  Stochastic control problems with delay are challenging due to the
path-dependent feature of the system and thus its intrinsic high dimensions. In
this paper, we propose and systematically study deep neural networks-based
algorithms to solve stochastic control problems with delay features.
Specifically, we employ neural networks for sequence modeling (\emph{e.g.},
recurrent neural networks such as long short-term memory) to parameterize the
policy and optimize the objective function. The proposed algorithms are tested
on three benchmark examples: a linear-quadratic problem, optimal consumption
with fixed finite delay, and portfolio optimization with complete memory.
Particularly, we notice that the architecture of recurrent neural networks
naturally captures the path-dependent feature with much flexibility and yields
better performance with more efficient and stable training of the network
compared to feedforward networks. The superiority is even evident in the case
of portfolio optimization with complete memory, which features infinite delay.
