The density of a passage time for a renewal-reward process perturbed by a diffusion.

Summary Let $X$ be a mixed process, sum of a brownian motion and a renewal-reward process, and $\tau_{x}$ be the first passage time of a fixed level $x<0$ by $X$. We prove that $\tau_x$ has a density and we give a formula for it. Links with ruin theory are presented. Our result may be computed in classical settings (for a Lévy or Sparre Andersen process) and also in a non markovian context with possible positive and negative jumps. Some numerical applications illustrate the interest of this density formula.