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

TL;DR

This paper establishes precise conditions based on the characteristics of a bivariate Lévy process that determine when the generalized Ornstein-Uhlenbeck process has zero probability of ruin over an infinite horizon, highlighting the influence of dependence between components.

Contribution

It provides the first necessary and sufficient criteria for no ruin in the GOU process, explicitly incorporating the dependence structure of the underlying Lévy process.

Findings

Conditions for zero ruin probability derived

Dependence between Lévy process components affects ruin risk

Structural insights into the GOU process's lower bound

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.