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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References
Anh, V. V., Yu, Z. G., Wanliss, J. A. and Watson, S. M. (2005). Prediction of magnetic storm events using the $D_st$ index. Nonlinear Process. Geophys. 12, 799--806.
Ayache, A. and Lévy Véhel, J. (2004). On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion. Stoch. Process. Appl. 111, 119--156.
Ayache, A., Benassi, A., Cohen, S. and Lévy Véhel, J. (2005). Regularity and identification of generalized multifractional Gaussian processes. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 1857), Springer, Berlin, pp. 290--312.
Benassi, A., Cohen, S. and Istas, J. (1998). Identifying the multifractional function of a Gaussian process. Statist. Prob. Lett. 39, 337--345.
Benassi, A., Bertrand, P., Cohen, S. and Istas, J. (1999). Identification d'un processus gaussien multifractionnaire avec des ruptures sur la fonction d'échelle. C. R. Acad. Sci. Paris Sér. I Math. 329, 435--440.
Benassi, A., Bertrand, P., Cohen, S. and Istas, J. (2000). Identification of the Hurst index of a step fractional Brownian motion. Statist. Infer. Stoch. Process. 3, 101--111.
Carbone, A., Castelli, G. and Stanley, H. E. (2004). Time-dependent Hurst exponent. Physica A 344, 267--271.
Chung, K. L. (2001). A Course in Probability Theory. Academic Press, San Diego, CA.
Elliott, R. J. and van der Hoek, J. (2003). A general fractional white noise theory and applications to finance. Math. Finance 13, 301--330.
Janson, S. (1997). Gaussian Hilbert Spaces. Cambridge University Press.
Mandelbrot, B. B. and Fisher, A. and Calvet, L. (1997). A multifractal model of asset returns. CFDP 1164. Available at http://cowles.econ.yale.edu/P/cd/dy1997.htm.
Murali Krishna, P., Gadre, V. M. and Desai, V. B. (2003). Multifractal Based Network Traffic Modeling. Kluwer, Boston, MA.
Peltier, R. and Lévy Véhel, J. (1995). Multifractional Brownian motion: definition and preliminary results. Res. Rept. 2645, INRIA.
Sreenivasan, K. R. (1991). Fractals and multifractals in fluid turbulence. Ann. Rev. Fluid Mech. 23, 539--600.
Stoev, S. A. and Taqqu, M. S. (2006). How rich is the class of multifractional Brownian motions? Stoch. Process. Appl. 116, 200--221.
Watkins, N. W. \et (2005). Towards synthesis of solar wind and geomagnetic scaling exponents: a fractional Lévy motion model. Space Sci. Rev. 121, 271--284.
Yu, Z.-G., Anh, V. and Lau, K.-S. (2003). Multifractal and correlation analyses of protein sequences from complete genomes. Phys. Rev. E 68, 021913.
