Optimal investment and reinsurance of an insurer with model uncertainty

We introduce a novel approach to optimal investment-reinsurance problems of an insurance company facing model uncertainty via a game theoretic approach. The insurance company invests in a capital market index whose dynamics follow a geometric Brownian motion. The risk process of the company is governed by either a compound Poisson process or its diffusion approximation. The company can also transfer a certain proportion of the insurance risk to a reinsurance company by purchasing reinsurance. The optimal investment-reinsurance problems with model uncertainty are formulated as two-player, zero-sum, stochastic differential games between the insurance company and the market. We provide verification theorems for the Hamilton-Jacobi-Bellman-Isaacs (HJBI) solutions to the optimal investment-reinsurance problems and derive closed-form solutions to the problems.