×

Smoothness for the collision local time of two multidimensional bifractional Brownian motions. (English) Zbl 1274.60119

Summary: Let \(B^{H_{i},K_i}=\{B^{H_{i},K_i}_t,\, t\geq 0 \}\), \(i=1,2\) be two independent, \(d\)-dimensional bifractional Brownian motions with respective indices \(H_i\in (0,1)\) and \(K_i\in (0,1]\). Assume \(d\geq 2\). One of the main motivations of this paper is to investigate smoothness of the collision local time \[ \ell _T=\int _{0}^{T}\delta (B_{s}^{H_{1},K_1}-B_{s}^{H_{2},K_2})\,\text{d} s, \quad T>0, \] where \(\delta \) denotes the Dirac delta function. By an elementary method we show that \(\ell _T\) is smooth in the sense of Meyer-Watanabe if and only if \(\min \{H_{1}K_1,H_{2}K_2\}<{1}/{(d+2)}\).

MSC:

60G15 Gaussian processes
60G18 Self-similar stochastic processes
60G22 Fractional processes, including fractional Brownian motion
60J65 Brownian motion
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] L. An, L. Yan: Smoothness for the collision local time of fractional Brownian motion. Preprint, 2010.
[2] C. Chen, L. Yan: Remarks on the intersection local time of fractional Brownian motions. Stat. Probab. Lett. 81 (2011), 1003–1012. · Zbl 1225.60062 · doi:10.1016/j.spl.2011.01.021
[3] K. Es-Sebaiy, C.A. Tudor: Multidimensional bifractional Brownian motion: Itô and Tanaka formulas. Stoch. Dyn. 7 (2007), 365–388. · Zbl 1139.60321 · doi:10.1142/S0219493707002050
[4] Ch. Houdré, J. Villa: An example of infinite dimensional quasi-helix. Stochastic models. Seventh symposium on probability and stochastic processes, June 23–28, 2002, Mexico City, Mexico. Selected papers. Providence, RI: American Mathematical Society (AMS), Contemp. Math. 336 (2003), 195–201.
[5] Y. Hu: Self-intersection local time of fractional Brownian motion – via chaos expansion. J. Math, Kyoto Univ. 41 (2001), 233–250. · Zbl 1008.60091
[6] Y. Hu: Integral transformations and anticipative calculus for fractional Brownian motions. Mem. Am. Math. Soc. 825 (2005). · Zbl 1072.60044
[7] Y. Jiang, Y. Wang: Self-intersection local times and collision local times of bifractional Brownian motions. Sci. China, Ser. A 52 (2009), 1905–1919. · Zbl 1181.60059 · doi:10.1007/s11425-009-0081-z
[8] I. Kruk, F. Russo, C.A. Tudor: Wiener integrals, Malliavin calculus and covariance measure structure. J. Funct. Anal. 249 (2007), 92–142. · Zbl 1126.60046 · doi:10.1016/j.jfa.2007.03.031
[9] P. Lei, D. Nualart: A decomposition of the bifractional Brownian motion and some applications. Stat. Probab. Lett. 79 (2009), 619–624. · Zbl 1157.60313 · doi:10.1016/j.spl.2008.10.009
[10] Y. Mishura: Stochastic Calculus for Fractional Brownian Motions and Related Processes. Lecture Notes in Mathematics 1929. Springer, Berlin, 2008.
[11] D. Nualart, S. Ortiz-Latorre: Intersection local time for two independent fractional Brownian motions. J. Theor. Probab. 20 (2007), 759–767. · Zbl 1154.60028 · doi:10.1007/s10959-007-0106-x
[12] D. Nualart: The Malliavin Calculus and Related Topics. 2nd ed. Probability and Its Applications. Springer, Berlin, 2006. · Zbl 1099.60003
[13] F. Russo, C.A. Tudor: On bifractional Brownian motion. Stochastic Processes Appl. 116 (2006), 830–856. · Zbl 1100.60019 · doi:10.1016/j.spa.2005.11.013
[14] G. Shen, L. Yan: Smoothness for the collision local times of bifractional Brownian motions. Sci. China, Math. 54 (2011), 1859–1873. · Zbl 1236.60042 · doi:10.1007/s11425-011-4228-3
[15] C.A. Tudor, Y. Xiao: Sample path properties of bifractional Brownian motion. Bernoulli 13 (2007), 1023–1052. · Zbl 1132.60034 · doi:10.3150/07-BEJ6110
[16] S. Watanabe: Lectures on Stochastic Differential Equations and Malliavin Calculus. Lectures on Mathematics and Physics. Mathematics, 73. Tata Institute of Fundamental Research. Springer, Berlin, 1984. · Zbl 0546.60054
[17] L. Yan, J. Liu, C. Chen: On the collision local time of bifractional Brownian motions. Stoch. Dyn. 9 (2009), 479–491. · Zbl 1180.60034 · doi:10.1142/S0219493709002749
[18] L. Yan, B. Gao, J. Liu: The Bouleau-Yor identity for a bi-fractional Brownian motion. To appear in Stochastics 2012. · Zbl 1319.60083
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.