Smoothness for the collision local time of two multidimensional bifractional Brownian motions

Shen, Guangjun ; Yan, Litan ; Chen, Chao

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 4, pp. 969-989

Abstract (VO)

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 $$ \ell _T=\int _{0}^{T}\delta (B_{s}^{H_{1},K_1}-B_{s}^{H_{2},K_2}) {\rm d} s, \qquad T>0, $$ 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 $\min \{H_{1}K_1,H_{2}K_2\}{1}/{(d+2)}$.

Détail

DOI : 10.1007/s10587-012-0077-7

Classification : 60G15, 60G18, 60G22, 60J55, 60J65
Keywords: bifractional Brownian motion; collision local time; intersection local time; chaos expansion

Bibliographie

[1] An, L., Yan, L.: Smoothness for the collision local time of fractional Brownian motion. Preprint, 2010.

[2] Chen, C., Yan, L.: Remarks on the intersection local time of fractional Brownian motions. Stat. Probab. Lett. 81 (2011), 1003-1012. | DOI | MR | Zbl

[3] Es-Sebaiy, K., Tudor, C. A.: Multidimensional bifractional Brownian motion: Itô and Tanaka formulas. Stoch. Dyn. 7 (2007), 365-388. | DOI | MR | Zbl

[4] Houdré, Ch., Villa, J.: 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. | DOI | MR | Zbl

[5] Hu, Y.: Self-intersection local time of fractional Brownian motion - via chaos expansion. J. Math, Kyoto Univ. 41 (2001), 233-250. | DOI | MR

[6] Hu, Y.: Integral transformations and anticipative calculus for fractional Brownian motions. Mem. Am. Math. Soc. 825 (2005). | MR | Zbl

[7] Jiang, Y., Wang, Y.: Self-intersection local times and collision local times of bifractional Brownian motions. Sci. China, Ser. A 52 (2009), 1905-1919. | DOI | MR | Zbl

[8] Kruk, I., Russo, F., Tudor, C. A.: Wiener integrals, Malliavin calculus and covariance measure structure. J. Funct. Anal. 249 (2007), 92-142. | DOI | MR | Zbl

[9] Lei, P., Nualart, D.: A decomposition of the bifractional Brownian motion and some applications. Stat. Probab. Lett. 79 (2009), 619-624. | DOI | MR | Zbl

[10] Mishura, Y.: Stochastic Calculus for Fractional Brownian Motions and Related Processes. Lecture Notes in Mathematics 1929. Springer, Berlin (2008). | MR

[11] Nualart, D., Ortiz-Latorre, S.: Intersection local time for two independent fractional Brownian motions. J. Theor. Probab. 20 (2007), 759-767. | DOI | MR | Zbl

[12] Nualart, D.: The Malliavin Calculus and Related Topics. 2nd ed. Probability and Its Applications. Springer, Berlin (2006). | MR | Zbl

[13] Russo, F., Tudor, C. A.: On bifractional Brownian motion. Stochastic Processes Appl. 116 (2006), 830-856. | DOI | MR | Zbl

[14] Shen, G., Yan, L.: Smoothness for the collision local times of bifractional Brownian motions. Sci. China, Math. 54 (2011), 1859-1873. | DOI | MR | Zbl

[15] Tudor, C. A., Xiao, Y.: Sample path properties of bifractional Brownian motion. Bernoulli 13 (2007), 1023-1052. | DOI | MR | Zbl

[16] Watanabe, S.: Lectures on Stochastic Differential Equations and Malliavin Calculus. Lectures on Mathematics and Physics. Mathematics, 73. Tata Institute of Fundamental Research. Springer, Berlin (1984). | MR | Zbl

[17] Yan, L., Liu, J., Chen, C.: On the collision local time of bifractional Brownian motions. Stoch. Dyn. 9 (2009), 479-491. | DOI | MR | Zbl

[18] Yan, L., Gao, B., Liu, J.: The Bouleau-Yor identity for a bi-fractional Brownian motion. (to appear) in Stochastics 2012.