On Integral of a Semi-Markov Diffusion Process

Abstract

Let (X(t)) (t ≥ 0) be a semi-Markov diffusion process. The process (J(T )) (T ≥ 0) equal to the integral of (X(t)) on interval [0, T ) is studied. The relation between one-dimensional differential equation of the second order of elliptical type and asymptotics of a solution to Dirichlet problem on an interval with length tending to zero is established. This relation is used to derive a differential equation for the Laplace transform of the semi-Markov generating function of the process (J(t)).