脉冲和正则控制下的最优注资:一种混合策略

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

本文用漂移Brown运动表示公司的现金流,研究了公司的最优注资问题.基于实际情况,本文假设市场上有两种注资类型:脉冲注资和正则注资,同时假设这两种注资都需要支付比例成本,且每次脉冲注资还需支付固定成本.公司决策者要确定公司的注资策略,就需要确定正则注资率(有最大值限制)、注资的时间和脉冲注资量.从控制公司成本的角度出发,决策者需在现金流为正的约束下,寻找最小化注资成本的注资策略.因此,决策者面临一个脉冲和正则控制的混合问题,本文得到了该问题的值函数和最优控制策略,发现最优的注资策略是与模型参数相关的混合注资策略,同时也分析了模型参数对值函数和最优注资策略的敏感性.

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.