First passage time law for some Lévy processes with compound Poisson: Existence of a density

Abstract

Let ($X_t, t \geq 0)$ be a Lévy process with compound Poisson process and $τ_x$ be the first passage time of a fixed level $x > 0$ by ($X_t, t \geq 0$). We prove that the law of $τ_x$ has a density (defective when $\mathbb{E}(X_{1})\ < 0)$ with respect to the Lebesgue measure.