Social Optima of Linear Forward-Backward Stochastic System

TL;DR

This paper studies a large population linear quadratic stochastic system driven by forward-backward stochastic differential equations, proposing decentralized strategies and verifying their asymptotic social optimality.

Contribution

It introduces a novel decentralized control approach for large population systems using FBSDEs and Riccati equations, distinct from mean field game methods.

Findings

Derived an auxiliary LQ control problem with decentralized information

Obtained decentralized strategies via FBSDE consistency conditions

Verified asymptotic social optimality of the proposed strategies

Abstract

A linear quadratic (LQ) stochastic optimization system involving large population, which is driven by forward-backward stochastic differential equation (FBSDE), is investigated in this paper. Agents cooperate with each other to minimize the so-called social objective, which is rather different from mean field (MF) game. Employing forward-backward person-by-person optimality principle, we derive an auxiliary LQ control problem by decentralized information. A decentralized strategy is obtained by virtue of an MF-type forward-backward stochastic differential equation consistency condition. Applying Riccati equation decoupling method, we solve the consistency condition system. We also verify the asymptotic social optimality in this framework.