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Modelling NASDAQ series by sparse multifractional Brownian motion. (English) Zbl 1241.62143

Summary: The objective of this paper is to compare the performance of different estimators of the Hurst index for multifractional Brownian motion (mBm), namely, the Generalized Quadratic Variation (GQV) Estimator, Wavelet Estimator and Linear Regression GQV Estimator. These estimators are used in the real financial data set Nasdaq time series from 1971 to the 3rd quarter of 2009. Firstly, we review definitions, properties and statistical studies of fractional Brownian motion (fBm) and mBm. Secondly, a numerical artifact is observed: when we estimate the time varying Hurst index \(H(t)\) for an mBm, sampling fluctuation gives the impression that \(H(t)\) is itself a stochastic process, even when \(H(t)\) is constant. To avoid this artifact, we introduce sparse modelling for mBm and apply it to Nasdaq time series.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62M05 Markov processes: estimation; hidden Markov models
91G70 Statistical methods; risk measures
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
62J05 Linear regression; mixed models
91B84 Economic time series analysis
65C60 Computational problems in statistics (MSC2010)
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