On some inequalities for the optional projection and the predictable projection of a discrete parameter process
Annales mathématiques Blaise Pascal, Volume 29 (2022) no. 1, pp. 149-185.

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 ].

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 ].

Published online:
DOI: 10.5802/ambp.409
Classification: 60G07, 46E30
Keywords: Optional projection, Predictable projection, Banach function space
Kikuchi, Masato 1

1 Department of Mathematics University of Toyama 3190 Gofuku, Toyama 930-8555 JAPAN
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Kikuchi, Masato. On some inequalities for the optional projection and the predictable projection of a discrete parameter process. Annales mathématiques Blaise Pascal, Volume 29 (2022) no. 1, pp. 149-185. doi : 10.5802/ambp.409. http://www.numdam.org/articles/10.5802/ambp.409/

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