Abstract
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.
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Acknowledgments
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.
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Buckdahn, R., Li, J. & Ma, J. A Stochastic Maximum Principle for General Mean-Field Systems. Appl Math Optim 74, 507–534 (2016). https://doi.org/10.1007/s00245-016-9394-9
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DOI: https://doi.org/10.1007/s00245-016-9394-9