Startseite Moduli of continuity of the local time of a class of sub-fractional Brownian motions
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Moduli of continuity of the local time of a class of sub-fractional Brownian motions

  • Mohamed Ait Ouahra EMAIL logo und Raby Guerbaz
Veröffentlicht/Copyright: 27. Oktober 2017
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

The aim of this paper is to establish sharp estimates for the moduli of continuity of the local time of a class of sub-fractional Brownian motions. We also investigate the continuity of their local times with respect to the self-similarity index.


Communicated by Vyacheslav L. Girko


A Appendix

Proof of Lemma 3.2.

Since in the Gaussian case 𝔼(X/𝔉2) is the orthogonal projection on 𝔉2 in the L2(Ω) sense, we obtain that X-𝔼(X/𝔉2) is independent of 𝔉2. Therefore,

Var(X/𝔉2)=𝔼[[X-𝔼(X/𝔉2)]2/𝔉2]=𝔼[X-𝔼(X/𝔉2)]2.

Now since

(A.1)𝔼[X-𝔼(X/𝔉2)]2𝔼[X-U]2for all U𝔉2,

and, moreover, 𝔼(X/𝔉1) is 𝔉1-measurable and 𝔉1𝔉2, we obtain that 𝔼(X/𝔉1) is 𝔉2-measurable. Hence, by using (A.1), we get

𝔼[X-𝔼(X/𝔉2)]2𝔼[X-𝔼(X/𝔉1)]2,

which proves the lemma. ∎

References

[1] X. Bardina and D. Bascompte, Weak convergence towards two independent Gaussian processes from a unique Poisson process, Collect. Math. 61 (2010), no. 2, 191–204. 10.1007/BF03191241Suche in Google Scholar

[2] S. M. Berman, Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J. 23 (1973/74), 69–94. 10.1512/iumj.1974.23.23006Suche in Google Scholar

[3] T. Bojdecki, L. G. Gorostiza and A. Talarczyk, Sub-fractional Brownian motion and its relation to occupation times, Statist. Probab. Lett. 69 (2004), no. 4, 405–419. 10.1016/j.spl.2004.06.035Suche in Google Scholar

[4] B. Boufoussi, M. Dozzi and R. Guerbaz, Path properties of a class of locally asymptotically self similar processes, Electron. J. Probab. 13 (2008), no. 29, 898–921. 10.1214/EJP.v13-505Suche in Google Scholar

[5] J. Cuzick and J. P. DuPreez, Joint continuity of Gaussian local times, Ann. Probab. 10 (1982), no. 3, 810–817. 10.1214/aop/1176993789Suche in Google Scholar

[6] D. Geman and J. Horowitz, Occupation densities, Ann. Probab. 8 (1980), no. 1, 1–67. 10.1214/aop/1176994824Suche in Google Scholar

[7] M. Jolis and N. Viles, Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion, J. Theoret. Probab. 20 (2007), no. 2, 133–152. 10.1007/s10959-007-0054-5Suche in Google Scholar

[8] I. Mendy, On the local time of sub-fractional Brownian motion, Ann. Math. Blaise Pascal 17 (2010), no. 2, 357–374. 10.5802/ambp.288Suche in Google Scholar

[9] D. Monrad and H. Rootzén, Small values of Gaussian processes and functional laws of the iterated logarithm, Probab. Theory Related Fields 101 (1995), no. 2, 173–192. 10.1007/BF01375823Suche in Google Scholar

[10] L. D. Pitt, Local times for Gaussian vector fields, Indiana Univ. Math. J. 27 (1978), no. 2, 309–330. 10.1512/iumj.1978.27.27024Suche in Google Scholar

[11] J. Ruiz de Chávez and C. Tudor, A decomposition of sub-fractional Brownian motion, Math. Rep. (Bucur.) 11(61) (2009), no. 1, 67–74. Suche in Google Scholar

[12] A. Sghir, The generalized sub-fractional Brownian motion, Commun. Stoch. Anal. 7 (2013), no. 3, 373–382. 10.31390/cosa.7.3.02Suche in Google Scholar

[13] C. A. Tudor and Y. Xiao, Sample path properties of bifractional Brownian motion, Bernoulli 13 (2007), no. 4, 1023–1052. 10.3150/07-BEJ6110Suche in Google Scholar

[14] Y. Xiao, Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields, Probab. Theory Related Fields 109 (1997), no. 1, 129–157. 10.1007/s004400050128Suche in Google Scholar

[15] Y. Xiao, Strong local nondeterminism and sample path properties of Gaussian random fields, Asymptotic Theory in Probability and Statistics with Applications, Adv. Lect. Math. (ALM) 2, International Press, Somerville (2008), 136–176. Suche in Google Scholar

Received: 2017-4-10
Accepted: 2017-5-10
Published Online: 2017-10-27
Published in Print: 2017-12-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 5.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/rose-2017-0017/html
Button zum nach oben scrollen