On the collision local time of sub-fractional Brownian motions

Let SHi={StHi,t≥0}, i=1,2i=1,2, be two independent sub-fractional Brownian motions with respective indices Hi∈(0,1)Hi∈(0,1). We consider the so-called collision local time ℓT=∫0Tδ(StH1−StH2)dt,T>0, where δδ denotes the Dirac delta function. By an elementary method we show that ℓTℓT is smooth in the sense of Meyer and Watanabe if and only if min{H1,H2}<1/3min{H1,H2}<1/3.