TY - GEN
T1 - The master equation for large population equilibriums
AU - Carmona, Rene A.
AU - Delarue, François
N1 - Publisher Copyright:
© Springer International Publishing Switzerland 2014.
PY - 2014
Y1 - 2014
N2 - 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.
AB - 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.
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U2 - 10.1007/978-3-319-11292-3_4
DO - 10.1007/978-3-319-11292-3_4
M3 - Conference contribution
AN - SCOPUS:84919448149
T3 - Springer Proceedings in Mathematics and Statistics
SP - 77
EP - 128
BT - Stochastic Analysis and Applications 2014
A2 - Crisan, Dan
A2 - Hambly, Ben
A2 - Zariphopoulou, Thaleia
PB - Springer New York LLC
T2 - Conference on Stochastic Analysis and Applications, 2013
Y2 - 23 September 2013 through 27 September 2013
ER -