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On the Ruin Probability of the Generalised Ornstein-Uhlenbeck Process in the Cramér Case

For a bivariate \Levy process $(ξ_t,η_t)_{t\ge 0}$ and initial value $V_0$ define the Generalised Ornstein-Uhlenbeck (GOU) process \[ V_t:=e^{ξ_t}\Big(V_0+\int_0^t e^{-ξ_{s-}}\ud η_s\Big),\quad t\ge0,\] and the associated stochastic integral process \[Z_t:=\int_0^t e^{-ξ_{s-}}\ud η_s,\quad t\ge0.\] Let $T_z:=\inf\{t>0:V_t<0\mid V_0=z\}$ and $ψ(z):=P(T_z<\infty)$ 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 $ψ(z)$ and the distribution of $T_z$ as $z\to\infty$, under very general, easily checkable, assumptions, when $ξ$ satisfies a Cramér condition.