Risk modelling on liquidations with Lévy processes

In classical ruin theory, the time of ruin is defined as the time when the surplus of an insurance portfolio falls below zero. This simplification of a single barrier, however, needs careful adaptations to imitate the real-world liquidation process. Inspired by [7] and [24], this paper adopts a three-barrier model to describe the financial stress leading to bankruptcy of an insurer. The financial status of the insurer is classified into three states, namely, the solvent, the insolvent, and the liquidated. The insurer’s surplus processes at the states of solvent and insolvent are modeled by two spectrally negative Lévy processes, which have been taken as good candidates to model insurance risks in the recent literature. Accordingly, the time of liquidation is defined in this three-barrier model. By adopting the techniques of excursions in fluctuation theory, we obtain the joint distribution of the time of liquidation, the surplus at liquidation, and the historical high of the surplus until liquidation, which generalizes the known results on the classical expected discounted penalty function from [16]. The results have semi-explicit expressions in terms of the scale functions and the Lévy triplets associated with the two underlying Lévy processes. The special case when the two underlying Lévy processes coincide with each other or differ from each other by a constant drift term is also studied, and our results are expressed compactly via only the scale functions. The corresponding results are consistent with the classic works of literature on Parisian ruin with (or without) a lower barrier in [4,22], and [14]. Numerical examples are provided to illustrate the underlying features of liquidation ruin.