The master equation for large population equilibriums

Rene A. Carmona, François Delarue

Research output: Chapter in Book/Report/Conference proceedingConference contribution

34 Scopus citations

Abstract

We use a simple N-player stochastic gamewith idiosyncratic and common noises to introduce the concept of Master Equation originally proposed by Lions in his lectures at the Collège de France. Controlling the limit N →∞of the explicit solution of the N-player game, we highlight the stochastic nature of the limit distributions of the states of the players due to the fact that the random environment does not average out in the limit, and we recast the Mean Field Game (MFG) paradigm in a set of coupled Stochastic Partial Differential Equations (SPDEs). The first one is a forward stochastic Kolmogorov equation giving the evolution of the conditional distributions of the states of the players given the common noise. The second is a form of stochastic Hamilton Jacobi Bellman (HJB) equation providing the solution of the optimization problem when the flow of conditional distributions is given. Being highly coupled, the system reads as an infinite dimensional Forward Backward Stochastic Differential Equation (FBSDE). Uniqueness of a solution and its Markov property lead to the representation of the solution of the backward equation (i.e. the value function of the stochastic HJB equation) as a deterministic function of the solution of the forward Kolmogorov equation, function which is usually called the decoupling field of the FBSDE. The (infinite dimensional) PDE satisfied by this decoupling field is identified with the master equation.We also show that this equation can be derived for other large populations equilibriums like those given by the optimal control of McKean-Vlasov stochastic differential equations. The paper is written more in the style of a review than a technical paper, and we spend more time motivating and explaining the probabilistic interpretation of the Master Equation, than identifying the most general set of assumptions under which our claims are true.

Original languageEnglish (US)
Title of host publicationStochastic Analysis and Applications 2014
EditorsDan Crisan, Ben Hambly, Thaleia Zariphopoulou
PublisherSpringer New York LLC
Pages77-128
Number of pages52
ISBN (Electronic)9783319112916
DOIs
StatePublished - 2014
EventConference on Stochastic Analysis and Applications, 2013 - Oxford, United Kingdom
Duration: Sep 23 2013Sep 27 2013

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume100
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

OtherConference on Stochastic Analysis and Applications, 2013
CountryUnited Kingdom
CityOxford
Period9/23/139/27/13

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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