This study examines the optimal investment and dividend problem for an insurer with CRRA preference. The insurer's goal is to maximize the expected discounted accumulated utility from dividend before ruin and the insurer subjects to high gain tax payment. Both the surplus process and the financial market are modulated by an external Markov chain. Using the weak dynamic programming principle (WDPP), we prove that the value function of our control problem is the unique viscosity solution to coupled Hamilton-Jacobi-Bellman (HJB) equations with first derivative constraints. Solving an auxiliary problem without regime switching, we prove that, it is optimal for the insurer in a multiple-regime market to adopt the policies in the same way as in a single-regime market. The regularity of the viscosity solution on its domain is proved and thus the HJB equations admits classical solution. A numerical scheme for the value function is provided by the Markov chain approximation method, two numerical examples are given to illustrate the impact of the high gain tax and regime switching on the optimal policies.

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This article deals with an optimal dividend, reinsurance and capital injection control problem in the diffusion risk model. Under the objective of maximizing the insurance company's value, we aim at finding the joint optimal control strategy. We assume that there exist both the fixed and proportional costs in control processes and the excess-of-loss reinsurance is "expensive". We derive the closed-form solutions of the value function and optimal strategy by using stochastic control methods. Some economic interpretations of the obtained results are also given.

<p style='text-indent:20px;'>This paper discusses the optimal dividend and capital injection problems when an insurance company has two lines of business with common shock dependence. Suppose that the manager of the company has time-inconsistent preference, which can be described by a quasi-hyperbolic discount function. The value of the company is measured by the expected discounted dividend payments minus the expected discounted costs of capital injection. The goal is to find out the optimal strategies for maximizing the value of the company. By using the stochastic control techniques, we solve the problems in all cases of time-consistent preference manager, naive manager and sophisticated manager, respectively. The closed-form solutions of the value functions are presented. Our results show that the sophisticated manager is inclined to pay out dividends earlier than the naive manager, time-inconsistent preference manager is more likely to pay dividends in advance than time-consistent preference manager. Furthermore, we provide some numerical analysis to reveal the sensitivity of the optimal dividend strategies to the dependence intensity and the cost of capital injection. The results show that, as the cost of capital injection increases, sophisticated manager will give up capital injection first, followed by naive manager. In addition, we also find that: when the cost of capital injection is low, managers think that the value of the company that undertakes two kinds of insurance with high correlation is higher; while, managers have opposite evaluations, when the cost of capital injection is high.</p>