Optimal Per-Loss Reinsurance for a Risk Model with a Thinning-Dependence Structure

Abstract

In this paper, we consider the optimal reinsurance problem for a risk model with a thinning-dependence structure, where the stochastic sources related to claim occurrence are classified into different groups, and each group may cause a claim in each insurance class with some probability. We assume that the insurer can manage the risk by purchasing per-loss reinsurance, and their aim is to maximize the expected utility of the terminal wealth. By using the technique of stochastic control, we obtain the corresponding Hamilton–Jaccobi–Bellman equation. From the perspective of game theory, we derive the closed-form expression of the optimal strategy for each class of business, which is actually the best response to other given strategies. We also investigate the necessary conditions for optimal strategies and transfer the original optimization problem into a system of equations. Furthermore, we prove that the solution of the system of equations always exists, but may not be unique, and we also study some features of the optimal strategies in special cases and derive several interesting results. Finally, some numerical examples are given to show the impacts of some important parameters on the optimal strategies.

Keywords

• per-loss reinsurance
• thinning dependence
• stochastic optimal control
• the expected utility
• best response