Gaussian lower bounds for the density via Malliavin calculus

Tien Dung Nguyen

doi:10.5802/crmath.13

Probabilités

Gaussian lower bounds for the density via Malliavin calculus
[Bornes inférieures gaussiennes pour la densité par le calcul de Malliavin]

Tien Dung Nguyen ^1,²

¹ Department of Mathematics, FPT University, Hoa Lac High Tech Park, Hanoi, Vietnam
² Department of Mathematics, VNU University of Science, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, 084 Vietnam

Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 79-87.

Résumé
Abstract

Dans cet article, à partir d’une formule connue, nous utilisons une idée simple pour obtenir une nouvelle représentation de la densité des variables aléatoires différentiables de Malliavin. Cette nouvelle représentation est particulièrement utile pour trouver des bornes inférieures de la densité.

In this paper, based on a known formula, we use a simple idea to get a new representation for the density of Malliavin differentiable random variables. This new representation is particularly useful for finding lower bounds for the density.

Reçu le : 2019-08-11
Révisé le : 2019-11-25
Accepté le : 2019-12-20
Publié le : 2020-03-18

DOI : 10.5802/crmath.13

Classification : 60G15, 60H07

Affiliations des auteurs :

Tien Dung Nguyen ^{1, 2}

¹ Department of Mathematics, FPT University, Hoa Lac High Tech Park, Hanoi, Vietnam
² Department of Mathematics, VNU University of Science, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, 084 Vietnam

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Gaussian lower bounds for the density via {Malliavin} calculus},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {79--87},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {1},
     year = {2020},
     doi = {10.5802/crmath.13},
     language = {en},
}

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Tien Dung Nguyen. Gaussian lower bounds for the density via Malliavin calculus. Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 79-87. doi : 10.5802/crmath.13. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.13/

Bibliographie
Cité par

[1] Mireia Besalú; Arturo Kohatsu-Higa; Samy Tindel Gaussian-type lower bounds for the density of solutions of SDEs driven by fractional Brownian motions, Ann. Probab., Volume 44 (2016) no. 1, pp. 399-443 | DOI | MR | Zbl

[2] Nguyen Tien Dung The density of solutions to multifractional stochastic Volterra integro-differential equations, Nonlinear Anal., Theory Methods Appl., Volume 130 (2016), pp. 176-189 | MR | Zbl

[3] Shigeo Kusuoka; Daniel W. Strook Applications of the Malliavin calculus. III, J. Fac. Sci., Univ. Tokyo, Sect. I A (1987) no. 2, pp. 391-442 | MR | Zbl

[4] Paul Malliavin; Eulalia Nualart Density minoration of a strongly non-degenerated random variable, J. Funct. Anal., Volume 256 (2009) no. 12, pp. 4197-4214 | DOI | MR | Zbl

[5] Tien Dung Nguyen; Nicolas Privault; Giovanni Luca Torrisi Gaussian estimates for the solutions of some one-dimensional stochastic equations, Potential Anal., Volume 43 (2015) no. 2, pp. 289-311 | DOI | MR | Zbl

[6] Ivan Nourdin; Frederi G. Viens Density formula and concentration inequalities with Malliavin calculus, Electron. J. Probab., Volume 14 (2009) no. 78, pp. 2287-2309 | DOI | MR | Zbl

[7] David Nualart The Malliavin calculus and related topics, Probability and Its Applications, Springer, 2006 | Zbl

[8] Eulalia Nualart Exponential divergence estimates and heat kernel tail, C. R. Math. Acad. Sci. Paris, Volume 338 (2004) no. 1, pp. 77-80 | DOI | MR | Zbl

[9] Martina Zähle Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Relat. Fields, Volume 111 (1998) no. 3, pp. 333-374 | MR | Zbl

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