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Mathematical Problems in Engineering
Volume 2013, Article ID 794130, 10 pages
Research Article

Identification of Nonstandard Multifractional Brownian Motions under White Noise by Multiscale Local Variations of Its Sample Paths

Industrial Engineering Department, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 133-791, Republic of Korea

Received 18 July 2013; Accepted 9 August 2013

Academic Editor: Ming Li

Copyright © 2013 Kwang-Il Ahn and Kichun Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Hurst exponent and variance are two quantities that often characterize real-life, high-frequency observations. Such real-life signals are generally measured under noise environments. We develop a multiscale statistical method for simultaneous estimation of a time-changing Hurst exponent and a variance parameter in a multifractional Brownian motion model in the presence of white noise. The method is based on the asymptotic behavior of the local variation of its sample paths which applies to coarse scales of the sample paths. This work provides stable and simultaneous estimators of both parameters when independent white noise is present. We also discuss the accuracy of the simultaneous estimators compared with a few selected methods and the stability of computations with regard to adapted wavelet filters.