Indefinite Mean-Field Stochastic Linear-Quadratic Optimal Control

Yuan Hua Ni, Ji Feng Zhang, Xun Li

Research output: Journal article publicationJournal articleAcademic researchpeer-review

54 Citations (Scopus)


This paper is concerned with the discrete-time indefinite mean-field linear-quadratic optimal control problem. The so-called mean-field type stochastic control problems refer to the problem of incorporating the means of the state variables into the state equations and cost functionals, such as the mean-variance portfolio selection problems. A dynamic optimization problem is called to be nonseparable in the sense of dynamic programming if it is not decomposable by a stage-wise backward recursion. The classical dynamic-programming-based optimal stochastic control methods would fail in such nonseparable situations as the principle of optimality no longer applies. In this paper, we show that both the well-posedness and the solvability of the indefinite mean-field linear-quadratic problem are equivalent to the solvability of two coupled constrained generalized difference Riccati equations and a constrained linear recursive equation. We characterize the optimal control set completely, and obtain a set of necessary and sufficient conditions on the mean-variance portfolio selection problem. The results established in this paper offer a more accurate solution scheme in tackling directly the issue of nonseparability and deriving the optimal policies analytically for the mean-variance-type portfolio selection problems.
Original languageEnglish
Article number6995939
Pages (from-to)1786-1800
Number of pages15
JournalIEEE Transactions on Automatic Control
Issue number7
Publication statusPublished - 1 Jul 2015


  • Indefinite stochastic linear-quadratic optimal control
  • Mean-field theory
  • Multi-period mean-variance portfolio selection

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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