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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.

In this paper, we consider the dividends and equity issuances control problem in the perturbed dual model under a penalty for ruin. Transaction costs are incurred by these business activities: dividend is taxed and fixed costs are generated by equity issuance. The objective is to maximize the expected present value of dividends minus the discounted costs of equity issuances and the discounted penalty until the ruin time. We find the joint optimal dividend and equity issuance strategy by solving the control problems of two categories of suboptimal model. Furthermore, we derive the explicit closed solutions for the value functions and the optimal strategies when the jumps are mixed exponential. In particular, we investigate the effects of the penalty, the proportional and fixed transaction costs, the expense rate and the force of interest on the optimal strategies by numerical calculations, and give some interesting economic insights.

We assume that the asset value process of some company is directly related to its stock price dynamics, which can be modeled by geometric Brownian motion. The company can control its asset by paying dividends and injecting capitals, of course both procedures imply proportional and fixed costs for the company. To maximize the expected present value of the dividend payments minus the capital injections until the time of bankruptcy, which is defined as the first time when the asset value falls below the regulation requirement $m $, we seek to find the joint optimal dividend payment and capital injection strategy. By solving the Quasi-variational inequalities, the optimal control problem is addressed, which depends on the parameters of the model and the costs. The sensitivities of transaction costs (such as tax, consulting fees) to the optimal strategy, the expected growth rate and volatility of the firm asset value are also examined, some interesting economic insights are included.

The optimal investment and consumption problem is investigated for an insurance company, which is subject to the payment of high-watermark fee from profit. The objective of insurance company is to maximize the expected cumulated discount utility up to ruin time. The consumption behavior considered in this paper can be viewed as dividend payment of the insurance company. It turns out that the value function of the proposed problem is the viscosity solution to the associated HJB equation. The regularity of the viscosity is discussed and some asymptotic results are provided. With the help of the smooth properties of viscosity solutions, we complete the verification theorem of the optimal control policies and the potential applications of the main result are discussed.

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 Levy 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 Levy processes. Besides, numerical examples are studied to illustrate our results.

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.

In this paper we consider the dividend payments and capital injections control problem in a dual risk model. Such a model might be appropriate for a company that specializes in inventions and discoveries, which pays costs continuously and has occasional profits. The objective is to maximize the expected present value of the dividends minus the discounted costs of capital injections. This paper can be considered as an extension of Yao et al. (2010), we include fixed transaction costs incurred by capital injections in this paper. This leads to an impulse control problem. Using the techniques of quasi-variational inequalities (QVI), this optimal control problem is solved. Numerical solutions are provided to illustrate the idea and methodologies, and some interesting economic insights are included.

In contrast with the classical Cramer–Lundberg model where the premium process is a linear function of time, we consider the ruin probability under the risk model where the aggregate premium consists of both a compound Poisson process and a linear process of time. Moreover, a constant interest force is also taken into account in our model. We restrict ourselves to the case where the claim size is heavy-tailed, i.e., the equilibrium distribution function of the claim size belongs to a wide subclass of the subexponential distribution. An asymptotic formula for the ruin probability is obtained by using the similar method of Kalashnikov and Konstantinides (2000). The asymptotic formula we get here is the same as the one in Asmussen (1998), Kluppelberg and Stadtmuller (1998), and Kalashnikov and Konstantinides (2000) which did not consider the stochastic premium.