Quantitative Convergence for Displacement Monotone Mean Field Games with Controlled Volatility
研究了带共同噪声和受控波动率的平均场博弈的收敛问题，利用最大原理和位移单调性得到前向-后向混沌传播，从而获得开环纳什均衡的定量收敛结果，首次适用于共同噪声受控的博弈。
We study the convergence problem for mean field games with common noise and controlled volatility. We adopt the strategy recently put forth by Laurière and the second author, using the maximum principle to recast the convergence problem as a question of “forward-backward propagation of chaos” (i.e., (conditional) propagation of chaos for systems of particles evolving forward and backward in time). Our main results show that displacement monotonicity can be used to obtain this propagation of chaos, which leads to quantitative convergence results for open-loop Nash equilibria for a class of mean field games. Our results seem to be the first (quantitative or qualitative) that apply to games in which the common noise is controlled. The proofs are relatively simple and rely on a well-known technique for proving wellposedness of forward-backward stochastic differential equations, which is combined with displacement monotonicity in a novel way. To demonstrate the flexibility of the approach, we also use the same arguments to obtain convergence results for a class of infinite horizon discounted mean field games. Funding: J. Jackson is supported by the National Science Foundation [Grant DGE1610403]. L. Tangpi is partially supported by the National Science Foundation [Grants DMS-2005832 and DMS-2143861].