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Motivated by a zero-intelligence approach, the aim of this paper is toconnect the microscopic (discrete price and volume), mesoscopic (discrete priceand continuous volume) and macroscopic (continuous price and volume) frameworksfor the modelling of limit order books, with a view to providing a naturalprobabilistic description of their behaviour in a high to ultra high-frequencysetting. Starting with a microscopic framework, we first examine the limitingbehaviour of the order book process when order arrival and cancellation ratesare sent to infinity and when volumes are considered to be of infinitesimalsize. We then consider the transition between this mesoscopic model and amacroscopic model for the limit order book, obtained by letting the tick sizetend to zero. The macroscopic limit can then be described using reflected SPDEswhich typically arise in stochastic interface models. We then use financialdata to discuss a possible calibration procedure for the model and illustratenumerically how it can reproduce observed behaviour of prices. This could thenbe used as a market simulator for short-term price prediction or for testingoptimal execution strategies.

Limit order books, diffusion approximations and reflected SPDEs: from microscopic to macroscopic models