Smoothness for the collision local time of two multidimensional bifractional Brownian motions

Abstract

Let \({B^{{H_i},{K_i}}} = \{ B_t^{{H_i},{K_i}},t \ge 0\} ,{\rm{ }}i = 1,2\) be two independent, d-dimensional bifractional Brownian motions with respective indices Hi ∈ (0, 1) and Ki ∈ (0, 1]. Assume d ⩾ 2. One of the main motivations of this paper is to investigate smoothness of the collision local time $${l_T} = \int_0^T {\delta (B_s^{{H_1},{K_1}} - B_s^{{H_2},{K_2}}} )ds,{\rm{ }}T > 0,$$ , where δ denotes the Dirac delta function. By an elementary method we show that lT is smooth in the sense of Meyer-Watanabe if and only if min{H1K1,H2K2} < 1/(d + 2).