High-dimensional Stochastic Newsvendor Problem And Its Deep Learning Resolution

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,this paper proposes an improved algorithm to solve the nonlinear HJB equations,based on the idea of deep learning to solve ordinary PDE.The algorithm introduces the projection of the first derivative and the second derivative of the value function respectively,thus avoiding the gradient explosion problem caused by the nonlinear characteristics of the partial differential equation in the training phase.Based on the deep learning algorithm,this paper solves the nonlinear HJB equation with dimensions up to 6 dimensions.This paper then analyzes the impact of financial hedging on the retailer’s(newsvendor)decision and his final income distribution.In addition,this paper further considers the Stackelberg game between manufacturers and retailers,so as to analyze the impact of financial hedging with Stackelberg game on the internal decisionmaking equilibrium and final income distribution of the supply chain.In a special case,this paper compares the optimal solution based on deep learning algorithm with the analytical solution.The relative error is less than 4%,which shows the accuracy of the improved deep learning algorithm.Finally,this paper uses Monte Carlo simulation method to simulate the wealth distribution at the end of four cases.The numerical results show that appropriate financial hedging can effectively reduce supply chain risk.