Drift Control of High-Dimensional Reflected Brownian Motion: A Computational Method Based on Neural Networks

Motivated by applications in queueing theory, we consider a stochastic control problem whose state space is the d-dimensional positive orthant. The controlled process Z evolves as a reflected Brownian motion whose covariance matrix is exogenously specified, as are its directions of reflection from the orthant’s boundary surfaces. A system manager chooses a drift vector 𝜃⁡(𝑡) at each time t based on the history of Z, and the cost rate at time t depends on both 𝑍⁡(𝑡) and 𝜃⁡(𝑡). In our initial problem formulation, the objective is to minimize expected discounted cost over an infinite planning horizon, after which we treat the corresponding ergodic control problem. Extending the earlier work by Han et al. [Han J, Jentzen A, Weinan E (2018) Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci. USA 115(34):8505–8510], we develop and illustrate a simulation-based computational method that relies heavily on deep neural network technology. For the test problems studied thus far, our method is accurate to within a fraction of 1% and is computationally feasible in dimensions up to at least 𝑑=30.