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Limits for weighted p-variations and likewise functionals of fractional diffusions with drift

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  • León, José
  • Ludeña, Carenne

Abstract

Let Xt be the pathwise solution of a diffusion driven by a fractional Brownian motion with Hurst constant H>1/2 and diffusion coefficient [sigma](t,x). Consider the successive increments of this solution, [Delta]Xi=Xi/n-X(i-1)/n. Using a cylinder approximation for the solution Xt, our main result yields that if 1/2 [infinity] where W is a Wiener process which is independent of BH and CH,p is a constant which depends on H and on p. In the place of p-variations we may consider functions that satisfy an almost multiplicative structure such as even polynomials or polynomials of absolute values. By considering second order increments of the discrete sample Xi we obtain analogous results for the whole interval 1/2

Suggested Citation

  • León, José & Ludeña, Carenne, 2007. "Limits for weighted p-variations and likewise functionals of fractional diffusions with drift," Stochastic Processes and their Applications, Elsevier, vol. 117(3), pages 271-296, March.
    • Handle: RePEc:eee:spapps:v:117:y:2007:i:3:p:271-296
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    References listed on IDEAS

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    1. Jean Jacod, 2000. "Non‐parametric Kernel Estimation of the Coefficient of a Diffusion," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(1), pages 83-96, March.
    2. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
    3. Gloter, A. & Hoffmann, M., 2004. "Stochastic volatility and fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 113(1), pages 143-172, September.
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    2. Barndorff-Nielsen, Ole E. & Corcuera, José Manuel & Podolskij, Mark, 2009. "Power variation for Gaussian processes with stationary increments," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1845-1865, June.

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