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A class of micropulses and antipersistent fractional Brownian motion

Author

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

Abstract

We begin with stochastic processes obtained as sums of "up-and-down" pulses with random moments of birth [tau] and random lifetime w determined by a Poisson random measure. When the pulse amplitude [var epsilon] --> 0, while the pulse density [delta] increases to infinity, one obtains a process of "fractal sum of micropulses." A CLT style argument shows convergence in the sense of finite dimensional distributions to a Gaussian process with negatively correlated increments. In the most interesting case the limit is fractional Brownian motion (FBM), a self-affine process with the scaling constant . The construction is extended to the multidimensional FBM field as well as to micropulses of more complicated shape.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:spapps:v:60:y:1995:i:1:p:1-18
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    Cited by:

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    3. Luis Gil-Alana, 2003. "Stochastic behavior of nominal exchange rates," Atlantic Economic Journal, Springer;International Atlantic Economic Society, vol. 31(2), pages 159-173, June.
    4. Jumarie, Guy, 2005. "Merton's model of optimal portfolio in a Black-Scholes Market driven by a fractional Brownian motion with short-range dependence," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 585-598, December.
    5. Segnon, Mawuli & Lux, Thomas, 2013. "Multifractal models in finance: Their origin, properties, and applications," Kiel Working Papers 1860, Kiel Institute for the World Economy (IfW).
    6. Christian Fischer & Luis Gil-Alana, 2009. "The nature of the relationship between international tourism and international trade: the case of German imports of Spanish wine," Applied Economics, Taylor & Francis Journals, vol. 41(11), pages 1345-1359.
    7. L. A. Gil-Alana, 2003. "A fractional integration analysis of the population in some OECD countries," Journal of Applied Statistics, Taylor & Francis Journals, vol. 30(10), pages 1147-1159.
    8. 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.
    9. Guglielmo Maria Caporale & Luis Gil‐Alana, 2014. "Long‐Run and Cyclical Dynamics in the US Stock Market," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 33(2), pages 147-161, March.
    10. Francis Ahking, 2010. "Non-parametric tests of real exchange rates in the post-Bretton Woods era," Empirical Economics, Springer, vol. 39(2), pages 439-456, October.
    11. Rea, William & Oxley, Les & Reale, Marco & Brown, Jennifer, 2013. "Not all estimators are born equal: The empirical properties of some estimators of long memory," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 93(C), pages 29-42.
    12. Luis Gil-Alana, 2004. "The dynamics of the real exchange rates in Europe: a comparative study across countries using fractional integration," Applied Economics Letters, Taylor & Francis Journals, vol. 11(7), pages 429-432.
    13. Diebold, Francis X. & Inoue, Atsushi, 2001. "Long memory and regime switching," Journal of Econometrics, Elsevier, vol. 105(1), pages 131-159, November.
    14. Barros, Carlos P. & Gil-Alana, Luis A. & Wanke, Peter, 2016. "Energy production in Brazil: Empirical facts based on persistence, seasonality and breaks," Energy Economics, Elsevier, vol. 54(C), pages 88-95.
    15. Gil-Alana, Luis A., 2004. "Modelling the U.S. interest rate in terms of I(d) statistical models," The Quarterly Review of Economics and Finance, Elsevier, vol. 44(4), pages 475-486, September.
    16. Gourieroux, Christian & Jasiak, Joann, 2001. "Memory and infrequent breaks," Economics Letters, Elsevier, vol. 70(1), pages 29-41, January.
    17. Guglielmo Maria Caporale & Luis Gil-Alana, 2004. "Long range dependence in daily stock returns," Applied Financial Economics, Taylor & Francis Journals, vol. 14(6), pages 375-383.

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