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

Abstract

For a bivariate L\'evy process $(\xi_t,\eta_t)_{t\geq 0}$ the generalised Ornstein-Uhlenbeck (GOU) process is defined as V_t:=e^{\xi_t}(z+\int_0^t e^{-\xi_{s-}}d\eta_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 L\'evy 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.