# A stochastic partial differential equation model for limit order book dynamics

## TL;DR

This paper introduces a new class of stochastic PDE models for limit order book dynamics that are analytically tractable, allowing for efficient estimation and capturing key market features.

## Contribution

It develops a novel SPDE framework with conditions for finite-dimensional realization, enabling practical modeling of limit order book dynamics with low-dimensional Markov processes.

## Key findings

- Model reproduces statistical properties of price changes
- Analyzes role of parameters in price and order book dynamics
- Provides intuitive financial interpretation of variables

## Abstract

We propose an analytically tractable class of models for the dynamics of a limit order book, described through a stochastic partial differential equation (SPDE) 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 estimation and computation methods. We study two examples of parsimonious models in this class: a two-factor model and a model with mean-reverting order book depth. For each model we analyze in detail the role of different parameters, 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.