Optimal stochastic impulse and regular control for capital injections: A hybrid strategy

Abstract

This paper investigates an optimal capital injection problem for a large company whose cash flow is described by a drifted Brownian motion. We firstly bring forward the fact that the decision maker can choose the types of capital injection including regular or/and impulse injections in practice. The decision maker has the option to decide the capital injection rate (with a maximum restriction) of regular injection and the capital injection time and the amount of impulse injection as well. We assume that both types of capital injections have proportional costs, and an additional fixed cost occurs associated with each impulse capital injection. From the point of view of cost control, the purpose of the decision maker is to find a strategy to minimize the cost of capital injections, with the minimal requirement that the firm has a positive cash flow. It leads to a mixed stochastic impulse and regular control problem. Under this criterion, we obtain the value function and the optimal control strategy which is a hybrid type of impulse and regular capital injections. At last, we also show the sensibilities of model parameters to the optimal strategy and the value function.