Stochastic differential equations with jumps and stochastic flows of diffeomorphisms
Stochastic differential equations with jumps and stochastic flows of diffeomorphisms
复制标题
具有跳跃和微分同胚随机流的随机微分方程
DOI：
10.1007/978-4-431-68532-6_13
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发表时间：
1996
影响因子：
2.2
通讯作者：
H. Kunita
中科院分区：
文献类型：
--
作者：
H. Kunita
自20世纪40年代伊藤的基础工作以来，随机微分方程组理论得到了广泛的研究。Elworth，Bismut，Ikeda-Watanabe，Kurita，Meyer等人在1980年左右研究了随机微分方程解的流动性质，证明了在方程系数的Lipschitz条件下，由布朗运动或连续半鞅驱动的随机微分方程解允许一类同胚的随机流.此外，如果系数是光滑的，则它允许一种形式的微分同胚随机流。详情见库尼塔的书[11]。在这篇文章中，我们将主要讨论由Levy过程或带跳的半鞅驱动的随机微分方程，并讨论解的流动性质。在我们介绍SDE之前，让我们简单地回顾一下由布朗运动或连续半环驱动的SDE与同胚的随机流之间的关系。
After fundamental works of K. Ito in 1940s, theory of stochastic differential equations (SDE) has been studied extensively. The flow property of the solution of SDE was studied around 1980 by Elworthy, Bismut, Ikeda-Watanabe, Kunita, Meyer etc. It was proved that under the Lipschitz condition of the coefficients of the equation, the solution of any SDE driven by a Brownian motion or a continuous semimartingale admits a version of a stochasic flows of homeomorphisms. Further if the coefficients are smooth, it admits a version of a stochastic flow of diffeomorphisms. Details are found in Kunita’s book [11]. In this paper, we will be mainly concerned with SDE driven by a Levy process or a semimartingale with jumps and discuss the flow property of the solutions. Before we introduce our SDE, let us briefly recall the relation between SDE driven by a Brownian motion or a continuous semimartingle and a stochastic flow of homeomorphisms.