Optimal Investment-Reinsurance Design Under Asymmetric Nash Bargaining in a Square Root Factor Process for Jump-Diffusion Risk Model

This paper studies an investment-reinsurance contract between an insurer and a reinsurer with asymmetric bargaining power. We assume that the surplus of the insurer follows a jump-diffusion process. To reduce the risk of claims, the insurer can purchase proportional reinsurance, with the reinsurance premium calculated based on the expected value principle. The surplus of the insurer and the reinsurer can be allocated to a financial market consisting of a risk-free asset and a risky asset, respectively. The price processes of the insurer's and reinsurer's risky assets satisfy different square root factor processes. To consider the benefits of both the insurer and the reinsurer, the optimization problem is formulated as an asymmetric Nash bargaining game. To maximize the weighted product of the expected exponential utility of the terminal wealth of both parties, explicit expressions for the Pareto-optimal strategy and the corresponding value function are derived by employing stochastic control techniques and the Hamilton-Jacobi-Bellman (HJB) equation. In addition, we provide equilibrium strategies under several special cases, including cases where only the insurer or the reinsurer is considered, as well as models under the CEV (Constant Elasticity of Variance) model and the Heston stochastic volatility model. Finally, numerical examples illustrate the impact of bargaining power on optimal strategy.