Applications of robust stochastic control in optimal inventory management

Abstract

This paper investigates the general approach that applies the concept of equivalent measures and the change of measure to build robust impulse control models based on the existing optimal impulse control framework. Two dynamic decision making processes are studied in the presence of stochastic uncertainty for ambiguity-averse decision makers. In the case of inventory management, the management team concerns about the model uncertainty and seeks for an optimal robust control that minimizes the cost while maintains the adaptability of the business throughout the approximately infinite time span of operation. In the case of portfolio selection, the investor concerns about the model uncertainty and seeks for an trading strategy that maximizes expected utility of consumption during the finite investment horizon. For both cases, we introduce the notion of relative entropy in recognizing the imposed penalty due to model uncertainty. However, different formulations of the preference function are deployed for the models, given the fact that their value functions are of different forms. We formulate each problems into a set of quasi-variational inequities (QVI). Applying perturbation techniques to the QVIs associated with the inventory problem, we can derive an inventory control strategy that approximates the optimal strategy of the robust optimization problem under the presence of stochastic volatility. For the portfolio selection problem, we simplify it to an alternative problem and the numerical method to solve such problem could be an interest of future research work. In the end, we discuss numerical results to illustrate the practical use of theoretical results and draw economic explanations from the robust control policy.