De Finetti’s Control Problem with Parisian Ruin for Spectrally Negative Lévy Processes

Abstract

We consider de Finetti's stochastic control problem when the (controlled) process is allowed to spend time under the critical level. More precisely, we consider a generalized version of this control problem in a spectrally negative Lévy model with exponential Parisian ruin. We show that, under mild assumptions on the Lévy measure, an optimal strategy is formed by a barrier strategy and that this optimal barrier level is always less than the optimal barrier level when classical ruin is implemented. Also, we give necessary and sufficient conditions for the barrier strategy at level zero to be optimal.

Minimizing the Discounted Probability of Exponential Parisian Ruin via Reinsurance

“…By taking the limit of resulting inequality 1 Dassios and Wu [16] first extended the concept of ruin to this Parisian type of ruin and computed it for a classical risk model with exponential claims and for a diffusion approximation of the classical risk model. Recently, Parisian ruin has been actively studied by, for example, Czarna and Palmowski [14], Loeffen, Czarna, and Palmowski [24], Guerin and Renaud [19], Czarna and Palmowski [15], Baurdoux, Pardo, Pérez, and Renaud [5], and Renaud [26]. Most of this research focuses on calculating the probability of Parisian ruin via excursion theory of spectrally negative Lévy processes and their associated scale functions; none of the above-mentioned research controls the probability of Parisian ruin.…”