Social optima of linear forward-backward stochastic system

¹ School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China
² School of Management, Shandong University, Jinan, Shandong 250100, China
³ Department of Mathematics and SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China

^* Corresponding author: wangshujun@sdu.edu.cn

Received: 29 April 2024
Accepted: 18 August 2025

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.