  We propose a model for the dynamics of a limit order book in a liquid market
where buy and sell orders are submitted at high frequency. We derive a
functional central limit theorem for the joint dynamics of the bid and ask
queues and show that, when the frequency of order arrivals is large, the
intraday dynamics of the limit order book may be approximated by a Markovian
jump-diffusion process in the positive orthant, whose characteristics are
explicitly described in terms of the statistical properties of the underlying
order flow. This result allows to obtain tractable analytical approximations
for various quantities of interest, such as the probability of a price increase
or the distribution of the duration until the next price move, conditional on
the state of the order book. Our results allow for a wide range of
distributional assumptions and temporal dependence in the order flow and apply
to a wide class of stochastic models proposed for order book dynamics,
including models based on Poisson point processes, self-exciting point
processes and models of the ACD-GARCH family.
