Optimal investment and reinsurance policies for the Cram{\'e}r-Lundberg risk model under monotone mean-variance preference

TL;DR

This paper derives explicit optimal investment and reinsurance strategies for an insurance company under a monotone mean-variance criterion, using dynamic programming and HJBI equations, with practical numerical illustrations.

Contribution

It introduces a novel MMV-based optimization framework for insurance risk management and explicitly solves for optimal strategies using HJBI equations.

Findings

Explicit closed-form solutions for optimal strategies.

Derivation of the MMV efficient frontier.

Numerical example demonstrating practical application.

Abstract

In this paper, an optimization problem for the monotone mean-variance(MMV) criterion is considered in the perspective of the insurance company. The MMV criterion is an amended version of the classical mean-variance(MV) criterion which guarantees the monotonicity of the utility function. With this criterion we study the optimal investment and reinsurance problem which is formulated as a zero-sum game between the insurance company and an imaginary player. We apply the dynamic programming principle to obtain the corresponding Hamilton-Jacobi-Bellman-Isaacs(HJBI) equation. As the main conclusion of this paper, by solving the HJBI equation explicitly, the closed forms of the optimal strategy and the value function are obtained. Moreover, the MMV efficient frontier is also provided. At the end of the paper, a numerical example is presented.