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Bismut-Elworthy-Li Formulae for Bessel Processes

  • Henri Elad Altman
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2215)

Abstract

In this article we are interested in the differentiability property of the Markovian semi-group corresponding to the Bessel processes of nonnegative dimension. More precisely, for all δ ≥ 0 and T > 0, we compute the derivative of the function \(x \mapsto P^{\delta }_{T} F (x) \), where \((P^{\delta }_{t})_{t \geq 0}\) is the transition semi-group associated to the δ-dimensional Bessel process, and F is any bounded Borel function on \(\mathbb {R}_{+}\). The obtained expression shows a nice interplay between the transition semi-groups of the δ—and the (δ + 2)-dimensional Bessel processes. As a consequence, we deduce that the Bessel processes satisfy the strong Feller property, with a continuity modulus which is independent of the dimension. Moreover, we provide a probabilistic interpretation of this expression as a Bismut-Elworthy-Li formula.

Keywords

Bismut-Elworthy-Li formula Strong Feller property Bessel processes 

Notes

Acknowledgements

I would like to thank Lorenzo Zambotti, my Ph.D. advisor, for all the time he patiently devotes in helping me with my research. I would also like to thank Thomas Duquesne and Nicolas Fournier, who helped me solve a technical problem, as well as Yves Le Jan for a helpful discussion on the Bessel flows of low dimension, and Lioudmila Vostrikova for answering a question on this topic.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université Pierre et Marie Curie, LPMAParisFrance

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