Abstract
In this paper, we consider the local time and the self-intersection local time for a bifractional Brownian motion, and the collision local time for two independent bifractional Brownian motions. We mainly prove the existence and smoothness of the self-intersection local time and the collision local time, through the strong local nondeterminism of bifractional Brownian motion, L 2 convergence and Chaos expansion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bonami A, Estrade A. Anisotropic analysis of some Gaussian models. J Fourier Anal Appl, 9: 215–236: (2003)
Cheridito P. Gaussian moving averages, semimartingales and option pricing. Stochastic Process Appl, 109: 47–68: (2004)
Benson D A, Meerschaert M M, Baeumer B. Aquifer operator-scaling and the effect on solute mixing and disperson. Water Resour Res, 42: 1–18: (2006)
Rosen J. The intersection local time of fractional Brownian motion in the plane. J Multivariate Anal, 23(1): 37–46: (1987)
Xiao Y M, Zhang T S. Local time of fractional Brownian sheets. Probab Theory Related Fields, 124: 121–139: (2002)
Ayache A, Wu D S, Xiao Y M. Joint continuity of the local times of fractional Brownian sheets. Ann Inst H Poincaré Probab Statist, 44(4): 727–748: (2008)
Nualart D, Vives J. Chaos expansion and local time. Publ Mat, 36(2): 827–836: (1992)
Imkeller P, Abreu V, Vives J. Chaos expansions of double intersection local time of Brownian motion in Rd and renormalization. Stochastic Process Appl, 56: 1–34: (1995)
Hu Y Z. Self-intersection of fractional Brownian motions.via chaos expansion. J Math Kyoto Univ, 41(2): 233–250: (2001)
Hu Y Z, Nualart D. Renormalized Self-intersection local time for fractional Brownian motion. Ann Probab, 33(3): 948–983: (2005)
Jiang Y M, Wang Y J. On the collision local time of fractional Brownian motions. Chin Ann Math Ser B, 28(3): 311–320: (2007)
Houdré C, Villa J. An example of infinite-dimensional quasi-helix. Stoch Models, 336: 195–201: (2003)
Russo F, Tudor C A. On the bifractional Brownian motion. Stochastic Process Appl, 5: 830–856: (2006)
Tudor C A, Xiao Y M. Sample path properties of bifractional Brownian motion. Bernoulli, 14: 1023–1052: (2007)
Simon B. The P(φ)2 Euclidean Field Theory. Princeton: Princeton University Press, 1974
Meyer P A. Quantum Probability for Probabilists. Lecture Notes in Mathematics, Vol. 1538. Berlin: Springer-Verlag, 1993
Watanabe S. Lectures on Stochastic Differetial Equation andMalliavin Calculus. Berlin-New York: Springer-Verlag, 1984
Berman S M. Local nondeterminism and local times of Gaussian processes. Indiana Univ Math J, 23: 69–94: (1973)
Xiao Y M. Strong local nondeterminism of Gaussian random fields and its applications. In: Lai T L, Shao Q M, Qian L, eds. Asymptotic Theory in Probability and Statistics with Applications. Beijing: Higher Education Press, 2007, 136–176
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by National Natural Science Foundation of China (Grant No. 10871103)
Rights and permissions
About this article
Cite this article
Jiang, Y., Wang, Y. Self-intersection local times and collision local times of bifractional Brownian motions. Sci. China Ser. A-Math. 52, 1905–1919 (2009). https://doi.org/10.1007/s11425-009-0081-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-009-0081-z
Keywords
- bifractional Brownian motion
- self-intersection local time
- collision local time
- strong local nondeterminism