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

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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 and 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.
[2]	P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications, Cambridge University Press, 2010. doi: 10.1017/CBO9780511845079.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2^nd edition, Springer, 1991. doi: 10.1007/978-1-4612-0949-2.
[10]	J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer, 2016. doi: 10.1007/978-3-319-31089-3.
[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.
[12]	T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310. doi: 10.4171/RMI/240.
[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.
[14]	D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.
[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.
[16]	P. E. Protter, Stochastic Integration and Differential Equations, 2^nd edition, Springer, 2005. doi: 10.1007/978-3-662-10061-5.
[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.
[18]	H. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6 (1978), 19-41. doi: 10.1214/aop/1176995608.

show all references

References:

[1]	H. Doss, Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré, 13 (1977), 99-125.
[2]	P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications, Cambridge University Press, 2010. doi: 10.1017/CBO9780511845079.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2^nd edition, Springer, 1991. doi: 10.1007/978-1-4612-0949-2.
[10]	J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer, 2016. doi: 10.1007/978-3-319-31089-3.
[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.
[12]	T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310. doi: 10.4171/RMI/240.
[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.
[14]	D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.
[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.
[16]	P. E. Protter, Stochastic Integration and Differential Equations, 2^nd edition, Springer, 2005. doi: 10.1007/978-3-662-10061-5.
[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.
[18]	H. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6 (1978), 19-41. doi: 10.1214/aop/1176995608.

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