Linear quadratic mean-field-game of backward stochastic differential systems

Kai Du, Jianhui Huang, Zhen Wu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

18 Citations (Scopus)


This paper is concerned with a dynamic game of N weakly-coupled linear backward stochastic differential equation (BSDE) systems involving mean-field interactions. The backward mean-field game (MFG) is introduced to establish the backward decentralized strategies. To this end, we introduce the notations of Hamiltonian-type consistency condition (HCC) and Riccati-type consistency condition (RCC) in BSDE setup. Then, the backward MFG strategies are derived based on HCC and RCC respectively. Under mild conditions, these two MFG solutions are shown to be equivalent. Next, the approximate Nash equilibrium of derived MFG strategies are also proved. In addition, the scalar-valued case of backward MFG is solved explicitly. As an illustration, one example from quadratic hedging with relative performance is further studied.

Original languageEnglish
Pages (from-to)653-678
Number of pages26
JournalMathematical Control and Related Fields
Issue number3-4
Publication statusPublished - Aug 2018


  • Backward mean-field game (BMFG)
  • Backward stochastic differential equation (BSDE)
  • Hamiltonian-type consistency condition (HCC)
  • Riccati-type consistency condition (RCC)
  • ɛ-Nash equilibrium

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics


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