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Analysis of nondegenerate Wiener-Poisson functionals and its applications to Itô’s SDE with jumps

Abstract

We define three types of nondegenerate Wiener-Poisson functionals. Then, for each type we show that the (weighted) characteristic function of nondegenerate functional is of polynomial decay. Our discussion is based on the analysis of Wiener-Poisson functional (Malliavin calculus) developed by Ishikawa and Kunita (2006). Then we apply the decay property to solutions of Itô’s SDE with jumps. We show that if the SDE is nondegenerate, the law of the solution has a rapidly decreasing C -density function. Further, we show that transition function P s,t ϕ(x) of the jump-diffusion determined by the SDE is extended to a tempered distribution Φ such that P s,t Φ(x) is a C -function of x, through which we show that the transition density function p s,t (x, y) is a C -function of x and y, and is rapidly decreasing with respect to y.

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Correspondence to Hiroshi Kunita.

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Kunita, H. Analysis of nondegenerate Wiener-Poisson functionals and its applications to Itô’s SDE with jumps. Sankhya A 73, 1–45 (2011). https://doi.org/10.1007/s13171-011-0009-x

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  • DOI: https://doi.org/10.1007/s13171-011-0009-x

AMS (2000) subject classification

  • Primary 60H07, 60J75

Keywords and phrases

  • Malliavin calculus
  • Wiener-Poisson space
  • jump diffusion
  • density functions