On the Time Discretization of the Feynman-Kac Forward-Backward Stochastic Differential Equations for Value Function Approximation

Novel numerical estimators are proposed for the forward-backward stochastic differential equations (FBSDE) appearing in the Feynman-Kac representation of the value function. In contrast to the current numerical method approaches based on discretization of the continuous-time FBSDE results, we propose a converse approach, by first obtaining a discrete-time approximation of the on-policy value function, and then developing a discrete-time result which resembles the continuous-time counterpart. This approach yields improved numerical estimators in the function approximation phase, and demonstrates enhanced error analysis for those value function estimators. Numerical results and error analysis are demonstrated on a scalar nonlinear stochastic optimal control problem, and they show improvements in the performance of the proposed estimators in comparison with the state-of-the-art methodologies.