A bivariate risk model with mutual deficit coverage

TL;DR

This paper analyzes a bivariate risk model where two insurance companies mutually cover each other's deficits, deriving survival probabilities and boundary transforms, especially when claims are independent, using Wiener-Hopf factorization.

Contribution

It introduces a novel bivariate risk model with mutual deficit coverage and derives explicit survival probability expressions using Wiener-Hopf techniques.

Findings

Derived boundary transforms for survival probabilities.

Explicit formulas when claims are independent.

Extended analysis to non-mutual reinsurance case.

Abstract

We consider a bivariate Cramer-Lundberg-type risk reserve process with the special feature that each insurance company agrees to cover the deficit of the other. It is assumed that the capital transfers between the companies are instantaneous and incur a certain proportional cost, and that ruin occurs when neither company can cover the deficit of the other. We study the survival probability as a function of initial capitals and express its bivariate transform through two univariate boundary transforms, where one of the initial capitals is fixed at 0. We identify these boundary transforms in the case when claims arriving at each company form two independent processes. The expressions are in terms of Wiener-Hopf factors associated to two auxiliary compound Poisson processes. The case of non-mutual (reinsurance) agreement is also considered.