On differentiation of functionals containing the first exit of a diffusion process from a domain

Abstract

One of the problems arising in the differentiation of functionals of random diffusion processes in domains with absorbing boundaries is to compute parametric derivatives for the functionals containing the first exit time τ from the domain for the underlying diffusion process. Earlier work [S. A. Gusev, Numer. Anal. Appl., 1 (2008), pp. 314-331] proposed a method for solving this problem under some condition of existence of mean square derivatives for τ with respect to the parameter; this condition was restrictive and difficult to verify. In this paper, we show that this condition can be waived under some mild assumptions.