EXISTENCE-UNIQUENESS PROBLEMS FOR INFINITE DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH DELAYS

Daoyi Xu; Xiaohu Wang; Zhiguo Yang

doi:10.11948/2012034

2012 Volume 2 Issue 4

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Daoyi Xu, Xiaohu Wang, Zhiguo Yang. EXISTENCE-UNIQUENESS PROBLEMS FOR INFINITE DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH DELAYS[J]. Journal of Applied Analysis & Computation, 2012, 2(4): 449-463. doi: 10.11948/2012034

Citation:

Daoyi Xu, Xiaohu Wang, Zhiguo Yang. EXISTENCE-UNIQUENESS PROBLEMS FOR INFINITE DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH DELAYS[J]. Journal of Applied Analysis & Computation, 2012, 2(4): 449-463. doi: 10.11948/2012034

EXISTENCE-UNIQUENESS PROBLEMS FOR INFINITE DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH DELAYS

1 College of Mathematics, Sichuan University, Chengdu 610064, China;
2 College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068, China

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Abstract

Abstract

The main aim of this paper is to develop the basic theory of a class of infinite dimensional stochastic differential equations with delays (IDSDEs) under local Lipschitz conditions. Firstly, we establish a global existenceuniqueness theorem for the IDSDEs under the global Lipschitz condition in C without the linear growth condition. Secondly, the non-continuable solution for IDSDEs is given under the local Lipschitz condition in C. Then, the classical Itĉ's formula is improved and a global existence theorem for IDSDEs is obtained. Our new theorems give better results while conditions imposed are much weaker than some existing results. For example, we need only the local Lipschitz condition in C but neither the linear growth condition nor the continuous condition on the time t. Finally, two examples are provided to show the effectiveness of the theoretical results.
- Infinite dimensional stochastic differential equations /
- delay /
- existence /
- uniqueness
MSC: 34K30;34K50;60H15

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References

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