American Institute of Mathematical Sciences

August  2009, 24(3): 1005-1023. doi: 10.3934/dcds.2009.24.1005

Existence theorems for periodic Markov process and stochastic functional differential equations

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

Received  October 2007 Revised  July 2008 Published  April 2009

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.
Citation: Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005
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