## Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

### Upper bounds for the density of solutions to stochastic differential equations driven by fractional Brownian motions

#### Abstract

In this paper we study upper bounds for the density of solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/3$. We show that under some geometric conditions, in the regular case $H>1/2$, the density of the solution satisfies the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case $H>1/3$ and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper sub-Gaussian bound.

#### Résumé

Dans ce papier nous étudions des bornes supérieures pour la densité d’une solution déquation différentielle conduite par un mouvement brownien fractionnaire d’indice de Hurst $H>1/3$. Nous montrons, que sous certaines conditions géomètriques, dans le cas régulier $H>1/2$, la densité de la solution satisfait l’inégalité de log-Sobolev, l’inégalité de concentration gaussienne et admet une borne supérieure gaullienne. Dans le cas $H>1/3$ et sous la même condition géomètrique, nous montrons que la densité est infiniment différentiable et admet une borne supérieure sous-gaussienne.

#### Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 1 (2014), 111-135.

Dates
First available in Project Euclid: 1 January 2014

https://projecteuclid.org/euclid.aihp/1388545268

Digital Object Identifier
doi:10.1214/12-AIHP522

Mathematical Reviews number (MathSciNet)
MR3161525

Zentralblatt MATH identifier
1286.60051

#### Citation

Baudoin, Fabrice; Ouyang, Cheng; Tindel, Samy. Upper bounds for the density of solutions to stochastic differential equations driven by fractional Brownian motions. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 1, 111--135. doi:10.1214/12-AIHP522. https://projecteuclid.org/euclid.aihp/1388545268

#### References

• [1] F. Baudoin and L. Coutin. Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Process. Appl. 117 (2007) 550–574.
• [2] F. Baudoin and M. Hairer. A version of Hörmander’s theorem for the fractional Brownian motion. Probab. Theory Related Fields 139 (2007) 373–395.
• [3] F. Baudoin and C. Ouyang. Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions. Stochastic. Process. Appl. 121 (2011) 759–792.
• [4] F. Baudoin, E. Nualart, C. Ouyang and S. Tindel. Work in progress. Preprint, 2012.
• [5] M. Besalú and D. Nualart. Estimates for the solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H\in(\frac{1}{3},\frac{1}{2})$. Stoch. Dyn. 11 (2011) 243–263.
• [6] M. Capitaine, E. Hsu and M. Ledoux. Martingale representation and logarithmic Sobolev inequality. Electron. Com. Probab. 2 (1997) 71–81.
• [7] T. Cass and P. Friz. Densities for rough differential equations under Hörmander condition. Ann. Math. To appear.
• [8] T. Cass, P. Friz and N. Victoir. Non-degeneracy of Wiener functionals arising from rough differential equations. Trans. Amer. Math. Soc. 361 (2009) 3359–3371.
• [9] T. Cass, C. Litterer and T. Lyons. Integrability estimates for Gaussian rough differential equations. Arxiv preprint, 2011.
• [10] A. Chronopoulou and S. Tindel. On inference for fractional differential equations. Arxiv preprint, 2011.
• [11] L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108–140.
• [12] R. C. Dalang and E. Nualart. Potential theory for hyperbolic SPDEs. Ann. Probab. 32(3A) (2004) 2099–2148.
• [13] A. Davie. Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express. 2007 (2007) abm009.
• [14] A. Deya, A. Neuenkirch and S. Tindel. A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 48(2) (2012) 518–550.
• [15] P. Driscoll. Smoothness of density for the area process of fractional Brownian motion. Arxiv preprint, 2010.
• [16] P. Friz and N. Victoir. Multidimensional Stochastic Processes Seen as Rough Paths. Cambridge Univ. Press, Cambridge, 2010.
• [17] M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004) 86–140.
• [18] M. Gubinelli and S. Tindel. Rough evolution equations. Ann. Probab. 38 (2010) 1–75.
• [19] M. Hairer. Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann. Probab. 33 (2005) 703–758.
• [20] M. Hairer and A. Ohashi. Ergodicity theory of SDEs with extrinsic memory. Ann. Probab. 35 (2007) 1950–1977.
• [21] M. Hairer and N. S. Pillai. Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths. Preprint, 2011.
• [22] E. Hsu. Stochastic Analysis on Manifolds. Graduate Series in Mathematics 38. Amer. Math. Soc., Providence, RI, 2002.
• [23] Y. Hu and D. Nualart. Differential equations driven by Hölder continuous functions of order greater than $1/2$. Abel Symp. 2 (2007) 349–413.
• [24] S. Kou and X. Sunney-Xie. Generalized Langevin equation with fractional Gaussian noise: Subdiffusion within a single protein molecule. Phys. Rev. Lett. 93 (2004) 18.
• [25] M. Ledoux. The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI, 2001.
• [26] T. Lyons and Z. Qian. System Control and Rough Paths. Oxford Univ. Press, Oxford, 2002.
• [27] A. Millet and M. Sanz-Solé. Large deviations for rough paths of the fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Stat. 42 (2006) 245–271.
• [28] A. Neuenkirch, I. Nourdin, A. Rößler and S. Tindel. Trees and asymptotic developments for fractional diffusion processes. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 157–174.
• [29] A. Neuenkirch, S. Tindel and J. Unterberger. Discretizing the Lévy area. Stochastic Process. Appl. 120 (2010) 223–254.
• [30] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Probability and Its Applications. Springer-Verlag, Berlin, 2006.
• [31] D. Nualart and A. Rǎşcanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55–81.
• [32] D. Nualart and B. Saussereau. Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Stochastic Process. Appl. 119 (2009) 391–409.
• [33] J. Szymanski and M. Weiss. Elucidating the origin of anomalous diffusion in crowded fluids. Phys. Rev. Lett. 103 (2009) 3.
• [34] V. Tejedor, O. Bénichou, R. Voituriez, R. Jungmann, F. Simmel, C. Selhuber-Unkel, L. Oddershede and R. Metzle. Quantitative analysis of single particle trajectories: Mean maximal excursion method. Biophysical J. 98 (2010) 1364–1372.
• [35] A. S. Üstünel. Analysis on Wiener space and applications. Arxiv preprint, 2010.
• [36] M. Zähle. Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111 (1998) 333–374.