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The Master Equation for large population equilibria

Presented by: 
R Carmona Princeton University
Wednesday 8th October 2014 - 17:15 to 17:45
INI Seminar Room 1
We use a simple N-player stochastic game with idiosyncratic and common noises to illustrate the introduction of the concept of Master Equation originally proposed by Lasry & Lions. 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 \textit{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.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons