Higher-order derivative of intersection local time for two independent fractional Brownian motions

In this article, we obtain sharp conditions for the existence of the high order derivatives ($k$-th order) of intersection local time ${\widehat{\alpha }}^{(k)}(0)$ of two independent d-dimensional fractional Brownian motions $B_t^{H_1}$ and ${\tilde{B}}_s^{H_2}$ with Hurst parameters $H_1$ and $H_2$, respectively. We also study their exponential integrability.