Parisian ruin with random deficit-dependent delays for spectrally negative Lévy processes

Speaker:  Duy Phat Nguyen
Affiliation: University of Melbourne

Abstract

Collective risk theory uses mathematical models to describe an insurer's vulnerability to ruin. Suppose that the dynamics of an insurance company’s reserve is given by a stochastic process {X_t} starting at X_0 > 0. Traditionally, ruin occurs when {X_t} turns negative. Parisian ruin, introduced by Dassios and Wu (2008), occurs when {X_t} drops below zero and continuously stays below zero for longer than a prespecified period. This period is called the delayed (or grace) period.

In this talk, we discuss three results related to Parisian ruin problems for spectrally negative Lévy processes. We first discuss Parisian ruin with deficit-dependent delays for bounded variation processes. We obtained the probability of Parisian ruin and the joint Laplace transform of the Parisian ruin time and the deficit at that time. Next, we introduce the concept of “epsilon-Parisian ruin”, which extends the Parisian ruin problem with deficit-dependent delays to unbounded variation processes. Finally, we consider Parisian ruin with arbitrary delays independent of the deficit in the setting of the compound Poisson risk process. We prove a continuity theorem that allows us to approximate the probability of Parisian ruin.