The covariation for Banach space valued processes and applications
AbstractThis article focuses on a new concept of quadratic variation for processes taking values in a Banach space B and a corresponding covariation. This is more general than the classical one of Métivier and Pellaumail. Those notions are associated with some subspace ? 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 ¯V0-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. 2010 Math Subject Classification: 60G22, 60H05, 60H07, 60H15, 60H30, 26E20, 35K90 46G05
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Bibliographic InfoPaper provided by Centre d'Études des Politiques Économiques (EPEE), Université d'Evry Val d'Essonne in its series Documents de recherche with number 13-01.
Length: 45 pages
Date of creation: Jan 2013
Date of revision:
Calculus via regularization; Infinite dimensional analysis; Tensor analysis; Clark-Ocone formula; Dirichlet processes; Itô formula; Quadratic variation; Stochastic partial differential equations; Kolmogorov equation;
Other versions of this item:
- Cristina Girolami & Giorgio Fabbri & Francesco Russo, 2014. "The covariation for Banach space valued processes and applications," Metrika, Springer, vol. 77(1), pages 51-104, January.
This paper has been announced in the following NEP Reports:
- NEP-ALL-2013-04-13 (All new papers)
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- Giorgio FABBRI & Francesco RUSSO, 2012. "Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control," Discussion Papers (IRES - Institut de Recherches Economiques et Sociales) 2012017, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
- Gozzi, Fausto & Russo, Francesco, 2006. "Weak Dirichlet processes with a stochastic control perspective," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1563-1583, November.
- Christian Bender & Tommi Sottinen & Esko Valkeila, 2008. "Pricing by hedging and no-arbitrage beyond semimartingales," Finance and Stochastics, Springer, vol. 12(4), pages 441-468, October.
- Russo, Francesco & Tudor, Ciprian A., 2006. "On bifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 830-856, May.
- Rosanna Coviello & Cristina Di Girolami & Francesco Russo, 2011. "On stochastic calculus related to financial assets without semimartingales," Papers 1102.2050, arXiv.org.
- Errami, Mohammed & Russo, Francesco, 2003. "n-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes," Stochastic Processes and their Applications, Elsevier, vol. 104(2), pages 259-299, April.
- Gozzi, Fausto & Russo, Francesco, 2006. "Verification theorems for stochastic optimal control problems via a time dependent Fukushima-Dirichlet decomposition," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1530-1562, November.
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