Optimal impulse dividend and capital injection model with proportional and fixed transaction costs

Abstract

In this paper, a hybrid strategy with optimal proportional reinsurance, dividends, and capital injections is discussed by a diffusion model with impulse process of discrete dividend events and discrete capital injections. Due to the presence of fixed transaction costs, the problem is formulated as a more complicated impulse stochastic control problem. By using methodologies from impulse control theory and the techniques of translation transformation, we obtain the explicit solutions of the value function and the corresponding optimal strategy. Firstly, the original problem is transformed into two types of suboptimal impulse control problems, one of which is “the case without capital injections” and the other one of which is “the case with compulsory capital injections.” Then, by verification theorem and the expressions of value functions of these two suboptimal problems, we verify that the optimal solution of the original problem is either “keeping bankruptcy without injections” or “never bankruptcy with injections.” It depends on the parameters of the model.