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Alternative micropulses and fractional Brownian motion

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  • Cioczek-Georges, R.
  • Mandelbrot, B. B.

Abstract

We showed in an earlier paper (1995a) that negatively correlated fractional Brownian motion (FBM) can be generated as a fractal sum of one kind of micropulses (FSM). That is, FBM of exponent is the limit (in the sense of finite-dimensional distributions) of a certain sequence of processes obtained as sums of rectangular pulses. We now show that more general pulses yield a wide range of FBMs: either negatively (as before) or positively () correlated. We begin with triangular (conical and semi-conical) pulses. To transform them into micropulses, the base angle is made to decrease to zero, while the number of pulses, determined by a Poisson random measure, is made to increase to infinity. Then we extend our results to more general pulse shapes.

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  • Cioczek-Georges, R. & Mandelbrot, B. B., 1996. "Alternative micropulses and fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 64(2), pages 143-152, November.
    • Handle: RePEc:eee:spapps:v:64:y:1996:i:2:p:143-152
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    1. Cioczek-Georges, R. & Mandelbrot, B. B., 1995. "A class of micropulses and antipersistent fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 60(1), pages 1-18, November.
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    Cited by:

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    3. Jumarie, Guy, 2009. "From Lagrangian mechanics fractal in space to space fractal Schrödinger’s equation via fractional Taylor’s series," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1590-1604.
    4. Nuugulu, Samuel M & Gideon, Frednard & Patidar, Kailash C, 2021. "A robust numerical scheme for a time-fractional Black-Scholes partial differential equation describing stock exchange dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    5. M. Çağlar, 2004. "A Long-Range Dependent Workload Model for Packet Data Traffic," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 92-105, February.
    6. Jumarie, Guy, 2006. "Fractionalization of the complex-valued Brownian motion of order n using Riemann–Liouville derivative. Applications to mathematical finance and stochastic mechanics," Chaos, Solitons & Fractals, Elsevier, vol. 28(5), pages 1285-1305.
    7. Mine Caglar, 2011. "Stock Price Processes with Infinite Source Poisson Agents," Papers 1106.6300, arXiv.org.
    8. Serge Cohen & Murad S. Taqqu, 2004. "Small and Large Scale Behavior of the Poissonized Telecom Process," Methodology and Computing in Applied Probability, Springer, vol. 6(4), pages 363-379, December.
    9. Jumarie, Guy, 2007. "Lagrangian mechanics of fractional order, Hamilton–Jacobi fractional PDE and Taylor’s series of nondifferentiable functions," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 969-987.
    10. Hermine Biermé & Anne Estrade & Ingemar Kaj, 2010. "Self-similar Random Fields and Rescaled Random Balls Models," Journal of Theoretical Probability, Springer, vol. 23(4), pages 1110-1141, December.
    11. Jumarie, Guy, 2008. "Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 271-287, February.
    12. Luis G. Gorostiza & Reyla A. Navarro & Eliane R. Rodrigues, 2004. "Some Long-Range Dependence Processes Arising from Fluctuations of Particle Systems," RePAd Working Paper Series lrsp-TRS401, Département des sciences administratives, UQO.

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