Necessary and sufficient conditions for near-optimality in stochastic control of FBSDEs

We consider control problems for systems governed by a nonlinear forward backward stochastic differential equation (FBSDE). We establish necessary as well as sufficient conditions for near optimality, satisfied by all near optimal controls. These conditions are described by two adjoint processes, corresponding to the forward and backward components and a nearly maximum condition on the Hamiltonian. The proof of the main result is based on Ekeland's variational principle and some estimates on the state and the adjoint processes with respect to the control variable. As is well known, optimal controls may fail to exist even in simple cases. This justifies the use of near optimal controls, which exist under minimal assumptions and are sufficient in most practical cases. Moreover, since there are many nearly optimal controls, it is possible to choose suitable ones, that are convenient for implementation.