Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

TL;DR

This paper investigates the probability of two interconnected insurance companies' reserves leaving the positive quadrant, providing exact formulas for Poisson claims and asymptotic analysis for general renewal claims.

Contribution

It derives a closed-form expression for the ruin probability under Poisson claims and analyzes asymptotic behavior for general renewal processes with light-tail claims.

Findings

Exact ruin probability formula for Poisson claims

Asymptotic ruin probability behavior for large reserves

Analysis of exit times from the quadrant in risk models

Abstract

Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process, we obtain a closed form expression for the ultimate ruin probability. In the general case, we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cram\'{e}r light-tail assumption on the claim size distribution.