Combined Optimal Stopping and Mixed Regular-Singular Control of Jump Diffusions

In this paper, we examine a model that maximises dividend payments for an insurance company with a debt liability. We assume that the company has a policy to reinvest a proportion of its surplus cash before paying dividends to shareholders. We model the dynamics of the cash reserves as a jump-diffusion process. Combined optimal stopping and mixed regular-singular control of the jump-diffusion process is presented and investigated. In the paper, we show that when the premium rate？？is less than the liability rate , then the company should not get into business and the optimal dividend policy is to immediately pay out the initial cash reserve as dividends to shareholders. For the case , we show that the optimal risk management depends on the current level of the cash reserves. We demonstrate that the company’s optimal dividend policy is to pay out as dividends surplus cash above a predetermined threshold. We also present numerical examples to illustrate the results obtained.