scholarly journals Controlled diffusion Mean Field Games with common noise and McKean-Vlasov second order backward SDEs

2021 ◽  
Vol 66 (4) ◽  
pp. 774-805
Author(s):  
Adrien Barrasso ◽  
Adrien Barrasso ◽  
Nizar Touzi ◽  
Nizar Touzi
Keyword(s):  
Mean Field ◽  
Second Order ◽  
Mean Field Games ◽  
Controlled Diffusion ◽  
Common Noise ◽  
Backward Sdes

Рассматривается игра среднего поля с одним внешним шумом, коэффициент диффузии которого включает управление. Доказывается существование слабого решения с релаксацией при некоторых условиях на коэффициент диффузии. Далее, показывается, что при отсутствии этого внешнего шума игра среднего поля описывается обратным стохастическим дифференциальным уравнением типа Маккина-Власова второго порядка.

The Second-Order Master Equation

2019 ◽  
pp. 128-158
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Master Equation ◽  
Mean Field ◽  
Second Order ◽  
Well Posedness ◽  
Mean Field Game ◽  
First Order ◽  
Common Noise ◽  
System P ◽  
The Mean

This chapter investigates the second-order master equation with common noise, which requires the well-posedness of the mean field game (MFG) system. It also defines and analyzes the solution of the master equation. The chapter explains the forward component of the MFG system that is recognized as the characteristics of the master equation. The regularity of the solution of the master equation is explored through the tangent process that solves the linearized MFG system. It also analyzes first-order differentiability and second-order differentiability in the direction of the measure on the same model as for the first-order derivatives. This chapter concludes with further description of the derivation of the master equation and well-posedness of the stochastic MFG system.

scholarly journals An entropy minimization approach to second-order variational mean-field games

2019 ◽  
Vol 29 (08) ◽  
pp. 1553-1583 ◽  
Author(s):  
Jean-David Benamou ◽  
Guillaume Carlier ◽  
Simone Di Marino ◽  
Luca Nenna
Keyword(s):  
Efficient Algorithm ◽  
Mean Field ◽  
Time Discretization ◽  
Second Order ◽  
Mean Field Games ◽  
Time Step ◽  
Entropy Minimization ◽  
Convergence Results ◽  
Quadratic Hamiltonian ◽  
Minimization Approach

We propose an entropy minimization viewpoint on variational mean-field games with diffusion and quadratic Hamiltonian. We carefully analyze the time discretization of such problems, establish [Formula: see text]-convergence results as the time step vanishes and propose an efficient algorithm relying on this entropic interpretation as well as on the Sinkhorn scaling algorithm.

scholarly journals On mean field games with common noise and McKean-Vlasov SPDEs

2019 ◽  
Vol 37 (4) ◽  
pp. 522-549 ◽  
Author(s):  
Vassili N. Kolokoltsov ◽  
Marianna Troeva
Keyword(s):  
Mean Field ◽  
Mean Field Games ◽  
Common Noise

Mean Field Game System with a Common Noise

2019 ◽  
pp. 85-127
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Existence And Uniqueness ◽  
Mean Field ◽  
Mean Field Games ◽  
Random Functions ◽  
Common Noise ◽  
The Common ◽  
The Mean ◽  
Random Flow ◽  
Uniqueness Of A Solution ◽  
Classical Strategy

This chapter talks about the unique solvability of the mean field games (MFGs) system with common noise. In terms of a game with a finite number of players, the common noise describes some noise that affects all the players in the same way, so that the dynamics of one given particle reads a certain master equation. It explains the use of the standard convention from the theory of stochastic processes that consists in indicating the time parameter as an index in random functions. Using a continuation like argument instead of the classical strategy based on the Schauder fixed-point theorem, this chapter investigates the existence and uniqueness of a solution. It discusses the effect of the common noise in randomizing the MFG equilibria so that it becomes a random flow of measures.

scholarly journals Asymptotic analysis of mean field games with small common noise

2018 ◽  
Vol 106 (3-4) ◽  
pp. 205-232
Author(s):  
Saran Ahuja ◽  
Weiluo Ren ◽  
Tzu-Wei Yang
Keyword(s):  
Asymptotic Analysis ◽  
Mean Field ◽  
Mean Field Games ◽  
Common Noise
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