Stochastic Maximum Principle of Near-Optimal Control of Fully Coupled Forward-Backward Stochastic Differential Equation

Abstract

This paper first makes an attempt to investigate the near-optimal control of systems governed by fully nonlinear coupled forward-backward stochastic differential equations (FBSDEs) under the assumption of a convex control domain. By Ekeland’s variational principle and some basic estimates for state processes and adjoint processes, we establish the necessary conditions for any $\epsilon$-near optimal control in a local form with an error order of exact ${\epsilon }^{1/2}$. Moreover, under additional convexity conditions on Hamiltonian function, we prove that an $\epsilon$-maximum condition in terms of the Hamiltonian in the integral form is sufficient for near-optimality of order ${\epsilon }^{1/2}$.