Risk Theory in a Finite Customer-Pool Setting

M. Mandjes, Daniël T. Rutgers

2026, v.32, Issue 2, 165-204

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 Lévy process. We begin by analyzing this general model, in which the inter-arrival times of the major clients 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.

doi:10.61102/1024-2953-mprf.2026.32.2.001

Keywords: Ruin theory, Cramér-Lundberg model, finite client pool, Laplace trans-forms, tail asymptotics, overshoot distribution

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