Finite Time Ruin Probabilities for Tempered Stable Insurance Risk Processes

TL;DR

This paper analyzes the probability of ruin within finite time for tempered stable Lévy processes in insurance risk, providing explicit formulas, numerical methods, and discussing implications for risk modeling.

Contribution

It introduces explicit Laplace transform-based formulas for ruin probabilities in tempered stable processes and compares them with simulations, highlighting limitations of inverse Gaussian models.

Findings

Explicit ruin time distribution formulas derived

Numerical approximations validated against importance sampling

Inverse Gaussian process may have drawbacks as a risk reserve model

Abstract

We study the probability of ruin before time $t$ for the family of tempered stable L\'evy insurance risk processes, which includes the spectrally positive inverse Gaussian processes. Numerical approximations of the ruin time distribution are derived via the Laplace transform of the asymptotic ruin time distribution, for which we have an explicit expression. These are benchmarked against simulations based on importance sampling using stable processes. Theoretical consequences of the asymptotic formulae are found to indicate some potential drawbacks to the use of the inverse Gaussian process as a risk reserve process. We offer as alternatives natural generalizations which fall within the tempered stable family of processes.