Applied Mathematics & Optimization

, Volume 79, Issue 1, pp 181–204 | Cite as

Stochastic Control of Memory Mean-Field Processes

  • Nacira Agram
  • Bernt ØksendalEmail author


By a memory mean-field process we mean the solution \(X(\cdot )\) of a stochastic mean-field equation involving not just the current state X(t) and its law \(\mathcal {L}(X(t))\) at time t,  but also the state values X(s) and its law \(\mathcal {L}(X(s))\) at some previous times \(s<t.\) Our purpose is to study stochastic control problems of memory mean-field processes. We consider the space \(\mathcal {M}\) of measures on \(\mathbb {R}\) with the norm \(|| \cdot ||_{\mathcal {M}}\) introduced by Agram and Øksendal (Model uncertainty stochastic mean-field control. arXiv:1611.01385v5, [2]), and prove the existence and uniqueness of solutions of memory mean-field stochastic functional differential equations. We prove two stochastic maximum principles, one sufficient (a verification theorem) and one necessary, both under partial information. The corresponding equations for the adjoint variables are a pair of (time-advanced backward stochastic differential equations (absdes), one of them with values in the space of bounded linear functionals on path segment spaces. As an application of our methods, we solve a memory mean–variance problem as well as a linear–quadratic problem of a memory process.


Mean-field stochastic differential equation Law process Memory Path segment spaces Random probability measures Stochastic maximum principle Operator-valued absde Mean–variance problem 

Mathematics Subject Classification

60H05 60H20 60J75 93E20 91G80 91B70 



This research was carried out with support of the Norwegian Research Council, within the Research Project Challenges in Stochastic Control, Information and Applications (STOCONINF), Project Number 250768/F20.


  1. 1.
    Agram, N.: Stochastic optimal control of McKean–Vlasov equations with anticipating law. arXiv:1604.03582
  2. 2.
    Agram, N., Øksendal, B.: Model uncertainty stochastic mean-field control. arXiv:1611.01385v5
  3. 3.
    Agram, N., Røse, E.E.: Optimal control of forward–backward mean-field stochastic delay systems. arXiv:1412.5291
  4. 4.
    Anderson, D., Djehiche, B.: A maximum principle for SDEs of mean-field type. Appl. Math. Optim. 63, 341–356 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Banos, D.R., Cordoni, F., Di Nunno, G., Di Persio, L., Røse, E.E.: Stochastic systems with memory and jumps. arXiv:1603.00272
  6. 6.
    Buckdahn, R., Li, J., Peng, S.: Mean-field backward stochastic differential equations and related partial differential equations. Stoch. Process. Appl. 119, 3133–3154 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cardaliaguet, P.: Notes on Mean Field Games. Technical report (2010)Google Scholar
  8. 8.
    Carmona, R., Delarue, F.: Control of McKean–Vlasov dynamics versus mean field games. Math. Financ. Econ. 7(2), 131–166 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Carmona, R., Delarue, F.: Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics. arXiv:1303.5835v1
  10. 10.
    Chen, L., Wu, Z.: Maximum principle for the stochastic optimal control problem with delay and application. Automatica 46, 1074–1080 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dahl, K., Mohammed, S., Øksendal, B., Røse, 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
  12. 12.
    Framstad, N.C., Øksendal, B., Sulem, A.: A sufficient maximum principle for optimal control of jump diffusions and applications to finance. J. Optim. Theory Appl. 124(2), 511–512 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hu, Y., Øksendal, B.: Partial information linear quadratic control for jump diffusions. SIAM J. Control Optim. 47(4), 1744–1761 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hu, Y., Øksendal, B., Sulem, A.: Singular mean-field control games with applications to optimal harvesting and investment problems. Stoch. Anal. Appl. arXiv:1406.1863 (to appear)
  15. 15.
    Jeanblanc, M., Lim, T., Agram, N.: Some existence results for advanced backward stochastic differential equations with a jump time. ESAIM Proc. Surv. 60, 84–106 (2017)zbMATHGoogle Scholar
  16. 16.
    Lions, P.-L.: Cours au Collège de France: Théorie des jeux à champs moyens (2013)Google Scholar
  17. 17.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mohammed, S.E.A.: Stochastic Functional Differential Equations. Research Notes in Mathematics, vol. 99. Pitman (Advanced Publishing Program), Boston (1984)Google Scholar
  19. 19.
    Øksendal, B., Sulem, A.: Optimal control of predictive mean-field equations and applications to finance. In: Benth, F.E., Di Nunno, G. (eds) Stochastics of Environmental and Financial Economics, pp. 301–320. Springer, Oslo (2015)Google Scholar
  20. 20.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Peng, S., Yang, Z.: Anticipated backward stochastic differential equations. Ann. Probab. 37(3), 877–902 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Røse, E.E.: Optimal control for mean-field SDEs with jumps and delay. Manuscript, August. University of Oslo (2013)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsloOsloNorway

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