Deep-Control of Memory via Stochastic Optimal Control and Deep Learning

In this survey work, we introduce Stochastic Differential Delay Equations and their impacts on Stochastic Optimal Control problems. We observe time delay in the dynamics of a state process that may correspond to inertia or memory in a financial system. For such systems, we demonstrate two special approaches to handle delayed control problems by applying the Dynamic Programming Principle. Moreover, we clarify the technical challenges rising as a consequence of the conflict between the path-dependent, infinite-dimensional nature of the problem and the necessity of the Markov property. Furthermore, we present two different Deep Learning algorithms to solve targeted delayed control tasks and illustrate the results for a complete memory portfolio optimization problem.