Bellman equation and viscosity solutions for mean-field stochastic control problem

Abstract

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 in [32], 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 McKeanVlasov 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.MSC Classification: 93E20, 60H30, 60K35.

Dynamic Programming for Optimal Control of Stochastic McKean--Vlasov Dynamics

“…This leads to a characterization of the solution in terms of an adjoint backward stochastic differential equation (BSDE) coupled with a forward SDE, and we refer to [19] for a theory of BSDE of McKean-Vlasov type. Alternatively, dynamic programming approach for the control of McKean-Vlasov dynamics has been considered in [6], [7], [32] for specific McKean-Vlasov dynamics and under a density assumption on the probability law of the state process, and then analyzed in a general framework in [36] (without noise W 0 ), where the problem is reformulated into a deterministic control problem involving the marginal distribution process.…”

Discrete Time McKean–Vlasov Control Problem: A Dynamic Programming Approach

“…We end this section by mentioning that it would be possible to allow the coefficients in (2.1) and (2.3) to depend more generally on the joint law of the state and control. This is actually considered in the continuoustime version [14] of the McKean-Vlasov control problem.…”

“…The case of continuous time McKean-Vlasov equations requires more technicalities and mathematical tools, and will be addressed in [14]. The discrete time framework has been also considered in [9] for LQ problem, and arises naturally in situations where signal values are available only at certain times.…”

Krasovskii–Subbotin Approach to Mean Field Type Differential Games

“…The link between these two approaches is discussed in [21]. Solutions of Hamilton-Jacobi PDEs in the space of probabilities in the framework of the extrinsic approach were studied in [13], [16], [20], [21], [28]. The intrinsic sub-and superdifferentials were also used for this class of equations (see [14], [15], [26]).…”