Applied Mathematics & Optimization

, Volume 74, Issue 3, pp 507–534 | Cite as

A Stochastic Maximum Principle for General Mean-Field Systems

  • Rainer Buckdahn
  • Juan LiEmail author
  • Jin Ma


In this paper we study the optimal control problem for a class of general mean-field stochastic differential equations, in which the coefficients depend, nonlinearly, on both the state process as well as of its law. In particular, we assume that the control set is a general open set that is not necessary convex, and the coefficients are only continuous on the control variable without any further regularity or convexity. We validate the approach of Peng (SIAM J Control Optim 2(4):966–979, 1990) by considering the second order variational equations and the corresponding second order adjoint process in this setting, and we extend the Stochastic Maximum Principle of Buckdahn et al. (Appl Math Optim 64(2):197–216, 2011) to this general case.


Stochastic control Maximum principle Mean-field SDE McKean–Vlasov equation 

Mathematics Subject Classification

93E20 60H30 60H10 91B28 



Rainer Buckdahn is supported in part by the ANR Project CAESARS (ANR-15-CE05-0024). Juan Li is supported in part by the NSF of P.R.China (No. 11222110), NSFC-RS (No. 11661130148), 111 Project (No. B12023). Jin Ma is supported in part by US NSF Grant #1106853.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de Bretagne-OccidentaleBrest CedexFrance
  2. 2.School of Mathematics and StatisticsShandong UniversityJinanPeople’s Republic of China
  3. 3.School of Mathematics and StatisticsShandong University, WeihaiWeihaiPeople’s Republic of China
  4. 4.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

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