MULTIFRACTIONAL PROPERTIES OF STOCK INDICES DECOMPOSED BY FILTERING THEIR POINTWISE HÖLDER REGULARITY

We propose a decomposition of financial time series into Gaussian subsequences characterized by a constant Hölder exponent. In (multi)fractal models this condition is equivalent to the subsequences themselves being stationarity. For the different subsequences, we study the scaling of the variance and the bias that is generated when the Hölder exponent is re-estimated using traditional estimators. The results achieved by both analyses are shown to be strongly consistent with the assumption that the price process can be modeled by the multifractional Brownian motion, a nonstationary process whose Hölder regularity changes from point to point.

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PATH PROPERTIES OF THE LINEAR MULTIFRACTIONAL STABLE MOTION

Fractals ◽

10.1142/s0218348x05002775 ◽

2005 ◽

Vol 13 (02) ◽

pp. 157-178 ◽

Cited By ~ 19

Author(s):

STILIAN STOEV ◽

MURAD S. TAQQU

Keyword(s):

Heavy Tails ◽

Hölder Regularity ◽

Self Similarity ◽

Similarity Parameter ◽

Stable Motion ◽

Multifractional Brownian Motion ◽

Sample Paths ◽

Hölder Exponents ◽

Fractional Stable Motion ◽

Holder Regularity

The linear multifractional stable motion (LMSM) processes Y = {Y(t)}t∈ℝ is an α-stable (0 < α < 2) stochastic process, which exhibits local self-similarity, has heavy tails and can have skewed distributions. The process Y is obtained from the well-known class of linear fractional stable motion (LFSM) processes by replacing their self-similarity parameter H by a function of time H(t). We show that the paths of Y(t) are bounded on bounded intervals only if 1/α ≤ H(t) < 1, t ∈ ℝ. In particular, if 0 < α ≤ 1, then Y has everywhere discontinuous paths, with probability one. On the other hand, Y has a version with continuous paths if H(t) is sufficiently regular and 1/α < H(t), t ∈ ℝ. We study the Hölder regularity of the sample paths when these are continuous and establish almost sure bounds on the pointwise and uniform pointwise Hölder exponents of the (random) function Y(t,ω), t ∈ ℝ, in terms of the function H(t) and its corresponding Hölder exponents. The Gaussian multifractional Brownian motion (MBM) processes are LMSM processes when α = 2. We obtain some new results on the Hölder regularity of their paths.

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PATHWISE IDENTIFICATION OF THE MEMORY FUNCTION OF MULTIFRACTIONAL BROWNIAN MOTION WITH APPLICATION TO FINANCE

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024905002937 ◽

2005 ◽

Vol 08 (02) ◽

pp. 255-281 ◽

Cited By ~ 30

Author(s):

SERGIO BIANCHI

Keyword(s):

Brownian Motion ◽

Null Hypothesis ◽

Financial Time Series ◽

Memory Function ◽

Scaling Law ◽

Time Dependent ◽

Financial Time ◽

Price Process ◽

Multifractional Brownian Motion ◽

Class Of Estimators

We extend and adapt a class of estimators of the parameter H of the fractional Brownian motion in order to estimate the (time-dependent) memory function of a multifractional process. We provide: (a) the estimator's distribution when H ∈ (0,3/4); (b) the confidence interval under the null hypothesis H = 1/2; (c) a scaling law, independent on the value of H, discriminating between fractional and multifractional processes. Furthermore, assuming as a model for the price process the multifractional Brownian motion, empirical evidence is offered which is able to conciliate the inconsistent results achieved in estimating the intensity of dependence in financial time series.

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The Mañé–Conze–Guivarc’h lemma for intermittent maps of the circle

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385708000837 ◽

2009 ◽

Vol 29 (5) ◽

pp. 1603-1611 ◽

Cited By ~ 4

Author(s):

IAN D. MORRIS

Keyword(s):

Existence Of Solutions ◽

Continuous Solution ◽

Functional Inequality ◽

Continuous Functions ◽

Ergodic Average ◽

Hölder Regularity ◽

Hölder Exponent ◽

Continuous Solutions ◽

Sharp Estimates ◽

Holder Regularity

AbstractWe study the existence of solutions g to the functional inequality f≤g ∘T−g+β, where f is a prescribed continuous function, T is a weakly expanding transformation of the circle having an indifferent fixed point, and β is the maximum ergodic average of f. Using a method due to T. Bousch, we show that continuous solutions g always exist when the Hölder exponent of f is close to 1. In the converse direction, we construct explicit examples of continuous functions f with low Hölder exponent for which no continuous solution g exists. We give sharp estimates on the best possible Hölder regularity of a solution g given the Hölder regularity of f.

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Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

Journal of Applied Probability ◽

10.1017/s0021900200003041 ◽

2007 ◽

Vol 44 (02) ◽

pp. 393-408 ◽

Cited By ~ 1

Author(s):

Allan Sly

Keyword(s):

Stochastic Process ◽

Brownian Motion ◽

White Noise ◽

Gaussian Process ◽

Discrete Data ◽

Hölder Exponent ◽

Scaling Properties ◽

Multifractional Brownian Motion ◽

Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

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