Ergodicity and diffusivity of Markovian order book models: a general framework

TL;DR

This paper introduces a comprehensive Markovian framework for order book modeling that captures the dynamics of price formation and the link between microscopic order flows and macroscopic asset behavior, including ergodicity and diffusivity.

Contribution

It proposes a novel order book representation and decomposes the modeling into two sub-problems, incorporating state-dependent order flow and non-instantaneous price impact, with proofs of ergodicity and convergence to Brownian motion.

Findings

System is ergodic under general assumptions.

Price process converges to a Brownian motion after rescaling.

Framework captures both microscopic and macroscopic features.

Abstract

We present a general Markovian framework for order book modeling. Through our approach, we aim at providing a tool enabling to get a better understanding of the price formation process and of the link between microscopic and macroscopic features of financial assets. To do so, we propose a new method of order book representation, and decompose the problem of order book modeling into two sub-problems: dynamics of a continuous-time double auction system with a fixed reference price; interactions between the double auction system and the reference price movements. State dependency is included in our framework by allowing the order flow intensities to depend on the order book state. Furthermore, contrary to most existing models, the impact of the order book updates on the reference price dynamics is not assumed to be instantaneous. We first prove that under general assumptions, our system is ergodic. Then we deduce the convergence towards a Brownian motion of the rescaled price process.