Afrika Matematika

, Volume 29, Issue 1–2, pp 149–174 | Cite as

Optimal control of forward–backward mean-field stochastic delayed systems

  • Nacira Agram
  • Elin Engen Røse


We study methods for solving stochastic control problems of systems offorward–backward mean-field equations with delay, in finite and infinite time horizon.Necessary and sufficient maximum principles under partial information are given. The results are applied to solve a mean-field recursive utility optimal problem.


Optimal control Stochastic delay equation Mean-field Stochastic maximum principle Hamiltonian Advanced backward stochastic equation Partial information 

Mathematics Subject Classification

Primary 93EXX 93E20 60J75 34K50 Secondary 60H10 60H20 49J55 


  1. 1.
    Agram, N., Haadem, S., Øksendal, B., Proske, F.: A maximum principle for infinite horizon delay equations. SIAM J. Math. Anal. 45, 2499–2522 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Agram, N., Øksendal, B.: Infinite horizon optimal control of forward-backward stochastic differential equations with delay. Comput. Appl. Math. 259, 336–349 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Anderson, A., Djehiche, B.: A maximum principle for SDEs of mean-field type. Appl. Math. Optim. 63, 341–356 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Carmona, R., Delarue, F., Lachapelle, A.: Control of Mckean–Vlasov dynamics versus mean field games. Math. Financial Econ. 7(2), 131 166 (2013)MathSciNetMATHGoogle Scholar
  5. 5.
    Dahl, K.R., Mohammed, S.-E.A., Øksendal, B., Røse, E.E.: Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives. J. Funct. Anal. (2016). doi: 10.1016/j.jfa.2016.04.031
  6. 6.
    Duffie, D., Epstein, L.G.: Stochastic differential utility. Econometrica 60(2), 353–394 (1992)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hu, Y., Øksendal, B., Sulem, A.: Singular mean-field control games. Stoch. Anal. Appl. 35(5), 823–851 (2017)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kunita, K.: Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms. In: Real and stochastic analysis, Trends Math., pp. 305–373. Birkhauser, Boston (2004)Google Scholar
  9. 9.
    Meng, Q., Shen, Y.: Optimal control of mean-field jump-diffusion systems with delay: a stochastic maximum principle approach. J. Comput. Appl. Math. 279, 13–30 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Mohammed, S.E.A. : Stochastic Functional Differential Equations, Research Notes in Mathematics, vol. 99. Pitman (Advanced Publishing Program), Boston (1984)Google Scholar
  11. 11.
    Øksendal, B., Sulem, A., Zhang, T.: Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations. Adv. Appl. Probab. 43, 572–596 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Øksendal, B., Sulem, A.: Risk minimization in financial markets modeled by Itô-Lévy processes. Afr. Mat. (2014). doi: 10.1007/s13370-014-0248-9
  13. 13.
    Pardoux, E.: BSDE’s, weak convergence and homogenizations of semilinear PDE’s. In: Clark, F.H., Stern, R.J. (eds.) Nonlinear Analysis, Differential Equations and Control, pp. 503–549. Kluwer Academic, Dordrecht (1999)CrossRefGoogle Scholar
  14. 14.
    Peng, S.: Backward stochastic differential equations and applications to optimal control. Appl. Math. Optim. 27, 125–144 (1993). (978-1-4673-5534-6/13/31.00c 2013 IEEE) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Touboul, J.: Propagation of chaos in neural fields. Ann. Appl. Probab. 24(3), 1298–1328 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsloOsloNorway
  2. 2.University of BiskraBiskraAlgeria

Personalised recommendations