×

Remarks on the intersection local time of fractional Brownian motions. (English) Zbl 1225.60062

The existence of the so-called intersection local time
\[ I(B^H,\widetilde B^H)\equiv\int_0^T\int_0^T\delta(B_t^H-\widetilde B_s^H)\,ds\,dt, \]
where \(\delta(x)\) is the Dirac delta function was given by D. Nualart and S. Ortiz-Latorre [J. Theor. Probab. 20, No. 4, 759–767 (2007; Zbl 1154.60028)].
The authors give another proof for the existence of the random variable \(I(B^H,\widetilde B^H)\) in \(L^2\). They also discuss necessary and sufficient conditions for the intersection local time process to be smooth in the sense of Meyer-Watanabe.

MSC:

60G22 Fractional processes, including fractional Brownian motion
60G15 Gaussian processes
60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus

Citations:

Zbl 1154.60028
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] An, L., Yan, L., 2009. Smoothness for the collision local time of fractional Brownian motion. Preprint.; An, L., Yan, L., 2009. Smoothness for the collision local time of fractional Brownian motion. Preprint.
[2] Bass, R. F.; Chen, X., Self-intersection local time: critical exponent, large deviations, and laws of the iterated logarithm, Ann. Probab., 32, 3221-3247 (2004) · Zbl 1075.60097
[3] Berman, S. M., Self-intersections and local nondeterminism of Gaussian processes, Ann. Probab., 19, 160-191 (1991) · Zbl 0728.60037
[4] Biagini, F.; Hu, Y.; Øksendal, B.; Zhang, T., (Stochastic Calculus for Fractional Brownian Motion and Applications. Stochastic Calculus for Fractional Brownian Motion and Applications, Probability and its Application (2008), Springer: Springer Berlin) · Zbl 1157.60002
[5] Hu, Y., Integral transformations and anticipative calculus for fractional Brownian motion, Mem. Amer. Math. Soc., 175, 825 (2005) · Zbl 1072.60044
[6] Hu, Y.; Nualart, D., Renormalized self-intersection local time of fractional Brownian motion, Ann. Probab., 33, 948-983 (2005) · Zbl 1093.60017
[7] Jiang, Y.; Wang, Y., On the collision local time of fractional Brownian motion, Chin. Ann. Math., 28, 311-320 (2007) · Zbl 1124.60036
[8] Meyer, P. A., (Quantum for Probabilists. Quantum for Probabilists, Lecture Notes in Mathmatics, vol. 1538 (1993), Springer: Springer Heidelberg) · Zbl 0773.60098
[9] Mishura, Y. S., (Stochastic Calculus for Fractional Brownian Motion and Related Processes. Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes in Math., vol. 1929 (2008)) · Zbl 1138.60006
[10] Nualart, D., The Malliavin Calculus and Related Topics (2006), Springer: Springer New York · Zbl 1099.60003
[11] Nualart, D.; Ortiz-Latorre, S., Intersection local time for two independent fractional Brownian motions, J. Theoret. Probab., 20, 759-767 (2007) · Zbl 1154.60028
[12] Rosen, J., The intersection local time of fractional Brownian motion in the plane, J. Multivariate Anal., 23, 37-46 (1987) · Zbl 0633.60057
[13] Watanabe, S., (Stochachastic Differential Equation and Malliavin Calcus. Stochachastic Differential Equation and Malliavin Calcus, Tata Institute of Fundamental Research (1984), Spring: Spring New York) · Zbl 0546.60054
[14] Wu, D.; Xiao, Y., Regularity of intersection local times of fractional Brownian motions, J. Theoret. Probab., 23, 972-1001 (2010) · Zbl 1217.60030
[15] Xiao, Y., Sample path properties of anisotropic Gaussian random fields, Lecture Notes in Math., 1962, 145-212 (2009) · Zbl 1167.60011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.