Indefinite Linear Quadratic Mean Field Social Control Problems With Multiplicative Noise

This article studies uniform stabilization and social optimality for linear quadratic (LQ) mean field control problems with multiplicative noise, where agents are coupled via dynamics individual costs. The state weights in cost functionals not limited to be positive semidefinite. leads an indefinite LQ problem, which may still well-posed due deep nature of noise. We first obtain a set forward–backward stochastic differential equations (FBSDEs) from variational analysis, construct feedback by decoupling the FBSDEs. By virtue solutions two Riccati equations, we design decentralized laws, is further shown asymptotically optimal. Some equivalent conditions given systems help matrix inequalities. A numerical example illustrate effectiveness proposed laws.