A martingale bound for the entropy associated with a trimmed filtration on
Volume 1, Issue 2 (2014), pp. 151–165
Pub. online: 2 February 2015
Type: Research Article
Open Access
Open Access
Received
22 December 2014
22 December 2014
Revised
18 January 2015
18 January 2015
Accepted
19 January 2015
19 January 2015
Published
2 February 2015
2 February 2015
Abstract
Using martingale methods, we provide bounds for the entropy of a probability measure on ${\mathbb{R}}^{d}$ with the right-hand side given in a certain integral form. As a corollary, in the one-dimensional case, we obtain a weighted log-Sobolev inequality.
References
Capitaine, M., Hsu, E., Ledoux, M.: Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Electron. Commun. Probab. 2, 71–81 (1997) MR1484557. doi:10.1214/ECP.v2-986
Carlen, E., Pardoux, E.: Differential calculus and integration by parts on Poisson space. In: Stochastics, Algebra and Analysis in Classical and Quantum Dynamics, pp. 63–73. Kluwer, Dordrecht (1990) MR1052702
Clark, I.M.C.: The representation of functionals of Brownian motion by stochastic integrals. Ann. Math. Stat. 41(4), 46–52 (1970) MR0270448
Elliott, R.J.: Stochastic Calculus and Applications. Appl. Math., vol. 18, Springer-Verlag, Berlin, Heidelberg, New York (1982) MR0678919
Elliott, R.J., Tsoi, A.H.: Integration by parts for Poisson processes. J. Multivar. Anal. 44, 179–190 (1993) MR1219202. doi:10.1006/jmva.1993.1010
Gong, F.-Zh., Ma, Zh.-M.: Martingale representation and log-Sobolev inequality on loop space. C. R. Acad. Sci. Paris 326, 749–753 (1998) MR1641711. doi:10.1016/S0764-4442(98)80043-8
Koshevoy, G.A., Mosler, K.: Zonoid trimming for multivariate distributions. Ann. Stat. 25(5), 1998–2017 (1997) MR1474078. doi:10.1214/aos/1069362382
Ledoux, M.: Concentration of measure and logarithmic Sobolev inequalities. In: Seminaire de Probabilités XXXIII. Lect. Notes Math., vol. 1709, pp. 120–219 (1999) MR1767995. doi:10.1007/BFb0096511
Nualart, D.: Analysis in Wiener space and anticipating stochastic calculus, Springer, Berlin, Heidelberg, Lect. Notes Math., vol. 1690, pp. 123–227 (1998) MR1668111. doi:10.1007/BFb0092538
Ocone, D.: Malliavin’s calcululs and stochastic integral representation of functionals of diffusion processes. Stochastics 8, 161–185 (1984) MR0749372. doi:10.1080/17442508408833299
