Let $$B^H$$ B H be a fractional Brownian motion with Hurst index $$\frac{1}{2}<H<1$$ 1 2 < H < 1 and the weighted local time $${\mathscr {L}}^H(x,t)$$ L H ( x , t ) . In this paper, we consider the process $$\begin{aligned} {\mathcal S}^{H,\alpha }_t(a):=- \frac{2}{\alpha (1-\alpha )}\left( \left( B^H_t-a\right) _{+}^{1-\alpha }- (-a)_{+}^{1-\alpha }\right) + \frac{2}{\alpha }\int _0^t\left( B^H_s-a\right) _{+}^{-\alpha }dB^H_s \end{aligned}$$ S t H , α ( a ) : = - 2 α ( 1 - α ) B t H - a + 1 - α - ( - a ) + 1 - α + 2 α ∫ 0 t B s H - a + - α d B s H with $$t\ge 0$$ t ≥ 0 and $$a\in {\mathbb R}$$ a ∈ R , where $$0<\alpha <\frac{1-H}{2H}$$ 0 < α < 1 - H 2 H and the integral is the Skorohod integral. We show that $$-\frac{\alpha }{\Gamma (1-\alpha )}{\mathcal S}^{H,\alpha }_t(\cdot )$$ - α Γ ( 1 - α ) S t H , α ( · ) coincides with the left-handed Marchaud fractional derivative $${\mathscr {D}}^\alpha _{-}{\mathscr {L}}^H(\cdot ,t)$$ D - α L H ( · , t ) of order $$\alpha $$ α , and the “occupation formula” $$\begin{aligned} \int _{\mathbb R}{\mathcal S}^{H,\alpha }_t(a)f(a)da=2H\alpha ^{-1}\Gamma (1-\alpha ) \int _0^t({\mathscr {D}}_{+}^\alpha f)\left( B^H_s\right) s^{2H-1}ds \end{aligned}$$ ∫ R S t H , α ( a ) f ( a ) d a = 2 H α - 1 Γ ( 1 - α ) ∫ 0 t ( D + α f ) B s H s 2 H - 1 d s holds for all Hölder continuous functions f of order $$\beta >\alpha $$ β > α with compact support, where $${\mathscr {D}}^\alpha _{+}$$ D + α denotes the right-handed Marchaud fractional derivative of order $$\alpha $$ α .