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ON BOUNCING GEOMETRIC BROWNIAN MOTIONS

Published online by Cambridge University Press:  27 December 2018

Xin Liu Open the ORCID record for Xin Liu [Opens in a new window] ,
Vidyadhar G. Kulkarni  and
Qi Gong
Xin Liu
Affiliation:
School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC 29634, USA E-mail: [email protected]
Vidyadhar G. Kulkarni
Affiliation:
Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA E-mail: [email protected]; [email protected]
Qi Gong
Affiliation:
Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA E-mail: [email protected]; [email protected]
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Abstract

A pair of bouncing geometric Brownian motions (GBMs) is studied. The bouncing GBMs behave like GBMs except that, when they meet, they bounce off away from each other. The object of interest is the position process, which is defined as the position of the latest meeting point at each time. We study the distributions of the time and position of their meeting points, and show that the suitably scaled logarithmic position process converges weakly to a standard Brownian motion as the bounce size δ→0. We also establish the convergence of the bouncing GBMs to mutually reflected GBMs as δ→0. Finally, applying our model to limit order books, we derive a simple and effective prediction formula for trading prices.

Keywords

diffusion approximationsgeometric brownian motionsinverse gaussian distributionslimit order booksmutually reflected brownian motionsnormal inverse gaussian distributionsreflected brownian motionsrenewal reward processesscaling limits
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 4 , October 2019 , pp. 591 - 617
Copyright
Copyright © Cambridge University Press 2018 

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