On integral of a semi-Markov diffusion process

A semi-Markov diffusion process $(X(t))$ $(t\ge0)$ is considered. The process $(J(t))$ $(t\ge0)$ equals to integral of the process $(X(t))$ 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 of Dirichlet problem on an interval with length tending to zero is derived. This relation is used for deriving a differential equation Laplace transform for the semi-Markov generating function of the process $(J(t))$.