6. Bismut-Elworthy-Li Formulae for Bessel Processes

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.