  This paper develops algorithms for high-dimensional stochastic control
problems based on deep learning and dynamic programming. Unlike classical
approximate dynamic programming approaches, we first approximate the optimal
policy by means of neural networks in the spirit of deep reinforcement
learning, and then the value function by Monte Carlo regression. This is
achieved in the dynamic programming recursion by performance or hybrid
iteration, and regress now methods from numerical probabilities. We provide a
theoretical justification of these algorithms. Consistency and rate of
convergence for the control and value function estimates are analyzed and
expressed in terms of the universal approximation error of the neural networks,
and of the statistical error when estimating network function, leaving aside
the optimization error. Numerical results on various applications are presented
in a companion paper (arxiv.org/abs/1812.05916) and illustrate the performance
of the proposed algorithms.
