A probabilistic representation of the derivative of a one dimensional killed diffusion semigroup

Abstract

We provide a probabilistic representation for the derivative of the semigroup corresponding to a diffusion process killed at the boundary of a half interval. In particular, we show that the derivative of the semi-group can be expressed as the expected value of a functional of a reflected diffusion process. Furthermore, as an application, we obtain a Bismut-Elworthy-Li formula which is also valid at the boundary.