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

Understand

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 pro\-bability measures recently introduced by P.L.
• Lions in [32], and a special It{\^o} formula for flows of probability measures, we derive the (dynamic programming) Bellman equation for mean-field stochastic control problem, and prove a veri\-fication theorem in our McKean-Vlasov framework.