Forward-Backward Rapidly-Exploring Random Trees for Stochastic Optimal Control

TL;DR

This paper introduces a novel numerical approach combining RRT and LSMC techniques to efficiently solve forward-backward stochastic differential equations in stochastic optimal control, showing improved convergence.

Contribution

It presents a new method integrating RRT with entropy-weighted LSMC for solving FBSDEs in stochastic control, enhancing accuracy and convergence.

Findings

Significant convergence improvements over previous methods.

Effective handling of non-quadratic running costs.

Successful application to linear and nonlinear problems.

Abstract

We propose a numerical method for the computation of the forward-backward stochastic differential equations (FBSDE) appearing in the Feynman-Kac representation of the value function in stochastic optimal control problems. By the use of the Girsanov change of probability measures, it is demonstrated how a rapidly-exploring random tree (RRT) method can be utilized for the forward integration pass, as long as the controlled drift terms are appropriately compensated in the backward integration pass. Subsequently, a numerical approximation of the value function is proposed by solving a series of function approximation problems backwards in time along the edges of the constructed RRT. Moreover, a local entropy-weighted least squares Monte Carlo (LSMC) method is developed to concentrate function approximation accuracy in regions most likely to be visited by optimally controlled trajectories. The results of the proposed methodology are demonstrated on linear and nonlinear stochastic optimal control problems with non-quadratic running costs, which reveal significant convergence improvements over previous FBSDE-based numerical solution methods.