A two-dimensional ruin problem on the positive quadrant

TL;DR

This paper analyzes the joint ruin probabilities of two insurance companies sharing claims and premiums, deriving explicit formulas using Laplace transforms for exponential claims and characterizing joint ruin times.

Contribution

It introduces a new model for joint ruin analysis of two insurance entities and provides explicit formulas for ruin probabilities and times using Laplace transforms.

Findings

Explicit Laplace transform of joint ruin probability derived.

Closed-form expression for joint ruin probability with exponential claims.

Characterization of the Laplace transform of joint ruin time provided.

Abstract

In this paper we study the joint ruin problem for two insurance companies that divide between them both claims and premia in some specified proportions (modeling two branches of the same insurance company or an insurance and re-insurance company). Modeling the risk processes of the insurance companies by Cram\'{e}r-Lundberg processes we obtain the Laplace transform in space of the probability that either of the insurance companies is ruined in finite time. Subsequently, for exponentially distributed claims, we derive an explicit analytical expression for this joint ruin probability by explicitly inverting this Laplace transform. We also provide a characterization of the Laplace transform of the joint ruin time.