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

Forward-backward stochastic differential equations have recently become a key focus in the computational field, and their role in continuous-time stochastic optimal control and reinforcement learning has grown increasingly prominent.

In this paper, we develop an optimization-based framework for solving coupled forward-backward stochastic differential equations, which naturally arise in stochastic optimal control through the stochastic maximum principle and related Hamiltonian systems.

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 the proposed objective and its theoretical interpretation using two analytical test cases, and further illustrate its numerical applicability on a nonlinear stochastic optimal control problem with up to 1000 dimensions.