Linear-Quadratic Delayed Mean-Field Social Optimization

TL;DR

This paper studies a linear quadratic stochastic optimization problem with delay in a large population setting, focusing on social cooperation rather than competitive mean field games, and derives decentralized strategies with proven asymptotic optimality.

Contribution

It introduces a novel delayed person-by-person optimality principle and develops a decentralized control strategy using anticipated forward-backward stochastic differential delay equations.

Findings

Derived a decentralized control strategy for delayed mean-field social optimization.

Proved asymptotic social optimality of the proposed strategy.

Employed a delay-aware discounting method to solve the consistency system.

Abstract

A linear quadratic (LQ) stochastic optimization problem with delay involving weakly-coupled large population is investigated in this paper. Different to classic mean field (MF) game, here agents cooperate with each other to minimize the so-called \emph{social} objective. With the aid of \emph{delayed person-by-person optimality} principle, one arrives at an auxiliary LQ delayed control problem by decentralized information. A decentralized strategy is obtained by feat of an MF type anticipated forward-backward stochastic differential delay equation (AFBSDDE) consistency condition. The discounting method with delay feature is employed to solve the consistency condition system. Finally, by some estimates of AFBSDDEs we derive the asymptotic social optimality.