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 $\left(\Omega ,\Sigma ,\mathrm{P}\right)$ be a nonatomic probability space. If $ℱ={\left({ℱ}_{n}\right)}_{n\in {ℤ}_{+}}$ is a filtration of $\Omega$ and if $f={\left({f}_{n}\right)}_{n\in {ℤ}_{+}}$ is a stochastic process on $\Omega$ such that ${f}_{n}$ is integrable for all $n\in {ℤ}_{+}$, the optional projection ${}^{\phantom{\rule{0.83328pt}{0ex}}O\left(ℱ\right)}f={\left({}^{\phantom{\rule{0.83328pt}{0ex}}O\left(ℱ\right)}{f}_{n}\right)}_{n\in {ℤ}_{+}}$ of $f={\left({f}_{n}\right)}_{n\in {ℤ}_{+}}$ is defined by ${}^{\phantom{\rule{0.83328pt}{0ex}}O\left(ℱ\right)}{f}_{n}=\mathrm{E}\phantom{\rule{0.83328pt}{0ex}}\left[$|.] Given a Banach function space $X$ over $\Omega$ and $r\in \left[1,\infty \right)$, let $X\left[{\ell }_{r}\right]$ denote the Banach space consisting of all processes $f={\left({f}_{n}\right)}_{n\in {ℤ}_{+}}$ such that ${\left({\sum }_{n=0}^{\infty }{|{f}_{n}\phantom{\rule{0.0pt}{0ex}}|}^{r}\right)}^{1/r}\in X$, and let ${\parallel f\phantom{\rule{0.0pt}{0ex}}\parallel }_{X\left[{\ell }_{r}\right]}^{}={∥{\left({\sum }_{n=0}^{\infty }{|{f}_{n}\phantom{\rule{0.0pt}{0ex}}|}^{r}\phantom{\rule{0.0pt}{0ex}}\right)}^{1/r}\phantom{\rule{0.0pt}{0ex}}∥}_{X}^{}$ for $f={\left({f}_{n}\right)}_{\in {ℤ}_{+}}\in X\left[{\ell }_{r}\right]$. One of the main results gives a necessary and sufficient condition on $X$ for the inequality ${\parallel {}^{\phantom{\rule{0.83328pt}{0ex}}O\left(ℱ\right)}f\phantom{\rule{0.0pt}{0ex}}\parallel }_{X\left[{\ell }_{r}\right]}^{}\le C{\parallel f\phantom{\rule{0.0pt}{0ex}}\parallel }_{X\left[{\ell }_{r}\right]}^{}$ to be valid for all $f={\left({f}_{n}\right)}_{n\in {ℤ}_{+}}\in X\left[{\ell }_{r}\right]$.

Soit $\left(\Omega ,\Sigma ,\mathrm{P}\right)$ un espace de probabilité non atomique. Si $ℱ=\left({ℱ}_{n}\right)$ est une filtration de $\Omega$ et si $f={\left({f}_{n}\right)}_{n\in Z}$ est un processus stochastique sur $\Omega$ tel que ${f}_{n}$ est intégrable pour tout $n\in {ℤ}_{+}$, la projection optionnelle ${}^{\phantom{\rule{0.83328pt}{0ex}}O\left(ℱ\right)}f={\left({}^{\phantom{\rule{0.83328pt}{0ex}}O\left(ℱ\right)}{f}_{n}\right)}_{n\in {ℤ}_{+}}$ de $f={\left({f}_{n}\right)}_{n\in {ℤ}_{+}}$ est définie par ${}^{\phantom{\rule{0.83328pt}{0ex}}O\left(ℱ\right)}{f}_{n}=\mathrm{E}\phantom{\rule{0.83328pt}{0ex}}\left[$|.] Étant donné un espace de fonction de Banach $X$ sur $\Omega$ et $r\in \left[1,\infty \right)$, on laisse $X\left[{\ell }_{r}\right]$ désigner l’espace de Banach constitué de tous les processus $f={\left({f}_{n}\right)}_{n\in {ℤ}_{+}}$ tels que ${\left({\sum }_{n=0}^{\infty }{|{f}_{n}\phantom{\rule{0.0pt}{0ex}}|}^{r}\right)}^{1/r}\in X$, et on laisse ${\parallel f\phantom{\rule{0.0pt}{0ex}}\parallel }_{X\left[{\ell }_{r}\right]}^{}={∥{\left({\sum }_{n=0}^{\infty }{|{f}_{n}\phantom{\rule{0.0pt}{0ex}}|}^{r}\phantom{\rule{0.0pt}{0ex}}\right)}^{1/r}\phantom{\rule{0.0pt}{0ex}}∥}_{X}^{}$ pour $f={\left({f}_{n}\right)}_{\in {ℤ}_{+}}\in X\left[{\ell }_{r}\right]$. L’un des principaux résultats donne une condition nécessaire et suffisante sur $X$ pour que l’inégalité ${\parallel {}^{\phantom{\rule{0.83328pt}{0ex}}O\left(ℱ\right)}f\phantom{\rule{0.0pt}{0ex}}\parallel }_{X\left[{\ell }_{r}\right]}^{}\le C{\parallel f\phantom{\rule{0.0pt}{0ex}}\parallel }_{X\left[{\ell }_{r}\right]}^{}$ soit valable pour tout $f={\left({f}_{n}\right)}_{n\in {ℤ}_{+}}\in X\left[{\ell }_{r}\right]$.

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

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