Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction

We consider a problem of risk control and dividend optimization for a financial corporation facing a constant liability payment. More specifically we investigate the case of excess-of-loss reinsurance for an insurance company. In this scheme the insurance company diverts a part of its premium stream to another company, the reinsurer, in exchange for an obligation to pick up that amount of each claim which exceeds a certain level a. The objective of the insurer is to maximize the expected present value of total future dividend pay-outs. We consider cases when there is restriction on the rate of dividend pay-outs and when there is no restriction. In both cases we describe explicitly the optimal return function as well as the optimal policy.