pp-variation of an integral functional driven by fractional Brownian motion

Let (BtH)0⩽t⩽T be a one-dimensional fractional Brownian motion with Hurst parameter H∈(0,1)H∈(0,1). We study the functionals A1(t,x)=∫0t1[0,∞)(x−BsH)s2H−1ds,A2(t,x)=∫0t1[0,∞)(x−BsH)ds. We show that there exists a constant pH∈(1,2)pH∈(1,2) depending only on HH such that the pp-variation of Aj(t,BtH)−∫0tLj(s,BsH)dBsH (j=1,2j=1,2) is zero if p>pHp>pH, where L1,L2L1,L2 are the local time and weighted local time of BHBH, respectively. This extends the illustrated result for Brownian motion.