Exact conditions for no ruin for the generalised Ornstein-Uhlenbeck process

Abstract

For a bivariate Lévy process $({\xi }_t,{\eta }_t{)}_{t\ge 0}$ the generalised Ornstein-Uhlenbeck (GOU) process is defined as V_t:=e^{ξ_t}(z+\int_0^t e^{-ξ_{s-}}dη_s), t\ge0, where $z\in \mathbb{R}.$ We define necessary and sufficient conditions under which the infinite horizon ruin probability for the process is zero. These conditions are stated in terms of the canonical characteristics of the Lévy process and reveal the effect of the dependence relationship between $\xi$ and $\eta .$ We also present technical results which explain the structure of the lower bound of the GOU.

24 pages