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

Fabrice Baudoin, Cheng Ouyang, and Samy Tindel

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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

Permanent link to this document
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


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