The quadratic covariation for a weighted fractional Brownian motion

Let $B^{a,b}$ be a weighted fractional Brownian motion with indices $a,b$ satisfying $a>-1,-1<b<0,\mathrm{\mid }b\mathrm{\mid }<1+a$. In this paper, motivated by the asymptotic property $$E[(B_{s+\epsilon }^{a,b}-B_s^{a,b}{)}^2]=O({\epsilon }^{1+b})\sim ̸{\epsilon }^{1+a+b}=E[(B_{\epsilon }^{a,b}{)}^2](\epsilon \to 0)$$ for all $s>0$, we consider the generalized quadratic covariation $[f(B^{a,b}),B^{a,b}{]}^{(a,b)}$ defined by $$[f(B^{a,b}),B^{a,b}{]}_t^{(a,b)}=\underset{\epsilon \downarrow 0}{\lim }\frac{1+a+b}{{\epsilon }^{1+b}}{\int }_{\epsilon }^{t+\epsilon }\{f(B_{s+\epsilon }^{a,b})-f(B_s^{a,b})\}(B_{s+\epsilon }^{a,b}-B_s^{a,b})s^{b}ds,$$ provided the limit exists uniformly in probability. We construct a Banach space $\mathcal{H}$ of measurable functions such that the generalized quadratic covariation exists in $L^2(\mathrm{\Omega })$ and the generalized Bouleau-Yor identity $$[f(B^{a,b}),B^{a,b}{]}_t^{(a,b)}=-\frac{1}{(1+b)\mathbb{B}(a+1,b+1)}{\int }_{\mathbb{R}}f(x){\mathcal{L}}^{a,b}(dx,t)$$ holds for all $f\in \mathcal{H}$, where ${\mathcal{L}}^{a,b}(x,t)={\int }_0^t\delta (B_s^{a,b}-x)d{s}^{1+a+b}$ is the weighted local time of $B^{a,b}$ and $\mathbb{B}(\cdot ,\cdot )$ is the Beta function.