We study the Mandelbrot-van Ness representation of fractional Brownian motion $ B^H = (B^H_t)_{t \geq 0} $ with Hurst parameter $ H \in (0,1) $ and show that for arbitrary fixed $ t \geq 0 $ the mapping $ (0,1) \ni H \mapsto B_t^H \in \mathbb{R} $ is almost surely infinitely differentiable. Thus, the sample paths of fractional Brownian motion are smooth with respect to $ H $. As a byproduct we obtain that scalar stochastic differential equations are differentiable with respect to the Hurst parameter of the driving fractional Brownian motion.
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