Robust optimal reinsurance in minimizing the penalized expected time to reach a goal

This paper studies a robust optimal reinsurance problem for an ambiguity-averse insurer, who does not have perfect information in the drift term of insurance process. The objective is to minimize the robust value involving the expected time to reach a given capital level before ruin and a penalization of model ambiguity. By using the techniques of stochastic control theory and exponential transformation, we derive the closed-form expressions of the optimal reinsurance strategy and the associated value function for the risk model with cheap reinsurance. For the non-cheap reinsurance, we prove that there exists a "safe level" such that the optimization problem becomes a trivial one when the initial surplus is below this safe level. Therefore, for this case, we focus on solving the corresponding boundary-value problems when the initial surplus is greater than the safe level, and the value function is obtained explicitly as well. Furthermore, we investigate the influence of model ambiguity in theory. Some properties and numerical examples are also presented to show the impact of model parameters on the optimal results.