Ergodicity and Diffusivity of Markovian Order Book Models: A General Framework

Abstract

We present a general Markovian framework for order book modeling. Through our approach, we aim at providing a tool enabling us 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 subproblems: 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 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 dynamic is not assumed to be instantaneous. We first prove under general assumptions the ergodicity of our system. Then we deduce the convergence towards a Brownian motion of the rescaled price process.