Ruin probabilities for two collaborating insurance companies

TL;DR

This paper derives formulas for ruin probabilities of two collaborating insurance companies using spectrally one-sided Levy processes with broken linear drift, providing new insights into their joint risk behavior.

Contribution

It introduces explicit formulas for the supremum distribution of Levy processes with broken drift, applicable to calculating ruin probabilities for two companies sharing claims and premiums.

Findings

Formulas for ruin probabilities with broken linear drift

Explicit supremum distribution for specific Levy processes

Laplace transform identities for supremum of Levy processes

Abstract

In this note we find a formula for the supremum distribution of spectrally positive or negative L\'evy 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 L\'evy process, $\alpha$-stable L\'evy process and Brownian motion. Moreover we obtain identities for Laplace transform of the distribution for the supremum of L\'evy processes with randomly broken drift and on random intervals.