Bismut-Elworthy-Li Formulae for Bessel Processes

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


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.


Bismut-Elworthy-Li formula Strong Feller property Bessel processes 



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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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université Pierre et Marie Curie, LPMAParisFrance

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