This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first...

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.

We analyze a stochastic optimal control problem, where the state process follows a McKean-Vlasov dynamics and the diffusion coefficient can be degenerate. We prove that its value function $V$ admits a nonlinear Feynman-Kac representation in terms of a class of forward-backward stochastic differential equations, with an autonomous forward process. We exploit this probabilistic representation to rigorously prove the dynamic programming principle (DPP) for $V$ . The Feynman-Kac representation we obtain has an important role beyond its intermediary role in obtaining our main result: in fact it would be useful in developing probabilistic numerical schemes for $V$ . The DPP is important in obtaining a characterization of the value function as a solution of a nonlinear partial differential equation (the so-called Hamilton-Jacobi-Belman equation), in this case on the Wasserstein space of measures. We should note that the usual way of solving these equations is through the Pontryagin maximum principle, which requires some convexity assumptions. There were attempts in using the dynamic programming approach before, but these works assumed a priori that the controls were of Markovian feedback type, which helps write the problem only in terms of the distribution of the state process (and the control problem becomes a deterministic problem). In this paper, we will consider open-loop controls and derive the dynamic programming principle in this most general case. In order to obtain the Feynman-Kac representation and the randomized dynamic programming principle, we implement the so-called randomization method, which consists of formulating a new McKean-Vlasov control problem, expressed in weak form taking the supremum over a family of equivalent probability measures. One of the main results of the paper is the proof that this latter control problem has the same value function $V$ of the original control problem.

Recently proposed numerical algorithms for solving high-dimensional nonlinear partial differential equations (PDEs) based on neural networks have shown their remarkable performance. We review some ...