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

Stefan Koch; Andreas Neuenkirch

doi:10.3934/dcdsb.2018334

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2019, Volume 24, Issue 8: 3865-3880. Doi: 10.3934/dcdsb.2018334

This issue Previous Article Construction of a contraction metric by meshless collocation Next Article Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation

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

Stefan Koch^1, and
Andreas Neuenkirch^1, ,

Institut für Mathematik, Universität Mannheim, B6, 26, 68131 Mannheim, Germany

^* 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

Received: March 2018

Revised: June 2018

Early access: January 2019

Published: August 2019

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

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Abstract

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

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