Stochastic maximum principle for generalized mean-field delay control problem

TL;DR

This paper establishes the existence and uniqueness of solutions for generalized mean-field delay stochastic differential equations and derives a stochastic maximum principle for related control problems, incorporating distribution-dependent states.

Contribution

It introduces a stochastic maximum principle for mean-field delay control problems with distribution-dependent states, using Fréchet derivatives and backward stochastic differential equations.

Findings

Proved existence and uniqueness of GMFDSDEs and MFABSDEs.

Derived a stochastic maximum principle for the control problem.

Provided sufficient conditions for optimality under additional assumptions.

Abstract

In this paper, we first give the existence and uniqueness theorems for generalized mean-filed delay stochastic differential equations (GMFDSDEs) and mean-field anticipated backward stochastic differential equations (MFABSDEs). Then we study the stochastic maximum principle for generalized mean-filed delay control problem. Since the state is distribution-depending, we define the adjoint equation as a MFABSDE, in which, all the derivatives of coefficients are in Fr\'echet sense. We deduce the stochastic maximum principle, and also obtain, under some additional assumptions, a sufficient condition for the optimality of the control.