
Contents
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2.1 Notations 2.1 Notations
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2.2 Derivatives 2.2 Derivatives
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2.2.1 First-Order Derivatives 2.2.1 First-Order Derivatives
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2.2.2 Second-Order Derivatives 2.2.2 Second-Order Derivatives
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2.2.3 Comments on the Notions of Derivatives 2.2.3 Comments on the Notions of Derivatives
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2.3 Assumptions 2.3 Assumptions
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2.4 Statement of the Main Results 2.4 Statement of the Main Results
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2.4.1 First-Order Master Equation 2.4.1 First-Order Master Equation
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2.4.2 The Mean Field Game System with Common Noise 2.4.2 The Mean Field Game System with Common Noise
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2.4.3 Second-Order Master Equation 2.4.3 Second-Order Master Equation
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2.4.4 The Convergence of the Nash System for N Players 2.4.4 The Convergence of the Nash System for N Players
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Two Presentation of the Main Results
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Published:August 2019
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Abstract
This chapter collects the main results of the master equation for the convergence of the Nash system. It explains the notation used, specifies the notion of derivatives in the space of measures, and describes the assumptions on the data. One of the striking features of the master equation is that it involves derivatives of the unknown with respect to the measure. This chapter also discusses the link between the two notions of derivatives, which have been used in the mean field game (MFG) theory. The main result states that the master equation has a unique classical solution under the regularity and monotonicity assumptions on H, F, and G. Once the master equation has a solution, this solution can be used to build approximate solutions for the Nash system with N-players.
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