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The Mandelbrot-van Ness fractional Brownian motion is infinitely differentiable with respect to its Hurst parameter

  • * Corresponding author: Andreas Neuenkirch

    * Corresponding author: Andreas Neuenkirch

Dedicated to Peter Kloeden on the occasion of his 70th birthday: a great mathematician and inspiring mentor

The first author is supported by the DFG RTG 1953 Statistical Modeling of Complex Systems and Processes

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  • 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.

    Mathematics Subject Classification: Primary: 60G22; Secondary: 60H10.


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