A Stochastic Partial Differential Equation Model for Limit Order Book Dynamics

Abstract

We propose an analytically tractable class of models for the dynamics of a limit order book, described through a stochastic partial differential equation with multiplicative noise for the order book centered at the mid-price, along with stochastic dynamics for the mid-price which is consistent with the order flow dynamics. We provide conditions under which the model admits a finite-dimensional realization driven by a (low-dimensional) Markov process, leading to efficient methods for estimation and computation. We study two examples of parsimonious models in this class: a two-factor model and a model in which the order book depth is mean reverting. For each model we perform a detailed analysis of the role of different parameters, study the dynamics of the price, order book depth, volume, and order imbalance, provide an intuitive financial interpretation of the variables involved, and show how the model reproduces statistical properties of price changes, market depth, and order flow in limit order markets.