Conditions for certain ruin for the generalised Ornstein–Uhlenbeck process and the structure of the upper and lower bounds

For a bivariate Lévy process ( ξ t , η t ) t ≥ 0 the generalised Ornstein–Uhlenbeck (GOU) process is defined as V t ≔ e ξ t ( z + ∫ 0 t e − ξ s − d η s ) , t ≥ 0 , where z ∈ R . We present conditions on the characteristic triplet of ( ξ , η ) which ensure certain ruin for the GOU. We present a detailed analysis on the structure of the upper and lower bounds and the sets of values on which the GOU is almost surely increasing, or decreasing. This paper is the sequel to Bankovsky and Sly (2008) [2], which stated conditions for zero probability of ruin, and completes a significant aspect of the study of the GOU.