Bismut-Elworthy-Li formulae for Bessel processes

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 $\delta \ge 0$ and $T>0$, we compute the derivative of the function $x\mapsto P_T^{\delta }F(x)$, where $(P_t^{\delta }{)}_{t\ge 0}$ is the transition semi-group associated to the $\delta$ - 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 $\delta$ - and the $(\delta +2)$-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.