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In this paper we study the combined optimal dividend, capital injection and reinsurance problems in a dynamic setting. The reinsurance premium is assumed to be calculated via the variance principle instead of the expected value principle. The proportional and fixed transaction costs and the salvage value at bankruptcy are included in the model. In both cases of unrestricted dividend rate and restricted dividend rate, we obtain the closed-form solutions of the value function and the optimal joint strategies, which depend on the transaction costs and the profitability in future.

This paper investigates an optimal dividend and capital injection problem in the dual model with a random horizon. Both fixed and proportional costs from the transactions of capital injection are considered. The objective is to maximize the total value of the expected discounted dividends and the penalized discounted capital injections during the horizon, which is described by the minimum of the time of ruin and an exponential random variable. By the fluctuation theory of Lévy processes, the optimal dividend and capital injection strategy is obtained. We also find that the optimal return function can be expressed in terms of the scale functions of Lévy processes. Besides, numerical examples are studied to illustrate our results.

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