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Applied Mathematics & Optimization

, Volume 63, Issue 3, pp 341–356 | Cite as

A Maximum Principle for SDEs of Mean-Field Type

  • Daniel Andersson
  • Boualem Djehiche
Article

Abstract

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 mean-variance portfolio selection problem.

Keywords

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

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References

  1. 1.
    Ahmed, N.U., Ding, X.: Controlled McKean-Vlasov equations. Commun. Appl. Anal. 5(2), 183–206 (2001) MathSciNetzbMATHGoogle Scholar
  2. 2.
    Basak, S., Chabakauri, G.: Dynamic mean-variance asset allocation. In: EFA 2007 Ljubljana Meetings; AFA 2009 San Francisco Meetings Paper. SSRN: http://ssrn.com/abstract=965926 (2009)
  3. 3.
    Bensoussan, A.: Lectures on stochastic control. In: Mitter, S.K., Moro, A. (eds.) Nonlinear Filtering and Stochastic Control. Springer Lecture Notes in Mathematics, vol. 972. Springer, Berlin (1982) CrossRefGoogle Scholar
  4. 4.
    Björk, T., Murgoci, A.: A general theory of Markovian time inconsistent stochastic control problems. Preprint (2008) Google Scholar
  5. 5.
    Buckdahn, R., Li, J., Peng, S.: Mean-field backward stochastic differential equations and related partial differential equations. Stoch. Process. Appl. 119(10), 3133–3154 (2007) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Buckdahn, R., Djehiche, B., Li, J., Peng, S.: Mean-field backward stochastic differential equations. A limit approach. Ann. Probab. 37(4), 1524–1565 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chighoub, F., Djehiche, B., Mezerdi, B.: The stochastic maximum principle in optimal control of degenerate diffusions with non-smooth coefficients. Random Oper. Stoch. Equ. 17, 35–53 (2008) MathSciNetGoogle Scholar
  8. 8.
    Framstad, N.C., Sulem, A., Øksendal, B.: Sufficient stochastic maximum principle for optimal control of jump diffusions and applications to finance. J. Optim. Theory Appl. 121(1), 77–98 (2004) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Huang, M., Malhamé, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–252 (2006) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Jourdain, B., Méléard, S., Woyczynski, W.: Nonlinear SDEs driven by Lévy processes and related PDEs. Alea 4, 1–29 (2008) zbMATHGoogle Scholar
  11. 11.
    Kantorovich, L.B., Rubinstein, G.S.: On the space of completely additive functions. Vestn. Leningr. Univ., Mat. Meh. Astron. 13(7), 52–59 (1958) Google Scholar
  12. 12.
    Lasry, J.M., Lions, P.L.: Mean-field games. Jpn. J. Math. 2, 229–260 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Sznitman, A.S.: Topics in propagation of chaos. In: Ecôle de Probabilites de Saint Flour, XIX-1989. Lecture Notes in Math, vol. 1464, pp. 165–251. Springer, Berlin (1989) CrossRefGoogle Scholar
  14. 14.
    Yong, J., Zhou, X.Y.: Stochastic Control: Hamiltonian Systems and HJB Equations. Springer, New York (1999) zbMATHGoogle Scholar
  15. 15.
    Zhou, X.Y., Li, D.: Continuous-time mean-variance portfolio selection: a stochastic LQ framework. Appl. Math. Optim. 42, 19–33 (2000) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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