Publication Cover
Stochastics
An International Journal of Probability and Stochastic Processes
Volume 91, 2019 - Issue 4
65
Views
3
CrossRef citations to date
0
Altmetric
Articles

Continuity in law of some additive functionals of bifractional Brownian motion

, &
Pages 613-628 | Received 27 Apr 2017, Accepted 08 Jan 2019, Published online: 17 Jan 2019

References

  • M. Ait Ouahra, B. Boufoussi, and E. Lakhel, Théorémes limites pour certaines fonctionnelles associées aux processus stables dans une classe d'espaces de Besov Standard, Stoch. Stoch. Rep. 74 (2002), pp. 411–427. doi: 10.1080/10451120290024883
  • S.M. Berman, Local nondeterminism and local times of gaussian processes, Indiana. Univ. Math. J. 23 (1973), pp. 69–94. doi: 10.1512/iumj.1974.23.23006
  • N.H. Bingham, C.M. Goldie, and J.L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.
  • B. Boufoussi, Espaces de Besov: caractérisations et applications, Thése de L'Université de Henri Poincaré Nancy, France, 1994.
  • B. Boufoussi, M. Dozzi, and R. Guerbaz, On the local time of the multifractional Brownian motion, Stoch. Stoch. Reports 78 (2006), pp. 33–49. MR0239652. doi: 10.1080/17442500600578073
  • B. Boufoussi, and E.H. Lakhel, Un résultat d'approximation d'une EDPS hyperbolique en norme de Besov anisotropique, C. R. Acad. Sci. Paris. Ser. I 330 (2000), pp. 883–888. doi: 10.1016/S0764-4442(00)00279-2
  • Z. Ciesielski, G. Kerkyacharian, and B. Roynette, Quelques espaces fonctionnels associés á des processus gaussiens, Studia Matimatica. 107(2) (1993), pp. 171–204. doi: 10.4064/sm-107-2-171-204
  • J.F. Coeurjolly, Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths, Stat. Inf. Stoch. Process. 4(2) (2001), pp. 199–227. doi: 10.1023/A:1017507306245
  • M. Eddahbi and J. Vives, Chaotic expansion and smoothness of some functionals of the fractional Brownian motion, J. Math. Kyoto Univ. (JMKYAZ). 43(2) (2003), pp. 349–368. doi: 10.1215/kjm/1250283731
  • W. Ehm, Sample function properties of multi-parameter stable processes, Z. Wahrsch. Verw. Gebiete 56 (1981), pp. 195–228. doi: 10.1007/BF00535741
  • K. Es-Sebaiy and C.A. Tudor, Multidimensional bifractional Brownian motion. Itô and Tanaka formulas, Stoch. Dyn. 7(3) (2007), pp. 366–388. doi: 10.1142/S0219493707002050
  • P.J. Fitzsimmons and R.K. Getoor, Limit theorems and variation properties for fractional derivatives of the local time of a stable process, Ann. Inst. H. Poincaré. 28(2) (1992), pp. 311–333.
  • D. Geman and J. Horowitz, Occupation densities, Ann. Prob. 8(1) (1980), pp. 1–67. doi: 10.1214/aop/1176994824
  • C. Houdré and J. Villa, An example of infinite dimensional quasi-helix, Contemp. Math. (Amer. Math. Soc.) 336 (2003), pp. 195–201. doi: 10.1090/conm/336/06034
  • J. Istas and G. Lang, Quadratic variations and estimation of the Hólder index of a Gaussian process, Ann. de l'institut Henri Poincaré (B) Prob. Stat. 33(4) (1997), pp. 407–436. doi: 10.1016/S0246-0203(97)80099-4
  • M. Jolis and N. Viles, Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion, J. Theoretical. Probab. 20(2) (2007), pp. 133–152. doi: 10.1007/s10959-007-0054-5
  • J.P. Kahane, Hélices et quasi-hélices, Adv. Math. 7B (1981), pp. 417–433.
  • A. Kamont, Isomorphism of some anisotropic Besov and sequence spaces, Studia. Math, 110(2) (1994), pp. 169–189. doi: 10.4064/sm-110-2-169-189
  • J. Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104 (1962), pp. 64–78.
  • P. Lei and D. Nualart, A decomposition of the bifractional Brownian motion and some applications, Statist. Probab. Lett. 79 (2009), pp. 619–624. doi: 10.1016/j.spl.2008.10.009
  • C. Lacaux and J.M. Loubes, Hurst exponent estimation of fractional Lévy motions, ALEA Latin Amer. J. Prob. Math. Stat. Inst. Nac. de Matemática Pura e Apl. 3 (2007), pp. 143–164.
  • I. Mendy, On the local time of sub-fractional Brownian motion, Ann. Math. Blaise. Pascal. 17(2) (2010), pp. 357–374. doi: 10.5802/ambp.288
  • H. Ouahhabi and A. Sghir, Regularities and limit theorems of some additive functionals of symmetric stable process in some anisotropic Besov spaces, Portugal. Math.(N.S.) 69(4) (2012), pp. 321–339. Fasc. doi: 10.4171/PM/1920
  • F. Russo and C. Tudor, On the bifractional Brownian motion, Stoch. Process. Their Appl. 116(5) (2006), pp. 830–856. doi: 10.1016/j.spa.2005.11.013
  • S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science, Yverdon, 1993.
  • N.R. Shieh, Limits theorems for local times of fractional Brownian motions and some other self-similar processes, J. Math. Kyoto Univ. 36(4) (1996), pp. 641–652. doi: 10.1215/kjm/1250518444
  • C.A. Tudor and F.G. Viens, Variations of the fractional Brownian motion via Malliavin calculus. Preprint. 2008.
  • C.A. Tudor and Y. Xiao, Sample path properties of bifractional Brownian motion, Bernoulli 13(4) (2007), pp. 1023–1052. doi: 10.3150/07-BEJ6110
  • D. Wu and Y. Xiao, Geometric properties of fractional Brownian sheets, J. Fourier Anal. Appl. 13 (2007), pp. 1–37. doi: 10.1007/s00041-005-5078-y
  • D. Wu and Y. Xiao, Continuity in the Hurst index of the local times of anisotropic gaussian random fields, Stoch. Proc. Their Appl. 119 (2009), pp. 1823–1844. doi: 10.1016/j.spa.2008.09.001
  • Y. Xiao, Sample path properties of anisotropic Gaussian random fields, A Minicourse Stoch. Partial Differ. Equ. Lect. Notes Math. 1962 (2007), pp. 145–212.
  • L. Yan, B. Gao, and J. Liu, The Bouleau-Yor identity for a bifractional Brownian motion, Stoch: Int. J. Prob. Stoch. Proc. 86(3) (2014), pp. 382–414. doi: 10.1080/17442508.2013.797424
  • T. Yamada, On some limit theorems for occupation times of one-dimensional Brownian motion and its continuous additive functionals locally of zero energy, J. Math. Kyoto Univ. 26(2) (1986), pp. 309–322. doi: 10.1215/kjm/1250520924

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.