic resultson the existence of periodic solution of stochastic differential equations withoutdelays. But, the above results can not be used to check the existence of peri-odic solution of general stochastic delay differential equations. In [15], Xu et