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Fractional Brownian motion as a weak limit of Poisson shot noise processes--with applications to finance

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  • Klüppelberg, Claudia
  • Kühn, Christoph

Abstract

We consider Poisson shot noise processes that are appropriate to model stock prices and provide an economic reason for long-range dependence in asset returns. Under a regular variation condition we show that our model converges weakly to a fractional Brownian motion. Whereas fractional Brownian motion allows for arbitrage, the shot noise process itself can be chosen arbitrage-free. Using the marked point process skeleton of the shot noise process we construct a corresponding equivalent martingale measure explicitly.

Suggested Citation

  • Klüppelberg, Claudia & Kühn, Christoph, 2004. "Fractional Brownian motion as a weak limit of Poisson shot noise processes--with applications to finance," Stochastic Processes and their Applications, Elsevier, vol. 113(2), pages 333-351, October.
    • Handle: RePEc:eee:spapps:v:113:y:2004:i:2:p:333-351
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    References listed on IDEAS

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    1. Granger, C. W. J., 1980. "Long memory relationships and the aggregation of dynamic models," Journal of Econometrics, Elsevier, vol. 14(2), pages 227-238, October.
    2. Tommi Sottinen, 2001. "Fractional Brownian motion, random walks and binary market models," Finance and Stochastics, Springer, vol. 5(3), pages 343-355.
    3. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    4. C. W. J. Granger & Roselyne Joyeux, 1980. "An Introduction To Long‐Memory Time Series Models And Fractional Differencing," Journal of Time Series Analysis, Wiley Blackwell, vol. 1(1), pages 15-29, January.
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    Cited by:

    1. Garzón, J. & Gorostiza, L.G. & León, J.A., 2009. "A strong uniform approximation of fractional Brownian motion by means of transport processes," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3435-3452, October.
    2. Mine Caglar, 2011. "Stock Price Processes with Infinite Source Poisson Agents," Papers 1106.6300, arXiv.org.
    3. Irle, Albrecht & Prelle, Claas, 2008. "A note on arbitrage under transaction costs," Kiel Working Papers 1450, Kiel Institute for the World Economy (IfW Kiel).
    4. Vasily E. Tarasov, 2019. "On History of Mathematical Economics: Application of Fractional Calculus," Mathematics, MDPI, vol. 7(6), pages 1-28, June.
    5. Erhan Bayraktar & Ulrich Horst & Ronnie Sircar, 2007. "A Limit Theorem for Financial Markets with Inert Investors," Papers math/0703831, arXiv.org.
    6. Christian Bender & Tommi Sottinen & Esko Valkeila, 2010. "Fractional processes as models in stochastic finance," Papers 1004.3106, arXiv.org.
    7. 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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