The ruin problem for L\'evy-driven linear stochastic equations with applications to actuarial models with negative risk sums

TL;DR

This paper analyzes the asymptotic behavior of ruin probabilities in insurance models driven by Lévy processes, providing exact decay rates under minimal assumptions, which is crucial for risk management.

Contribution

It establishes the precise asymptotic form of ruin probabilities for Lévy-driven linear SDEs in actuarial models, with minimal assumptions on the price process.

Findings

Ruin probability asymptotically behaves as Cu^{-eta} for large initial capital u.

The asymptotic form holds under the non-arithmetic law of V_T without additional conditions.

Provides explicit decay rate linked to the root of the cumulant-generating function.

Abstract

We study the asymptotic of the ruin probability for a process which is the solution of linear SDE defined by a pair of independent L\'evy processes. Our main interest is the model describing the evolution of the capital reserve of an insurance company selling annuities and investing in a risky asset. Let $\beta>0$ be the root of the cumulant-generating function $H$ of the increment of the log price process $V$. We show that the ruin probability admits the exact asymptotic $Cu^{-\beta}$ as the initial capital $u\to\infty$ assuming only that the law of $V_T$ is non-arithmetic without any further assumptions on the price process.