Metrika

, Volume 77, Issue 1, pp 51–104 | Cite as

The covariation for Banach space valued processes and applications

  • Cristina Di Girolami
  • Giorgio Fabbri
  • Francesco Russo
Article

Abstract

This article focuses on a recent concept of covariation for processes taking values in a separable Banach space \(B\) and a corresponding quadratic variation. The latter is more general than the classical one of Métivier and Pellaumail. Those notions are associated with some subspace \(\chi \) of the dual of the projective tensor product of \(B\) with itself. We also introduce the notion of a convolution type process, which is a natural generalization of the Itô process and the concept of \(\bar{\nu }_0\)-semimartingale, which is a natural extension of the classical notion of semimartingale. The framework is the stochastic calculus via regularization in Banach spaces. Two main applications are mentioned: one related to Clark–Ocone formula for finite quadratic variation processes; the second one concerns the probabilistic representation of a Hilbert valued partial differential equation of Kolmogorov type.

Keywords

Calculus via regularization Infinite dimensional analysis Clark–Ocone formula Itô formula Quadratic variation Stochastic partial differential equations Kolmogorov equation 

Mathematics Subject Classification (2010)

60G22 60H05 60H07 60H15 60H30 26E20 35K90 46G05 

Notes

Acknowledgments

The research was supported by the ANR Project MASTERIE 2010 BLAN-0121-01. The second named author was partially supported by the Post-Doc Research Grant of Unicredit & Universities and his research has been developed in the framework of the center of excellence LABEX MME-DII (ANR-11-LABX-0023-01). The authors are grateful to two anonymous Referees for reading carefully the paper and helping us in improving its quality.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Cristina Di Girolami
    • 1
    • 2
  • Giorgio Fabbri
    • 3
  • Francesco Russo
    • 4
  1. 1.Dipartimento di Economia AziendaleUniversità G.D’Annunzio di PescaraPescaraItaly
  2. 2.Laboratoire Manceau de Mathématiques, Département de Mathématiques, Faculté des Sciences et TechniquesUniversité du MaineLe Mans Cedex 9France
  3. 3.Département d’EconomieEPEE, Université d’Evry-Val-d’Essonne (TEPP, FR-CNRS 3126)Evry cedexFrance
  4. 4.ENSTA ParisTech, Unité de Mathématiques appliquéesPalaiseauFrance

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