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The Mandelbrot-van Ness fractional Brownian motion is infinitely differentiable with respect to its Hurst parameter
Institut für Mathematik, Universität Mannheim, B6, 26, 68131 Mannheim, Germany |
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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H. Doss,
Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré, 13 (1977), 99-125.
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P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications, Cambridge University Press, 2010.
doi: 10.1017/CBO9780511845079.![]() ![]() ![]() |
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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. |
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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. |
[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, 2nd 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, 2nd 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, 2nd 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, 2nd 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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