 
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 interarrivals 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 WienerHopf factorization of generator
matrices.
Keywords: Two sided exit problem, phase{type distributions, semiMarkov em
bedding, WienerHopf factorization of matrices, renewal process.
AMS Subject Classication: 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 integrodierential equations or renewal equations for the
functions of interest and solving them via a double Laplace  Stieltjes transform
