Rational expectations: an approach of anticipated linear-quadratic social optima

Abstract

Motivated by rational expectations theory, this paper studies a class of stochastic linear-quadratic dynamic optimisation problems involving a large number of weakly-coupled heterogeneous agents. Different from well-studied mean field games, these agents formalise a team with cooperation to minimise some social cost functional. Moreover, the state here evolves by some anticipated backward stochastic differential equation in which the terminal instead initial condition is specified and the anticipated terms are involved. Applying a so-called anticipated person-by-person optimality principle, we construct an auxiliary control problem for each agent based on decentralised information. The decentralised social strategy is derived by a class of new consistency condition systems, which are mean-field-type anticipated forward–backward stochastic differential delay equations (AFBSDDEs). The well-posedness of such consistency condition system is obtained via discounting method. The corresponding asymptotic social optimality is also verified as the population size goes to infinity.