On the expected uniform error of geometric Brownian motion approximated by the Lévy-Ciesielski construction
On the expected uniform error of geometric Brownian motion approximated by the Lévy-Ciesielski construction
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06.2017Publisher
Institut für Numerische Simulation (INS)Publisher Location
BonnMetadata
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Citable Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11835
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 error. In particular, we show for geometric Brownian motion that at level N , at which there are d = 2N points evaluated on the Brownian path, the expected uniform error has an upper bound of order O(√N/2N ), or equivalently, O(√lnd/d). This upper bound matches the known order for the expected uniform error of the standard Brownian motion. We apply the result to an option pricing example.
Classification (DDC)
510 Mathematik 518 Numerische AnalysisFirst Publication
https://ins.uni-bonn.de/publication/preprintsRelated Publications
https://doi.org/10.48550/arXiv.1706.00915Part of Series
INS Preprints ; 1706Brown, Bruce; Griebel, Michael; Kuo, Frances Y.; Sloan, Ian H.: On the expected uniform error of geometric Brownian motion approximated by the Lévy-Ciesielski construction. In: INS Preprints, 1706.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11835
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11835
@unpublished{handle:20.500.11811/11835,
author = {{Bruce Brown} and {Michael Griebel} and {Frances Y. Kuo} and {Ian H. Sloan}},
title = {On the expected uniform error of geometric Brownian motion approximated by the Lévy-Ciesielski construction},
publisher = {Institut für Numerische Simulation (INS)},
year = 2017,
month = jun,
INS Preprints},
volume = 1706,
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 error. In particular, we show for geometric Brownian motion that at level N , at which there are d = 2N points evaluated on the Brownian path, the expected uniform error has an upper bound of order O(√N/2N ), or equivalently, O(√lnd/d). This upper bound matches the known order for the expected uniform error of the standard Brownian motion. We apply the result to an option pricing example.},
url = {https://hdl.handle.net/20.500.11811/11835}
}
author = {{Bruce Brown} and {Michael Griebel} and {Frances Y. Kuo} and {Ian H. Sloan}},
title = {On the expected uniform error of geometric Brownian motion approximated by the Lévy-Ciesielski construction},
publisher = {Institut für Numerische Simulation (INS)},
year = 2017,
month = jun,
INS Preprints},
volume = 1706,
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 error. In particular, we show for geometric Brownian motion that at level N , at which there are d = 2N points evaluated on the Brownian path, the expected uniform error has an upper bound of order O(√N/2N ), or equivalently, O(√lnd/d). This upper bound matches the known order for the expected uniform error of the standard Brownian motion. We apply the result to an option pricing example.},
url = {https://hdl.handle.net/20.500.11811/11835}
}
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