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Volume 2, Number 4, 2012, Pages 449-463                                                                DOI:10.11948/2012034
Existence-Uniqueness Problems For Infinite Dimensional Stochastic Differential Equations With Delays
Daoyi Xu,Xiaohu Wang,Zhiguo Yang
Keywords:Infinite dimensional stochastic differential equations, delay, existence, uniqueness
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 existence-uniqueness 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 Itô'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.
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