On the local time of sub-fractional Brownian motion

Mendy, Ibrahima

Annales mathématiques Blaise Pascal, Tome 17 (2010) no. 2, pp. 357-374.

$S^H=\{S_t^H,t\ge 0\}$ be a sub-fractional Brownian motion with $H\in (0,1)$. 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].

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

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

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

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

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

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

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

[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., Volume 32 (2007), pp. 161-172 | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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, Volume 109 (1997), pp. 129-157 | DOI | MR | Zbl