Online 1. Mean field games with common noise [electronic resource] [2015]
 Ahuja, Saran.
 2015.
 Description
 Book — 1 online resource.
 Summary

Mean Field Games (MFG) are a limit of stochastic differential games with a large number of identical players. They were proposed and first studied by Lasry and Lions and independently by Caines, Huang, and Malhame in 2006. They have attracted a lot of interest in the past decades due to their application in many fields. By assuming independence among each agent, taking the limit as N goes to infinity reduces a problem to a fullycoupled system of forwardbackward partial differential equations (PDE). The backward one is a HamiltonJacobiBellman (HJB) equation for the value function of each player while the forward one is the FokkerPlanck (FP) equation for the evolution of the players distribution. This limiting system is more tractable and one can use its solution to approximate the Nash equilibrium strategy of Nplayer games. In this thesis, we consider the MFG model in the presence of common noise, relaxing the usual independence assumption of individual random noise. The presence of common noise clearly adds an extra layer of complexity to the problem as the distribution of players now evolves stochastically. Our first task is proving existence and uniqueness of a Nash equilibrium strategy for this game, showing wellposedness of MFG with common noise. We use a probabilistic approach, namely the Stochastic Maximum Principle (SMP), instead of a PDE approach. This approach gives us a forwardbackward stochastic differential equation (FBSDE) of McKeanVlasov type instead of coupled HJBFP equations. This was first done by Carmona and Delarue in the case of no common noise and we extend their results to MFG with common noise. We are able to extend their results under a linearconvexity framework and a weak monotonicity assumption on the cost functions. In addition to wellposedness results, we also prove the Markov property of McKeanVlasov FBSDE by proving the existence of a decoupling function. In the second part of this thesis, we consider MFG models when the common noise is small. For simplicity, we assume a quadratic running cost function while keeping a general terminal cost function satisfying the same assumptions as in the first part. Our goal is to give an approximation of Nash equilibrium of this game using the solution from the original MFG with no common noise, which could be described through a finitedimensional system of PDEs. We characterize the first order approximation terms as the solution to a linear FBSDE of meanfield type. We then show that the solution to this FBSDE is a centered Gaussian process with respect to the common noise. By assuming regularity of the decoupling function of the 0MFG problem, we can find an explicit solution showing that they are in the form of a stochastic integral with respect to the common noise with the integrands adapted to the information from the 0MFG only. We then are able to compute the covariance function explicitly.
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