Integration with Respect to Fractional Local Time with Hurst Index 1/2 < H < 1
Integration with Respect to Fractional Local Time with Hurst Index 1/2 < H < 1
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DOI：
10.1007/s11118-008-9108-2
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发表时间：
2008-03
影响因子：
1.1
通讯作者：
Litan Yan;Junfeng Liu;Xiangfeng Yang
中科院分区：
文献类型：
--
作者：
Litan Yan;Junfeng Liu;Xiangfeng Yang
Let be the weighted local time of fractional Brownian motionBHwith Hurst index 1/2 <H< 1. In this paper, we use Young integration to study the integral of determinate functions As an application, we investigate theweighted quadratic covariationdefined by $$ \left[f\big(B^H\big),B^H\right]^{(W)}_t:=\lim_{n\to \infty}2H\sum_{k=0}^{n-1} k^{2H-1}\left\{f\big(B^H_{t_{k+1}}\big)-f\big(B^H_{t_{k}}\big)\right\} \left(B^H_{t_{k+1}}-B^H_{t_{k}}\right), $$ where the limit is uniform in probability andtk=kt/n. We show that it exists and providedfis of boundedp-variation with. Moreover, we extend this result to the time-dependent case. These allow us to write the fractional Itô formula for new classes of functions.
Let be the weighted local time of fractional Brownian motionBHwith Hurst index 1/2 <H< 1. In this paper, we use Young integration to study the integral of determinate functions As an application, we investigate theweighted quadratic covariationdefined by $$ \left[f\big(B^H\big),B^H\right]^{(W)}_t:=\lim_{n\to \infty}2H\sum_{k=0}^{n-1} k^{2H-1}\left\{f\big(B^H_{t_{k+1}}\big)-f\big(B^H_{t_{k}}\big)\right\} \left(B^H_{t_{k+1}}-B^H_{t_{k}}\right), $$ where the limit is uniform in probability andtk=kt/n. We show that it exists and providedfis of boundedp-variation with. Moreover, we extend this result to the time-dependent case. These allow us to write the fractional Itô formula for new classes of functions.