Motivated by applications in queueing theory, we consider a stochasticcontrol problem whose state space is the d-dimensional positive orthant. Thecontrolled process Z evolves as a reflected Brownian motion whose covariancematrix is exogenously specified, as are its directions of reflection from theorthant's boundary surfaces. A system manager chooses a drift vector θ(t) ateach time t based on the history of Z, and the cost rate at time t depends onboth Z(t) and θ(t). In our initial problem formulation, the objective is tominimize expected discounted cost over an infinite planning horizon, afterwhich we treat the corresponding ergodic control problem. Extending theearlier work by Han et al. [Han J, Jentzen A, Weinan E (2018) Solvinghigh-dimensional partial differential equations using deep learning. Proc.Natl. Acad. Sci. USA 115(34):8505-8510], we develop and illustrate asimulation-based computational method that relies heavily on deep neuralnetwork technology. For the test problems studied thus far, our method isaccurate to within a fraction of 1% and is computationally feasible indimensions up to at least d = 30.
