A Mathematical Approach to Order Book Modelling

We present a mathematical study of the order book as a multidimensional continuous- time Markov chain where the order flow is modeled by independent Poisson processes. Our aim is to bridge the gap between the microscopic description of price formation (agent- based modeling), and the Stochastic Differential Equations approach used classically to describe price evolution at macroscopic time scales. To do this, we rely on the theory of infinitesimal generators and Foster-Lyapunov stability criteria for Markov chains. We motivate our approach using an elementary example where the spread is kept constant ("perfect market making"). Then we compute the infinitesimal generator associated with the order book in a general setting, and link the price dynamics to the instantaneous state of the order book. In the last section, we prove that the order book is ergodic - in particular it has a stationary distribution - that it converges to its stationary state exponentially fast, and that the large-scale limit of the price process is a Brownian motion.