  The approximation of solutions to second order Hamilton--Jacobi--Bellman
(HJB) equations by deep neural networks is investigated. It is shown that for
HJB equations that arise in the context of the optimal control of certain
Markov processes the solution can be approximated by deep neural networks
without incurring the curse of dimension. The dynamics is assumed to depend
affinely on the controls and the cost depends quadratically on the controls.
The admissible controls take values in a bounded set.
