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Local estimation of the Hurst index of multifractional Brownian motion by increment ratio statistic method

Published online by Cambridge University Press:  17 May 2013

Pierre Raphaël Bertrand ,
Mehdi Fhima  and
Arnaud Guillin
Pierre Raphaël Bertrand
Affiliation:
INRIA Saclay, 91893 Orsay Cedex, France Laboratoire de Mathématiques, UMR CNRS 6620 & Université de Clermont-Ferrand 2, France. [email protected]
Mehdi Fhima
Affiliation:
Laboratoire de Mathématiques, UMR CNRS 6620 & Université de Clermont-Ferrand 2, France. [email protected]
Arnaud Guillin
Affiliation:
Laboratoire de Mathématiques, UMR CNRS 6620 & Université de Clermont-Ferrand 2, France. [email protected]
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Abstract

We investigate here the central limit theorem of the increment ratio statistic of amultifractional Brownian motion, leading to a CLT for the time varying Hurst index. Theproofs are quite simple relying on Breuer–Major theorems and an original freezingof time strategy. A simulation study shows the goodness of fit of thisestimator.

Keywords

Increment ratio statisticfractional Brownian motionlocal estimationmultifractional Brownian motionwavelet series representation
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 307 - 327
Copyright
© EDP Sciences, SMAI, 2013

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References

P. Abry, P. Flandrin, M.S. Taqqu and D. Veitch, Self-similarity and long-range dependence through the wavelet lens, in Theory and applications of long-range dependenc. Birkhauser, Boston (2003).
Arcones, M.A., Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 (1994) 2242–2274. Google Scholar
Ayache, A. and Taqqu, M.S., Rate optimality of wavelet series approximations of fractional Brownian motions. J. Fourier Anal. Appl. 9 (2003) 451–471. Google Scholar
Ayache, A. and Taqqu, M.S., Multifractional process with random exponent. Publ. Math. 49 (2005) 459–486. Google Scholar
Ayache, A., Bertrand, P. and Lévy-Véhel, J., A central limit theorem for the generalized quadratic variation of the step fractional Brownian motion. Stat. Inference Stoch. Process. 10 (2007) 1–27. Google Scholar
Bardet, J.M. and Bertrand, P.R., Definition, properties and wavelet analysis of multiscale fractional Brownian motions. Fractals 15 (2007) 73–87. Google Scholar
Bardet, J.M. and Bertrand, P.R., Identification of the multiscale fractional Brownian motion with biomechanical applications. J. Time Ser. Anal. 28 (2007) 1–52. Google Scholar
Bardet, J.M. and Bertrand, P.R., A nonparametric estimator of the spectral density of a continuous-time Gaussian process observed at random times. Scand. J. Stat. 37 (2010) 458–476. Google Scholar
Bardet, J.M. and Surgailis, D., Nonparametric estimation of the local hurst function of multifractional Gaussian processes, Stoch. Proc. Appl. 123 (2013) 1004–1045. Google Scholar
Bardet, J.M. and Surgailis, D., Measuring roughness of random paths by increment ratios. Bernoulli 17 (2011) 749–780. Google Scholar
Bégyn, A., Functional limit theorems for generalized quadratic variations of Gaussian processes. Stoch. Proc. Appl. 117 (2007) 1848–1869. Google Scholar
Benassi, A., Jaffard, S. and Roux, D., Gaussian processes and pseudodifferential elliptic operators. Rev. Mat. Iberoam. 13 (1997) 19–81. Google Scholar
Benassi, A., Cohen, S. and Istas, J., Identifying the multifractional function of a Gaussian process. Stat. Probab. Lett. 39 (1998) 337–345. Google Scholar
Bertrand, P.R., Hamdouni, A. and Khadhraoui, S., Modelling NASDAQ series by sparse multifractional Brownian motion. Method. Comput. Appl. Probab. 14 (2012) 107–124. Google Scholar
Biermé, H., Bonami, A. and Leon, J., Central limit theorems and quadratic variations in terms of spectral density. Electronic Journal of Probability 16 (2011) 362–395. Google Scholar
Pa. Billingsley, Probability and measure, 2nd edition. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York (1986).
Bružaitė, K. and Vaičiulis, M., The increment ratio statistic under deterministic trends. Lith. Math. J. 48 (2008) 256–269. Google Scholar
G. Chan and A.T.A. Wood, Simulation of multifractal Brownian motions, Proc. of Computational Statistics (1998) 233–238.
Cheridito, P., Arbitrage in fractional Brownian motion models. Finance Stoch. 7 (2003) 533–553. Google Scholar
Coeurjolly, J.F., Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 (2001) 199–227. Google Scholar
Coeurjolly, J.-F., Identification of multifractional Brownian motions. Bernoulli 11 (2005) 987–1008. Google Scholar
S. Cohen, From self-similarity to local self-similarity: the estimation problem, Fractal: Theory and Applications in Engineering, edited by M. Dekking, J. Lévy Véhel, E. Lutton and C. Tricot. Springer Verlag (1999).
H. Cramèr and M.R. Leadbetter, Stationary and Related Stochastic Processes. Sample Function Properties and Their Applications, Wiley and Sons, London (1967).
M. Fhima, Ph.D. thesis (2011) in preparation.
Guyon, X. and Leon, J., Convergence en loi des h-variations d’un processus Gaussien stationnaire. Ann. Inst. Henri Poincaré 25 (1989) 265–282. Google Scholar
Istas, J. and Lang, G., Quadratic variations and estimation of the hölder index of a Gaussian process. Ann. Inst. Henri Poincaré 33 (1997) 407–436. Google Scholar
Kolmogorov, A.N., Wienersche spiralen und einige andere interessante kurven im hilbertschen raum. C.R. (Doklady) Acad. URSS (N.S.) 26 (1940) 115–118. Google Scholar
J. Lévy-Véhel and R.F. Peltier, Multifractional Brownian motion: definition and preliminary results. Techn. Report RR-2645, INRIA (1996).
Mandelbrot, B. and Van Ness, J., Fractional Brownian motions, fractional noises and applications. SIAM Review 10 (1968) 422–437. Google Scholar
Meyer, Y., Sellan, F. and Taqqu, M.S., Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motions. J. Fourier Anal. Appl. 5 (1999) 465–494. Google Scholar
Nourdin, I. and Peccati, G., Stein’s method on wiener chaos. Probab. Theory Relat. Fields 145 (2009) 75–118. Google Scholar
I. Nourdin, G. Peccati and M. Podolskij, Quantitative Breuer-Major theorems, HAL: hal-00484096, version 2 (2010).
Peccati, G. and Tudor, C.A., Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probabilités XXXVIII, Lecture Notes Math. 1857 (2005) 247–262. Google Scholar
G. Samorodnitsky and M.S. Taqqu, Stable non-Gaussian random processes. Chapman & Hall (1994).
Stoev, A.S. and Taqqu, M.S., How rich is the class of multifractional brownian motions. Stoch. Proc. Appl. 116 (2006) 200–221. Google Scholar
Stoncelis, M. and Vaičiulis, M., Numerical approximation of some infinite Gaussian series and integrals. Nonlinear Anal.: Modelling and Control 13 (2008) 397–415. Google Scholar
Surgailis, D., Teyssière, G. and Vaičiulis, M., The increment ratio statistic. J. Multivar. Anal. 99 (2008) 510–541. Google Scholar
Yaglom, A.M., Some classes of random fields in n-dimensional space, related to stationary random processes. Theory Probab. Appl. 2 (1957) 273–320. Google Scholar