Optimal Reinsurance for Gerber-Shiu Functions in the Cramer-Lundberg Model

TL;DR

This paper develops a method to determine optimal reinsurance strategies in the Cramer-Lundberg model by solving a Hamilton-Jacobi-Bellman equation, enhancing the understanding of minimizing Gerber-Shiu functions.

Contribution

It introduces a stochastic control framework for optimal reinsurance in the Cramer-Lundberg model and proves existence and uniqueness of solutions.

Findings

Existence and uniqueness of the solution to the control problem

Numerical examples with different claim tail behaviors

Insights on asymptotic behavior of the solutions

Abstract

Complementing existing results on minimal ruin probabilities, we minimize expected discounted penalty functions (or Gerber-Shiu functions) in a Cramer-Lundberg model by choosing optimal reinsurance. Reinsurance strategies are modelled as time dependant control functions, which leads to a setting from the theory of optimal stochastic control and ultimately to the problem's Hamilton-Jacobi-Bellman equation. We show existence and uniqueness of the solution found by this method and provide numerical examples involving light and heavy tailed claims and also give a remark on the asymptotics.