On the ruin probability of the generalised ornstein-uhlenbeck process in the cramér case

For a bivariate Lévy process (ξ_t, η_t)t≤0 and initial value V₀, define the generalised Ornstein-Uhlenbeck (GOU) process V_t:= e^ξ_t (V₀ + ∫^t₀ e^−ξ_s− dη_s), t ≤ 0, and the associated stochastic integral process Z_t:= ∫^t₀ e^−ξ_s− dη_s, t ≤ 0. Let T_z:= inf(t > 0: V_t < 0 | V₀ = z) and Ψ(z):= P(Tz < ∞) for z ≤ 0 be the ruin time and infinite horizon ruin probability of the GOU process. Our results extend previous work of Nyrhinen (2001) and others to give asymptotic estimates for Ψ(z) and the distribution of Tz as z →∞, under very general, easily checkable, assumptions, when ξ satisfies a Cramér condition.