Existence theorems for periodic Markov process and  stochasticfunctional differential equations

Daoyi Xu; Yumei Huang; Zhiguo Yang

doi:10.3934/dcds.2009.24.1005

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2009, Volume 24, Issue 3: 1005-1023. Doi: 10.3934/dcds.2009.24.1005

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Existence theorems for periodic Markov process and stochastic functional differential equations

1.
Yangtze center of Mathematics, Sichuan University, Chengdu, Sichuan 610064
2.
College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066

Received: October 2007

Revised: July 2008

Published: September 2009

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Abstract

In this paper, an effective existence theorem for periodic Markov process is first established. Using the theorem, we consider a class of periodic $It\hat{o}$ stochastic functional differential equations, and some sufficient conditions for the existence of periodic solution of the equations are given. To overcome the difficulties created by the special features possessed by the periodic stochastic differential equations with delays, as one will see, several lemmas are introduced. These existence theorems are rather general and therefore have great power in applications. Especially, our results are natural generalization of some classical periodic theorems on the model without stochastic perturbation. An example is worked out to demonstrate the advantages of our results.

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Mathematics Subject Classification: Primary: 34K50, 37H10, 60H10; Secondary: 34K13.

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