On the ruin time distribution for a Sparre Andersen process with exponential claim sizes

TL;DR

This paper derives a closed-form series representation for the ruin time distribution in a Sparre Andersen risk model with exponential claims, extending previous results to more general inter-claim time distributions.

Contribution

It provides a new explicit series formula for ruin time distribution in Sparre Andersen processes with exponential claims, generalizing earlier Erlang-based results.

Findings

Closed-form series representation for ruin time distribution.

Extension of previous results to gamma and mixed exponential inter-claim times.

Applicable to a broader class of inter-claim time distributions.

Abstract

We derive a closed-form (infinite series) representation for the distribution of the ruin time for the Sparre Andersen model with exponentially distributed claims. This extends a recent result of Dickson et al. (2005) for such processes with Erlang inter-claim times. We illustrate our result in the cases of gamma and mixed exponential inter-claim time distributions.