On the collision local time of sub-fractional Brownian motions

Abstract

Let S H i = { S t H i , t ≥ 0 } , i = 1 , 2 , be two independent sub-fractional Brownian motions with respective indices H i ∈ ( 0 , 1 ) . We consider the so-called collision local time ℓ T = ∫ 0 T δ ( S t H 1 − S t H 2 ) d t , T > 0 , where δ denotes the Dirac delta function. By an elementary method we show that ℓ T is smooth in the sense of Meyer and Watanabe if and only if min { H 1 , H 2 } < 1 / 3 .