Optimality of a barrier strategy in a spectrally negative L\'evy model with a level-dependent intensity of bankruptcy

TL;DR

This paper investigates a stochastic control problem for spectrally negative Lévy processes with a level-dependent bankruptcy rate, establishing the optimality of a barrier strategy and analyzing its sensitivity through numerical methods.

Contribution

It proves the existence and optimality of a barrier strategy in a Lévy model with level-dependent bankruptcy intensity, extending previous results.

Findings

Barrier strategy is optimal under mild conditions.

Analytical properties of Omega scale functions are derived.

Numerical analysis shows the impact of bankruptcy rate on the strategy.

Abstract

We consider de Finetti's stochastic control problem for a spectrally negative L\'evy process in an Omega model. In such a model, the (controlled) process is allowed to spend time under the critical level but is then subject to a level-dependent intensity of bankruptcy. First, before considering the control problem, we derive some analytical properties of the corresponding Omega scale functions. Second, we prove that exists a barrier strategy that is optimal for this control problem under a mild assumption on the L\'evy measure. Finally, we analyse numerically the impact of the bankruptcy rate function on the optimal strategy.