A mathematical approach to order book modelling

Abstract

We present a mathematical study of the order book as a multidimensional continuous-time Markov chain where the order flow is modelled by independent Poisson processes. Our aim is to bridge the gap between the microscopic description of price formation (agent-based modelling), and the Stochastic Differential Equations approach used classically to describe price evolution in macroscopic time scales. To do this we rely on the theory of infinitesimal generators. 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 the stationarity of the order book and give some hints about the behaviour of the price process in long time scales. © Springer-Verlag Italia 2011.