Optimal Dynamic Reinsurance

We consider a classical surplus process where the insurer can choose a different level of reinsurance at the start of each year. We assume the insurer’s objective is to minimise the probability of ruin up to some given time horizon, either in discrete or continuous time. We develop formulae for ruin probabilities under the optimal reinsurance strategy, i.e. the optimal retention each year as the surplus changes and the period until the time horizon shortens. For our compound Poisson process, it is not feasible to evaluate these formulae, and hence determine the optimal strategies, in any but the simplest cases. We show how we can determine the optimal strategies by approximating the (compound Poisson) aggregate claims distributions by translated gamma distributions, and, alternatively, by approximating the compound Poisson process by a translated gamma process.