SIAM Journal on Financial Mathematics, Volume 12, Issue 2, Page 744-787, January 2021.
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.

中文翻译：

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限价订单簿动力学的随机偏微分方程模型

SIAM 金融数学杂志，第 12 卷，第 2 期，第 744-787 页，2021 年 1 月。
我们为限价订单簿的动态提出了一类分析上易于处理的模型，通过随机偏微分方程描述，订单簿的乘法噪声以中间价格为中心，以及中间价格的随机动态是一致的与订单流动态。我们提供了模型允许由（低维）马尔可夫过程驱动的有限维实现的条件，从而产生有效的估计和计算方法。我们在这门课中研究了两个简约模型的例子：一个双因子模型和一个订单簿深度为均值回归的模型。对于每个模型，我们对不同参数的作用进行详细分析，研究价格、订单深度、交易量和订单不平衡的动态，