Optimal impulse control for dividend and capital injection with proportional reinsurance and exponential premium principle

This article supposes that a large insurance company can control its surplus process by reinsurance, paying dividends, or injecting capitals. The exponential premium principle and proportional reinsurance are adopted in business activities. We investigate the general situation that the company needs to pay both proportional and fixed costs for dividends and capital injections. The object of the company is to determine an optimal joint reinsurance-dividend-capital injection strategy for maximizing the expected present value of dividends less capital injections until the time of bankruptcy. In both cases of non cheap and cheap reinsurance, we obtain the explicit solutions for value function and optimal strategy.