The density of the ruin time for a renewal-reward process perturbed by a diffusion

Christophette Blanchet-Scalliet (1) , Diana Dorobantu (2) , Didier Rullière (2)

1 ICJ - Institut Camille Jordan
2 SAF - Laboratoire de Sciences Actuarielle et Financière

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.

Mots clés: Renewal-reward process; Brownian motion; Jump-diffusion process; Time of ruin.; Time of ruin

Domaines: Probabilités [math.PR]; Gestion des risques [q-fin.RM]

Christophette Blanchet-Scalliet, Diana Dorobantu, Didier Rullière. The density of the ruin time for a renewal-reward process perturbed by a diffusion. Applied Mathematics Letters, 2013, 26 (1), http://dx.doi.org/10.1016/j.aml.2012.04.003. ⟨10.1016/j.aml.2012.04.003⟩. ⟨hal-00625099v3⟩