First passage time law for some jump-diffusion processes : existence of a density

TL;DR

This paper proves that the first hitting time of a jump-diffusion process has a density with respect to Lebesgue measure, extending understanding of such processes' probabilistic properties.

Contribution

It establishes the existence of a density for the first passage time in jump-diffusion processes, including cases with negative drift where the density is defective.

Findings

The law of T_x has a density when E(X_1) >= 0.

The density is defective when E(X_1) < 0.

The result applies to processes combining Brownian motion with jumps.

Abstract

Let (Xt, t >= 0) be a diffusion process with jumps, sum of a Brownian motion with drift and a compound Poisson process. We consider T_x the first hitting time of a fixed level x > 0 by (Xt, t >= 0). We prove that the law of T_x has a density (defective when E(X1) < 0) with respect to the Lebesgue measure.