Constrained monotone mean--variance investment-reinsurance under the Cram\'er--Lundberg model with random coefficients

TL;DR

This paper develops an optimal investment-reinsurance strategy for insurers under a stochastic market model with jumps, using a monotone mean-variance criterion, and demonstrates the strategy's robustness in jump-diffusion settings.

Contribution

It extends existing mean-variance and MMV models to include jumps and random market coefficients, deriving explicit strategies under convex constraints.

Findings

Optimal strategies are robust to market jumps and randomness.

The MMV and mean-variance criteria yield the same strategies even with jumps.

Backward stochastic differential equations characterize the solutions.

Abstract

This paper studies an optimal investment-reinsurance problem for an insurer (she) under the Cram\'er--Lundberg model with monotone mean--variance (MMV) criterion. At any time, the insurer can purchase reinsurance (or acquire new business) and invest in a security market consisting of a risk-free asset and multiple risky assets whose excess return rate and volatility rate are allowed to be random. The trading strategy is subject to a general convex cone constraint, encompassing no-shorting constraint as a special case. The optimal investment-reinsurance strategy and optimal value for the MMV problem are deduced by solving certain backward stochastic differential equations with jumps. In the literature, it is known that models with MMV criterion and mean--variance criterion lead to the same optimal strategy and optimal value when the wealth process is continuous. Our result shows that the conclusion remains true even if the wealth process has compensated Poisson jumps and the market coefficients are random.