  Motivated by a zero-intelligence approach, the aim of this paper is to
connect the microscopic (discrete price and volume), mesoscopic (discrete price
and continuous volume) and macroscopic (continuous price and volume) frameworks
for the modelling of limit order books, with a view to providing a natural
probabilistic description of their behaviour in a high to ultra high-frequency
setting. Starting with a microscopic framework, we first examine the limiting
behaviour of the order book process when order arrival and cancellation rates
are sent to infinity and when volumes are considered to be of infinitesimal
size. We then consider the transition between this mesoscopic model and a
macroscopic model for the limit order book, obtained by letting the tick size
tend to zero. The macroscopic limit can then be described using reflected SPDEs
which typically arise in stochastic interface models. We then use financial
data to discuss a possible calibration procedure for the model and illustrate
numerically how it can reproduce observed behaviour of prices. This could then
be used as a market simulator for short-term price prediction or for testing
optimal execution strategies.
