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

This paper studies the functionals A1(t,x)=∫0t1[0,∞)(x−SsH)ds,A2(t,x)=∫0t1[0,∞)(x−SsH)s2H−1ds, where (StH)0≤t≤T is a one-dimension sub-fractional Brownian motion with index H∈(0,1)H∈(0,1). It shows that there exists a constant pH∈(1,2)pH∈(1,2) such that pp-variation of the process Aj(t,StH)−∫0tℒj(s,SsH)dSsH (j=1,2j=1,2) is equal to 0 if p>pHp>pH, where ℒjℒj, j=1,2j=1,2, are the local time and weighted local time of SHSH, respectively. This extends the classical results for Brownian motion.