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August  2019, 24(8): 3865-3880. doi: 10.3934/dcdsb.2018334

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

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

Received  March 2018 Revised  June 2018 Published  August 2019 Early access  January 2019

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

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.

Keywords: Fractional Brownian motion, Hurst parameter, continuous dependence, stochastic differential equation.
Mathematics Subject Classification: Primary: 60G22; Secondary: 60H10.
Citation: Stefan Koch, Andreas Neuenkirch. The Mandelbrot-van Ness fractional Brownian motion is infinitely differentiable with respect to its Hurst parameter. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3865-3880. doi: 10.3934/dcdsb.2018334
References:
[1]

H. Doss, Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré, 13 (1977), 99-125.   Google Scholar

[2] P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications, Cambridge University Press, 2010.  doi: 10.1017/CBO9780511845079.  Google Scholar
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J. E. Hutton and P. I. Nelson, Interchanging the order of differentiation and stochastic integration, Stochastic Process. Appl., 18 (1984), 371-377.  doi: 10.1016/0304-4149(84)90307-7.  Google Scholar

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M. Jolis and N. Viles, Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion, J. Theoret. Probab., 20 (2007), 133-152.  doi: 10.1007/s10959-007-0054-5.  Google Scholar

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M. Jolis and N. Viles, Continuity in the Hurst parameter of the law of the Wiener integral with respect to the fractional Brownian motion, Statist. Probab. Lett., 80 (2010), 566-572.  doi: 10.1016/j.spl.2009.12.011.  Google Scholar

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M. Jolis and N. Viles, Continuity in the Hurst parameter of the law of the symmetric integral with respect to the fractional Brownian motion, Stochastic Process. Appl., 120 (2010), 1651-1679.  doi: 10.1016/j.spa.2010.05.002.  Google Scholar

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I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edition, Springer, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

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J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer, 2016. doi: 10.1007/978-3-319-31089-3.  Google Scholar

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T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310.  doi: 10.4171/RMI/240.  Google Scholar

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B. B. Mandelbrot and J. W. van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review, 10 (1968), 422-437.  doi: 10.1137/1010093.  Google Scholar

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D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.   Google Scholar

[15]

D. Nualart and B. Saussereau, Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, Stochastic Process. Appl., 119 (2009), 391-409.  doi: 10.1016/j.spa.2008.02.016.  Google Scholar

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P. E. Protter, Stochastic Integration and Differential Equations, 2nd edition, Springer, 2005. doi: 10.1007/978-3-662-10061-5.  Google Scholar

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A. Richard and D. Talay, Noise sensitivity of functionals of fractional Brownian motion driven stochastic differential equations: Results and perspectives, Modern Problems of Stochastic Analysis and Statistics, 219–235, Springer Proc. Math. Stat., 208, Springer, 2017.  Google Scholar

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H. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6 (1978), 19-41.  doi: 10.1214/aop/1176995608.  Google Scholar

show all references

References:
[1]

H. Doss, Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré, 13 (1977), 99-125.   Google Scholar

[2] P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications, Cambridge University Press, 2010.  doi: 10.1017/CBO9780511845079.  Google Scholar
[3]

P. Friz and N. Victoir, Differential equations driven by Gaussian signals, Ann. Inst. H. Poincaré Probab. Statist., 46 (2010), 369-413.  doi: 10.1214/09-AIHP202.  Google Scholar

[4]

J. E. Hutton and P. I. Nelson, Interchanging the order of differentiation and stochastic integration, Stochastic Process. Appl., 18 (1984), 371-377.  doi: 10.1016/0304-4149(84)90307-7.  Google Scholar

[5]

M. Jolis and N. Viles, Continuity with respect to the Hurst parameter of the laws of the multiple fractional integrals, Stochastic Process. Appl., 117 (2007), 1189-1207.  doi: 10.1016/j.spa.2006.12.005.  Google Scholar

[6]

M. Jolis and N. Viles, Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion, J. Theoret. Probab., 20 (2007), 133-152.  doi: 10.1007/s10959-007-0054-5.  Google Scholar

[7]

M. Jolis and N. Viles, Continuity in the Hurst parameter of the law of the Wiener integral with respect to the fractional Brownian motion, Statist. Probab. Lett., 80 (2010), 566-572.  doi: 10.1016/j.spl.2009.12.011.  Google Scholar

[8]

M. Jolis and N. Viles, Continuity in the Hurst parameter of the law of the symmetric integral with respect to the fractional Brownian motion, Stochastic Process. Appl., 120 (2010), 1651-1679.  doi: 10.1016/j.spa.2010.05.002.  Google Scholar

[9]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edition, Springer, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[10]

J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer, 2016. doi: 10.1007/978-3-319-31089-3.  Google Scholar

[11]

T. Lyons, Differential equations driven by rough signals. Ⅰ. An extension of an inequality of L.C. Young, Math. Res. Lett., 1 (1994), 451-464.  doi: 10.4310/MRL.1994.v1.n4.a5.  Google Scholar

[12]

T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310.  doi: 10.4171/RMI/240.  Google Scholar

[13]

B. B. Mandelbrot and J. W. van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review, 10 (1968), 422-437.  doi: 10.1137/1010093.  Google Scholar

[14]

D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.   Google Scholar

[15]

D. Nualart and B. Saussereau, Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, Stochastic Process. Appl., 119 (2009), 391-409.  doi: 10.1016/j.spa.2008.02.016.  Google Scholar

[16]

P. E. Protter, Stochastic Integration and Differential Equations, 2nd edition, Springer, 2005. doi: 10.1007/978-3-662-10061-5.  Google Scholar

[17]

A. Richard and D. Talay, Noise sensitivity of functionals of fractional Brownian motion driven stochastic differential equations: Results and perspectives, Modern Problems of Stochastic Analysis and Statistics, 219–235, Springer Proc. Math. Stat., 208, Springer, 2017.  Google Scholar

[18]

H. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6 (1978), 19-41.  doi: 10.1214/aop/1176995608.  Google Scholar

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