Electronic Journal of Probability

On the distances between probability density functions

Vlad Bally and Lucia Caramellino

Full-text: Open access

Abstract

We give estimates of the distance between the densities of the laws of two functionals $F$ and $G$ on the Wiener space in terms of the Malliavin-Sobolev norm of $F-G.$ We actually consider a  more general framework which allows one to treat with similar (Malliavin type)methods functionals of a Poisson point measure (solutions of jump type stochastic equations). We use the above estimates in order to obtain a criterion which ensures that convergence in distribution implies convergence in total variation distance; in particular, if the functionals at hand are absolutely continuous, this implies convergence in $L^{1}$ of the densities.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 110, 33 pp.

Dates
Accepted: 11 December 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065752

Digital Object Identifier
doi:10.1214/EJP.v19-3175

Mathematical Reviews number (MathSciNet)
MR3296526

Zentralblatt MATH identifier
1307.60072

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Bally, Vlad; Caramellino, Lucia. On the distances between probability density functions. Electron. J. Probab. 19 (2014), paper no. 110, 33 pp. doi:10.1214/EJP.v19-3175. https://projecteuclid.org/euclid.ejp/1465065752


Export citation

References

  • Bally, Vlad. An elementary introduction to Malliavin calculus. Research report INRIA n. 00071868
  • Bally, Vlad; Caramellino, Lucia. Riesz transform and integration by parts formulas for random variables. Stochastic Process. Appl. 121 (2011), no. 6, 1332–1355.
  • Bally, Vlad; Caramellino, Lucia. Positivity and lower bounds for the density of Wiener functionals. Potential Anal. 39 (2013), no. 2, 141–168.
  • textscV. Bally, L. Caramellino (2013). Convergence in total variation and CLT for Wiener functionals. arXiv:1407.0896
  • Bally, Vlad; Clément, Emmanuelle. Integration by parts formula and applications to equations with jumps. Probab. Theory Related Fields 151 (2011), no. 3-4, 613–657.
  • Bally, Vlad; Clément, Emmanuelle. Integration by parts formula with respect to jump times for stochastic differential equations. Stochastic analysis 2010, 7–29, Springer, Heidelberg, 2011.
  • Bally, V.; Talay, D. The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Related Fields 104 (1996), no. 1, 43–60.
  • Bally, Vlad; Talay, Denis. The law of the Euler scheme for stochastic differential equations. II. Convergence rate of the density. Monte Carlo Methods Appl. 2 (1996), no. 2, 93–128.
  • Bichteler, Klaus; Gravereaux, Jean-Bernard; Jacod, Jean. Malliavin calculus for processes with jumps. Stochastics Monographs, 2. Gordon and Breach Science Publishers, New York, 1987. x+161 pp. ISBN: 2-88124-185-9
  • Bouleau, Nicolas; Hirsch, Francis. Dirichlet forms and analysis on Wiener space. de Gruyter Studies in Mathematics, 14. Walter de Gruyter & Co., Berlin, 1991. x+325 pp. ISBN: 3-11-012919-1
  • Fournier, Nicolas. Jumping SDEs: absolute continuity using monotonicity. Stochastic Process. Appl. 98 (2002), no. 2, 317–330.
  • Guyon, Julien. Euler scheme and tempered distributions. Stochastic Process. Appl. 116 (2006), no. 6, 877–904.
  • Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3
  • Malicet, D.; Poly, G. Properties of convergence in Dirichlet structures. J. Funct. Anal. 264 (2013), no. 9, 2077–2096.
  • Nourdin, Ivan; Peccati, Giovanni. Normal approximations with Malliavin calculus. From Stein's method to universality. Cambridge Tracts in Mathematics, 192. Cambridge University Press, Cambridge, 2012. xiv+239 pp. ISBN: 978-1-107-01777-1
  • Nourdin, Ivan; Poly, Guillaume. Convergence in total variation on Wiener chaos. Stochastic Process. Appl. 123 (2013), no. 2, 651–674.
  • Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5
  • Nourdin, Ivan; Nualart, David; Poly, Guillaume. Absolute continuity and convergence of densities for random vectors on Wiener chaos. Electron. J. Probab. 18 (2013), no. 22, 19 pp.