On the expected uniform error of Brownian motion approximated by the Lévy-Ciesielski construction
On the expected uniform error of Brownian motion approximated by the Lévy-Ciesielski construction
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04.2023Publisher
Institut für Numerische SimulationPublisher Location
BonnMetadata
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Citable Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11576
Abstract
It is known that the Brownian bridge or Lévy-Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. In the present article the focus is 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 O ( (ln d / d)^(1/2) ), matching the known orders. We apply the results to an option pricing example.
Classification (DDC)
510 Mathematik 518 Numerische AnalysisFirst Publication
https://ins.uni-bonn.de/publication/preprintsRelated Publications
https://doi.org/10.1017/S0004972723000850Part of Series
INS Preprints ; 2301Brown, Bruce; Griebel, Michael; Kuo, Frances Y.; Sloan, Ian H.: On the expected uniform error of Brownian motion approximated by the Lévy-Ciesielski construction. In: INS Preprints, 2301.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11576
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11576
@unpublished{handle:20.500.11811/11576,
author = {{Bruce Brown} and {Michael Griebel} and {Frances Y. Kuo} and {Ian H. Sloan}},
title = {On the expected uniform error of Brownian motion approximated by the Lévy-Ciesielski construction},
publisher = {Institut für Numerische Simulation},
year = 2023,
month = apr,
INS Preprints},
volume = 2301,
note = {It is known that the Brownian bridge or Lévy-Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. In the present article the focus is 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 O ( (ln d / d)^(1/2) ), matching the known orders. We apply the results to an option pricing example.},
url = {https://hdl.handle.net/20.500.11811/11576}
}
author = {{Bruce Brown} and {Michael Griebel} and {Frances Y. Kuo} and {Ian H. Sloan}},
title = {On the expected uniform error of Brownian motion approximated by the Lévy-Ciesielski construction},
publisher = {Institut für Numerische Simulation},
year = 2023,
month = apr,
INS Preprints},
volume = 2301,
note = {It is known that the Brownian bridge or Lévy-Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. In the present article the focus is 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 O ( (ln d / d)^(1/2) ), matching the known orders. We apply the results to an option pricing example.},
url = {https://hdl.handle.net/20.500.11811/11576}
}
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