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.

Résumé

Let $B^{H_i,K_i}=\{B_t^{H_i,K_i},t\ge 0\}$, $i=1,2$ be two independent, $d$-dimensional bifractional Brownian motions with respective indices $H_i\in (0,1)$ and $K_i\in (0,1]$. Assume $d\ge 2$. 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
Mots-clés : bifractional Brownian motion; collision local time; intersection local time; chaos expansion