An Optimization-Based Framework for Solving Forward-Backward Stochastic Differential Equations: Convergence Analysis and Error Bounds

TL;DR

This paper presents an optimization-based method for solving coupled forward-backward stochastic differential equations, providing convergence guarantees and error bounds, validated through analytical tests and high-dimensional stochastic control applications.

Contribution

It introduces an integral-form objective function and proves its equivalence to the solution error, enabling convergence analysis and error estimation for the first time.

Findings

Convergence of the proposed method is theoretically established.

Explicit bounds relate the objective function to solution error.

Successful numerical validation on high-dimensional stochastic control problems.

Abstract

In this paper, we develop an optimization-based framework for solving coupled forward-backward stochastic differential equations. We introduce an integral-form objective function and prove its equivalence to the error between consecutive Picard iterates. Our convergence analysis establishes that minimizing this objective generates sequences that converge to the true solution. We provide explicit upper and lower bounds that relate the objective value to the error between trial and exact solutions. We validate our approach using two analytical test cases and demonstrate its effectiveness by achieving numerical convergence in a nonlinear stochastic optimal control problem with up to 1000 dimensions.