On some inequalities for the optional projection and the predictable projection of a discrete parameter process

Masato Kikuchi

doi:10.5802/ambp.409

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]

Masato Kikuchi ¹

¹ Department of Mathematics University of Toyama 3190 Gofuku, Toyama 930-8555 JAPAN

Annales mathématiques Blaise Pascal, Tome 29 (2022) no. 1, pp. 149-185.

Résumé
Abstract

Soit $(Ω, Σ, P)$ un espace de probabilité non atomique. Si $ℱ = (ℱ_{n})$ est une filtration de $Ω$ et si $f = {(f_{n})}_{n \in Z}$ est un processus stochastique sur $Ω$ tel que $f_{n}$ est intégrable pour tout $n \in ℤ_{+}$ , la projection optionnelle $^{O (ℱ)} f = {(^{O (ℱ)} f_{n})}_{n \in ℤ_{+}}$ de $f = {(f_{n})}_{n \in ℤ_{+}}$ est définie par $^{O (ℱ)} f_{n} = E [$ |.] Étant donné un espace de fonction de Banach $X$ sur $Ω$ et $r \in [1, \infty)$ , on laisse $X [ℓ_{r}]$ désigner l’espace de Banach constitué de tous les processus $f = {(f_{n})}_{n \in ℤ_{+}}$ tels que ${(\sum_{n = 0}^{\infty} {| f_{n} |}^{r})}^{1 / r} \in X$ , et on laisse ${∥ f ∥}_{X [ℓ_{r}]}^{} = {∥{(\sum_{n = 0}^{\infty} {| f_{n} |}^{r})}^{1 / r}∥}_{X}^{}$ pour $f = {(f_{n})}_{\in ℤ_{+}} \in X [ℓ_{r}]$ . L’un des principaux résultats donne une condition nécessaire et suffisante sur $X$ pour que l’inégalité ${∥^{O (ℱ)} f ∥}_{X [ℓ_{r}]}^{} \leq C {∥ f ∥}_{X [ℓ_{r}]}^{}$ soit valable pour tout $f = {(f_{n})}_{n \in ℤ_{+}} \in X [ℓ_{r}]$ .

Let $(Ω, Σ, P)$ be a nonatomic probability space. If $ℱ = {(ℱ_{n})}_{n \in ℤ_{+}}$ is a filtration of $Ω$ and if $f = {(f_{n})}_{n \in ℤ_{+}}$ is a stochastic process on $Ω$ such that $f_{n}$ is integrable for all $n \in ℤ_{+}$ , the optional projection $^{O (ℱ)} f = {(^{O (ℱ)} f_{n})}_{n \in ℤ_{+}}$ of $f = {(f_{n})}_{n \in ℤ_{+}}$ is defined by $^{O (ℱ)} f_{n} = E [$ |.] Given a Banach function space $X$ over $Ω$ and $r \in [1, \infty)$ , let $X [ℓ_{r}]$ denote the Banach space consisting of all processes $f = {(f_{n})}_{n \in ℤ_{+}}$ such that ${(\sum_{n = 0}^{\infty} {| f_{n} |}^{r})}^{1 / r} \in X$ , and let ${∥ f ∥}_{X [ℓ_{r}]}^{} = {∥{(\sum_{n = 0}^{\infty} {| f_{n} |}^{r})}^{1 / r}∥}_{X}^{}$ for $f = {(f_{n})}_{\in ℤ_{+}} \in X [ℓ_{r}]$ . One of the main results gives a necessary and sufficient condition on $X$ for the inequality ${∥^{O (ℱ)} f ∥}_{X [ℓ_{r}]}^{} \leq C {∥ f ∥}_{X [ℓ_{r}]}^{}$ to be valid for all $f = {(f_{n})}_{n \in ℤ_{+}} \in X [ℓ_{r}]$ .

Publié le : 2022-09-23

DOI : 10.5802/ambp.409

Classification : 60G07, 46E30
Mots clés : Optional projection, Predictable projection, Banach function space

Affiliations des auteurs :

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

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     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {149--185},
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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/

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