Integration with respect to fractional local times with Hurst index $H$ greater than 1/2
| dc.creator | Yan, Litan | |
| dc.creator | Liu, Junfeng | |
| dc.creator | Yang, Xiangfeng | |
| dc.date | 2008-03-26 | |
| dc.date | 2008-12-04 | |
| dc.date.accessioned | 2026-07-07T12:08:57Z | |
| dc.date.available | 2026-07-07T12:08:57Z | |
| dc.description | Let ${\mathscr L}^H(x,t)=2H\int_0^tδ(B^H_s-x)s^{2H-1}ds$ be the weighted local time of fractional Brownian motion $B^H$ with Hurst index $1/2<H<1$. In this paper, we use Young integration to study the integral of determinate functions $\int_{\mathbb R}f(x){\mathscr L}^H(dx,t)$. As an application, we investigate the {\it weighted quadratic covariation} $[f(B^H),B^H]^{(W)}$ defined by $$ [f(B^H),B^H]^{(W)}_t:=\lim_{n\to \infty}2H\sum_{k=0}^{n-1} k^{2H-1}\{f(B^H_{t_{k+1}})-f(B^H_{t_{k}})\}(B^H_{t_{k+1}}-B^H_{t_{k}}), $$ where the limit is uniform in probability and $t_k=kt/n$. We show that it exists and $$ [f(B^H),B^H]^{(W)}_t=-\int_{\mathbb R}f(x){\mathscr L}^H(dx,t), $$ provided $f$ is of bounded $p$-variation with $1\leq p<\frac{2H}{1-H}$. Moreover, we extend this result to the time-dependent case. These allow us to write the fractional Itô formula for new classes of functions. | |
| dc.description | 21 pages | |
| dc.identifier | https://arxiv.org/abs/0803.3665 | |
| dc.identifier | http://arxiv.org/abs/0803.3665 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/209471 | |
| dc.subject | Probability | |
| dc.subject | Statistics Theory | |
| dc.subject | 60G15; 60H05; 60H07 | |
| dc.title | Integration with respect to fractional local times with Hurst index $H$ greater than 1/2 | |
| dc.type | text |