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Canonical Stochastic Differential Equations based on Lévy Processes and Their Supports

In: Stochastic Dynamics

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  • Hiroshi Kunita

Abstract

We study a canonical stochastic differential equation (SDE) with jumps driven by a Lévy process. The equation is defined through the canonical stochastic integral, which is different from the Itô integral or the Stratonovich one. A feature of the canonical SDE is that it is a coordinate free representation of an SDE with jumps and the solution admits a nice geometrical interpretation, making use of the integral curves of vector fields appearing in the equation. A main result of this paper is stated in Theorem 3.3, where we determine the support of the probability distribution of the solution of the canonical SDE. It is characterized through solutions of control systems with jumps for ordinary integro-differential equations. It can be regarded as an extension of the support theory for continuous SDE due to Stroock and Varadhan.

Suggested Citation

  • Hiroshi Kunita, 1999. "Canonical Stochastic Differential Equations based on Lévy Processes and Their Supports," Springer Books, in: Stochastic Dynamics, chapter 12, pages 283-304, Springer.
    • Handle: RePEc:spr:sprchp:978-0-387-22655-2_12
    DOI: 10.1007/0-387-22655-9_12
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