Bernoulli

  • Bernoulli
  • Volume 13, Number 3 (2007), 849-867.

Sample path properties of the local time of multifractional Brownian motion

Brahim Boufoussi, Marco Dozzi, and Raby Guerbaz

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Abstract

We establish estimates for the local and uniform moduli of continuity of the local time of multifractional Brownian motion, BH=(BH(t)(t), t∈ℝ+). An analogue of Chung’s law of the iterated logarithm is studied for BH and used to obtain the pointwise Hölder exponent of the local time. A kind of local asymptotic self-similarity is proved to be satisfied by the local time of BH.

Article information

Source
Bernoulli, Volume 13, Number 3 (2007), 849-867.

Dates
First available in Project Euclid: 7 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1186503490

Digital Object Identifier
doi:10.3150/07-BEJ6140

Mathematical Reviews number (MathSciNet)
MR2348754

Zentralblatt MATH identifier
1138.60032

Keywords
Chung-type law of iterated logarithm local asymptotic self-similarity multifractional Brownian motion local times

Citation

Boufoussi, Brahim; Dozzi, Marco; Guerbaz, Raby. Sample path properties of the local time of multifractional Brownian motion. Bernoulli 13 (2007), no. 3, 849--867. doi:10.3150/07-BEJ6140. https://projecteuclid.org/euclid.bj/1186503490


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References

  • Adler, R.J. (1990)., An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. IMS Lecture Notes-Monograph Series 12. Hayward, CA: IMS.
  • Ayache, A. (2001). Du mouvement Brownien fractionnaire au mouvement Brownien multifractionnaire., Techniques et Sciences Informatiques 20 1133--1152.
  • Benassi, A., Jaffard, S. and Roux, D. (1997). Elliptic Gaussian random processes., Rev. Mat. Iberoamericana 13 19-90.
  • Berman, S.M. (1969). Local times and sample function properties of stationary Gaussian processes., Trans. Amer. Math. Soc. 137 277--299.
  • Berman, S.M. (1973). Local nondeterminism and local times of Gaussian processes., Indiana Univ. Math. J. 23 69--94.
  • Boufoussi, B., Dozzi, M. and Guerbaz, R. (2006). On the local time of multifractional Brownian motion., Stochastics 78 33--49.
  • Cohen, S. (1999). From self-similarity to local self-similarity: the estimation problem. In M. Dekking, J. Lévy-Véhel, E. Lutton and C. Tricot (eds), Fractals: Theory and Applications in Engineering, pp. 3--16. London: Springer.
  • Csörgö, M., Lin, Z. and Shao, Q.M. (1995). On moduli of continuity for local times of Gaussian processes., Stoch. Proc. Appl. 58 1--21.
  • Davies, L. (1976). Local Hölder conditions for the local times of certain stationary Gaussian processes., Ann. Probab. 14 277--298.
  • Dozzi, M. (2003). Occupation density and sample path properties of $N$-parameter processes., Topics in Spatial Stochastic Processes. Lecture Notes in Math. 1802 127--166. Berlin: Springer.
  • Ehm, W. (1981). Sample function properties of multi-parameter stable processes., Z. Wahrsch. Verw. Gebiete 56 195--228.
  • Geman, D. and Horowitz, J. (1980). Occupation densities., Ann. Probab. 8 1--67.
  • Gihman, I.I. and Skorohod, A.V. (1974)., The Theory of Stohastic Processes. I. New York: Springer.
  • Kasahara, Y. and Kosugi, N. (1997). A limit theorem for occupation times of fractional Brownian motion., Stoch. Proc. Appl. 67 161--175.
  • Kesten, H. and Spitzer, F. (1979). A limit theorem related to a new class of self similar processes., Z. Wahrsch. Verw. Gebiete 50 5--25.
  • Kôno, N. (1977). Hölder conditions for the local times of certain Gaussian processes with stationary increments., Proc. Japan Acad. Ser. A Math. Sci. 53 84--87.
  • Lévy-Véhel, J. and Peltier, R.F. (1995). Multifractional Brownian motion: Definition and preliminary results. Technical Report RR-2645, INRIA.
  • Lim, S.C. (2001). Fractional Brownian motion and multifractional Brownian motion of Riemann--Liouville type., J. Phys. A 34 1301--1310.
  • Li, W.V. and Shao, Q.-M. (2001). Gaussian processes: Inequalities, small ball probabilities and applications. In C.R. Rao and D. Shanbhag (eds), Stochastic Processes: Theory and Methods. Handbook of Statistics 19, pp. 533--597. Amsterdam: North-Holland.
  • Monrad, D. and Rootzén, H. (1995). Small values of Gaussian processes and functional laws of the iterated logarithm., Probab. Theory Related Fields 101 173--192.
  • Xiao, Y. (2005). Strong local nondeterminism of Gaussian random fields and its applications. In T.-L. Lai, Q.-M. Shao and L. Qian (eds), Probability and Statistics with Applications. To appear.