Derivative for the intersection local time of fractional Brownian Motions

Let $B^{H_1}$ and ${\tilde{B}}^{H_2}$ be two independent fractional Brownian motions on $\mathbb{R}$ with respective indices $H_i\in (0,1)$ and $H_1\leq H_2$\mathrm{.}Inthispaper,weconsidertheirintersectionlocaltime$\ell_t(a)$. We show that ${\mathrm{\ell }}_t(a)$ is differentiable in the spatial variable if $\frac{1}{H_1}+\frac{1}{H_2}>3$, and we introduce the so-called {\it hybrid quadratic covariation} $[f(B^{H_1}-{\tilde{B}}^{H_2}),B^{H_1}{]}^{(HC)}$. When $H_1<\frac{1}{2}$, we construct a Banach space $\mathcal{H}$ of measurable functions such that the quadratic covariation exists in $L^2(\mathrm{\Omega })$ for all $f\in \mathcal{H}$, and the Bouleau-Yor type identity $$[f(B^{H_1}-{\tilde{B}}^{H_2}),B^{H_1}{]}_t^{(HC)}=-{\int }_{\mathbb{R}}f(a){\mathrm{\ell }}_t(da)$$ holds. When $H_1\ge \frac{1}{2}$, we show that the quadratic covariation exists also in $L^2(\mathrm{\Omega })$ and the above Bouleau-Yor type identity holds also for all H"older functions $f$ of order $\nu >\frac{2H_1-1}{H_1}$.