## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 24 (2019), paper no. 51, 54 pp.

### From the master equation to mean field game limit theory: a central limit theorem

François Delarue, Daniel Lacker, and Kavita Ramanan

#### Abstract

Mean field games (MFGs) describe the limit, as $n$ tends to infinity, of stochastic differential games with $n$ players interacting with one another through their common empirical distribution. Under suitable smoothness assumptions that guarantee uniqueness of the MFG equilibrium, a form of law of large of numbers (LLN), also known as propagation of chaos, has been established to show that the MFG equilibrium arises as the limit of the sequence of empirical measures of the $n$-player game Nash equilibria, including the case when player dynamics are driven by both idiosyncratic and common sources of noise. The proof of convergence relies on the so-called master equation for the value function of the MFG, a partial differential equation on the space of probability measures. In this work, under additional assumptions, we establish a functional central limit theorem (CLT) that characterizes the limiting fluctuations around the LLN limit as the unique solution of a linear stochastic PDE. The key idea is to use the solution to the master equation to construct an associated McKean-Vlasov interacting $n$-particle system that is sufficiently close to the Nash equilibrium dynamics of the $n$-player game for large $n$. We then derive the CLT for the latter from the CLT for the former. Along the way, we obtain a new multidimensional CLT for McKean-Vlasov systems. We also illustrate the broader applicability of our methodology by applying it to establish a CLT for a specific linear-quadratic example that does not satisfy our main assumptions, and we explicitly solve the resulting stochastic PDE in this case.

#### Article information

**Source**

Electron. J. Probab., Volume 24 (2019), paper no. 51, 54 pp.

**Dates**

Received: 27 April 2018

Accepted: 18 March 2019

First available in Project Euclid: 23 May 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1558576902

**Digital Object Identifier**

doi:10.1214/19-EJP298

**Mathematical Reviews number (MathSciNet)**

MR3954791

**Zentralblatt MATH identifier**

07068782

**Subjects**

Primary: 60F05: Central limit and other weak theorems 60H10: Stochastic ordinary differential equations [See also 34F05] 60H15: Stochastic partial differential equations [See also 35R60] 91A13: Games with infinitely many players 91A15: Stochastic games

Secondary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

mean field games master equation McKean-Vlasov interacting particle systems common noise fluctuations central limit theorem systemic risk

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Delarue, François; Lacker, Daniel; Ramanan, Kavita. From the master equation to mean field game limit theory: a central limit theorem. Electron. J. Probab. 24 (2019), paper no. 51, 54 pp. doi:10.1214/19-EJP298. https://projecteuclid.org/euclid.ejp/1558576902