A Modified Method of Successive Approximations for Stochastic Recursive Optimal Control Problems

TL;DR

This paper introduces a modified successive approximation method for stochastic recursive optimal control problems, incorporating second-order adjoint processes and BMO martingales to ensure convergence and near-optimality.

Contribution

It develops a new modified MSA algorithm for stochastic control problems with non-convex control domains, providing convergence analysis and near-optimality conditions.

Findings

Proves convergence of the modified MSA algorithm.

Establishes a logarithmic convergence rate in a special case.

Provides near-optimal control conditions for convex control domains.

Abstract

Based on the stochastic maximum principle for the partially coupled forward-backward stochastic control system (FBSCS for short), a modified method of successive approximations (MSA for short) is established for stochastic recursive optimal control problems. The second-order adjoint processes are introduced in the augmented Hamiltonian minimization step since the control domain is not necessarily convex. Thanks to the theory of bounded mean oscillation martingales (BMO martingales for short), we give a delicate proof of the error estimate and then prove the convergence of the modified MSA algorithm. In a special case, we obtain a logarithmic convergence rate. When the control domain is convex and compact, a sufficient condition which makes the control returned from the MSA algorithm be a near-optimal control is given for a class of linear FBSCSs.