A Stochastic Maximum Principle for General Mean-Field Systems
- 376 Downloads
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.
KeywordsStochastic control Maximum principle Mean-field SDE McKean–Vlasov equation
Mathematics Subject Classification93E20 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.
- 2.Bensoussan, A., Frehse, J., Yam, S.C.P.: Master equation in mean-field theorie, arXiv:1404.4150v1 (2014)
- 5.Buckdahn, R., Li, J., Ma, J. (2016), A Mean-field Stochastic Control Problem with Partial Observations, submittedGoogle Scholar
- 7.Buckdahn, R, Li, J., Peng, S., Rainer, C.: Mean-field stochastic differential equations and associated PDEs. Preprint, arXiv:1407.1215v1 (2014)
- 8.Bahlali, K., Mezerdi, M., Mezerdi, B.: Existence of optimal controls for systems governed by mean-field stochastic differential equations, preprint (2014)Google Scholar
- 9.Cardaliaguet, P.: Notes on Mean Field Games (from P.-L. Lions’ lectures at Collège de France). https://www.ceremade.dauphine.fr/~cardalia/ (2013)
- 10.Cardaliaguet, P.: Weak solutions for first order mean field games with local coupling. http://hal.archives-ouvertes.fr/hal-00827957 (2013)
- 15.Lions, P.-L.: Cours au Collège de France : Théorie des jeu à champs moyens. http://www.college-de-france.fr/default/EN/all/equder/audiovideo.jsp (2013)
- 19.Revus, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (2004)Google Scholar