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Original Articles

Variations and estimators for self-similarity parameter of sub-fractional Brownian motion via Malliavin calculus

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Pages 3276-3289 | Received 24 Jan 2013, Accepted 19 Jun 2013, Published online: 26 Jan 2015

References

  • Beran, J. (1994). Statistics for Long-Memory Processes. Boca Raton: Chapman and Hall.
  • Bojdecki, T., Gorostiza, L., Talarczyk, A. (2004a). Sub-fractional Brownian motion and its relation to occupation times. Statist. Probab. Lett. 69:405–419.
  • Bojdecki, T., Gorostiza, L., Talarczyk, A. (2004b). Fractional Brownian density process and its self-intersection local time of order k. J. Theoret. Probab. 69(5):717–739.
  • Bojdecki, T., Gorostiza, L., Talarczyk, A. (2006). Limit theorems for occupation time fluctuations of branching systems 1: long-range dependence. Stochastic. Process. Appl. 116:1–18.
  • Bojdecki, T., Gorostiza, L., Talarczyk, A. (2007). Some extension of fractional Brownian motion and sub-fractional Brownian motion related to particle systems. Elect. Comm. Probab. 12:161–172.
  • Chronopoulou, A., Tudor, C.A., Viens, F. (2009). Variations and Hurst index estimation for a Rosenblatt process using longer filters. Electro. J. Statist.1393–1435.
  • Coeurjolly, J.F. (2001). Estimating the parameters of a fractional Brownian motion by discrte variations of its sample paths. Stat. Infer. Stoch. Process. 4:199–227.
  • Dzhaparidze, K., van Zanten, H. (2004). A series expansion of fractional Brownian motion. Probab. Theory. Ral. Fields. 130:39–55.
  • Es-Sebaiy, K., Tudor, C.A. (2011). Non-central limit theorem for the cubic variation of a class of selfsimilar stochastic process. Theory Probab. Appl. 55(3):411–431.
  • León, J., Ludeña, C. (2007). Limits for weighted p-variations and likewise functionals of fractional diffusions with drift. Stochastic. Process. Appl. 117:271–296.
  • Liu, J., Yan, L. (2012). Remarks on asymptotic behavior of weighted quadratic variation of sub-fractional Brownian motion. J. Korean. Statist. Soc. 41:177–187.
  • Nualart, D. (2006). The Malliavin Calculus and Related Topics. 2nd ed. Berlin: Springer.
  • Nualart, D., Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33:177–193.
  • Nualart, D., Ortiz-Latorre, S. (2008). Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic. Process. Appl. 118:614–628.
  • Nourdin, I. (2008). Asymptotic behavior of weighted quadratic and cubic varitions of fractional Brownian motion. Ann. Probab. 36(6):2159–2175.
  • Nourdin, I., Peccati, G. (2008a). Weighted power variation of iterated Brownian motion. Electron. J. Probab. 13:1229–1256.
  • Nourdin, I., Peccati, G. (2008b). Stein’s method on Wiener chaos. Probab. Theory Rel. Fields. 145:75–118.
  • Nourdin, I. (2012). Lectures on Gaussian approximations with Malliavin calculus. to appear in Séminaire de Probabilités.
  • Peccati, G., Tudor, C.A. (2004). Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probabilités XXXIV:247–262.
  • Shen, G., Yan, L. (2011). Remarks on an integral functional driven by sub-fractional Brownian motion. J. Korean. Statist. Soc. 40:337–346.
  • Shen, G., Chen, C. (2012). Stochastic integration with respect to the sub-fractional Brownian motion with H ∈ (0, 1/2). Statist. Probab. Lett. 82:240–251.
  • Swanson, J. (2007). Variations of the solution to a stochastic heat equation. Ann. Probab. 35:2122–2159.
  • Tudor, C.A., Viens, F.G. (2008). Variations of the fractional Brownian motion via Malliavin calculus. to appear in Austral. J. Math. (preprint).
  • Tudor, C.A., Viens, F.G. (2009). Variations and estimators for self-similarity parameters through Malliavin calculus. Ann. Probab. 37(6):2093–2134.
  • Tudor, C. (2007). Some properties of the sub-fractional brownian motion. Stochastics 79:431–448.
  • Tudor, C. (2008a). Inner product spaces of integrands associated to subfractional Brownian motion. Statist. Probab. Lett. 78:2201–2209.
  • Tudor, C. (2008b). Multiple sub-fractional integral and some approximations. Appl. Anal. 87:311–323.
  • Tudor, C. (2008c). Some aspects of stochastic calculus for the sub-fractional Brownian motion. Analele Universităţii Bucureşti, Matematică Anal LV2. 87:199–230.
  • Tudor, C. (2009). On the Wiener integral with respect to a subfractional Brownian motion. J. Math. Anal. Appl. 351(1):456–468.
  • Tudor, C. (2011). Berry-Esséen bounds and almost sure CLT for the quadratic variation of the subfractional Brownian motion. J. Math. Anal. Appl. 375(2):667–676.
  • Yan, L., Shen, G. (2010). On the collision local time of sub-fractional Brownian motions. Statist. Probab. Lett. 80:296–308.
  • Yan, L., Shen, G., He, K. (2011). Itô’s formula for the sub-fractional Brownian motion. Commun. Stoch. Anal. 5(1):135–159.

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