Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle

This paper considers the optimal dividend problem with proportional reinsurance and capital injection for a large insurance portfolio. In particular, the reinsurance premium is assumed to be calculated via the variance principle instead of the expected value principle. Our objective is to maximize the expectation of the discounted dividend payments minus the discounted costs of capital injection. This optimization problem is studied in four cases depending on whether capital injection is allowed and whether there exist restrictions on dividend policies. In all cases, closed-form expressions for the value function and optimal dividend and reinsurance policies are obtained. From the results, we see that the optimal dividend distribution policy is of threshold type with a constant barrier, and that the optimal ceded proportion of risk exponentially decreases with the initial surplus and remains constant when the initial surplus exceeds the dividend barrier. Furthermore, we show that the optimization problem without capital injection is the limiting case of the one with capital injection when the proportional transaction cost goes to infinity.