An integral functional driven by fractional Brownian motion

Abstract

Let BH be a fractional Brownian motion with Hurst index 0<H<1 and the weighted local time ℒH(⋅,t). In this paper, we consider the integral process CtH(a)≔limε↓0∫0t1{|BsH−a|≥ε}2Hs2H−1BsH−ads≡−ℋℒH(⋅,t)(a),t≥0in L2(Ω) with a∈R, where ℋ denotes the Hilbert transform. We show that the Skorohod integral ∫0⋅log|BsH−a|dBsH exists in L2(Ω) and the fractional Yamada formula (BtH−a)log|BtH−a|−BtH+alog|a|−∫0tlog|BsH−a|dBsH=12CtH(a)holds for all a∈R,t≥0. Moreover, we introduce the next occupation type formula: ∫RCtH(a)g(a)da=2H∫0t(ℋg)(BsH)s2H−1dsfor all continuous functions g with compact support.