We study the McKean–Vlasov optimal control problem with common noise which allow the law of the control process to appear in the state dynamics under various formulations: strong and weak ones, Markovian or non-Markovian. By interpreting the controls as probability measures on an appropriate canonical space with two filtrations, we then develop the classical measurable selection, conditioning and concatenation arguments in this new context, and establish the dynamic programming principle under general conditions.
F. Djete acknowledges support from the région Île-de-France.
X. Tan acknowledges support from CUHK startup grant and CUHK Faculty of Science Direct Grant 2019-2020. This work also benefited from support of the ANR project PACMAN ANR–16–CE05–0027.
We are grateful to two anonymous reviewers for useful comments and suggestions.
Most parts of this work had been finished when M. F. Djete and X. Tan were still at University of Paris–Dauphine, whose support is greatly acknowledged.
"McKean–Vlasov optimal control: The dynamic programming principle." Ann. Probab. 50 (2) 791 - 833, March 2022. https://doi.org/10.1214/21-AOP1548