A maximum principle for SDEs of mean-field type

Daniel Andersson, Boualem Djehiche

Research output: Contribution to journalArticlepeer-review


We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the meanvariance portfolio selection problem.

Original languageEnglish (US)
Pages (from-to)341-356
Number of pages16
JournalApplied Mathematics and Optimization
Issue number3
StatePublished - Jun 2011


  • Maximum principle
  • McKean-Vlasov equation
  • Mean-field SDE
  • Stochastic control
  • Time inconsistent control

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics


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