Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation

We consider a model of a financial corporation which has to find an optimal policy balancing its risk and expected profits. The example treated in this paper is related to an insurance company with the risk control method known in the industry as excess-of-loss reinsurance. Under this scheme the insurance company divert part of its premium stream to another company in exchange of an obligation to pick up that amount of each claim which exceeds a certain level a. This reduces the risk but it also reduces the potential profit. The objective is to make a dynamic choice of a and find the dividend distribution policy, which maximizes the cumulative expected discounted dividend pay-outs. We use diffusion approximation for this optimal control problem, where two situations are considered: <p>(a) The rate of dividend pay-out are unrestricted and in this case mathematically the problem becomes a mixed singular-regular control problem for diffusion processes. Its analytical part is related to a free boundary (Stephan) problem for a linear second order differential equation. The optimal policy prescribes to reinsure using a certain retention level (depending on the reserve) and pay no dividends when the reserve is below some critical level $x_1$ and to pay out everything that exceeds $x_1$. Reinsurance will stop at a level $x_0\leq x_1$ depending on the claim size distribution. <p>(b) The rate of dividend pay-out is bounded by some positive constant $M<\infty$, in which case the problem becomes a regular control problem. Here the optimal policy is to reinsure at a certain rate and pay no dividends when the reserve is below $x_1$ and pay out at maximum rate when the reserve exceeds $x_1$. In this case reinsurance may or may not stop depending on the claim size distribution and the size of M, but in all cases the retention level will remain constant when the reserve exceeds $x_1$.