High-dimensional stochastic control models for newsvendor problems and deep learning resolution

Abstract

This paper studies continuous-time models for newsvendor problems with dynamic replenishment, financial hedging and Stackelberg competition. These factors are considered simultaneously and the high-dimensional stochastic control models are established. High-dimensional Hamilton-Jacobi-Bellman (HJB) equations are derived for the value functions. To circumvent the curse of dimensionality, a deep learning algorithm is proposed to solve the HJB equations. A projection is introduced in the algorithm to avoid the gradient explosion during the training phase. The deep learning algorithm is implemented for HJB equations derived from the newsvendor models with dimensions up to six. Numerical outcomes validate the algorithm’s accuracy and demonstrate that the high-dimensional stochastic control models can successfully mitigate the risk.