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Summary: The two barriers ruin problem via a Wiener Hopf
decomposition approach
Florin Avram Martijn R. Pistorius y Miguel Usabel z
abstract
Consider an insurance company whose capital U evolves as a risk processes with phase{
type inter-arrivals and claims. In this note we study the probability and severity of
ruin before the capital U reaches an upper barrier K > 0. The main tools we use are
Asmussen and Kella's embedding [5, 6] and Wiener-Hopf factorization of generator
matrices.
Keywords: Two sided exit problem, phase{type distributions, semi-Markov em-
bedding, Wiener-Hopf factorization of matrices, renewal process.
AMS Subject Classi cation: 60G40, 90A09.
1 Introduction
Consider an insurance company, whose capital is modeled by a positive drift
added to a pure jump process with negative jumps. The drift, say p, models the
premium income stream and the jumps stand for the claims the company re-
ceives. One is interested in the time and severity of ruin. The transform analytic
approach to this problems, going back to Cramer [12] and Sparre Andersen [1],
consists in formulating integro-di erential equations or renewal equations for the
functions of interest and solving them via a double Laplace - Stieltjes transform
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