Absolute ruin in the Ornstein-Uhlenbeck type risk model

TL;DR

This paper derives explicit formulas and conditions for absolute ruin probabilities in an Ornstein-Uhlenbeck risk model, extending classical results to more general claim processes and providing spectral and series representations.

Contribution

It introduces a novel approach linking ruin probabilities to Ornstein-Uhlenbeck processes and derives new spectral and series formulas for these probabilities.

Findings

Explicit expression for finite-time ruin probability using Ornstein-Uhlenbeck transition probabilities

Necessary and sufficient conditions for infinite-time ruin occurrence

Series expansions for ruin probabilities and Laplace transform of first-exit times

Abstract

We start by showing that the finite-time absolute ruin probability in the classical risk model with constant interest force can be expressed in terms of the transition probability of a positive Ornstein-Uhlenbeck type process, say X. Our methodology applies to the case when the dynamics of the aggregate claims process is a subordinator. From this expression, we easily deduce necessary and sufficient conditions for the infinite-time absolute ruin to occur. We proceed by showing that, under some technical conditions, the transition density of X admits a spectral type representation involving merely the limiting distribution of the process. As a by-product, we obtain a series expansions for the finite-time absolute ruin probability. On the way, we also derive, for the aforementioned risk process, the Laplace transform of the first-exit time from an interval from above. Finally, we illustrate our results by detailing some examples.