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Stochastic differential equations with jumps and stochastic flows of diffeomorphisms

In: Itô’s Stochastic Calculus and Probability Theory

Author

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  • Hiroshi Kunita

    (Kyushu University, Graduate School of Mahtematics)

Abstract

After fundamental works of K. Itô 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 Lévy 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.

Suggested Citation

  • Hiroshi Kunita, 1996. "Stochastic differential equations with jumps and stochastic flows of diffeomorphisms," Springer Books, in: Nobuyuki Ikeda & Shinzo Watanabe & Masatoshi Fukushima & Hiroshi Kunita (ed.), Itô’s Stochastic Calculus and Probability Theory, pages 197-211, Springer.
    • Handle: RePEc:spr:sprchp:978-4-431-68532-6_13
    DOI: 10.1007/978-4-431-68532-6_13
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