Wavelet analysis of a multifractional process in an arbitrary Wiener chaos
Authors:
A. Ayache and Y. Esmili
Journal:
Theor. Probability and Math. Statist. 98 (2019), 27-49
MSC (2010):
Primary 60G17, 60G22
DOI:
https://doi.org/10.1090/tpms/1061
Published electronically:
August 19, 2019
MathSciNet review:
3824677
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Additional Information
Abstract: The well-known multifractional Brownian motion (mBm) is the paradigmatic example of a continuous Gaussian process with non-stationary increments whose local regularity changes from point to point. In this article, using a wavelet approach, we construct a natural extension of mBm which belongs to a homogeneous Wiener chaos of an arbitrary order. Then, we study its global and local behavior.
References
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References
- B. Arras, On a class of self-similar processes with stationary increments in higher order Wiener chaoses, Stochastic Process. Appl. 124 (2014), no. 7, 2415–2441. MR 3192502
- A. Ayache, S. Jaffard, and M. S. Taqqu, Wavelet construction of generalized multifractional processes, Rev. Mat. Iberoam. 23 (2007), no. 1, 327–370. MR 2351137
- A. Ayache and M. S. Taqqu, Multifractional Processes with Random Exponent, Publ. Mat. 49 (2005), 459–486. MR 2177638
- A. Benassi, S. Cohen, and J. Istas, Identifying the multifractional function of a Gaussian process, Stat. Probab. Lett. 39 (1998), no. 4, 337–345. MR 1646220
- A. Benassi, S. Jaffard, and D. Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoam. 13 (1997), no. 1, 19–90. MR 1462329
- S. Bianchi, A. Pantanella, and A. Pianese, Modeling stock prices by multifractional Brownian motion: an improved estimation of the pointwise regularity, Quant. Finance 13 (2013), no. 8, 1317–1330. MR 3175906
- S. Bianchi and A. Pianese, Multifractional properties of stock indices decomposed by filtering their pointwise Hölder regularity, Int. J. Theor. Appl. Finance 11 (2008), no. 06, 567–595. MR 2455303
- S. Bianchi, Pathwise identification of the memory function of multifractional Brownian motion with application to finance, Int. J. Theor. Appl. Finance 8 (2005), no. 02, 255–281. MR 2130615
- I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1992. MR 1162107
- M. Dozzi and G. Shevchenko, Real harmonizable multifractional stable process and its local properties, Stochastic Process. Appl. 121 (2011), no. 7, 1509–1523. MR 2802463
- K. J. Falconer, Tangent fields and the local structure of random fields, J. Theoret. Probab. 15 (2002), no. 3, 731–750. MR 1922445
- K. J. Falconer, The local structure of random processes, J. Lond. Math. Soc. 67 (2003), no. 3, 657–672. MR 1967698
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- Y. Meyer, Wavelets and Operators, vol. 37, Cambridge University Press, 1992. MR 1228209
- T. Mori and H. Oodaira, The law of the iterated logarithm for self-similar processes represented by multiple Wiener integrals, Probab. Theory Related Fields 71 (1986), no. 3, 367–391. MR 824710
- D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, Berlin–Heidelberg, 2006. MR 2200233
- R.-F. Peltier and J. Lévy Véhel, Multifractional Brownian Motion: Definition and Preliminary Results, INRIA Research Report 2645 (1995).
- G. Shevchenko, Local times for multifractional square Gaussian processes, Bulletin of Taras Shevchenko National University of Kyiv, Series: Physics & Mathematics (2013).
- S. Stoev and M. S. Taqqu, Stochastic properties of the linear multifractional stable motion, Adv. in Appl. Probab. 36 (2004), no. 4, 1085–1115. MR 2119856
- S. Stoev and M. S. Taqqu, Path properties of the linear multifractional stable motion, Fractals 13 (2005), no. 2, 157–178. MR 2151096
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Additional Information
A. Ayache
Affiliation:
UMR CNRS 8524, Laboratoire Paul Painlevé, Université de Lille, Cité Scientifique, Bâtiment M2, 59655 Villeneuve d’Ascq, France
Email:
Antoine.Ayache@univ-lille.fr
Y. Esmili
Affiliation:
UMR CNRS 8524, Laboratoire Paul Painlevé, Université de Lille, Cité Scientifique, Bâtiment M2, 59655 Villeneuve d’Ascq, France
Email:
Yassine.Esmili@univ-lille.fr
Keywords:
Wiener chaos,
self-similar processes,
modulus of continuity,
wavelet bases,
fractional processes
Received by editor(s):
February 8, 2018
Published electronically:
August 19, 2019
Additional Notes:
The authors are very grateful to the anonymous referee for his valuable comments which have led to improvements of the article. This work has been partially supported by the Labex CEMPI (ANR-11-LABX-0007-01) and the GDR 3475 (Analyse Multifractale).
Article copyright:
© Copyright 2019
American Mathematical Society
