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Analysis of the Discrete Ornstein-Uhlenbeck Process Caused by the Tick Size Effect

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Daniel Wei-Chung Miao
Daniel Wei-Chung Miao*
Affiliation:
National Taiwan University of Science and Technology
*
∗ Current address: Imperial College Business School, Tanaka Building, South Kensington Campus, London SW7 2AZ, UK. Email address: miao@mail.ntust.edu.tw
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Abstract

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This paper provides an analysis on a discrete version of the Ornstein-Uhlenbeck (OU) process which reflects the small discrete movements caused by the tick size effect. This discrete OU process is derived from matching the first two moments to those of the standard OU process in an infinitesimal sense. We discuss the distributional convergence from the discrete to the continuous processes, and show that the convergence speed is in the second order of the step (tick) size. We also provide some analytical results for the proposed discrete OU process itself, including the closed-form formula of the moment generating function and a full characterisation of the steady state distribution. These results enable us to examine the convergence order explicitly.

Keywords

Discrete Ornstein-Uhlenbeck processmoment matchingtick sizeconvergence ordermoment generating functionsteady state

MSC classification

Primary: 60G20: Generalized stochastic processes 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60G15: Gaussian processes 60J75: Jump processes
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1102 - 1116
Copyright
© Applied Probability Trust 

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