Volume 24, Number 1, January-March 2018
|Page(s)||437 - 461|
|Published online||19 January 2018|
Bellman equation and viscosity solutions for mean-field stochastic control problem★
Laboratoire de Probabilités et Modèles Aléatoires, CNRS, UMR 7599, Université Paris Diderot Case courrier 7012,
Avenue de France,
Paris Cedex 13, France
a Corresponding author: H. Pham - firstname.lastname@example.org and CREST-ENSAE
Revised: 28 January 2017
Accepted: 28 February 2017
We consider the stochastic optimal control problem of McKean−Vlasov stochastic differential equation where the coefficients may depend upon the joint law of the state and control. By using feedback controls, we reformulate the problem into a deterministic control problem with only the marginal distribution of the process as controlled state variable, and prove that dynamic programming principle holds in its general form. Then, by relying on the notion of differentiability with respect to probability measures recently introduced by [P.L. Lions, Cours au Collège de France: Théorie des jeux à champ moyens, audio conference 2006−2012], and a special Itô formula for flows of probability measures, we derive the (dynamic programming) Bellman equation for mean-field stochastic control problem, and prove a verification theorem in our McKean−Vlasov framework. We give explicit solutions to the Bellman equation for the linear quadratic mean-field control problem, with applications to the mean-variance portfolio selection and a systemic risk model. We also consider a notion of lifted viscosity solutions for the Bellman equation, and show the viscosity property and uniqueness of the value function to the McKean−Vlasov control problem. Finally, we consider the case of McKean−Vlasov control problem with open-loop controls and discuss the associated dynamic programming equation that we compare with the case of closed-loop controls.
Mathematics Subject Classification: 93E20 / 60H30 / 60K35
Key words: McKean−Vlasov SDEs / dynamic programming / Bellman Equation / Wasserstein space, viscosity solutions
© EDP Sciences, SMAI 2018
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