Optimal investment and reinsurance for the insurer and reinsurer with the joint exponential utility

In this paper, we consider the problem of optimal investment-reinsurance for the insurer and reinsurer under the stochastic volatility model. The surplus process of the insurer is described by a diffusion model. The insurer can purchase proportional reinsurance from the reinsurer and the premium charged by the insurer and reinsurer follows the variance principle. Both the insurer and reinsurer are allowed to invest in risk-free assets and risky assets, and the market price of risk depends on a Markovian, affine-form, and square-root stochastic factor process. Our goal is to maximize the joint exponential utility of the terminal wealth of the insurer, and reinsurer over a certain period of time. By solving the HJB equation, we obtain the optimal investment-reinsurance strategies, and present the proof of the verification theorem. Finally, we demonstrate a numerical analysis, and the economic implications of our findings are illustrated.