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
$\theta(t)$ at each time $t$ based on the history of $Z$, and the cost rate at
time $t$ depends on both $Z(t)$ and $\theta(t)$. 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 earlier work by Han et al. (Proceedings of the
National Academy of Sciences, 2018, 8505-8510), we develop and illustrate a
simulation-based computational method that relies heavily on deep neural
network technology. For test problems studied thus far, our method is accurate
to within a fraction of one percent, and is computationally feasible in
dimensions up to at least $d=30$.

通过深度神经网络技术开发和展示了一种基于模拟的计算方法来解决随机控制问题，将其准确性扩展到 $d=30$ 的维度。