Long-Time Behavior of a Hawkes Process--Based Limit Order Book

International audienceHawkes processes provide a natural framework to model dependenciesbetween the intensities of point processes. In the contextof order-driven financial markets, the relevance of such dependencieshas been amply demonstrated from an empirical, as well as theoretical,standpoint. In this work, we build on previous empirical and numericalstudies and introduce a mathematical model of limit order books basedon Hawkes processes with exponential kernels. After proving a generalstationarity result, we focus on the long-time behaviour of the limit orderbook and the corresponding dynamics of the suitably rescaled price.A formula for the asymptotic (in time) volatility of the price dynamicsinduced by that of the order book is obtained, involving the average offunctions of the various order book events under the stationary distribution