Mean-Field Stochastic Linear Quadratic Control Problem with Random Coefficients

.In this paper, we first prove that the mean-field stochastic linear quadratic (MFSLQ) control problem with random coefficients has a unique optimal control and derive a preliminary stochastic maximum principle to characterize this optimal control by an optimality system. However, because of the term of the form \(E[A_1(· )^ Y(· )]\) in the adjoint equation, which cannot be represented in the form \(E[A_1(· )^ ]E [Y(· )]\), we cannot solve this optimality system explicitly. To this end, we decompose the MFSLQ control problem into two problems without the mean-field terms, and one of them is a constrained problem. The constrained SLQ control problem is solved explicitly by an extended Lagrange multiplier method developed in this article.Keywordsextended Lagrange multiplier methodmean-field controllinear quadratic control problemrandom coefficientRiccati equationMSC codes49N1060H1093E20