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On local regularity of multifractional Brownian motion: Hurst function estimation using a first difference increment estimator
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2015 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

Multifractional Brownian motion is a type of stochastic process with time-varying regularity. The main focus of this thesis is estimation of the pointwise regularity of such processes. The need for accurate estimation of the pointwise regularity is necessary as time series are assumed to be multifractional Brownian motion. Utilizing local properties of the multifractional Brownian motion, and a pointwise approximation method, this thesis proposes an estimator that overcomes the problem of separate estimation of a scaling constant known to be present in empirical data. It is shown by Monte Carlo simulation that the estimator manages to capture the behavior of the time-varying Hurst function for three cases of mBm processes. However, the pointwise estimates are volatile when considering a single trajectory. In an attempt to address this issue, a smoothing spline approach is applied. The smoothed single trajectory pointwise estimates shows better accuracy than the original pointwise estimates. For comparison purposes an estimator based on the Increment Ratio Statistic is introduced.

Abstract [sv]

Multifraktionell Browniansk rörelseär en typ av stokastisk process med tidsberoende regelbundenhet. I modelleringssyfteär precisa estimat av denna punktvisa regelbundenheten av intresse. Genom lokala egenskaper hos den multifraktionella Brownianska rörelsen och en punktvis approximationsmetod så föreslår denna uppsats en estimator somöverkommer behovet av separat estimering av en skalningskonstant. Denna skalningskonstant har tidigare visats förekomma i empiriska tidsserier. Den föreslagna estimatorn lyckas i genomsnitt fånga Hurstfunktionen i tre olika fall, vilket visas med Monte Carlo-simulering. När enskilda trajektorier beaktas kan det konstateras att de punktvisa estimatenär volatila. I ett försök attöka tillförlitligheten appliceras smoothing splines. Denna metod medför att de punktvisa estimaten får mindre variansän originalestimaten. En estimator baserad på Increment Ratio statistikan introduceras i jämförelsesyfte.

Place, publisher, year, edition, pages
2015. , 30 p.
National Category
URN: urn:nbn:se:umu:diva-105775OAI: diva2:828153
Available from: 2015-11-04 Created: 2015-06-29 Last updated: 2015-11-04Bibliographically approved

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