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Integration with Respect to Fractional Local Time with Hurst Index 1/2 < H < 1

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Abstract

Let be the weighted local time of fractional Brownian motion B H with Hurst index 1/2 < H < 1. In this paper, we use Young integration to study the integral of determinate functions As an application, we investigate the weighted quadratic covariation \([f\big(B^H\big),B^H]^{(W)}\) defined by

$$ \left[f\big(B^H\big),B^H\right]^{(W)}_t:=\lim_{n\to \infty}2H\sum_{k=0}^{n-1} k^{2H-1}\left\{f\big(B^H_{t_{k+1}}\big)-f\big(B^H_{t_{k}}\big)\right\} \left(B^H_{t_{k+1}}-B^H_{t_{k}}\right), $$

where the limit is uniform in probability and t k  = kt/n. We show that it exists and

provided f is of bounded p-variation with \(1\leq p<\frac{2H}{1-H}\). Moreover, we extend this result to the time-dependent case. These allow us to write the fractional Itô formula for new classes of functions.

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References

  1. E. Alós, Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29, 766–801 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Biagini, F., Hu, Y., Øksendal, B., Zhang, T.: Stochastic Calculus for Fractional Brownian Motion and Applications. Probability and its Application. Springer, Berlin (2008)

    Google Scholar 

  3. Bouleau, N., Yor, M.: Sur la variation quadratique des temps locaux de certaines semimartingales. C. R. Acad. Sci. Paris Sér. I Math. 292, 491–494 (1981)

    MATH  MathSciNet  Google Scholar 

  4. Coutin, L., Nualart, D., Tudor, C.A.: Tanaka formula for the fractional Brownian motion. Stochastic Process. Appl. 94, 301–315 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Eisenbaum, N.: Integration with respect to local time. Potential Anal. 13, 303–328 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Eisenbaum, N.: Local time Cspace stochastic calculus for Lévy processes. Stochastic Process. Appl. 116, 757–778 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Feng, C.R., Zhao, H.Z.: Two-parameters p, q-variation paths and integrations of local times. Potential Anal. 25, 165–204 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Föllmer, H., Protter, Ph., Shiryayev, A.N.: Quadratic covariation and an extension of Itô’s formula. Bernoulli 1, 149–169 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Geman, D., Horowitz, J.: Occupation densities. Ann. Probab. 8, 1–67 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gradinaru, M., Nourdin, I.: Weighted power variations of fractional Brownian motion and application to approximating schemes. Preprint (2007)

  11. Gradinaru, M., Nourdin, I., Russo, F., Vallois, P.: 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, 781–806 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gradinaru, M., Russo, F., Vallois, P.: Generalized covariations, local time and Stratonovich Itôs formula for fractional Brownian motion with Hurst index \(H\geq \frac{1}{4}\). Ann. Probab. 31, 1772–1820 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hu, Y., Økesendal, B., Salopek, D.M.: Weighted local time for fractional Brownian motion and applications to finance. Stoch. Anal. Appl. 23, 15–30 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hu, Y.: Integral transformations and anticipative calculus for fractional Brownian motions. Mem. Am. Math. Soc. 175(825) (2005)

  15. Klein, R., Giné, E.: On quadratic variation of processes with Gaussian increments. Ann. Probab. 3, 716–721 (1975)

    Article  MATH  Google Scholar 

  16. Nourdin, I.: Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion. Ann. Probab. (2008, in press)

  17. Nourdin, I., Nualart, D., Tudor, C.A.: Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Preprint (2007)

  18. Mishura, Y.S.: Stochastic calculus for fractional Brownian motion and related processes. Lecture Notes in Math. 1929, (2008)

  19. Mishura, Y.S., Valkeila, E.: An extension of the Lévy characterization to fractional Brownian motion. Preprint (2007)

  20. Moret, S., Nualart, D.: Quadratic covariation and Itô’s formula for smooth nondegenerate martingales. J. Theor. Probab. 13, 193–224 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Nualart, D.: Malliavin Calculus and Related Topics, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  22. Nualart, D., Taqqu, M.S.: Wick-Itô formula for Gaussian processes. Stoch. Anal. Appl. 24, 599–614 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Peskir, G.: A change-of-variable formula with local time on curves. J. Theoret. Probab. 18, 499–535 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Russo, F., Vallois, P.: Itô formula for \({\mathcal C}^1\)-functions of semimartingales. Probab. Theory Related Fields 104, 27–41 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  25. Russo, F., Vallois, P.: Elements of stochastic calculus via regularization. Sémin. Probab. XL, 147–185 (2007)

    Article  MathSciNet  Google Scholar 

  26. Yan, L., Yang, X.: Some remarks on local time-space calculus. Statist. Probab. Lett. 77, 1600–1607 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Young, L.C.: An inequality of Hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282 (1936)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, College of Science, Donghua University, 2999 North Renmin Rd., Songjiang, Shanghai, 201620, People’s Republic of China

    Litan Yan & Xiangfeng Yang

  2. Department of Mathematics, East China University of Science and Technology, 130 Mei Long Rd., Xuhui, Shanghai, 200237, People’s Republic of China

    Junfeng Liu

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  1. Litan Yan
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  2. Junfeng Liu
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Correspondence to Litan Yan.

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The Project-sponsored by NSFC (10571025) and the Key Project of Chinese Ministry of Education (No.106076).

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Yan, L., Liu, J. & Yang, X. Integration with Respect to Fractional Local Time with Hurst Index 1/2 < H < 1. Potential Anal 30, 115–138 (2009). https://doi.org/10.1007/s11118-008-9108-2

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