Fractional Brownian motion with variable Hurst parameter: definition and properties. (English) Zbl 1333.60077
The author introduces a class of Gaussian processes generalizing the fractional Brownian motion with Hurst index \(H\in (1/2,1).\) Any measurable function attaining values in the interval \((1/2,1)\) can be chosen as a variable Hurst parameter for this new class. These processes allow modeling of phenomena where long-range dependence is present and self-similarity changes as the phenomenon evolves. Sample path properties of these processes are investigated. Fokker-Plank-type equations for such processes are derived. It is found that the regularity properties of the chosen Hurst function directly correspond to the regularity of the sample paths of the process.
Reviewer’s comments: Such processes are also called locally self-similar. P. Gonçalves and P. Flandrin [in: Progress in wavelet analysis and applications. Proceedings of the 3rd international conference on wavelets and applications, Toulouse, France, 1992. Gif-sur-Yvette: Editions Frontières, 271–276 (1993; Zbl 0878.94024)] proposed such processes. Y.-Z. Wang et al. [J. Stat. Plann. Inference 99, No. 1, 91–110 (2001; Zbl 0989.62045)] studied estimations of the self-similarity index for locally self-similar processes (cf. [B. L. S. Prakasa Rao, Statistical inference for fractional diffusion processes. Wiley Series in Probability and Statistics. Chichester: John Wiley & Sons (2010; Zbl 1211.62143)]).
Reviewer’s comments: Such processes are also called locally self-similar. P. Gonçalves and P. Flandrin [in: Progress in wavelet analysis and applications. Proceedings of the 3rd international conference on wavelets and applications, Toulouse, France, 1992. Gif-sur-Yvette: Editions Frontières, 271–276 (1993; Zbl 0878.94024)] proposed such processes. Y.-Z. Wang et al. [J. Stat. Plann. Inference 99, No. 1, 91–110 (2001; Zbl 0989.62045)] studied estimations of the self-similarity index for locally self-similar processes (cf. [B. L. S. Prakasa Rao, Statistical inference for fractional diffusion processes. Wiley Series in Probability and Statistics. Chichester: John Wiley & Sons (2010; Zbl 1211.62143)]).
Reviewer: B. L. S. Prakasa Rao (Hyderabad)
MSC:
| 60G22 | Fractional processes, including fractional Brownian motion |
| 60G15 | Gaussian processes |
| 60G17 | Sample path properties |
| 60G18 | Self-similar stochastic processes |
Keywords:
fractional Brownian motion; variable Hurst parameter; Gaussian processes; self-similarity; sample path regularityReferences:
| [1] | Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Dover, New York (1965) · Zbl 0171.38503 |
| [2] | Alos, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29(2), 766-801 (2001) · Zbl 1015.60047 · doi:10.1214/aop/1008956692 |
| [3] | Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2009) · Zbl 1200.60001 · doi:10.1017/CBO9780511809781 |
| [4] | Ayache, A., Cohen, S., Véhel, J.: The covariance structure of multifractional Brownian motion, with application to long range dependence. In: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 6, pp. 3810-3813. IEEE (2000) · Zbl 0984.82032 |
| [5] | Benassi, A., Roux, D., Jaffard, S.: Elliptic Gaussian random processes. Rev. Mat. Iberoam. 13(1), 19-88 (1997) · Zbl 0880.60053 · doi:10.4171/RMI/217 |
| [6] | Benson, D., Wheatcraft, S., Meerschaert, M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36(6), 1403-1412 (2000) · doi:10.1029/2000WR900031 |
| [7] | Biagini, F., Hu, Y., Øksendal, B., Zhang, T.: Stochastic calculus for fractional Brownian motion and applications. Springer, Berlin (2007) · Zbl 1157.60002 |
| [8] | Boufoussi, B., Dozzi, M., Marty, R.: Local time and Tanaka formula for a Volterra-type multifractional Gaussian process. Bernoulli 16(4), 1294-1311 (2010) · Zbl 1213.60075 · doi:10.3150/10-BEJ261 |
| [9] | Cambanis, S., Rajput, B.: Some zero-one laws for Gaussian processes. Ann. Probab. 1(2), 304-312 (1973) · Zbl 0262.60017 · doi:10.1214/aop/1176996982 |
| [10] | Chronopoulou, A., Viens, F.: Estimation and pricing under long-memory stochastic volatility. Ann. Finance 8(2), 379-403 (2012) · Zbl 1298.91160 · doi:10.1007/s10436-010-0156-4 |
| [11] | Comte, F., Renault, E.: Long memory in continuous-time stochastic volatility models. Math. Finance 8(4), 291-323 (1998) · Zbl 1020.91021 · doi:10.1111/1467-9965.00057 |
| [12] | Conway, J.B.: A course in functional analysis. Springer, Berlin (1985) · Zbl 0558.46001 · doi:10.1007/978-1-4757-3828-5 |
| [13] | Decreusefond, L.: Stochastic integration with respect to Volterra processes. Ann. de l’Inst. Henri Poincare (B) Probab. Stat. 41(2), 123-149 (2005) · Zbl 1071.60040 · doi:10.1016/j.anihpb.2004.03.004 |
| [14] | Dekking, F., Lévy Véhel, J., Lutton, E., Tricot, C., et al.: Fractals: theory and applications in engineering. Springer, Berlin (1999) · Zbl 0936.00006 · doi:10.1007/978-1-4471-0873-3 |
| [15] | Doob, J.: Stochastic processes. Wiley, New York (1962) |
| [16] | Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. arXiv preprint arXiv:0805.3823 (2008) · Zbl 1030.26004 |
| [17] | Hahn, M., Kobayashi, K., Ryvkina, J., Umarov, S.: On time-changed Gaussian processes and their associated Fokker-Planck-Kolmogorov equations. Electron. Commun. Probab. 16, 150-164 (2010) · Zbl 1231.60029 |
| [18] | Janczura, J., Wyłomańska, A.: Subdynamics of financial data from fractional Fokker-Planck equation. Munich personal RePEc archive (2009) |
| [19] | Janson, S.: Gaussian Hilbert Spaces. Cambridge University Press, Cambridge (1997) · Zbl 0887.60009 · doi:10.1017/CBO9780511526169 |
| [20] | Jennane, R., Ohley, W., Majumdar, S., Lemineur, G.: Fractal analysis of bone X-ray tomographic microscopy projections. Trans. Med. Imaging 20(5), 443-449 (2001) · doi:10.1109/42.925297 |
| [21] | Kolmogorov, A.: Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. CR (Dokl.) Acad. Sci. URSS 26, 115-118 (1940) · JFM 66.0552.03 |
| [22] | Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer, Berlin (2011) · Zbl 1226.60003 |
| [23] | Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422-437 (1968) · Zbl 0179.47801 |
| [24] | Meerschaert, M., Scheffler, H.: Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41(3), 623-638 (2004) · Zbl 1065.60042 · doi:10.1239/jap/1091543414 |
| [25] | Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3 |
| [26] | Peltier, R.F., Véhel, J.L.: Multifractional Brownian motion: definition and preliminary results. Inria research report no. 2645 (1995) |
| [27] | Perrin, E., Harba, R., Iribarren, I., Jennane, R.: Piecewise fractional Brownian motion. IEEE Trans. Signal Process. 53(3), 1211-1215 (2005) · Zbl 1370.60067 · doi:10.1109/TSP.2004.842209 |
| [28] | Prudnikov, A., Brychkov, Y., Marichev, O.: Integrals and Series. Gordon and Breach, London (1986) · Zbl 0606.33001 |
| [29] | Ralchenko, K., Shevchenko, G.: Path properties of multifractal Brownian motion. Theory Probab. Math. Stat. 80, 119-130 (2010) · Zbl 1224.60080 · doi:10.1090/S0094-9000-2010-00799-X |
| [30] | Ryvkina, J.: Fractional Brownian motion with variable Hurst parameter. Ph.D. thesis, Tufts University (2013) · Zbl 1333.60077 |
| [31] | Véhel, J., Riedi, R.: Fractional Brownian motion and data traffic modeling: the other end of the spectrum. Fractals Eng. 97, 185-202 (1997) · doi:10.1007/978-1-4471-0995-2_15 |
| [32] | Zaslavsky, G.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371(6), 461-580 (2002) · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9 |
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