Mean Field Games of Optimal Trading with an SPDE Model of the Limit Order Book

Abstract

We develop a mean field game model for optimal trading in which the interaction among traders is mediated by the dynamics of the limit order book. The bid- and ask-side order densities are modelled as stochastic processes with values in infinite-dimensional function spaces that solve stochastic partial differential equations (SPDEs) with stochastic order submission rates and multiplicative infinite-dimensional noise. This extends an existing limit order book framework to capture the aggregate impact of the strategic traders on the order densities. Trading activity affects the order densities, which determine the market depths near the best quotes and hence the mid-price, and the mid-price in turn affects the traders' decisions. We establish the well-posedness and regularity of the resulting density-price system and use it to formulate an $N$-player optimal trading game in which the traders' market orders enter the book through their aggregate order flow. We then study the corresponding mean field game as the number of traders tends to infinity, characterize the mean field equilibrium as a fixed point of the associated consistency condition, and prove its existence and uniqueness under suitable conditions. Finally, we show that the mean field equilibrium induces an $ε_N$-Nash equilibrium for the finite-player game, with $ε_N=O(N^{-1/2})$, and provide a numerical illustration of the equilibrium dynamics.