 
Summary: Ruin probabilities and decit for the renewal
risk model with phase{type interarrival times
F. Avram 1 and M. Usabel 2
1 Dept. of Mathematics, Universit e de Pau, France
2 Dept. of Business Adm., Universidad Carlos III de Madrid, Spain
Key words and phrases: Finite time ruin probability, exponentially killed ruin
probability, decit at ruin, Sparre Andersen Model, phase{type distributions,
Laplace transform
Abstract We provide below a generalization of Thorin's
formula(1971) for the double Laplace transform of the nite time
ruin probability, by considering also the decit at ruin; the model is
that of a Sparre Andersen (renewal) risk process with phase{type
interarrival times.
In the case when the claims distribution is of phase{type as well, we
obtain also an alternative formula for the single Laplace transform
in time (or \exponentially killed probability"), in terms of the roots
with positive real part of the Lundberg's equations, which
complements Asmussen's representation(1992) in terms of the roots
with negative real part.
1. Introduction
