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ON THE EXPECTED UNIFORM ERROR OF BROWNIAN MOTION APPROXIMATED BY THE LÉVY–CIESIELSKI CONSTRUCTION

Part of: Markov processes

Published online by Cambridge University Press:  24 August 2023

BRUCE BROWN Open the ORCID record for BRUCE BROWN [Opens in a new window] ,
MICHAEL GRIEBEL Open the ORCID record for MICHAEL GRIEBEL [Opens in a new window] ,
FRANCES Y. KUO Open the ORCID record for FRANCES Y. KUO [Opens in a new window]  and
IAN H. SLOAN
BRUCE BROWN
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia e-mail: bruce.brown@unsw.edu.au
MICHAEL GRIEBEL
Affiliation:
Institut für Numerische Simulation, Universität Bonn, Bonn, Germany and Fraunhofer Institute SCAI, Schloss Birlinghoven, Sankt Augustin, Germany e-mail: griebel@ins.uni-bonn.de
FRANCES Y. KUO
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia e-mail: f.kuo@unsw.edu.au
IAN H. SLOAN*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia
*
e-mail: i.sloan@unsw.edu.au
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Abstract

The Brownian bridge or Lévy–Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. We focus on the uniform error. In particular, we show constructively that at level N, at which there are $d=2^N$ points evaluated on the Brownian path, the uniform error and its square, and the uniform error of geometric Brownian motion, have upper bounds of order $\mathcal {O}(\sqrt {\ln d/d})$, matching the known orders. We apply the results to an option pricing example.

Keywords

Brownian motionBrownian bridgeLévy–Ciesielski constructionexpected uniform error

MSC classification

Primary: 60J65: Brownian motion

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 3 , June 2024 , pp. 581 - 593
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The authors acknowledge support of the Australian Research Council under project DP210100831. Michael Griebel acknowledges support from the Sydney Mathematical Research Institute.

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