On the local time of sub-fractional Brownian motion
Mendy, Ibrahima
Annales mathématiques Blaise Pascal, Tome 17 (2010), p. 357-374 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

S H ={S t H ,t≥0} be a sub-fractional Brownian motion with H∈(0,1). We establish the existence, the joint continuity and the Hölder regularity of the local time L H of S H . We will also give Chung’s form of the law of iterated logarithm for S H . This results are obtained with the decomposition of the sub-fractional Brownian motion into the sum of fractional Brownian motion plus a stochastic process with absolutely continuous trajectories. This decomposition is given by Ruiz de Chavez and Tudor [10].

Publié le : 2010-01-01
DOI : https://doi.org/10.5802/ambp.288
Classification:  60G15,  60G17,  60G18 
@article{AMBP_2010__17_2_357_0,
     author = {Mendy, Ibrahima},
     title = {On the local time of sub-fractional Brownian motion},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {17},
     year = {2010},
     pages = {357-374},
     doi = {10.5802/ambp.288},
     zbl = {pre05839427},
     mrnumber = {2778915},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AMBP_2010__17_2_357_0}
}

Mendy, Ibrahima. On the local time of sub-fractional Brownian motion. Annales mathématiques Blaise Pascal, Tome 17 (2010) pp. 357-374. doi : 10.5802/ambp.288. http://gdmltest.u-ga.fr/item/AMBP_2010__17_2_357_0/

[1] Adler, Robert J. The geometry of random fields, John Wiley & Sons Ltd., Chichester (1981) (Wiley Series in Probability and Mathematical Statistics) | MR 611857 | Zbl 0478.60059

[2] Adler, Robert J. An introduction to continuity, extrema, and related topics for general Gaussian processes, Institute of Mathematical Statistics, Hayward, CA, Institute of Mathematical Statistics Lecture Notes—Monograph Series, 12 (1990) | MR 1088478 | Zbl 0747.60039

[3] Baraka, D.; Mountford, T.; Xiao, Y. Hölder properties of local times for fractional Brownian motions, Metrika, Tome 69 (2009) no. 2-3, pp. 125-152 | Article | MR 2481918

[4] Berman, S. M. Local times and sample function properties of stationary Gaussian processes, Trans. Amer. Math. Soc., Tome 137 (1969), pp. 277-299 | Article | MR 239652 | Zbl 0184.40801

[5] Berman, S. M. Gaussian processes with stationary increments: Local times and sample function properties, Ann. Math. Statist., Tome 41 (1970), pp. 1260-1272 | Article | MR 272035 | Zbl 0204.50501

[6] Berman, S. M. Local nondeterminism and local times of Gaussian processes, Indiana University Mathematical Journal, Tome 23 (1973), pp. 69-94 | Article | MR 317397 | Zbl 0264.60024

[7] Berman, Simeon M. Gaussian sample functions: Uniform dimension and Hölder conditions nowhere, Nagoya Math. J., Tome 46 (1972), pp. 63-86 | MR 307320 | Zbl 0246.60038

[8] Bojdecki, T. L. G.; Gorostiza, L. G.; Talarczyk, A. Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particule systems, Electron. Comm. Probab., Tome 32 (2007), pp. 161-172 | MR 2318163 | Zbl 1128.60025

[9] Boufoussi, B.; Dozzi, M.; Guerbaz, R. On the local time of the multifractional brownian motion, Stochastics and stochastic repports, Tome 78 (2006), pp. 33-49 | MR 2219711 | Zbl 1124.60061

[10] Ruiz De Chávez, J.; Tudor, C. A decomposition of sub-fractional Brownian motion, Math. Rep. (Bucur.), Tome 11(61) (2009) no. 1, pp. 67-74 | MR 2506510 | Zbl pre05635003

[11] Csörgő, Miklós; Lin, Zheng Yan; Shao, Qi Man On moduli of continuity for local times of Gaussian processes, Stochastic Process. Appl., Tome 58 (1995) no. 1, pp. 1-21 | Article | MR 1341551 | Zbl 0834.60088

[12] Ehm, W. Sample function properties of multi-parameter stable processes, Z. Wahrsch. Verw. Gebiete, Tome 56 (1981), pp. 195-228 | Article | MR 618272 | Zbl 0471.60046

[13] Geman, D.; Horowitz, J. Occupation densities, Annales of probability, Tome 8 (1980), pp. 1-67 | Article | MR 556414 | Zbl 0499.60081

[14] Guerbaz, R. Local time and related sample paths of filtered white noises, Annales Mathematiques Blaise Pascal, Tome 14 (2007), pp. 77-91 | Article | Numdam | MR 2298805 | Zbl 1144.60029

[15] Kôno, N. Hölder conditions for the local times of certain gaussian processes with stationary increments, Proceeding of the Japan Academy, Tome 53 (1977), pp. 84-87 | Article | MR 494453 | Zbl 0437.60057

[16] Kôno, N.; Shieh, N. R. Local times and related sample path proprieties of certain selfsimilar processes, J. Math. Kyoto Univ., Tome 33 (1993), pp. 51-64 | MR 1203890 | Zbl 0776.60054

[17] Lei, P.; Nualart, D. A decomposition of the bifractional Brownian motion and some applications, Statist. Probab. Lett, Tome 779 (2009), pp. 619-624 | Article | MR 2499385 | Zbl 1157.60313

[18] Li, W. V.; Shao, Q.-M. Gaussian processes: inequalities, small ball probabilities and applications, Stochastic processes: theory and methods, North-Holland, Amsterdam (Handbook of Statist.) Tome 19 (2001), pp. 533-597 | MR 1861734 | Zbl 0987.60053

[19] Monrad, D.; Rootzén, H. Small values of Gaussian processes and functional laws of the iterated logarithm, Probab. Th. Rel. Fields, Tome 101 (1995), pp. 173-192 | Article | MR 1318191 | Zbl 0821.60043

[20] Pitt, L. Local times for gaussian vector fields, Indiana Univ. Math. J., Tome 27 (1978), pp. 204-237 | Article | MR 471055 | Zbl 0382.60055

[21] Tudor, C. Some propreties of sub-fractional Brownian motion, Stochastics., Tome 79 (2007), pp. 431-448 | MR 2356519 | Zbl 1124.60038

[22] Tudor, C. Inner product spaces of integrands associated to sub-fractional Brownian motion, Statist. Probab. Lett., Tome 78 (2008), p. 2201-2209. | Article | MR 2458028 | Zbl pre05609596

[23] Xiao, Y. Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields, Probab. Th. Rel. fields, Tome 109 (1997), pp. 129-157 | Article | MR 1469923 | Zbl 0882.60035