Electronic Journal of Probability

Concentration inequalities for Stochastic Differential Equations with additive fractional noise

Maylis Varvenne

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Abstract

In this paper, we establish concentration inequalities both for functionals of the whole solution on an interval $[0,T]$ of an additive SDE driven by a fractional Brownian motion with Hurst parameter $H\in (0,1)$ and for functionals of discrete-time observations of this process. Then, we apply this general result to specific functionals related to discrete and continuous-time occupation measures of the process.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 124, 22 pp.

Dates
Received: 15 January 2019
Accepted: 2 November 2019
First available in Project Euclid: 9 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1573268588

Digital Object Identifier
doi:10.1214/19-EJP384

Subjects
Primary: 60G22: Fractional processes, including fractional Brownian motion 60H10: Stochastic ordinary differential equations [See also 34F05] 60E15: Inequalities; stochastic orderings

Keywords
concentration inequalities fractional Brownian motion occupation measures Stochastic Differential Equations

Rights
Creative Commons Attribution 4.0 International License.

Citation

Varvenne, Maylis. Concentration inequalities for Stochastic Differential Equations with additive fractional noise. Electron. J. Probab. 24 (2019), paper no. 124, 22 pp. doi:10.1214/19-EJP384. https://projecteuclid.org/euclid.ejp/1573268588


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References

  • [1] S. G. Bobkov and F. Götze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal., 163(1):1–28, 1999.
  • [2] Philippe Carmona, Laure Coutin, and Gérard Montseny. Stochastic integration with respect to fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist., 39(1):27–68, 2003.
    Zentralblatt MATH: 1016.60043
    Digital Object Identifier: doi:10.1016/S0246-0203(02)01111-1
  • [3] L. Decreusefond and A. S. Üstünel. Stochastic analysis of the fractional Brownian motion. Potential Anal., 10(2):177–214, 1999.
  • [4] H. Djellout, A. Guillin, and L. Wu. Transportation cost-information inequalities and applications to random dynamical systems and diffusions. The Annals of Probability, 32(3B):2702–2732, 2004.
    Zentralblatt MATH: 1061.60011
    Digital Object Identifier: doi:10.1214/009117904000000531
    Project Euclid: euclid.aop/1091813628
  • [5] Peter K. Friz and Nicolas B. Victoir. Multidimensional stochastic processes as rough paths, volume 120 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. Theory and applications.
  • [6] Mathieu Gourcy and Liming Wu. Logarithmic Sobolev inequalities of diffusions for the $L^{2}$ metric. Potential Anal., 25(1):77–102, 2006.
    Zentralblatt MATH: 1098.60027
    Digital Object Identifier: doi:10.1007/s11118-006-9009-1
  • [7] Toufik Guendouzi. Transportation inequalities for SDEs involving fractional Brownian motion and standard Brownian motion. Adv. Model. Optim., 14(3):615–634, 2012.
    Zentralblatt MATH: 1332.60083
  • [8] M. Ledoux. Concentration, transportation and functional inequalities. Preprint, 2002.
  • [9] Michel Ledoux. The concentration of measure phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001.
  • [10] K. Marton. Bounding $\overline{d} $-distance by informational divergence: a method to prove measure concentration. The Annals of Probability, 24(2):857–866, 1996.
  • [11] K. Marton. A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal., 6(3):556–571, 1996.
    Zentralblatt MATH: 0856.60072
    Digital Object Identifier: doi:10.1007/BF02249263
  • [12] Sebastian Riedel. Transportation-cost inequalities for diffusions driven by Gaussian processes. Electron. J. Probab., 22:Paper No. 24, 26, 2017.
    Zentralblatt MATH: 1373.60072
    Digital Object Identifier: doi:10.1214/17-EJP40
  • [13] Philippe Rigollet and Jan-Christian Hütter. High dimensional statistics. Lecture notes (MIT), 2017.
  • [14] Bruno Saussereau. Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion. Bernoulli, 18(1):1–23, 2012.
    Zentralblatt MATH: 1242.60056
    Digital Object Identifier: doi:10.3150/10-BEJ324
    Project Euclid: euclid.bj/1327068615
  • [15] M. Talagrand. Transportation cost for Gaussian and other product measures. Geom. Funct. Anal., 6(3):587–600, 1996.
    Zentralblatt MATH: 0859.46030
    Digital Object Identifier: doi:10.1007/BF02249265
  • [16] Li-ming Wu and Zheng-liang Zhang. Talagrand’s $T_{2}$-transportation inequality w.r.t. a uniform metric for diffusions. Acta Math. Appl. Sin. Engl. Ser., 20(3):357–364, 2004.
    Zentralblatt MATH: 1055.60009
    Digital Object Identifier: doi:10.1007/s10255-004-0175-x

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