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

For a bivariate Lévy process (ξt,ηt)t≥0(ξt,ηt)t≥0 the generalised Ornstein–Uhlenbeck (GOU) process is defined as Vt≔eξt(z+∫0te−ξs−dηs),t≥0, where z∈Rz∈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 ξξ and ηη. We also present technical results which explain the structure of the lower bound of the GOU.