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

Kikuchi, Masato

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

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

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

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

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