Let $B^{H_1}$ and ${\tilde{B}}^{H_2}$ be two independent fractional Brownian motions with respective indices $H_1,H_2\in (0,1)$ and $H_1\le H_2$. In this paper, we consider the derivative of their intersection local time $\ell (t,a)$. We show that the derivative exists in $L^p$ for all $p\in [1,+\mathrm{\infty })$ and it is joint Hölder continuous in space and time if $H_1+H_2>3H_1{H}_2$. In particular, we show that $\ell (t,a)$ is differentiable in the spatial variable a under the condition, and we introduce the so-called hybrid fractional quadratic covariation $[f(B^{H_1}-{\tilde{B}}^{H_2}),B^{H_1}{]}^{(HC)}$. When $0<H_1\le \frac{1}{2}$, we construct a Banach space $\mathcal{H}$ of measurable functions such that the quadratic covariation exists in $L^2(\mathrm{\Omega })$ and $f(B^{H_1}-{\tilde{B}}^{H_2}),B^{H_1}{]}_t^{(HC)}=-{\int }_{\mathbb{R}}f(a)\ell (t,\mathrm{d}a)$for all $f\in \mathcal{H}$. When $H_1>\frac{1}{2}$ a similar result is proved to hold for all Hölder functions f of order $\nu >\frac{2H_1-1}{H_1}$.

让${乙}^{H_1}$和${\widetilde{乙}}^{H_2}$是具有各自索引的两个独立分数布朗运动$H_1,H_2\in (0,1)$和$H_1\le H_2$. 在本文中，我们考虑它们的交点本地时间的导数$\ell (吨,\mathrm{一种})$. 我们证明导数存在于${\mathrm{大号}}^p$对全部$p\in [1,+\mathrm{\infty })$并且它在空间和时间上是联合 Hölder 连续的，如果$H_1+H_2>3H_1{H}_2$. 特别是，我们表明$\ell (吨,\mathrm{一种})$条件下空间变量a是可微的，我们引入所谓的混合分数二次协变 $[F({乙}^{H_1}-{\widetilde{乙}}^{H_2}),{乙}^{H_1}{]}^{(HC)}$. 什么时候$0<H_1\le \frac{1}{2}$, 我们构造一个 Banach 空间$\mathcal{H}$的可测量函数，使得二次协变存在于${\mathrm{大号}}^2(\mathrm{\Omega })$和$F({乙}^{H_1}-{\widetilde{乙}}^{H_2}),{乙}^{H_1}{]}_{吨}^{(HC)}=-{\int }_{\mathbb{R}}F(\mathrm{一种})\ell (吨,\mathrm{d}\mathrm{一种})$对全部$F\in \mathcal{H}$. 什么时候$H_1>\frac{1}{2}$类似的结果被证明对所有阶的 Hölder 函数f成立$\nu >\frac{2H_1-1}{H_1}$.