Theory of Probability and Mathematical Statistics
Wavelet analysis of a multifractional process in an arbitrary Wiener chaos
A. Ayache, Y. Esmili
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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.
Keywords: Wiener chaos, self-similar processes, modulus of continuity, wavelet bases, fractional processes.
Bibliography: 1. 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.
2. A. Ayache, S. Jaffard, M. S. Taqqu, Wavelet construction of generalized multifractional processes , Rev. Mat. Iberoam., 23 (2007), no. 1, 327-370.
3. A. Ayache, M. S. Taqqu, Multifractional Processes with Random Exponent, Publ. Mat., 49 (2005), 459-486.
4. A. Benassi, S. Cohen, J. Istas, Identifying the multifractional function of a Gaussian process, Stat. Probab. Lett., 39 (1998), no. 4, 337-345.
5. A. Benassi, S. Jaffard, D. Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoam., 13 (1997), no. 1, 19-90.
6. S. Bianchi, A. Pantanella, A. Pianese, Modeling stock prices by multifractional Brownian motion: an improved estimation of the pointwise regularity, Quant. Finance, 13 (2013), no. 8, 1317-1330.
7. S. Bianchi and A. Pianese, Multifractional properties of stock indices decomposed by filtering their pointwise Holder regularity, Int. J. Theor. Appl. Finance, 11 (2008), no. 06, 567-595.
8. 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.
9. I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1992.
10. M. Dozzi, G. Shevchenko, Real harmonizable multifractional stable process and its local properties , Stochastic Process. Appl., 121 (2011), no. 7, 1509-1523.
11. K. J. Falconer, Tangent fields and the local structure of random fields, J. Theoret. Probab. 15 (2002), no. 3, 731-750.
12. K. J. Falconer, The local structure of random processes, J. Lond. Math. Soc., 67 (2003), no. 3, 657-672.
13. S. Janson, Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics, Cambridge University Press, 1997.
14. A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, Comptes Rendus (Doklady) de l'Academie des Sciences de l'URSS (N.S.), 26 (1940), 115-118.
15. P.-G. Lemarie, Y. Meyer, Ondelettes et bases hilbertiennes, Rev. Mat. Iberoam., 2 (1986), no.1-2, 1-18.
16. B. Mandelbrot, J. W. Van Ness, Fractional Brownian Motions, Fractional Noises and Applications, SIAM Review, 10 (1968), no. 4, 422-437.
17. Y. Meyer, Wavelets and operators, vol. 37, Cambridge University Press, 1992.
18. T. Mori, 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.
19. D. Nualart, The Malliavin calculus and related topics, Springer-Verlag, Berlin Heidelberg, 2006.
20. R.-F. Peltier, J. Levy Vehel, 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. S. Stoev, M. S. Taqqu, Stochastic properties of the linear multifractional stable motion, Adv. in Appl. Probab., 36 (2004), no. 4, 1085-1115.
23. S. Stoev, M. S. Taqqu, Path properties of the linear multifractional stable motion, Fractals, 13 (2005), no. 2, 157-178.
