Clark-Ocone type formula for non-semimartingales with finite quadratic variation

Cristina Di Girolami; Francesco Russo

doi:10.1016/j.crma.2010.11.032

inria-00484993, version 2

Clark-Ocone type formula for non-semimartingales with finite quadratic variation

Cristina Di Girolami ()^a^, 1, 2, Francesco Russo () ^2, 3

Comptes-Rendus de l'Académie des Sciences, Série 1, Mathématiques 349, 3-4 (2011) 209-214

Résumé : We provide a suitable framework for the concept of finite quadratic variation for processes with values in a separable Banach space $B$ using the language of stochastic calculus via regularizations, introduced in the case $B= \R$ by the second author and P. Vallois. To a real continuous process $X$ we associate the Banach valued process $X(\cdot)$, called {\it window} process, which describes the evolution of $X$ taking into account a memory $\tau>0$. The natural state space for $X(\cdot)$ is the Banach space of continuous functions on $[-\tau,0]$. If $X$ is a real finite quadratic variation process, an appropriated Itô formula is presented, from which we derive a generalized Clark-Ocone formula for non-semimartingales having the same quadratic variation as Brownian motion. The representation is based on solutions of an infinite dimensional PDE.

a – Università LUISS Guido Carli
1 : Libera Universita Internazionale degli Studi Sociali Guido Carli di Roma (Luiss Guido Carli)
Libera Università Internazionale degli Studi Sociali
2 : Unité de Mathématiques Appliquées (UMA)
ENSTA ParisTech
3 : MATHFI (INRIA Rocquencourt)
INRIA – École des Ponts ParisTech (ENPC) – Université Paris-Est Créteil Val-de-Marne (UPEC)

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Contributeur : Francesco Russo <>
Soumis le : Mardi 26 Octobre 2010, 10:53:35
Dernière modification le : Vendredi 2 Mars 2012, 19:00:04

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arXiv : 1005.3608

DOI : 10.1016/j.crma.2010.11.032

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%% inria-00484993, version 2
%% http://hal.inria.fr/inria-00484993
@article{digirolami:inria-00484993,
    hal_id = {inria-00484993},
    url = {http://hal.inria.fr/inria-00484993},
    title = {{Clark-Ocone type formula for non-semimartingales with finite quadratic variation}},
    author = {Di Girolami, Cristina and Russo, Francesco},
    abstract = {{We provide a suitable framework for the concept of finite quadratic variation for processes with values in a separable Banach space $B$ using the language of stochastic calculus via regularizations, introduced in the case $B= \R$ by the second author and P. Vallois. To a real continuous process $X$ we associate the Banach valued process $X(\cdot)$, called {\it window} process, which describes the evolution of $X$ taking into account a memory $\tau>0$. The natural state space for $X(\cdot)$ is the Banach space of continuous functions on $[-\tau,0]$. If $X$ is a real finite quadratic variation process, an appropriated It{\^o} formula is presented, from which we derive a generalized Clark-Ocone formula for non-semimartingales having the same quadratic variation as Brownian motion. The representation is based on solutions of an infinite dimensional PDE.}},
    keywords = {Calculus via regularization; Infinite dimensional analysis; Clark-Ocone formula; Dirichlet processes; It{\^o} formula; Quadratic variation; Hedging theory without semimartingales.},
    language = {Anglais},
    affiliation = {Libera Universita Internazionale degli Studi Sociali Guido Carli di Roma - Luiss Guido Carli , Unit{\'e} de Math{\'e}matiques Appliqu{\'e}es - UMA , MATHFI - INRIA Rocquencourt},
    publisher = {Elsevier},
    pages = {209-214},
    journal = {Comptes-Rendus de l'Acad{\'e}mie des Sciences, S{\'e}rie 1, Math{\'e}matiques},
    volume = {349},
    number = {3-4 },
    audience = {internationale },
    doi = {10.1016/j.crma.2010.11.032 },
    year = {2011},
    month = Feb,
    pdf = {http://hal.inria.fr/inria-00484993/PDF/CRASSParis26OctSent.pdf},
}

%F inria-00484993, version 2
%0 Journal Article
%U http://hal.inria.fr/inria-00484993
%2 MATH:MATH_PR
%T Clark-Ocone type formula for non-semimartingales with finite quadratic variation
%A Di Girolami, Cristina
%A Russo, Francesco
%+ Libera Universita Internazionale degli Studi Sociali Guido Carli di Roma - Luiss Guido Carli , Unité de Mathématiques Appliquées - UMA , MATHFI - INRIA Rocquencourt
%X We provide a suitable framework for the concept of finite quadratic variation for processes with values in a separable Banach space $B$ using the language of stochastic calculus via regularizations, introduced in the case $B= \R$ by the second author and P. Vallois. To a real continuous process $X$ we associate the Banach valued process $X(\cdot)$, called {\it window} process, which describes the evolution of $X$ taking into account a memory $\tau>0$. The natural state space for $X(\cdot)$ is the Banach space of continuous functions on $[-\tau,0]$. If $X$ is a real finite quadratic variation process, an appropriated Itô formula is presented, from which we derive a generalized Clark-Ocone formula for non-semimartingales having the same quadratic variation as Brownian motion. The representation is based on solutions of an infinite dimensional PDE.
%2 Nous présentons un cadre adéquat pour le concept de variation quadratique finie lorsque le processus de référence est à valeurs dans un espace de Banach séparable $B$. Le langage utilisé est celui de l'intégrale via régularisations introduit dans le cas réel par le second auteur et P. Vallois. A un processus réel continu $X$, nous associons le processus $X(\cdot)$, appelé processus {\it fenêtre}, qui à l'instant $t$, garde en mémoire le passé jusqu'à $t-\tau$. L'espace naturel d' évolution pour $X(\cdot)$ est l'espace de Banach $B$ des fonctions continues définies sur $[-\tau,0]$. Si $X$ est un processus réel à variation quadratique finie, nous énonçons une formule d'Itô appropriée de laquelle nous déduisons une formule de Clark-Ocone relative à des non-semimartingales réelles ayant la même variation quadratique que le mouvement brownien. La représentation est basée sur des solutions d'une EDP infini-dimensionnelle.
%Z MSC 2010 60H05;60H07; 60H30;91G80
%G Anglais
%J Comptes-Rendus de l'Académie des Sciences, Série 1, Mathématiques
%8 2011-02-00
%2 internationale
%I Elsevier
%V 349
%N 3-4
%P 209-214
%R 10.1016/j.crma.2010.11.032
%K Calculus via regularization; Infinite dimensional analysis; Clark-Ocone formula; Dirichlet processes; Itô formula; Quadratic variation; Hedging theory without semimartingales.
%2 9202

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