A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria

By Jean-François Chassagneux, D. Crisan, Franccois Delarue. Memoirs of the American Mathematical Society, 2014

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We analyze a class of nonlinear partial differential equations (PDEs) defined on R d × P 2 ( R d ) , \mathbb {R}^d \times \mathcal {P}_2(\mathbb {R}^d), where P 2 ( R d ) \mathcal {P}_2(\mathbb {R}^d) is the Wasserstein space of probability measures on R d \mathbb {R}^d with a finite second-order moment. We show that such equations admit a classical solutions for sufficiently small time intervals. Under additional constraints, we prove that their solution can be extended to arbitrary large intervals. These nonlinear PDEs arise in the recent developments in the theory of large population stochastic control. More precisely they are the so-called master equations corresponding to asymptotic equilibria for a large population of controlled players with mean-field interaction and subject to minimization constraints. The results in the paper are deduced by exploiting this connection. In particular, we study the differentiability with respect to the initial condition of the flow generated by a forward-backward stochastic system of McKean-Vlasov type. As a byproduct, we prove that the decoupling field generated by the forward-backward system is a classical solution of the corresponding master equation. Finally, we give several applications to mean-field games and to the control of McKean-Vlasov diffusion processes.

Subfields: Finance, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/b50f41c998230b15330e649338f1bbd0a4af617c

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Citations in the corpus

Viscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controls (2018, background): “al example is nonlinear: again for ν∈ P2(Rd) and for ξwith distribution ν, ∂tV(t,ν) + 1 2 Eν tr(∂x∂νV(t,ν,ξ)) +f Eν ∂νV(t,ν,ξ) = 0,V(T,ν) = Eν g(ξ) .(2.32) Based on Chassagneux, Crisan, & Delarue [12], Saporito & Zhang [35] showed that it has a classical solution when Tis small. Here we use ”
Weak quantitative propagation of chaos via differential calculus on the space of measures (2019, methodology): “This leads in particular to the notion of a fully C2 function that will be of great interest for us (see [9]).”
Mean field games master equations with nonseparable Hamiltonians and displacement monotonicity (2021, methodology): “In the context of control problems of McKean-Vlasov type, displacement convexity assumptions appeared first in [24] and [27].”
Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, background): “Nonlinear analogues of these backward PDEs appear in the guise of the master equation from the theory of mean field games [20, 15] and in mean field control problems [60, 17].”
On the convergence of closed-loop Nash equilibria to the mean field game limit (2018, background): “Furthermore, it is very challenging to produce a classical solution of the master equation, and this has been accomplished so far only in quite restricted settings [10, 20, 30].”
Strong well posedness of McKean–Vlasov stochastic differential equations with Hölder drift (2015, methodology): “Since from the arguments of [CCD14] we have that, for all (t, x, μ) ∈ [0, T ]×R×P2(R), the solution of the PDE (4.”