Control on Hilbert Spaces and Application to Some Mean Field Type Control Problems

Abstract

We propose a new approach to studying classical solutions of the Bellman equation and Master equation for mean field type control problems, using a novel form of the "lifting" idea introduced by P.-L. Lions. Rather than studying the usual system of Hamilton-Jacobi/Fokker-Planck PDEs using analytic techniques, we instead study a stochastic control problem on a specially constructed Hilbert space, which is reminiscent of a tangent space on the Wasserstein space in optimal transport. On this Hilbert space we can use classical control theory techniques, despite the fact that it is infinite dimensional. A consequence of our construction is that the mean field type control problem appears as a special case. Thus we preserve the advantages of the lifiting procedure, while removing some of the difficulties. Our approach extends previous work by two of the coauthors, which dealt with a deterministic control problem for which the Hilbert space could be generic [2].

Mean Field Game Master Equations with Anti-monotonicity Conditions

“…In particular, [17] studied the local well-posedness of the master equations not only for MFGs involving homogeneous minor players but also for MFGs with a major player. It is much more challenging to obtain a global classical solution, we refer to Buckdahn-Li-Peng-Rainer [14], Chassagneux-Crisan-Delarue [23], Cardaliaguet-Delarue-Lasry-Lions [19], Carmona-Delarue [22], Gangbo-Meszaros-Mou-Zhang [32] and, in the realm of potential MFGs, Bensoussan-Graber-Yam [8,9], Gangbo-Meszaros [31]. We also refer to Mou-Zhang [43], Bertucci [12], and Cardaliaguet-Souganidis [20] for global weak solutions which require much weaker regularity on the data, and Bayraktar-Cohen [3], Bertucci-Lasry-Lions [13], Cecchin-Delarue [25], Bertucci [11] for classical or weak solutions of finite state mean field game master equations.…”

“…for any square integrable random variables ξ 1 , ξ 2 . Another type of monotonicity condition,originating in Ahuja [1] and was later sparsely used in the literature, see Ahuja-Ren-Yang [2] and [8,9,31,32],…”

Mean Field Games Master Equations with Non-separable Hamiltonians and Displacement Monotonicity

“…Such a sufficient condition is typically a sort of monotonicity condition, provides uniqueness of solutions to the underlying mean field game system (a phenomenon that heuristically corresponds to the non-crossing of generalized characteristics of the master equation). For a non-exhaustible list of results on the global in time well-posedness theory of mean field games master equations in various settings, we refer the reader to [15,17,18,19], and in the realm of potential mean field games, to [7,8,21]. We also refer to [30] for global existence and uniqueness of weak solutions and to [4,5,6,10] for classical solutions of finite state mean field games master equations.…”

“…However, let us emphasize that this master equation is fundamentally different from the master equation appearing in the theory of mean field games. In the framework of potential master equations and in particular in more classical infinite dimensional control problems on Hilbert spaces, the displacement convexity condition has been used in [7,8] 1 and [21] .…”

Well-posedness of mean field games master equations involving non-separable local Hamiltonians

“…Depending on the techniques used in these works to show the well-posedness of the corresponding master equations, one may group these results into three possible categories. We refer to a non-exhaustive list of works as follows: probabilistic ideas for problems including individual or common noise were used in [13,12,20,33]; variational techniques (based on optimal transport or optimal control theory in Hilbert spaces, for problems without noise or with individual noise) were exploited in [23,31,19,8]; and finally PDE techniques were used in [11,10] to attack problems with common or individual noise. In most of these references, a special hypothesis is assumed on the Hamiltonian H appearing in the master equation, namely it is such that the momentum variable it is separated from the measure variable, i.e.…”