Optimal Reinsurance and Dividend Under Model Uncertainty

In this paper, the authors analyze the optimal reinsurance and dividend problem with model uncertainty for an insurer. Here the model uncertainty represents possible deviations between the real market and the assumed model. In addition to the incorporation of model uncertainty into the traditional diffusion surplus process, the authors include a penalty function in the objective function. The proposed goal is to find the optimal reinsurance and dividend strategy that maximizes the expected discounted dividend before ruin in the worst case of all possible scenarios, namely, the worst market. Using a dynamic programming approach, the problem is reduced to solving a Hamilton-Jacob-Bellman-Isaac (HJBI) equation with singular control. This problem is more difficult than the traditional robust control or singular control problem. Here, the authors prove that the value function is the unique solution to this HJBI equation with singular control. Moreover, the authors present a verification theorem when a smooth solution can be found, and derive closed-form solution when the function in the objective function is specified.