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The master equation for large population equilibriums

Author(s): Carmona, Rene; Delarue, F

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Abstract: © Springer International Publishing Switzerland 2014. 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.
Publication Date: 1-Jan-2014
Citation: Carmona, R, Delarue, F. (2014). The master equation for large population equilibriums. Springer Proceedings in Mathematics and Statistics, 100 (77 - 128. doi:10.1007/978-3-319-11292-3_4
DOI: doi:10.1007/978-3-319-11292-3_4
ISSN: 2194-1009
EISSN: 2194-1017
Pages: 77 - 128
Type of Material: Journal Article
Journal/Proceeding Title: Springer Proceedings in Mathematics and Statistics
Version: Author's manuscript

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