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

Cristina Di Girolami; Francesco Russo

doi:10.1016/j.crma.2010.11.032

Probability Theory

Clark–Ocone type formula for non-semimartingales with finite quadratic variation
[Formule de Clark–Ocone generalisée pour des non-semimartingales à variation quadratique finie]

Présenté par : Marc Yor

Cristina Di Girolami ^1,² ; Francesco Russo ^2,³

¹ Luiss Guido Carli – Libera Università Internazionale degli Studi Sociali Guido Carli di Roma, Viale Pola 12, 00198 Roma, Italy
² ENSTA ParisTech, unité de mathématiques appliquées, 32, boulevard Victor, 75739 Paris cedex 15, France
³ INRIA Rocquencourt and Cermics École des ponts, projet MATHFI, domaine de Voluceau, BP 105, 78153 Le Chesnay cedex, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 209-214.

Résumé
Abstract

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. À un processus réel continu X, nous associons le processus $X (\cdot)$ , appelé processus fenêtre, qui à l'instant t, garde en mémoire le passé jusqu'à $t - τ$ . L'espace naturel d'évolution pour $X (\cdot)$ est l'espace de Banach B des fonctions continues définies sur $[- τ, 0]$ . 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.

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 window process, which describes the evolution of X taking into account a memory $τ > 0$ . The natural state space for $X (\cdot)$ is the Banach space of continuous functions on $[- τ, 0]$ . 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.

Reçu le : 2010-05-18
Accepté le : 2010-11-30
Publié le : 2011-01-07

DOI : 10.1016/j.crma.2010.11.032

Affiliations des auteurs :

Cristina Di Girolami ^{1, 2} ; Francesco Russo ^{2, 3}

¹ Luiss Guido Carli – Libera Università Internazionale degli Studi Sociali Guido Carli di Roma, Viale Pola 12, 00198 Roma, Italy
² ENSTA ParisTech, unité de mathématiques appliquées, 32, boulevard Victor, 75739 Paris cedex 15, France
³ INRIA Rocquencourt and Cermics École des ponts, projet MATHFI, domaine de Voluceau, BP 105, 78153 Le Chesnay cedex, France

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Cristina Di Girolami; Francesco Russo. Clark–Ocone type formula for non-semimartingales with finite quadratic variation. Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 209-214. doi : 10.1016/j.crma.2010.11.032. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.032/

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Cristina Di Girolami Notion of quadratic variation in Banach spaces, Stochastic Analysis and Applications, Volume 42 (2024) no. 4, pp. 674-701 | DOI:10.1080/07362994.2024.2369834 | Zbl:7942918
Andrea Cosso; Francesco Russo Crandall-Lions viscosity solutions for path-dependent PDEs: the case of heat equation, Bernoulli, Volume 28 (2022) no. 1, pp. 481-503 | DOI:10.3150/21-bej1353 | Zbl:1482.35072
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Andrea Cosso; Francesco Russo Functional and Banach space stochastic calculi: path-dependent Kolmogorov equations associated with the frame of a Brownian motion, Stochastics of environmental and financial economics. Centre of Advanced Study, Oslo, Norway, 2014–2015, Cham: Springer, 2016, pp. 27-80 | DOI:10.1007/978-3-319-23425-0_2 | Zbl:1338.60138
Cristina Di Girolami; Giorgio Fabbri; Francesco Russo The covariation for Banach space valued processes and applications, Metrika, Volume 77 (2014) no. 1, pp. 51-104 | DOI:10.1007/s00184-013-0472-6 | Zbl:1455.60073
Cristina Di Girolami; Francesco Russo Generalized covariation for Banach space valued processes, Itō formula and applications, Osaka Journal of Mathematics, Volume 51 (2014) no. 3, pp. 729-783 | Zbl:1308.60039
Heikki Tikanmäki Robust Hedging and Pathwise Calculus, Applied Mathematical Finance, Volume 20 (2013) no. 3, p. 287 | DOI:10.1080/1350486x.2012.725978
Cristina Di Girolami; Francesco Russo Generalized covariation and extended Fukushima decomposition for Banach space-valued processes: applications to windows of Dirichlet processes, Infinite Dimensional Analysis, Quantum Probability and Related Topics, Volume 15 (2012) no. 2, p. 1250007 | DOI:10.1142/s0219025712500075 | Zbl:1279.60067
Rosanna Coviello; Cristina di Girolami; Francesco Russo On stochastic calculus related to financial assets without semimartingales, Bulletin des Sciences Mathématiques, Volume 135 (2011) no. 6-7, pp. 733-774 | DOI:10.1016/j.bulsci.2011.06.008 | Zbl:1233.91337

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