Integration with Respect to Fractional Local Time with Hurst Index 1/2 < H < 1

Abstract

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 R f (x)L H (dx, t). As an application, we investigate the weighted quadratic covariationwhere the limit is uniform in probability and t k = kt/n. We show that it exists andprovided f is of bounded p-variation with 1 ≤ p < 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.

The generalized Bouleau-Yor identity for a sub-fractional Brownian motion

“…Moreover, this has be extended to fractional Brownian motion B H by Yan et al [25,27]. for all t ≥ 0.…”

“…In this section we study one parameter integral of local time [27], and the references therein. Moreover, this has be extended to fractional Brownian motion B H by Yan et al [25,27].…”

“…Moreover, these have also been extended to semimartingales by Bardina-Rovira [2], Eisenbaum [4,5], Elworthy et al [6], Feng-Zhao [8], Peskir [17], Rogers-Walsh [18], Yan-Yang [28]. For K = 1 and H = 1 2 , the process B is a standard fractional Brownian motion B H with Hurst index H. Yan et al [25,27] studied the integration with respect to local time of fractional Brownian motion, and the weighted quadratic covariation [f (B H ), B H ] (W ) of f (B H ) and B H . These deduce the fractional Itô formula for new classes of functions.…”

Remarks on an integral functional driven by sub-fractional Brownian motion

“…Recently, Yan, Yang, and Lu (2008) have considered the similar integral functional driven by fractional Brownian motion B H with Hurst index H ∈ (0, 1) which arises in the study of integration with respect to fractional local times of fractional Brownian motion (see Yan, Liu, and Yang (2008)). Inspired by these results, it seems interesting to study the similar integral functional driven by sub-fBm S H , a rather general class of self-similar Gaussian processes which do not have stationary increments.…”

Integration with respect to the G-Brownian local time

“…If f ∈ C 1 (R), (1.2) is the classical Itô formula. More works for the problem can be found in Eisenbaum [4,5], Elworthy et al [6], FengZhao [7,8], Moret-Nualart [13], Peskir [19], Russo-Vallois [20], Yan et al [21,22] and the references therein.…”