Stochastic integration in UMD Banach spaces

Neerven, J. M. A. M. van; Veraar, Mark; Weis, Lutz

doi:10.1214/009117906000001006

Cited by 204 publications

(432 citation statements)

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“…A stochastic calculus for Banach valued martingales was considered by [2,16] and references therein, generalizing the classical stochastic calculus of [5,11,7].…”

Section: Introductionmentioning

confidence: 99%

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

Girolami

Russo²

2011

Comptes Rendus. Mathématique

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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(·), called window process, which describes the evolution of X taking into account a memory τ > 0. The natural state space for X(·) 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. RésuméNous présentons un cadre adéquat pour le concept de variation quadratique finie lorsque le processus de référence està 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(·), appelé processus fenêtre, quià l'instant t, garde en mémoire le passé jusqu'à t − τ. L'espace naturel d'évolution pour X(·) est l'espace de Banach B des fonctions continues définies sur [−τ, 0]. Si X est un processus réelà variation quadratique finie, nousénonçons une formule d'Itô appropriée de laquelle nous déduisons une formule de ClarkOcone relativeà 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.Keywords: Calculus via regularization, Infinite dimensional analysis, Clark-Ocone formula, Itô formula, Quadratic variation, Hedging theory without semimartingales. 2010 MSC: 60H05, 60H07, 60H30, 91G80. Version française abrégéeDans cette Note nous développons un calcul stochastique via régularisation de type progressif (forward) lorsque le processus intégrateur X està valeurs dans un espace de Banach séparable B. Ceci est basé sur une notion sophistiquée de variation quadratique que nous appellerons χ-variation quadratique, où le symbole χ correspondà un sous-espace La théorie de l'intégration infini-dimensionnelle par rapportà des martingales (ou des semimartingales, [5,11,7]) n'est pas appliquable, même lorsque l'intégrateur est la fenêtre W(·) associée au mouvement brownien standard W. Au-delà des difficultés qui viennent du fait que C( [−τ, 0] n'est pas réflexif, W(·) n'est d'aucune manière une semimartingaleà valeurs dans C( [−τ, 0]). Motivés par des applications liéesà la couverture d'options dépendant de toute la trajectoire, nous discutons une formule de type Clark-Ocone visantà décomposer une classe significative h de v.a. dépendant de la trajectoire d'un processus X dont la variation quadratique vaut [X] t = t. Cette formule généralise des résultats inclus dans [15,1,3] visantà déterminer des formules de valor...

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“…A stochastic calculus for Banach valued martingales was considered by [2,16] and references therein, generalizing the classical stochastic calculus of [5,11,7].…”

Section: Introductionmentioning

confidence: 99%

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

Girolami

Russo²

2011

Comptes Rendus. Mathématique

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“…For more details we refer to [19,24] and the references therein. Let (A, Σ, µ) be a σ-finite measure space, H a Hilbert spaces and X a Banach space.…”

Section: Preliminariesmentioning

confidence: 99%

“…This follows from a simple application of the Kahane-Khintchine inequality; we refer to [24,Proposition 2.6] for the details. Here, H and X are allowed to be arbitrary Hilbert spaces and Banach spaces, respectively; the norm constants in the isomorphism are independent of H.…”

Section: Preliminariesmentioning

confidence: 99%

“…The latter was proven by Bourgain in [6], thereby starting the development of harmonic analysis for UMD-valued functions. In recent years, research in this field has accelerated as it appeared that its tools, and in particular square function estimates, are of fundamental importance in the study of the H ∞ -functional calculus (see [20]) and in stochastic analysis in UMD Banach spaces (see [24]). …”

Section: Introductionmentioning

confidence: 99%

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Conical square function estimates in UMD Banach spaces and applications to H ∞-functional calculi

2008

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Abstract. We study conical square function estimates for Banach-valued functions, and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-M c Intosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator A with certain off-diagonal bounds, such that A always has a bounded H ∞ -functional calculus on these spaces. This provides a new way of proving functional calculus of A on the Bochner spaces L p (R n ; X) by checking appropriate conical square function estimates, and also a conical analogue of Bourgain's extension of the Littlewood-Paley theory to the UMDvalued context. Even when X = C, our approach gives refined p-dependent versions of known results.

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“…We are inspired by the paper of van Neerven et al [30] in which they deal with the Wiener case. Here, the approach is based on a two-sided decoupling inequality which enables the authors to define the stochastic integral for random integrands by means of the integral for deterministic integrands.…”

Section: Introductionmentioning

confidence: 99%

Cylindrical fractional Brownian motion in Banach spaces

Issoglio

Riedle

2014

Stochastic Processes and their Applications

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Abstract. In this article we introduce cylindrical fractional Brownian motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical fractional Brownian motion is understood in the classical framework of cylindrical random variables and cylindrical measures. The developed stochastic integral for deterministic operator valued integrands is based on a series representation of the cylindrical fractional Brownian motion, which is analogous to the Karhunen-Loève expansion for genuine stochastic processes. In the last part we apply our results to study the abstract stochastic Cauchy problem in a Banach space driven by cylindrical fractional Brownian motion.

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Stochastic integration in UMD Banach spaces

Cited by 204 publications

References 27 publications

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

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

Conical square function estimates in UMD Banach spaces and applications to H ∞-functional calculi

Cylindrical fractional Brownian motion in Banach spaces

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