On some inequalities for the optional projection and the predictable projection of a discrete parameter process
[Sur quelques inégalités pour la projection optionnelle et la projection prévisible d’un processus de paramètre discret]
Annales mathématiques Blaise Pascal, Tome 29 (2022) no. 1, pp. 149-185.

Soit (Ω,Σ,P) un espace de probabilité non atomique. Si =( n ) est une filtration de Ω et si f=(f n ) nZ est un processus stochastique sur Ω tel que f n est intégrable pour tout n + , la projection optionnelle O() f=( O() f n ) n + de f=(f n ) n + est définie par O() f n =E[|.] Étant donné un espace de fonction de Banach X sur Ω et r[1,), on laisse X[ r ] désigner l’espace de Banach constitué de tous les processus f=(f n ) n + tels que ( n=0 |f n | r ) 1/r X, et on laisse f X[ r ] = n=0 |f n | r 1/r X pour f=(f n ) + X[ r ]. L’un des principaux résultats donne une condition nécessaire et suffisante sur X pour que l’inégalité O() f X[ r ] Cf X[ r ] soit valable pour tout f=(f n ) n + X[ r ].

Let (Ω,Σ,P) be a nonatomic probability space. If =( n ) n + is a filtration of Ω and if f=(f n ) n + is a stochastic process on Ω such that f n is integrable for all n + , the optional projection O() f=( O() f n ) n + of f=(f n ) n + is defined by O() f n =E[|.] Given a Banach function space X over Ω and r[1,), let X[ r ] denote the Banach space consisting of all processes f=(f n ) n + such that ( n=0 |f n | r ) 1/r X, and let f X[ r ] = n=0 |f n | r 1/r X for f=(f n ) + X[ r ]. One of the main results gives a necessary and sufficient condition on X for the inequality O() f X[ r ] Cf X[ r ] to be valid for all f=(f n ) n + X[ r ].

Publié le :
DOI : 10.5802/ambp.409
Classification : 60G07, 46E30
Mots clés : Optional projection, Predictable projection, Banach function space
Masato Kikuchi 1

1 Department of Mathematics University of Toyama 3190 Gofuku, Toyama 930-8555 JAPAN
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{AMBP_2022__29_1_149_0,
     author = {Masato Kikuchi},
     title = {On some inequalities for the optional projection and the predictable projection of a discrete parameter process},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {149--185},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
     volume = {29},
     number = {1},
     year = {2022},
     doi = {10.5802/ambp.409},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.409/}
}
TY  - JOUR
AU  - Masato Kikuchi
TI  - On some inequalities for the optional projection and the predictable projection of a discrete parameter process
JO  - Annales mathématiques Blaise Pascal
PY  - 2022
SP  - 149
EP  - 185
VL  - 29
IS  - 1
PB  - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.409/
DO  - 10.5802/ambp.409
LA  - en
ID  - AMBP_2022__29_1_149_0
ER  - 
%0 Journal Article
%A Masato Kikuchi
%T On some inequalities for the optional projection and the predictable projection of a discrete parameter process
%J Annales mathématiques Blaise Pascal
%D 2022
%P 149-185
%V 29
%N 1
%I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.409/
%R 10.5802/ambp.409
%G en
%F AMBP_2022__29_1_149_0
Masato Kikuchi. On some inequalities for the optional projection and the predictable projection of a discrete parameter process. Annales mathématiques Blaise Pascal, Tome 29 (2022) no. 1, pp. 149-185. doi : 10.5802/ambp.409. https://ambp.centre-mersenne.org/articles/10.5802/ambp.409/

[1] Hiroyuki Aoyama Lebesgue spaces with variable exponent on a probability space, Hiroshima Math. J., Volume 39 (2009) no. 2, pp. 207-216 | MR | Zbl

[2] Colin Bennett; Robert Sharpley Interpolation of operators, Pure and Applied Mathematics, 129, Academic Press Inc., 1988

[3] David W. Boyd Indices of function spaces and their relationship to interpolation, Can. J. Math., Volume 21 (1969), pp. 1245-1254 | DOI | MR | Zbl

[4] Donald L. Burkholder Distribution function inequalities for martingales, Ann. Probab., Volume 1 (1973), pp. 19-42 | MR | Zbl

[5] Donald L. Burkholder; Brent J. Davis; Richard F. Gundy Integral inequalities for convex functions of operators on martingales, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability Vol. II: Probability theory (1972), pp. 223-240 | Zbl

[6] Kong-Ming Chong; Norman M. Rice Equimeasurable rearrangements of functions, Queen’s Papers in Pure and Applied Mathematics, 28, Queen’s University, 1971

[7] Freddy Delbaen; Walter Schachermayer A general version of the fundamental theorem of asset pricing, Math. Ann., Volume 300 (1994) no. 3, pp. 463-520 | DOI | MR | Zbl

[8] Freddy Delbaen; Walter Schachermayer An inequality for the predictable projection of an adapted process, Séminaire de Probabilités, XXIX (Lecture Notes in Mathematics), Volume 1613 (1995), pp. 17-24 | DOI | Numdam | MR

[9] Claude Dellacherie; Paul-Andre Meyer Probabilités et potentiel, Chapitres V á VIII : Théorie des martingales, Actualités Scientifiques et Industrielles, 1385, Hermann, 1980

[10] Ronald A. DeVore; George G. Lorentz Constructive approximation, Grundlehren der Mathematischen Wissenschaften, 303, Springer, 1993 | DOI

[11] Stephen J. Dilworth Some probabilistic inequalities with applications to functional analysis, Banach spaces (Mérida, 1992) (Contemporary Mathematics), Volume 144 (1993), pp. 53-67 | DOI | MR | Zbl

[12] D. J. H. Garling Inequalities: a journey into linear analysis, Cambridge University Press, 2007

[13] William B. Johnson; Bernard Maurey; Gideon Schechtman; Lior Tzafriri Symmetric structures in Banach spaces, Mem. Am. Math. Soc., Volume 19 (1979) no. 217, pp. 1-298 | MR

[14] Masato Kikuchi Boundedness and convergence of martingales in rearrangement-invariant function spaces, Arch. Math., Volume 75 (2000) no. 4, pp. 312-320 | DOI | MR | Zbl

[15] Masato Kikuchi Characterization of Banach function spaces that preserve the Burkholder square-function inequality, Ill. J. Math., Volume 47 (2003) no. 3, pp. 867-882 | MR | Zbl

[16] Masato Kikuchi New martingale inequalities in rearrangement-invariant function spaces, Proc. Edinb. Math. Soc., II. Ser., Volume 47 (2004) no. 3, pp. 633-657 | DOI | MR | Zbl

[17] Masato Kikuchi On some mean oscillation inequalities for martingales, Publ. Mat., Barc., Volume 50 (2006) no. 1, pp. 167-189 | DOI | MR | Zbl

[18] Masato Kikuchi Uniform boundedness of conditional expectation operators on a Banach function space, Math. Inequal. Appl., Volume 16 (2013) no. 2, pp. 483-499 | MR | Zbl

[19] Masato Kikuchi On some martingale inequalities for mean oscillations in weak spaces, Ric. Mat., Volume 64 (2015) no. 1, pp. 137-165 | DOI | MR | Zbl

[20] Masato Kikuchi On Doob’s inequality and Burkholder’s inequality in weak spaces, Collect. Math., Volume 67 (2016) no. 3, pp. 461-483 | DOI | MR | Zbl

[21] Masato Kikuchi On martingale transform inequalities in certain quasi-Banach function spaces, Boll. Unione Mat. Ital., Volume 12 (2019) no. 3, pp. 485-514 | DOI | MR | Zbl

[22] H. König Eigenvalue distribution of compact operators, Operator Theory: Advances and Applications, 16, Birkhäuser, 1986 | DOI | Zbl

[23] Selim G. Kreĭn; Yuriĭ Ī. Petunīn; Evgeniĭ M. Semenov Interpolation of linear operators, Translations of Mathematical Monographs, 54, American Mathematical Society, 1982

[24] Dominique Lépingle Une inégalité de martingales, Séminaire de Probabilités, XII (Lecture Notes in Mathematics), Volume 649 (1978), pp. 134-137 | DOI | Numdam | Zbl

[25] Joram Lindenstrauss; Lior Tzafriri Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 97, Springer, 1979 | DOI

[26] Adam Osȩkowski Sharp L 1 ( q ) estimate for a sequence and its predictable projection, Stat. Probab. Lett., Volume 104 (2015), pp. 82-86 | DOI | MR | Zbl

[27] L. C. G. Rogers; David Williams Diffusions, Markov processes, and martingales. Vol. 1: Foundations, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1994

[28] Elias M. Stein Topics in harmonic analysis related to the Littlewood–Paley theory, Annals of Mathematics Studies, 63, Princeton University Press, 1970

Cité par Sources :