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

Abstract: We consider the existence and H\"{o}lder continuity conditions for the $k$-th order derivatives of self-intersection local time for $d$-dimensional fractional Brownian motion, where $k=(k_1,k_2,\cdots, k_d)$. Moreover, we show a limit theorem for the critical case with $H=\frac{2}{3}$ and $d=1$, which was conjectured by Jung and Markowsky (2014).