Chinese Annals of Mathematics, Series B

, Volume 40, Issue 4, pp 515–540 | Cite as

Forward and Backward Mean-Field Stochastic Partial Differential Equation and Optimal Control

  • Maoning Tang
  • Qingxin MengEmail author
  • Meijiao Wang


This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. The authors first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then they establish necessary and sufficient optimality conditions of the control problem in the form of Pontryagin’s maximum principles. To illustrate the theoretical results, the authors apply stochastic maximum principles to study the infinite-dimensional linear-quadratic control problem of mean-field type. Further, an application to a Cauchy problem for a controlled stochastic linear PDE of mean-field type is studied.


Mean-field Stochastic partial differential equation Backward stochastic partial differential equation Optimal control Maximum principle Adjoint equation 

2000 MR Subject Classification

60H15 35R60 93E20 


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  1. [1]
    Andersson, D. and Djehiche, B., A maximum principle for SDEs of mean-field type, Applied Mathematics and Optimization, 63, 2011, 341–356.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Bensoussan, A., Lectures on Stochastic Control, Nonlinear Filtering and Stochastic Control, S.K. Mitter, A. Moro (eds.), Springer Lecture Notes in Mathematics, 972, Springer-Verlag, Berlin, 1982.Google Scholar
  3. [3]
    Buckdahn, R., Djehiche, B., Li, J. and Peng, S., Mean-field backward stochastic differential equations: A limit approach, The Annals of Probability, 37, 2009, 1524–1565.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Buckdahn, R., Djehiche, B. and Li, J., A general stochastic maximum principle for SDEs of mean-field type, Applied Mathematics and Optimization, 64, 2011, 197–216.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Buckdahn, R., Li, J. and Peng, S., Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Processes and Their Applications, 119, 2009, 3133–3154.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Du, H., Huang, J. and Qin, Y., A stochastic maximum principle for delayed mean-field stochastic differential equations and its applications, IEEE Transactions on Automatic Control, 38, 2013, 3212–3217.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Du, K. and Meng, Q., A revisit to-theory of super-parabolic backward stochastic partial differential equations in rd, Stochastic Processes and Their Applications, 120(10), 2010, 1996–2015.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Du, K. and Tang, S., Strong solution of backward stochastic partial differential equations in C 2 domains, Probability Theory and Related Fields, 154(1), 2012, 255–285.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Du, K., Tang, S. and Zhang, Q., W m,p-solution (p ≥ 2) of linear degenerate backward stochastic partial differential equations in the whole space, Journal of Differential Equations, 254(7), 2013, 2877–2904.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Ekeland, I. and Temam, R., Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976.zbMATHGoogle Scholar
  11. [11]
    Elliott, R., Li, X. and Ni, Y. H., Discrete time mean-field stochastic linear-quadratic optimal control problems, Automatica, 49(11), 2013, 3222–3233.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Hu, Y. and Peng, S., Adapted solution of a backward semilinear stochastic evolution equation, Stochastic Analysis and Applications, 9(4), 1991, 445–459.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Kac, M., Foundations of kinetic theory, Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, 3, 1956, 171–197.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Li, J., Stochastic maximum principle in the mean-field controls, Automatica, 48, 2012, 366–373.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    McKean, H. P., A class of Markov processes associated with nonlinear parabolic equations, Proceedings of the National Academy of Sciences, 56, 1966, 1907–1911.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Meng, Q. and Shen, Y., Optimal control of mean-field jump-diffusion systems with delay: A stochastic maximum principle approach, Journal of Computational and Applied Mathematics, 279, 2015, 13–30.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Meyer-Brandis, T., Øksendal, B. and Zhou, X. Y., A mean-field stochastic maximum principle via Malliavin calculus, Stochastics, 84, 2012, 643–666.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Prévôt, C. and Rökner, M., A concise course on stochastic partial differen-tial equations, 1905, Springer-Verlag, Berlin, 2007.Google Scholar
  19. [19]
    Shen, Y., Meng, Q. and Shi, P., Maximum principle for mean-field jump diffusion stochastic delay differential equations and its application to finance, Automatica, 50(6), 2014, 1565–1579.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Shen, Y. and Siu, T. K., The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem, Nonlinear Analysis: Theory, Methods and Applications, 86, 2013, 58–73.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Wang, G., Zhang, C. and Zhang, W., Stochastic maximum principle for mean-field type optimal control under partial information, IEEE Transactions on Automatic Control, 59(2), 2014, 522–528.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Yong, J., Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM Journal on Control and Optimization, 51 (4), 2013, 2809–2838.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Yong, J. and Zhou, X. Y., Stochastic Control: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.CrossRefzbMATHGoogle Scholar

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© The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesHuzhou UniversityHuzhou, ZhejiangChina
  2. 2.Business SchoolUniversity of Shanghai for Science and TechnologyShanghaiChina

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