Ruin probabilities for two collaborating insurance companies

Abstract

In this note we find a formula for the supremum distribution of spectrally positive or negative Lévy processes with a broken linear drift. This gives formulas for ruin probabilities in the case when two insurance companies (or two branches of the same company) divide between them both claims and premia in some specified proportions. As an example we consider gamma Lévy process, $\alpha$-stable Lévy process and Brownian motion. Moreover we obtain identities for Laplace transform of the distribution for the supremum of Lévy processes with randomly broken drift and on random intervals.