Closed-form Ruin Probabilities in Classical Risk Models with Gamma Claims

Abstract

In this paper we provide three equivalent expressions for ruin probabilities in a Cram\'er-Lundberg model with gamma distributed claims. The results are solutions of integro-differential equations, derived by means of (inverse) Laplace transforms. All the three formulas have infinite series forms, two involving Mittag-Leffler functions and the third one involving moments of the claims distribution. This last result applies to any other claim size distributions that exhibits finite moments.