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Title: Limit order books, diffusion approximations and reflected SPDEs : from microscopic to macroscopic models
Author: Newbury, James
ISNI:       0000 0004 6494 0085
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2016
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Motivated by a zero-intelligence approach, the aim of this thesis is to unify the microscopic (discrete price and volume), mesoscopic (discrete price and continuous volume) and macroscopic (continuous price and volume) frameworks of limit order books, with a view to providing a novel yet analytically tractable description of their behaviour in a high to ultra high-frequency setting. Starting with the canonical microscopic framework, the first part of the thesis examines 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. Mathematically speaking, this amounts to establishing the weak convergence of a discrete-space process to a mesoscopic diffusion limit. This step is initially carried out in a reduced-form context, in other words, by simply looking at the best bid and ask queues, before the procedure is extended to the whole book. This subsequently leads us to the second part of the thesis, which is devoted to the transition between mesoscopic and macroscopic models of limit order books, where the general idea is to send the tick size to zero, or equivalently, to consider infinitely many price levels. The macroscopic limit is then described in terms of reflected SPDEs which typically arise in stochastic interface models. Numerical applications are finally presented, notably via the simulation of the mesocopic and macroscopic limits, which can be used as market simulators for short-term price prediction or optimal execution strategies.
Supervisor: Hambly, Ben Sponsor: University of Oxford
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics ; relative/absolute price grid ; invariant measure ; global/local order flow imbalance ; zero-intelligence model ; local time ; reflected SPDE ; stochastic interface model ; elastic Brownian motion ; diffusion approximation ; mesoscopic/macroscopic system ; heavy traffic limit ; limit order book