The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6

Ivan Nourdin (Université Nancy 1)
Anthony Réveillac (Humboldt-University)
Jason Swanson (University of Central Florida)

Abstract


Let $B$ be a fractional Brownian motion with Hurst parameter $H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums for $\int\!g(B(s))\,dB(s)$ do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of càdlàg functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent of $B$.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 2117-2162

Publication Date: December 14, 2010

DOI: 10.1214/EJP.v15-843

References

  1. K. Burdzy and J. Swanson (2010). A change of variable formula with Itô correction term. Ann. Probab. 38 (5), 1817-1869. MR2722787
  2. Cheridito, Patrick; Nualart, David. Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter $Hin(0,{1over2})$. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 6, 1049--1081. MR2172209 (2006m:60051)
  3. Donnelly, Peter; Kurtz, Thomas G. A countable representation of the Fleming-Viot measure-valued diffusion. Ann. Probab. 24 (1996), no. 2, 698--742. MR1404525 (98f:60162)
  4. Ethier, Stewart N.; Kurtz, Thomas G. Markov processes.Characterization and convergence.Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085 (88a:60130)
  5. Gradinaru, Mihai; Nourdin, Ivan; Russo, Francesco; Vallois, Pierre. $m$-order integrals and generalized Itô's formula: the case of a fractional Brownian motion with any Hurst index. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 4, 781--806. MR2144234 (2006a:60100)
  6. Malliavin, Paul. Stochastic analysis.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 313. Springer-Verlag, Berlin, 1997. xii+343 pp. ISBN: 3-540-57024-1 MR1450093 (99b:60073)
  7. Nourdin, Ivan; Nualart, David. Central limit theorems for multiple Skorokhod integrals. J. Theoret. Probab. 23 (2010), no. 1, 39--64. MR2591903
  8. I. Nourdin, D. Nualart and C. A. Tudor (2010). Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 46 (4), 1055-1079.
  9. Nourdin, Ivan; Réveillac, Anthony. Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: the critical case $H=1/4$. Ann. Probab. 37 (2009), no. 6, 2200--2230. MR2573556 (2010m:60086)
  10. Nualart, David. The Malliavin calculus and related topics.Second edition.Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5 MR2200233 (2006j:60004)
  11. Nualart, D.; Ortiz-Latorre, S. Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 (2008), no. 4, 614--628. MR2394845 (2009h:60053)
  12. Réveillac, Anthony. Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets. Stoch. Anal. Appl. 27 (2009), no. 1, 51--73. MR2473140 (2010e:60083)
  13. Russo, Francesco; Vallois, Pierre. Elements of stochastic calculus via regularization. Séminaire de Probabilités XL, 147--185, Lecture Notes in Math., 1899, Springer, Berlin, 2007. MR2409004 (2009d:60167)
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.