From the master equation to mean field game limit theory: a central limit theorem
From the master equation to mean field game limit theory: a central limit theorem
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DOI：
10.1214/19-ejp298
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发表时间：
2018-04
影响因子：
1.4
通讯作者：
F. Delarue;D. Lacker;K. Ramanan
中科院分区：
文献类型：
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作者：
F. Delarue;D. Lacker;K. Ramanan
平均场博弈 (MFG) 描述了随机微分博弈的极限，即 $n$ 趋于无穷大，其中 $n$ 玩家通过共同的经验分布相互交互。在保证 MFG 均衡唯一性的适当平滑性假设下，建立了一种大数定律 (LLN)，也称为混沌传播，以表明 MFG 均衡是作为 $n$ 玩家博弈纳什均衡的经验测量序列的极限而出现的，包括当玩家动态由特殊来源和共同来源驱动时的情况 噪音。收敛性证明依赖于 MFG 值函数的所谓主方程，即概率测度空间上的偏微分方程。在这项工作中，在额外的假设下，我们建立了一个函数中心极限定理（CLT），它将 LLN 极限周围的极限波动表征为线性随机偏微分方程的唯一解。关键思想是使用主方程的解来构建关联的 McKean-Vlasov 相互作用的 $n$ 粒子系统，该系统足够接近大 $n$ 的 $n$ 玩家博弈的纳什均衡动态。然后，我们从前者的 CLT 中推导出后者的 CLT。在此过程中，我们获得了 McKean-Vlasov 系统的新多维 CLT。我们还通过应用我们的方法为不满足我们主要假设的特定线性二次示例建立 CLT 来说明我们的方法的更广泛适用性，并且我们明确地求解了这种情况下产生的随机 PDE。
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.