Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-27T16:30:39.039Z Has data issue: false hasContentIssue false
  • English
  • Français

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: xliu9@clemson.edu
Vidyadhar G. Kulkarni
Affiliation:
Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA E-mail: vkulkarn@email.unc.edu; qgong@email.unc.edu
Qi Gong
Affiliation:
Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA E-mail: vkulkarn@email.unc.edu; qgong@email.unc.edu
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Abergel, F. & Jedidi, A. (2013). A mathematical approach to order book modeling. International Journal of Theoretical and Applied Finance 16(05): 1350025.Google Scholar
2.Barndorff-Nielsen, O.E. (1997). Processes of normal inverse Gaussian type. Finance and Stochastics 2(1): 41–68.Google Scholar
3.Barndorff-Nielsen, O.E., Mikosch, T., & Resnick, S. I. (2001). Lévy processes: theory and applications. Basel, Switzerland: Birkhäuser Basel.Google Scholar
4.Bayer, C., Horst, U., & Qiu, J. (2017). A functional limit theorem for limit order books with state dependent price dynamics. The Annals of Applied Probability 27(5): 2753–2806.Google Scholar
5.Bayraktar, E., Horst, U., & Sircar, R. (2007). Queueing theoretic approaches to financial price fluctuations. In Birge, J.R., & Linetsky, V. (eds.), Handbooks in Operations Research and Management Science 15: 637–677.Google Scholar
6.Becker-kern, P., Meerschaert, M.M., & Scheffler, H. (2004). Limit theorems for coupled continuous time random walks. The Annals of Probability 32(1B): 730–756.Google Scholar
7.Billingsley, P. (1999). Convergence of probability measures, 2nd ed. New York: Wiley.Google Scholar
8.Black, F. & Scholes, M. (1973). The pricing of options and corporate liabilities. The Journal of Political Economy 81(3): 637–654.Google Scholar
9.Brander-Nielsen, O.E. (1994). Normal inverse Gaussian processes and the modelling of stock returns. Research report. Department of Mathematical Sciences, Aarhus University.Google Scholar
10.Brander-Nielsen, O.E. (1996). Normal inverse Gaussian distributions and stochastic volatility modelling. Scandinavian Journal of Statistics 24: 1–13.Google Scholar
11.Brown, M. & Solomon, H. (1975). A second-order approximation for the variance of a renewal reward process. Stochastic Processes and their Applications 3(3): 301–314.Google Scholar
12.Burdzy, K. & Nualart, D. (2002). Brownian motion reflected on Brownian motion. Probability Theory and Related Fields 122(4): 471–493.Google Scholar
13.Casella, G. & Berger, R. (2002). Statistical inference. 2nd ed. Pacific Grove, CA: Duxbury Press.Google Scholar
14.Cont, R. & De Larrard, A. (2013). Price dynamics in a Markovian limit order market. SIAM Journal on Financial Mathematics 4(1): 1–25.Google Scholar
15.Cont, R. & Larrard, A.D. (2012. Order book dynamics in liquid markets: Limit theorems and diffusion approximations. Preprint, SSRN 1757861.Google Scholar
16.Cont, R., Stoikov, S., & Talreja, R. (2010). A stochastic model for order book dynamics. Operations Research 58(3): 549–563.Google Scholar
17.Decamps, M., De Schepper, A., & Goovaerts, M. (2006). A path integral approach to asset-liability management. Physica A: Statistical Mechanics and its Applications 363(2): 404–416.Google Scholar
18.Dupuis, P. & Ishii, H. (1991). On lipschitz continuity of the solution mapping to the skorokhod problem, with applications. Stochastics: An International Journal of Probability and Stochastic Processes 35(1): 31–62.Google Scholar
19.Horst, U. & Kreher, D. (2017). A weak law of large numbers for a limit order book model with fully state dependent order dynamics. SIAM Journal on Financial Mathematics 8(1): 314–343.Google Scholar
20.Horst, U. & Paulsen, M. (2017). A law of large numbers for limit order books. Mathematics of Operations Research 42(4): 1280–1312.Google Scholar
21.Iglehart, D.L. & Whitt, W. (1971). The equivalence of functional central limit theorems for counting processes and associated partial sums. The Annals of Mathematical Statistics 42(4): 1372–1378.Google Scholar
22.Jacod, J. & Shiryaev, A.N. (2003). Limit theorem for stochastic processes. 2nd ed. Berlin: Springer-Verlag.Google Scholar
23.Karatzas, I. & Shreve, S. (1991). Brownian motion and stochastic calculus 2nd ed. New York: Springer.Google Scholar
24.Karlin, S. & Taylor, H.M. (1975). A first course in stochastic processes 2nd ed. New York: Academic press.Google Scholar
25.Kotulski, M. (1995). Asymptotic distributions of continuous-time random walks: a probabilistic approach. Journal of Statistical Physics 81(3–4): 777–792.Google Scholar
26.Kruk, L. (2003). Functional limit theorems for a simple auction. Mathematics of Operations Research 28(4): 716–751.Google Scholar
27.Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3(1–2): 125–144.Google Scholar
28.Montroll, E.W. & Weiss, G.H. (1965). Random walks on lattices. II. Journal of Mathematical Physics 6(2): 167–181.Google Scholar
29.Rydberg, T.H. (1997). The normal inverse Gaussian Lévy process: simulation and approximation. Stochastic Models 13(4): 887–910.Google Scholar
30.Saisho, Y. & Tanaka, H. (1986). Stochastic differential equations for mutually reflecting Brownian balls. Osaka Journal of Mathematics 23(3): 725–740.Google Scholar
31.Seshadri, V. (1994). The inverse Gaussian distribution: a case study in exponential families. Clarendon Press.Google Scholar