Smoothness for the collision local times of bifractional Brownian motions. (English) Zbl 1236.60042

Summary: Let \(B^{H_i ,K_i } = \{ B_t^{H_i ,K_i } , \, t \geqslant 0\}\), \(i = 1, 2\) be two independent bifractional Brownian motions with respective indices \(H _{i } \in (0, 1)\) and \(K _{i } \in (0, 1]\). One of the main motivations of this paper is to investigate the smoothness of the collision local time, introduced by Y. Jiang and Y. Wang [Sci. China, Ser. A 52, No. 9, 1905–1919 (2009; Zbl 1181.60059)], \[ \ell _T = \int_0^T \delta (B_s^{H_1 ,K_1 } - B_s^{H_2 ,K_2 })\,ds,\;T > 0, \] 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 \(\min\{H _{1} K _{1}, H _{2} K _{2}\} < 1/3\).


60G22 Fractional processes, including fractional Brownian motion
60G15 Gaussian processes
60H07 Stochastic calculus of variations and the Malliavin calculus
60J55 Local time and additive functionals


Zbl 1181.60059
Full Text: DOI


[1] An L, Yan L. Smoothness for the collision local time of fractional Brownian motion. Preprint, 2010
[2] Berman S M. Self-intersections and local nondeterminism of Gaussian processes. Ann Probab, 1991, 19: 160–191 · Zbl 0728.60037 · doi:10.1214/aop/1176990539
[3] Biagini F, Hu Y, Øksendal B, et al. Stochastic Calculus for Fractional Brownian Motions and Applications. London: Springer-Verlag, 2008 · Zbl 1157.60002
[4] Es-sebaiy K, Tudor C A. Multidimensional bifractional Brownian motion: Itô and Tanaka formulas. Stoch Dyn, 2007, 7: 366–388 · Zbl 1139.60321
[5] Houdré C, Villa J. An example of infinite dimensional quasi-helix. Stoch Models, 2003, 336: 195–201 · Zbl 1046.60033 · doi:10.1090/conm/336/06034
[6] Hu Y. Integral Transformations and Anticipative Calculus for Fractional Brownian Motions. Mem Amer Math Soc, 175. Providence, RI: AMS, 2005 · Zbl 1072.60044
[7] Jiang Y, Wang Y. On the collision local time of fractional Brownian motions. Chinese Ann Math Ser B, 2007, 28: 311–320 · Zbl 1124.60036 · doi:10.1007/s11401-006-0029-3
[8] Jiang Y, Wang Y. Self-intersection local times and collision local times of bifractional Brownian motions. Sci China Ser A, 2009, 52: 1905–1919 · Zbl 1181.60059 · doi:10.1007/s11425-009-0081-z
[9] Kruk I, Russo F, Tudor C A. Wiener integrals, Malliavin calculus and covariance measure structure. J Funct Anal, 2007, 249: 92–142 · Zbl 1126.60046 · doi:10.1016/j.jfa.2007.03.031
[10] Meyer P A. Quantum for Probabilists. Lecture Notes in Math, 1538. Heidelberg: Springer, 1993 · Zbl 0773.60098
[11] Mishura Y. Stochastic calculus for fractional Brownian motions and related processes. Lecture Notes in Math, 1929. Berlin-Heidelberg: Springer-Verlag, 2008
[12] Nualart D. The Malliavin Calculus and Related Topics. Berlin: Springer-Verlag, 2006 · Zbl 1099.60003
[13] Russo F, Tudor C A. On the bifractional Brownian motion. Stochastic Process Appl, 2006, 5: 830–856 · Zbl 1100.60019 · doi:10.1016/j.spa.2005.11.013
[14] Tudor C A, Xiao Y. Sample path properties of bifractional Brownian motion. Bernoulli, 2007, 13: 1023–1052 · Zbl 1132.60034 · doi:10.3150/07-BEJ6110
[15] Varadhan S R S. Appendix to Euclidean quantum field theory, by K. Symanzik. In: Jost R, ed. Local Quantum Theory. New York: Academic Press, 1968
[16] Watanabe S. Stochachastic Differential Equation and Malliavin Calculus, Tata Inst Fund Res Stud Marth. New York: Springer, 1984
[17] Yan L, Chen C, Bian C. Integration with respect to local time of bifractional Brownian motion with HK &lt; 1/2. Preprint, 2010
[18] Yan L, Liu J, Bian C. Integration with respect to local time of bifractional Brownian motion with HK &gt; 1/2. Preprint, 2010
[19] Yan L, Liu J, Chen C. On the collision local time of bifractional Brownian motions. Stoch Dyn, 2009, 9: 479–491 · Zbl 1180.60034 · doi:10.1142/S0219493709002749
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.