Risk theory in a finite customer-pool setting

TL;DR

This paper develops a comprehensive risk model for a finite customer pool with major and minor clients, deriving explicit formulas for ruin probabilities and overshoot distributions, and analyzing tail behaviors under various claim size distributions.

Contribution

It introduces a novel finite customer-pool risk model with explicit ruin probability formulas and tail asymptotics, extending classical models to more realistic finite settings.

Findings

Closed-form Laplace transform of ruin probability.

Asymptotic tail behavior for phase-type and regularly varying claims.

Distribution of overshoot over exponential initial reserve.

Abstract

This paper investigates an insurance model with a finite number of major clients and a large number of small clients, where the dynamics of the latter group are modeled by a spectrally positive L\'evy process. We begin by analyzing this general model, in which the inter-arrival times are exponentially distributed (though not identically), and derive the closed-form Laplace transform of the ruin probability. Next, we examine a simplified version of the model involving only the major clients, and explore the tail asymptotics of the ruin probability, focusing on the cases where the claim sizes follow phase-type or regularly varying distributions. Finally, we derive the distribution of the overshoot over an exponentially distributed initial reserve, expressed in terms of its Laplace-Stieltjes transform.