Abstract: Studies of optimal reinsurance and/or investment decisions are becoming a significant portion of the mainstream research of insurance and actuarial science. In a regular framework, an insurer is assumed to purchase reinsurance contracts from a reinsurer to reduce the risk of random individual claims, while the insurer may invest in a financial market for a higher rate of return or to hedge the risk of the claims, under certain optimality rules. There is a voluminous literature examining optimal investment and reinsurance strategies for insurers with different objectives, as discussed in Wang et al.[Wang N, Zhang N, Jin Z, et al. (2019). Robust non-zero-sum investment and reinsurance game with default risk. Insurance:Mathematics and Economics, 84, 115-132]. However, the existing studies do not take into account the effect of interactions between an insurer and a reinsurer. In fact, economical and sociological studies have pointed out that human beings or firms always seek for partners, and that such cooperative behaviors have significant impacts on one's decision-making.
In this article, with considering the interests of both insurer's and reinsurer's, we study an investment and reinsurance game between an insurer and a reinsurer. The claim process is modeled by a Brownian motion with drift. The insurer is allowed to purchase the proportional reinsurance to mitigate the underlying risks of the claims; and both the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset whose price dynamics follows the famous Heston stochastic volatility model. The main objectives of the insurer and the reinsurer are to maximize the expected utility of a sum of weighted surplus at terminal time T. To this end. a nonzero-sum cooperative game model has been established.
In order to solve this established game model, dynamic programming principle has been utilized, and it is found that dynamic programming principle for this class of nonzero-sum game problems leads to a non-canonical fixed-point problem of coupled non-linear partial differential equations. Despite the complex structure, the existence of the Nash equilibrium reinsurance-investment strategies and the corresponding value functions of the insurer and the are established in a representative example of the constant absolute risk aversion utility function under a mild condition. Furthermore, closed-form expressions for the equilibrium reinsurance-investment strategies and value functions for both the insurer and the reinsurer are derived, and the pricing of insurance is analyzed by using the relationship between supply and demand of reinsurance contracts in the equilibrium market. Finally, numerical studies are also provided to illustrate the impact of parameters on the Nash equilibrium strategies. The obtained results serve as an interesting complementary case of the analogous analysis in nonzero-sum stochastic differential investment and reinsurance games.

Key words: investment-reinsurance, Heston model, Hamilton-Jacobi-Bellman (HJB) equation, Nash equilibrium