Moments of the ruin time in a L\'evy risk model

TL;DR

This paper derives formulas for the moments of ruin time in a Lévy risk model, analyzes their asymptotic behavior, and explicitly computes moments for specific claim distributions, distinguishing profitable and unprofitable scenarios.

Contribution

It provides new formulas for ruin time moments in Lévy risk models and explicit calculations for certain claim distributions, advancing risk process analysis.

Findings

Formulas for moments of ruin time derived

Asymptotic behavior of moments analyzed as initial capital increases

Explicit first two moments computed for phase-type and exponential claims

Abstract

We derive formulas for the moments of the ruin time in a L\'evy risk model and use these to determine the asymptotic behavior of the moments of the ruin time as the initial capital tends to infinity. In the special case of the perturbed Cram\'er-Lundberg model with phase-type or exponentially distributed claims, we explicitly compute the first two moments of the ruin time. All our considerations distinguish between the profitable and the unprofitable setting.