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

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.