Deep learning for high-dimensional continuous-time stochastic optimal control without explicit solution

This paper introduces the GPI-PINN algorithm, a novel numerical scheme for solving continuous-time stochastic optimal control problems in high dimensions when the optimal control does not admit an explicit solution. Combining Physics-Informed Neural Networks with an Actor-Critic structure built upon the Generalized Policy Iteration technique, this successive deep learning algorithm employs two separate neural networks to approximate both the value function and the multidimensional optimal control. This way, the GPI-PINN algorithm manages to achieve a global approximation of the optimal solution across all time and space, which can be evaluated online rapidly. The optimality and convergence of the scheme are demonstrated theoretically and its accuracy and efficacy are shown empirically based on two numerical examples. In particular, we generalize the standard Almgren-Chriss model arising from optimal liquidation in finance by allowing for a price impact model with fully nonlinear temporary and permanent impact functions and by considering a multidimensional setting with numerous co-integrated assets.