2010/10/25 by Frédéric Abergel, Abergel, Frederic, Aymen Jedidi +1 · 5 citations
Economics, Econometrics and Finance · #Complex Systems and Time Series Analysis #Economic theories and models #FOS: Economics and business #FOS: Mathematics #Probability (math.PR) #Trading and Market Microstructure (q-fin.TR)
paper · pdf · doi:10.48550/arxiv.1010.5136
openalex publication_date 2010/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by the desire 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, we present a mathematical study of the order book as a multidimensional continuous-time Markov chain and derive several mathematical results in the case of independent Poissonian arrival times. In particular, we show that the cancellation structure is an important factor ensuring the existence of a stationary distribution and the exponential convergence towards it. We also prove, by means of the functional central limit theorem (FCLT), that the rescaled-centered price process converges to a Brownian motion. We illustrate the analysis with numerical simulation and comparison against market data.