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In a recent paper, Yu ( 2021 ) define a-th order derivative of self-intersection local times with respect to  d -dimensional fractional Brownian motion. By the existence and Hölder continuity proved in Yu ( 2021 ), we study the smoothness of derivative of self-intersection local times with respect to fractional Brownian motion in this paper. We also consider the cases of intersection local times and collision local times, respectively.

Smoothness of higher order derivative of self-intersection local time for fractional Brownian motion