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Coupling Limit Order Books and Branching Random Walks

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  30 January 2018

Florian Simatos
Florian Simatos*
Affiliation:
Eindhoven University of Technology
*
∗ Postal address: Mathematics and Computer Science Department, Eindhoven University of Technology, MF 4.097a, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: f.simatos@tue.nl
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Abstract

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We consider a model for a one-sided limit order book proposed by Lakner, Reed and Stoikov (2013). We show that it can be coupled with a branching random walk and use this coupling to answer a nontrivial question about the long-term behavior of the price. The coupling relies on a classical idea of enriching the state space by artificially creating a filiation, in this context between orders of the book, which we believe has the potential of being useful for a broader class of models.

Keywords

Limit order bookbranching random walkbranching random walk with a barrier

MSC classification

Primary: 60J85: Applications of branching processes
Secondary: 60G55: Point processes

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 625 - 639
Copyright
© Applied Probability Trust 

References

Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover, Mineola, NY.Google Scholar
Biggins, J. D. (1976). The first- and last-birth problems for a multitype age-dependent branching process. Adv. Appl. Prob. 8, 446–459.Google Scholar
Biggins, J. D., Lubachevsky, B. D., Shwartz, A. and Weiss, A. (1991). A branching random walk with a barrier. Ann. Appl. Prob. 1, 573–581.Google Scholar
Cont, R., Stoikov, S. and Talreja, R. (2010). A stochastic model for order book dynamics. Operat. Res. 58, 549–563.CrossRefGoogle Scholar
Gould, M. D. et al. (2013). Limit order books. Quant. Finance 13, 1709–1742.Google Scholar
Hammersley, J. M. (1974). Postulates for subadditive processes. Ann. Prob. 2, 652–680.Google Scholar
Kendall, D. G. (1951). Some problems in the theory of queues. J. R. Statist. Soc. B 13, 151–173; discussion: 173–185.Google Scholar
Kingman, J. F. C. (1975). The first birth problem for an age-dependent branching process. Ann. Prob. 3, 790–801.Google Scholar
Lakner, P., Reed, J. and Stoikov, S. (2013). High frequency asymptotics for the limit order book. Submitted.Google Scholar
Yudovina, E. (2012). A simple model of a limit order book. Preprint. Available at http://uk.arxiv.org/abs/1205.7017.Google Scholar