Near-optimality Conditions for Relaxed and Strict Mean-field Singular FBSDEs

Abstract

In this paper, we investigate the relaxed and strict near-optimality conditions for mean-field singular FBSDEs, where the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. This makes the control problem time inconsistent in the sense that the Bellman's optimality principle does not hold. The purpose of this paper is to establish necessary and sufficient conditions of near-optimality for relaxed and strict mean-field singular controls. For strict mean-field singular FBSDEs, whose wellposedness is ensured under the twice continuously differentiable assumptions of coefficients. Then, the moment estimations of variational processes as well as firstorder and second-order adjoint processes are presented by using Burkholder-Davis-Gundy inequality. Further, by introducing Hamiltonian function via Ekeland's variational principle, the necessary near-optimality conditions are established. For relaxed mean-field singular FBSDEs, we first give the definition of admissible set of relaxed singular controls, then use the mapping defined by Dirac measure, we prove that the near-optimal problem of strict singular controls is a particular case of the nearoptimal problem of relaxed singular ones. Further, a well known chattering lemma is introduced. By virtue of this famous lemma addition with the stability of trajectories with respect to the control variable and dominated convergence theorem, necessary as well as sufficient near-optimality conditions for relaxed controls are established.