Minimizing the penalized probability of drawdown for a general insurance company under ambiguity aversion

We consider an optimal robust investment and reinsurance problem for a general insurance company which holds shares of an insurance company and a reinsurance company. It is assumed that the decision-maker is ambiguity-averse and does not have perfect information in drift terms of the investment and insurance risks. To capture the ambiguity aversion in the objective function, the criterion of this paper is to minimize a robust value involving the probability of drawdown and a penalization of model uncertainty. By using the technique of stochastic control theory and solving the corresponding boundary-value problems, the closed-form expressions of the optimal strategies are derived explicitly, and a new verification theorem is proved to show that a non-increasing solution to the Hamilton–Jacobi–Bellman equation is indeed our value function. Moreover, we examine theoretically how the level of ambiguity aversion affects the value function and optimal drift distortion. In the end, some numerical examples are exhibited to illustrate the influence of the different investment patterns on our optimal results.