Mixed equilibrium solution of time-inconsistent stochastic linear-quadratic problem

Yuan Hua Ni, Xun Li, Ji Feng Zhang, Miroslav Krstic

Research output: Journal article publicationJournal articleAcademic researchpeer-review

11 Citations (Scopus)


In this paper, we propose a novel equilibrium solution notion for the time-inconsistent stochastic linear-quadratic problem. This notion is called the mixed equilibrium solution, which consists of two parts: a pure-feedback-strategy part and an open-loop-control part. When the pure-feedback-strategy part is zero or the open-loop-control part does not depend on the initial state, the mixed equilibrium solution reduces to the open-loop equilibrium control and the feedback equilibrium strategy, respectively. Using a maximum-principle-like methodology with forward-backward stochastic difference equations, a necessary and sufficient condition is established to characterize the existence of a mixed equilibrium solution. Then, by decoupling the forward-backward stochastic difference equations, three sets of difference equations, which together portray the existence of a mixed equilibrium solution, are obtained. Moreover, the case with a fixed time-state initial pair and the case with all the initial pairs are separately investigated. Furthermore, an example is constructed to show that the mixed equilibrium solution exists for all the initial pairs, although neither the open-loop equilibrium control nor the feedback equilibrium strategy exists for some initial pairs.

Original languageEnglish
Pages (from-to)533-569
Number of pages37
JournalSIAM Journal on Control and Optimization
Issue number1
Publication statusPublished - 7 Feb 2019


  • Equilibrium solution
  • Forward-backward stochastic difference equation
  • Mean-field optimal control
  • Stochastic linear-quadratic optimal control
  • Time inconsistency

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics


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