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Existence of Density Functions for the Running Maximum of a Lévy-Itô Diffusion

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Abstract

In this note, by using Bismut’s approach to Malliavin calculus for jump processes, we obtain a criterion for the existence of density functions of the running maximum of Wiener-Poisson functionals. As an application, existence of density functions for the running maximum of a Lévy-Itô diffusion is proved.

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Acknowledgments

The authors are very grateful to the referee for detailed reports and corrections. They also would like to thank Professor Zhao Dong, Fengyu Wang and Xicheng Zhang for their valuable discussions and suggestions.

Y. Song is supported by National Natural Science Foundation of China (No.11501286) and Natural Science Foundation of Jiangsu Province (No.BK20150564). Y. Xie is supported by National Natural Science Foundation of China (No.11271169).

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, Jiangsu Province, 210093, People’s Republic of China

    Yulin Song

  2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu Province, 221116, People’s Republic of China

    Yingchao Xie

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  1. Yulin Song
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  2. Yingchao Xie
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Correspondence to Yulin Song.

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Song, Y., Xie, Y. Existence of Density Functions for the Running Maximum of a Lévy-Itô Diffusion. Potential Anal 48, 35–48 (2018). https://doi.org/10.1007/s11118-017-9625-y

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  • DOI: https://doi.org/10.1007/s11118-017-9625-y

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