Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T05:29:26.749Z Has data issue: false hasContentIssue false

A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

Published online by Cambridge University Press:  30 January 2018

Dawei Hong*
Affiliation:
Rutgers University
Shushuang Man*
Affiliation:
Southwest Minnesota State University
Jean-Camille Birget*
Affiliation:
Rutgers University
Desmond S. Lun*
Affiliation:
Rutgers University
*
Postal address: Center for Computational and Integrative Biology, Department of Computer Science, Rutgers University, Camden, NJ 08102, USA.
∗∗∗ Postal address: Department of Mathematics and Computer Science, Southwest Minnesota State University, Marshall, MN 56258, USA.
Postal address: Center for Computational and Integrative Biology, Department of Computer Science, Rutgers University, Camden, NJ 08102, USA.
Postal address: Center for Computational and Integrative Biology, Department of Computer Science, Rutgers University, Camden, NJ 08102, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

Type
Research Article
Copyright
© Applied Probability Trust 

References

Arcones, M. A. (1995). On the law of the iterated logarithm for Gaussian processes. J. Theoret. Prob. 8, 877903.Google Scholar
Ayache, A. and Taqqu, M. S. (2003). Rate optimality of wavelet series approximations of fractional Brownian motion. J. Fourier Anal. Appl. 9, 451471.Google Scholar
Bardina, X., Jolis, M. and Tudor, C. A. (2003). Weak convergence to the fractional Brownian sheet and other two-parameter Gaussian processes. Statist. Prob. Lett. 65, 317329.Google Scholar
Biagini, F., Hu, Y., Øksendal, B. and Zhang, T. (2008). Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London.Google Scholar
Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia, PA.Google Scholar
Davydov, Yu. A. (1970). The invariance principle for stationary processes. Teor. Vero. Primen. 15, 498509.Google Scholar
Delgado, R. and Jolis, M. (2000). Weak approximation for a class of Gaussian process. J. Appl. Prob. 37, 400407.Google Scholar
Dudley, R. M. (2002). Real Analysis and Probability. Cambridge University Press.Google Scholar
Dzhaparidze, K. and van Zanten, H. (2004). A series expansion of fractional Brownian motion. Prob. Theory Relat. Fields 130, 3955.Google Scholar
Dzhaparidze, K. and van Zanten, H. (2005). Optimality of an explicit series expansion of the fractional Brownian sheet. Statist. Prob. Lett. 71, 295301.CrossRefGoogle Scholar
Garzón, J., Gorostiza, L. G. and León, J. A. (2009). A strong uniform approximation of fractional Brownian motion by means of transport processes. Stoch. Process. Appl. 119, 34353452.Google Scholar
Kühn, T. and Linde, W. (2002). Optimal series representation of fractional Brownian sheets. Bernoulli 8, 669696.Google Scholar
Leighton, F. T. (1992). Introduction to Parallel Algorithms and Architectures. Morgan Kaufmann, San Mateo, CA.Google Scholar
Li, Y. and Dai, H. (2011). Approximations of fractional Brownian motion. Bernoulli 17, 11951216.Google Scholar
Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422437.CrossRefGoogle Scholar
Meyer, Y., Sellan, F. and Taqqu, M. S. (1999). Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion. J. Fourier Anal. Appl. 5, 465494.Google Scholar
Taqqu, M. S. (1974/75). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitsth. 31, 287302.Google Scholar
Veraar, M. (2012). The stochastic Fubini theorem revisited. Stochastics 84, 543551.Google Scholar