Finite Dimensional Approximations of Hamilton-Jacobi-Bellman Equations in Spaces of Probability Measures

We prove that viscosity solutions of Hamilton--Jacobi--Bellman (HJB) equations, corresponding either to deterministic optimal control problems for systems of $n$ particles or to stochastic optimal ...

[1]  Tokinaga Namba,et al.  Stability properties and large time behavior of viscosity solutions of Hamilton–Jacobi equations on metric spaces , 2016, Nonlinearity.

[2]  A. Bensoussan,et al.  On The Interpretation Of The Master Equation , 2015, 1503.07754.

[3]  Jianfeng Zhang,et al.  Viscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controls , 2018, The Annals of Applied Probability.

[4]  Sergio Mayorga.  Short time solution to the master equation of a first order mean field game , 2018, Journal of Differential Equations.

[5]  P. Cardaliaguet.  The Convergence Problem in Mean Field Games with Local Coupling , 2016, Applied Mathematics & Optimization.

[6]  Jin Feng,et al.  Hamilton–Jacobi equations in space of measures associated with a system of conservation laws , 2012 .

[7]  Huyên Pham,et al.  Dynamic Programming for Optimal Control of Stochastic McKean-Vlasov Dynamics , 2016, SIAM J. Control. Optim..

[9]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[10]  A. Bensoussan,et al.  Control problem on space of random variables and master equation , 2015, ESAIM: Control, Optimisation and Calculus of Variations.

[11]  François Delarue,et al.  Probabilistic Theory of Mean Field Games with Applications I: Mean Field FBSDEs, Control, and Games , 2018 .

[12]  Pierre Cardaliaguet,et al.  Long time behavior of the master equation in mean field game theory , 2017, Analysis & PDE.

[13]  Ugo Bessi.  Hamilton-Jacobi in metric spaces with a homological term , 2020, Communications in Partial Differential Equations.

[14]  Huyen Pham,et al.  Backward SDEs for optimal control of partially observed path-dependent stochastic systems: A control randomization approach , 2018, The Annals of Applied Probability.

[15]  Ugo Bessi.  The stochastic value function in metric measure spaces , 2016, 1612.06280.

[16]  Jin Feng,et al.  A Comparison Principle for Hamilton–Jacobi Equations Related to Controlled Gradient Flows in Infinite Dimensions , 2009 .

[17]  Y. Giga,et al.  Eikonal equations in metric spaces , 2011 .

[18]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[19]  W. Gangbo,et al.  Weak KAM Theory on the Wasserstein Torus with Multidimensional Underlying Space , 2014 .

[20]  Pierre Cardaliaguet,et al.  Splitting methods and short time existence for the master equations in mean field games , 2020, Journal of the European Mathematical Society.

[21]  H. Pham,et al.  Bellman equation and viscosity solutions for mean-field stochastic control problem , 2015, 1512.07866.

[22]  L. Szpruch,et al.  Weak quantitative propagation of chaos via differential calculus on the space of measures , 2019, The Annals of Applied Probability.

[23]  Hwa Kil Kim,et al.  Infinite horizon value functions in the Wasserstein spaces , 2013, 1310.3866.

[24]  Huyen Pham,et al.  Zero-sum stochastic differential games of generalized McKean–Vlasov type , 2018, Journal de Mathématiques Pures et Appliquées.

[25]  W. Gangbo,et al.  A partial Laplacian as an infinitesimal generator on the Wasserstein space , 2017, Journal of Differential Equations.

[26]  Ugo Bessi,et al.  Existence of Solutions of the Master Equation in the Smooth Case , 2016, SIAM J. Math. Anal..

[27]  Huyen Pham,et al.  Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem , 2016, Stochastic Processes and their Applications.

[28]  Wilfrid Gangbo,et al.  Hamilton-Jacobi equations in the Wasserstein space , 2008 .

[29]  M. Quincampoix,et al.  Hamilton Jacobi Isaacs equations for differential games with asymmetric information on probabilistic initial condition , 2018 .

[30]  Fausto Gozzi,et al.  Stochastic optimal control in infinite dimension : dynamic programming and HJB equations , 2017 .

[31]  A. Bensoussan,et al.  Mean Field Games and Mean Field Type Control Theory , 2013 .

[32]  K. Ramanan,et al.  From the master equation to mean field game limit theory: a central limit theorem , 2018, Electronic Journal of Probability.

[33]  Juan Li,et al.  Mean-field stochastic differential equations and associated PDEs , 2014, 1407.1215.

[34]  Atsushi Nakayasu.  On metric viscosity solutions for Hamilton-Jacobi equations of evolution type , 2014, 1407.7634.

[35]  T. Kurtz,et al.  Large Deviations for Stochastic Processes , 2006 .

[36]  W. Gangbo,et al.  Optimal transport and large number of particles , 2013 .

[37]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[38]  Hwa Kil Kim,et al.  Value functions in the Wasserstein spaces: finite time horizons , 2013, 1307.4667.

[39]  Jin Feng,et al.  Optimal control for a mixed flow of Hamiltonian and gradient type in space of probability measures , 2013 .

[40]  A. Bensoussan,et al.  Control on Hilbert Spaces and Application to Mean Field Type Control Theory , 2020 .

[41]  Wilfrid Gangbo,et al.  Existence of a solution to an equation arising from the theory of Mean Field Games , 2015 .

[42]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[43]  Wilfrid Gangbo,et al.  On differentiability in the Wasserstein space and well-posedness for Hamilton–Jacobi equations , 2019, Journal de Mathématiques Pures et Appliquées.

[44]  P. Lions,et al.  The Master Equation and the Convergence Problem in Mean Field Games , 2015, 1509.02505.

[45]  Richard C. Kraaij,et al.  A general convergence result for viscosity solutions of Hamilton-Jacobi equations and non-linear semigroups , 2019, Journal of Functional Analysis.

[46]  R. Carmona,et al.  Probabilistic Theory of Mean Field Games with Applications II: Mean Field Games with Common Noise and Master Equations , 2018 .

[47]  T. Mikami,et al.  A Hamilton–Jacobi PDE Associated with Hydrodynamic Fluctuations from a Nonlinear Diffusion Equation , 2019, Communications in Mathematical Physics.

[48]  Alain Bensoussan,et al.  The Master equation in mean field theory , 2014, 1404.4150.

[49]  W. Gangbo,et al.  Metric viscosity solutions of Hamilton–Jacobi equations depending on local slopes , 2015 .

[50]  L. Ambrosio,et al.  On a class of first order Hamilton–Jacobi equations in metric spaces , 2014 .

[51]  Qing Liu,et al.  Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces , 2017, Discrete & Continuous Dynamical Systems - A.

[52]  D. Crisan,et al.  A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria , 2014, Memoirs of the American Mathematical Society.

[53]  W. Gangbo,et al.  Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games , 2020, Communications on Pure and Applied Mathematics.