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

TL;DR

This paper establishes conditions on the characteristic triplet of a bivariate Lévy process that guarantee certain ruin for the generalized Ornstein-Uhlenbeck process, analyzing its bounds and monotonicity properties.

Contribution

It provides new criteria for ruin probability and detailed structural analysis of the GOU's bounds and monotonicity, extending previous zero-ruin probability results.

Findings

Conditions for certain ruin are derived.

Structural analysis of upper and lower bounds is provided.

Sets where the GOU is increasing or decreasing are characterized.

Abstract

For a bivariate \Levy process $(\xi_t,\eta_t)_{t\geq 0}$ the generalised Ornstein-Uhlenbeck (GOU) process is defined as \[V_t:=e^{\xi_t}(z+\int_0^t e^{-\xi_{s-}}\ud \eta_s), t\ge0,\]where $z\in\mathbb{R}.$ We present conditions on the characteristic triplet of $(\xi,\eta)$ 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 \cite{BankovskySly08}, which stated conditions for zero probability of ruin, and completes a significant aspect of the study of the GOU.