An explicit form of the fundamental solution of the master equation for a jump-diffusion Ornstein-Uhlenbeck process

Abstract

An integro-differential equation for the probability density of the generalized stochastic Ornstein-Uhlenbeck process with jump diffusion is considered. It is shown that for a certain ratio between the intensity of jumps and the speed of reversion, the fundamental solution can be found explicitly. The properties of this solution are investigated. The fundamental solution allows one to obtain explicit formulas for the densities at each moment of time.