On the Ruin Probability of the Generalised Ornstein-Uhlenbeck Process in the Cram\'er Case

TL;DR

This paper analyzes the ruin probability of a generalized Ornstein-Uhlenbeck process driven by a bivariate Lévy process, providing asymptotic estimates under Cramér conditions for large initial values.

Contribution

It extends previous results by deriving asymptotic estimates for ruin probabilities and ruin times of the GOU process under general conditions with Cramér assumptions.

Findings

Asymptotic estimates for ruin probability as initial value grows large.

Distributional results for the ruin time in the Cramér case.

General conditions under which the estimates hold.

Abstract

For a bivariate \Levy process $(\xi_t,\eta_t)_{t\ge 0}$ and initial value $V_0$ define the Generalised Ornstein-Uhlenbeck (GOU) process \[ V_t:=e^{\xi_t}\Big(V_0+\int_0^t e^{-\xi_{s-}}\ud \eta_s\Big),\quad t\ge0,\] and the associated stochastic integral process \[Z_t:=\int_0^t e^{-\xi_{s-}}\ud \eta_s,\quad t\ge0.\] Let $T_z:=\inf\{t>0:V_t<0\mid V_0=z\}$ and $\psi(z):=P(T_z<\infty)$ for $z\ge 0$ be the ruin time and infinite horizon ruin probability of the GOU. Our results extend previous work of Nyrhinen (2001) and others to give asymptotic estimates for $\psi(z)$ and the distribution of $T_z$ as $z\to\infty$, under very general, easily checkable, assumptions, when $\xi$ satisfies a Cram\'er condition.