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

TL;DR

This paper derives an explicit fundamental solution for the probability density of a jump-diffusion Ornstein-Uhlenbeck process, enabling direct computation of densities over time for this stochastic model.

Contribution

It provides an explicit form of the fundamental solution for the master equation of a jump-diffusion Ornstein-Uhlenbeck process, which was previously unknown.

Findings

Explicit fundamental solution derived for specific jump intensity and reversion ratio.

Properties of the solution analyzed and characterized.

Explicit formulas for densities at any time obtained.

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.