Journal of Applied Probability

A wavelet-based almost-sure uniform approximation of fractional Brownian motion with a parallel algorithm

Dawei Hong, Shushuang Man, Jean-Camille Birget, and Desmond S. Lun

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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.

Article information

J. Appl. Probab., Volume 51, Number 1 (2014), 1-18.

First available in Project Euclid: 25 March 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G22: Fractional processes, including fractional Brownian motion
Secondary: 65T60: Wavelets 65Y05: Parallel computation

Fractional Brownian motion wavelet expansion of stochastic integral almost-sure uniform approximation


Hong, Dawei; Man, Shushuang; Birget, Jean-Camille; Lun, Desmond S. A wavelet-based almost-sure uniform approximation of fractional Brownian motion with a parallel algorithm. J. Appl. Probab. 51 (2014), no. 1, 1--18. doi:10.1239/jap/1395771410.

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  • Arcones, M. A. (1995). On the law of the iterated logarithm for Gaussian processes. J. Theoret. Prob. 8, 877–903.
  • Ayache, A. and Taqqu, M. S. (2003). Rate optimality of wavelet series approximations of fractional Brownian motion. J. Fourier Anal. Appl. 9, 451–471.
  • 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, 317–329.
  • Biagini, F., Hu, Y., Øksendal, B. and Zhang, T. (2008). Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London.
  • Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia, PA.
  • Davydov, Yu. A. (1970). The invariance principle for stationary processes. Teor. Vero. Primen. 15, 498–509.
  • Delgado, R. and Jolis, M. (2000). Weak approximation for a class of Gaussian process. J. Appl. Prob. 37, 400–407.
  • Dudley, R. M. (2002). Real Analysis and Probability. Cambridge University Press.
  • Dzhaparidze, K. and van Zanten, H. (2004). A series expansion of fractional Brownian motion. Prob. Theory Relat. Fields 130, 39–55.
  • Dzhaparidze, K. and van Zanten, H. (2005). Optimality of an explicit series expansion of the fractional Brownian sheet. Statist. Prob. Lett. 71, 295–301.
  • 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, 3435–3452.
  • Kühn, T. and Linde, W. (2002). Optimal series representation of fractional Brownian sheets. Bernoulli 8, 669–696.
  • Leighton, F. T. (1992). Introduction to Parallel Algorithms and Architectures. Morgan Kaufmann, San Mateo, CA.
  • Li, Y. and Dai, H. (2011). Approximations of fractional Brownian motion. Bernoulli 17, 1195–1216.
  • Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437.
  • 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, 465–494.
  • Taqqu, M. S. (1974/75). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitsth. 31, 287–302.
  • Veraar, M. (2012). The stochastic Fubini theorem revisited. Stochastics 84, 543–551.