Optimal reinsurance to minimize the discounted probability of ruin under ambiguity

We solve an optimal robust reinsurance problem for an ambiguity-averse insurer, who worries about ambiguity in the rate of claim occurrence and who develops an optimal robust reinsurance strategy to minimize the penalized discounted probability of ruin, in which we discount for the time of ruin. Specifically, we minimize the expectation of e ^−δτ 1 _{τ<∞} , in which τ equals the time of ruin and δ measures the insurer's time value. Moreover, we penalize this expectation with an entropic term that accounts for the insurer's ambiguity concerning the claim rate. We suppose that the insurer is allowed to purchase per-claim reinsurance to transfer its risk exposure to a reinsurer. We derive the optimal robust reinsurance strategy and the corresponding value function by applying stochastic dynamic programming and by solving the corresponding boundary-value problem. We prove that, when the reinsurance premium is computed according to the mean–variance premium principle, the optimal robust reinsurance strategy is piecewise linear with respect to the claim size. Under the special case of the expected-value premium principle, excess-of-loss reinsurance is optimal, but under the special case of the variance premium principle, proportional reinsurance is not optimal due to the ambiguity. We also present and discuss two numerical examples to illustrate our results.