A mathematical framework for modelling order book dynamics

TL;DR

This paper introduces a comprehensive mathematical framework for modeling limit order book dynamics by combining stochastic order flow with deterministic market clearing, enabling flexible simulations and analysis of price behavior.

Contribution

It presents a novel unified framework that integrates order flow as a spatial point process with market clearing via mass transport, encompassing previous models and offering new analytical tools.

Findings

Framework includes existing models as special cases

Provides insights into order flow and price interplay

Enables simulation of various order book scenarios

Abstract

We present a general framework for modelling the dynamics of limit order books, built on the combination of two modelling ingredients: the order flow, modelled as a general spatial point process, and market clearing, modelled via a deterministic mass transport operator acting on distributions of buy and sell orders. At the mathematical level, this corresponds to a natural decomposition of the infinitesimal generator describing the evolution of the limit order book into two operators: the generator of the order flow and the clearing operator. Our model provides a flexible framework for modelling and simulating order book dynamics and studying various scaling limits of discrete order book models. We show that our framework includes previous models as special cases and yields insights into the interplay between order flow and price dynamics.