The Annals of Probability

Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics

René Carmona and François Delarue

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The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of McKean–Vlasov type. Motivated by the recent interest in mean-field games, we highlight the connection and the differences between the two sets of problems. We prove a new version of the stochastic maximum principle and give sufficient conditions for existence of an optimal control. We also provide examples for which our sufficient conditions for existence of an optimal solution are satisfied. Finally we show that our solution to the control problem provides approximate equilibria for large stochastic controlled systems with mean-field interactions when subject to a common policy.

Article information

Ann. Probab., Volume 43, Number 5 (2015), 2647-2700.

Received: March 2013
Revised: March 2014
First available in Project Euclid: 9 September 2015

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Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Stochastic control McKean–Vlasov diffusion stochastic Pontryagin principle mean-field interaction mean-field forward–backward stochastic differential equation


Carmona, René; Delarue, François. Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics. Ann. Probab. 43 (2015), no. 5, 2647--2700. doi:10.1214/14-AOP946.

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