Wavelet analysis of a multifractional process in an arbitrary Wiener chaos

Ayache, A.; Esmili, Y.

doi:10.1090/tpms/1061

Wavelet analysis of a multifractional process in an arbitrary Wiener chaos

Authors: A. Ayache and Y. Esmili
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 98 (2018).
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
Full-text PDF

Abstract | References | Similar Articles | 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.

[1] Benjamin 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, https://doi.org/10.1016/j.spa.2014.02.012
[2] Antoine Ayache, Stéphane Jaffard, and Murad S. Taqqu, Wavelet construction of generalized multifractional processes, Rev. Mat. Iberoam. 23 (2007), no. 1, 327–370. MR 2351137, https://doi.org/10.4171/RMI/497
[3] Antoine Ayache and Murad S. Taqqu, Multifractional processes with random exponent, Publ. Mat. 49 (2005), no. 2, 459–486. MR 2177638, https://doi.org/10.5565/PUBLMAT_49205_11
[4] Albert Benassi, Serge Cohen, and Jacques Istas, Identifying the multifractional function of a Gaussian process, Statist. Probab. Lett. 39 (1998), no. 4, 337–345. MR 1646220, https://doi.org/10.1016/S0167-7152(98)00078-9
[5] Albert Benassi, Stéphane Jaffard, and Daniel Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoamericana 13 (1997), no. 1, 19–90 (English, with English and French summaries). MR 1462329, https://doi.org/10.4171/RMI/217
[6] 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, https://doi.org/10.1080/14697688.2011.594080
[7] 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. 6, 567–595. MR 2455303, https://doi.org/10.1142/S0219024908004932
[8] Sergio Bianchi, Pathwise identification of the memory function of multifractional Brownian motion with application to finance, Int. J. Theor. Appl. Finance 8 (2005), no. 2, 255–281. MR 2130615, https://doi.org/10.1142/S0219024905002937
[9] Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107
[10] Marco Dozzi and Georgiy Shevchenko, Real harmonizable multifractional stable process and its local properties, Stochastic Process. Appl. 121 (2011), no. 7, 1509–1523. MR 2802463, https://doi.org/10.1016/j.spa.2011.03.012
[11] Kenneth J. Falconer, Tangent fields and the local structure of random fields, J. Theoret. Probab. 15 (2002), no. 3, 731–750. MR 1922445, https://doi.org/10.1023/A:1016276016983
[12] Kenneth J. Falconer, The local structure of random processes, J. London Math. Soc. (2) 67 (2003), no. 3, 657–672. MR 1967698, https://doi.org/10.1112/S0024610703004186
[13] Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. MR 1474726
[14] A. N. Kolmogoroff, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C. R. (Doklady) Acad. Sci. URSS (N.S.) 26 (1940), 115–118 (German). MR 0003441
[15] P. G. Lemarié and Y. Meyer, Ondelettes et bases hilbertiennes, Rev. Mat. Iberoamericana 2 (1986), no. 1-2, 1–18 (French). MR 864650, https://doi.org/10.4171/RMI/22
[16] Benoit B. Mandelbrot and John W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968), 422–437. MR 242239, https://doi.org/10.1137/1010093
[17] Yves Meyer, Wavelets and operators, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992. Translated from the 1990 French original by D. H. Salinger. MR 1228209
[18] Toshio Mori and Hiroshi Oodaira, The law of the iterated logarithm for self-similar processes represented by multiple Wiener integrals, Probab. Theory Relat. Fields 71 (1986), no. 3, 367–391. MR 824710, https://doi.org/10.1007/BF01000212
[19] David Nualart, The Malliavin calculus and related topics, 2nd ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233
[20] R.-F. Peltier and J. Lévy Véhel, Multifractional Brownian Motion: Definition and Preliminary Results, INRIA Research Report 2645 (1995).
[21] G. Shevchenko, Local times for multifractional square Gaussian processes, Bulletin of Taras Shevchenko National University of Kyiv, Series: Physics & Mathematics (2013).
[22] Stilian Stoev and Murad S. Taqqu, Stochastic properties of the linear multifractional stable motion, Adv. in Appl. Probab. 36 (2004), no. 4, 1085–1115. MR 2119856, https://doi.org/10.1239/aap/1103662959
[23] Stilian Stoev and Murad S. Taqqu, Path properties of the linear multifractional stable motion, Fractals 13 (2005), no. 2, 157–178. MR 2151096, https://doi.org/10.1142/S0218348X05002775