## Electronic Journal of Probability

### Concentration inequalities for Stochastic Differential Equations with additive fractional noise

Maylis Varvenne

#### 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
Accepted: 2 November 2019
First available in Project Euclid: 9 November 2019

https://projecteuclid.org/euclid.ejp/1573268588

Digital Object Identifier
doi:10.1214/19-EJP384

#### 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

#### 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.
• [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.
• [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.
• [7] Toufik Guendouzi. Transportation inequalities for SDEs involving fractional Brownian motion and standard Brownian motion. Adv. Model. Optim., 14(3):615–634, 2012.
• [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.
• [12] Sebastian Riedel. Transportation-cost inequalities for diffusions driven by Gaussian processes. Electron. J. Probab., 22:Paper No. 24, 26, 2017.
• [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.
• [15] M. Talagrand. Transportation cost for Gaussian and other product measures. Geom. Funct. Anal., 6(3):587–600, 1996.
• [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.