Ruin probabilities for a Lévy-driven generalised Ornstein–Uhlenbeck process

Abstract

We study the asymptotics of the ruin probability for a process which is the solution of a linear SDE defined by a pair of independent Levy processes. Our main interest is a 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 $V_{1}$ of the log-price process. 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.