Publication Cover
Stochastics
An International Journal of Probability and Stochastic Processes
Volume 94, 2022 - Issue 4
134
Views
0
CrossRef citations to date
0
Altmetric
Research Article

Bismut type derivative formulae and gradient estimate for multiplicative SDEs with fractional noises

&
Pages 493-518 | Received 25 Apr 2020, Accepted 20 Jul 2021, Published online: 28 Jul 2021

References

  • E. Alòs, O. Mazet, and D. Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab. 29 (2001), pp. 766–801.
  • A. Ayache, N.R. Shieh, and Y.M. Xiao, Multiparameter multifractional Brownian motion: Local nondeterminism and joint continuity of the local times, Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), pp. 1029–1054.
  • J. Bao, F.Y. Wang, and C. Yuan, Bismut formulae and applications for functional SPDEs, Bull. Sci. Math. 137 (2013), pp. 509–522.
  • J. Bao, F.Y. Wang, and C. Yuan, Derivative formula and Harnack inequality for degenerate functional SDEs, Stoch. Dyn. 13 (2013), pp. 1–22.
  • F. Baudoin and C. Ouyang, Gradient bounds for solutions of stochastic differential equations driven by fractional Brownian motions, Malliavin Calculus and Stochastic Analysis, Springer Proc. Math. Stat., Vol. 34, Springer, New York, 2013, pp. 413–426.
  • F. Baudoin, C. Ouyang, and S. Tindel, Upper bounds for the density of solutions of stochastic differential equations driven by fractional Brownian motions, Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), pp. 111–135.
  • S.M. Berman, An asymptotic bound for the tail of the distribution of the maximum of a Gaussian process, Ann. Inst. H. Poincar Probab. Statist. 21 (1985), pp. 47–57.
  • F. Biagini, Y. Hu, B. Øksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer-Verlag, London, 2008.
  • J.M. Bismut, Large Deviation and The Malliavin Calculus, Birkhäuser, Boston, MA, 1984.
  • B. Boufoussi and S. Hajji, Functional differential equations driven by a fractional Brownian motion, Comput. Math. Appl. 62 (2011), pp. 746–754.
  • L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields 122 (2002), pp. 108–140.
  • L. Decreusefond and A.S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potential Anal. 10 (1998), pp. 177–214.
  • C. Deng and S. Zhang, Log-Harnack inequalities for Markov semigroups generated by non-local Gruschin type operators, Math. Nachr. 291 (2018), pp. 1055–1074.
  • M.A. Diopa and M.J. Garrido-Atienza, Retarded evolution systems driven by fractional Brownian motion with Hurst parameter H>1/2, Nonlinear Anal. 97 (2014), pp. 15–29.
  • Z. Dong and Y. Xie, Ergodicity of linear SPDE driven by Lévy noise, J. Syst. Sci. Complex 23 (2010), pp. 137–152.
  • K.D. Elworthy and X.M. Li, Formulae for the derivatives of heat semigroups, J. Funct. Anal. 125 (1994), pp. 252–286.
  • X.L. Fan, Harnack inequality and derivative formula for SDE driven by fractional Brownian motion, Sci. China-Math. 561 (2013), pp. 515–524.
  • X.L. Fan, Bismut formulae and applications for stochastic (functional) differential equations driven by fractional Brownian motions, Stoch. Dyn. 17 (2017), pp. 19.
  • X.L. Fan, Derivative formulas and applications for degenerate SDEs with fractional noises, J. Theor. Probab. 32 (2019), pp. 1360–1381.
  • M. Ferrante and C. Rovira, Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H>1/2, Bernoulli 12 (2006), pp. 85–100.
  • A. Guillin and F.Y. Wang, Degenerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality, J. Differ. Equ. 253 (2012), pp. 20–40.
  • M. Hairer and N.S. Pillai, Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion, Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), pp. 601–628.
  • Y. Hu and D. Nualart, Differential equations driven by Hölder continuous functions of order greater than 1/2, Stochastic Analysis and Applications. Abel Symp., Vol. 2, Springer, Berlin, 2007, pp. 399–413.
  • Y. Hu, D. Nualart, and X. Song, A singular stochastic differential equations driven by fractional Brownian motion, Statist. Probab. Lett. 78 (2007), pp. 2075–2085.
  • T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998), pp. 215–310.
  • J. Mémin, Y. Mishura, and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett. 51 (2001), pp. 197–206.
  • Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2008.
  • A. Neuenkirch, I. Nourdin, and S. Tindel, Delay equations driven by rough paths, Electron. J. Probab.13 (2008), pp. 2031–2068.
  • A.F. Nikiforov and V.B. Uvarov, Special Functions of Mathematical Physics, Birkhäuser, Boston, 1988.
  • I. Nourdin and T. Simon, On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion, Statist. Probab. Lett. 76 (2006), pp. 907–912.
  • D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed., Springer-Verlag, Berlin, 2006.
  • D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), pp. 55–81.
  • D. Nualart and B. Saussereau, Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, Stoch. Process. Appl. 119 (2009), pp. 391–409.
  • E. Priola, Formulae for the derivatives of degenerate diffusion semigroups, J. Evol. Equ. 6 (2006), pp. 577–600.
  • S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yvendon, 1993.
  • B. Saussereau, Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion, Bernoulli 18 (2012), pp. 1–23.
  • F.Y. Wang, Derivative formula and gradient estimates for Gruschin type semigroups, J. Theor. Probab.27 (2012), pp. 80–95.
  • F.Y. Wang, Derivative formula and Harnack inequality for SDEs driven by Lévy processes, Stoch. Anal. Appl. 32 (2014), pp. 30–49.
  • F.Y. Wang and L. Xu, Derivative formulae and applications for hyperdissipative stochastic Navier-Stokes/Burgers equations, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15 (2012), pp. 1–19.
  • F.Y. Wang and L. Xu, Log-Harnack inequality for Gruschin type semigroups, Rev. Mat. Iberoam. 30 (2014), pp. 405–418.
  • F.Y. Wang and X.C. Zhang, Derivative formula and applications for degenerate diffusion semigroups, J. Math. Pures Appl. 99 (2013), pp. 726–740.
  • L.T. Yan, The fractional derivative for fractional Brownian local time, Math. Z. 283 (2016), pp. 437–468.
  • L.C. Young, An inequality of the Hölder type connected with Stieltjes integration, Acta Math. 67 (1936), pp. 251–282.
  • M. Zähle, Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields 111 (1998), pp. 333–374.
  • X.C. Zhang, Stochastic flows and Bismut formulas for stochastic Hamiltonian systems, Stochastic Process. Appl. 120 (2010), pp. 1929–1949.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.