Dynamic reinsurance design with heterogeneous beliefs under the mean-variance framework

TL;DR

This paper develops a dynamic reinsurance model considering different beliefs between insurer and reinsurer, incorporating moral hazard, and deriving complex optimal contracts using advanced mathematical techniques.

Contribution

It introduces a novel framework for reinsurance design under belief heterogeneity and moral hazard within the mean-variance setting, extending traditional models.

Findings

Optimal contracts are more complex than standard proportional and excess-of-loss types.

Heterogeneous beliefs significantly influence reinsurance strategies.

Numerical examples demonstrate the impact of belief differences on optimal solutions.

Abstract

This paper investigates the dynamic reinsurance design problem under the mean-variance criterion, incorporating heterogeneous beliefs between the insurer and the reinsurer, and introducing an incentive compatibility constraint to address moral hazard. The insurer's surplus process is modeled using the classical Cram\'er-Lundberg risk model, with the option to invest in a risk-free asset. To solve the extended Hamilton-Jacobi-Bellman (HJB) system, we apply the partitioned domain optimization technique, transforming the infinite-dimensional optimization problem into a finite-dimensional one determined by several key parameters. The resulting optimal reinsurance contracts are more complex than the standard proportional and excess-of-loss contracts commonly studied in the mean-variance literature with homogeneous beliefs. By further assuming specific forms of belief heterogeneity, we derive the parametric solutions and obtain a clear optimal equilibrium solution. Finally, we compare our results with models where the insurer and reinsurer share identical beliefs or where the incentive compatibility constraint is relaxed. Numerical examples are provided to illustrate the impact of belief heterogeneity on optimal reinsurance strategies.