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

Abstract

We study the optimal control of general stochastic McKean-Vlasov equation. Such problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles (players) in mean-field interaction under common noise. Our first main result is to state a dynamic programming principle for the value function in the Wasserstein space of probability measures, which is proved from a flow property of the conditional law of the controlled state process. Next, by relying on the notion of differentiability with respect to probability measures due to P.L. Lions [35], and Itô's formula along a flow of conditional measures, we derive the dynamic programming Hamilton-Jacobi-Bellman equation, and prove the viscosity property together with a uniqueness result for the value function. Finally, we solve explicitly the linear-quadratic stochastic McKean-Vlasov control problem and give an application to an interbank systemic risk model with common noise.MSC Classification: 93E20, 60H30, 60K35.

Krasovskii–Subbotin Approach to Mean Field Type Differential Games

“…Thus, the first condition of Corollary 4.3 is valid. Further, passing to the limit in (27) we obtain the second condition. To show the third condition, notice that by construction of ν (see (28)) and (24)…”

Optimal Stopping of McKean--Vlasov Diffusions via Regression on Particle Systems

“…are generally dependent, but, have the same distribution for each i. As such (14) is indeed can be viewed as an estimator of C * j (cf. (4)).…”

Zero-sum stochastic differential games of generalized McKean–Vlasov type

“…McKean-Vlasov (McKV) control problem (also called mean-field type control problem) has been knowing a surge of interest with the emergence of mean-field game (MFG) theory, see [21], [5], [22], [10]. Such a problem was originally motivated by large population stochastic control under mean-field interaction in the limiting case where the number of agents tends to infinity; now various applications can be found in economics, finance, and also in social sciences for modeling motion of socially interacting individuals and herd behavior.…”