  We investigate optimal order execution problems in discrete time with
instantaneous price impact and stochastic resilience. First, in the setting of
linear transient price impact we derive a closed-form recursion for the optimal
strategy, extending the deterministic results from Obizhaeva and Wang (J
Financial Markets, 2013). Second, we develop a numerical algorithm based on
dynamic programming and deep learning for the case of nonlinear transient price
impact as proposed by Bouchaud et al. (Quant. Finance, 2004). Specifically, we
utilize an actor-critic framework that constructs two neural-network (NN)
surrogates for the value function and the feedback control. The flexible
scalability of NN functional approximators enables parametric learning, i.e.,
incorporating several model or market parameters as part of the input space.
Precise calibration of price impact, resilience, etc., is known to be extremely
challenging and hence it is critical to understand sensitivity of the execution
policy to these parameters. Our NN learner organically scales across multiple
input dimensions and is shown to accurately approximate optimal strategies
across a wide range of parameter configurations. We provide a fully
reproducible Jupyter Notebook with our NN implementation, which is of
independent pedagogical interest, demonstrating the ease of use of NN
surrogates in (parametric) stochastic control problems.
