The covariation for Banach space valued processes and applications

Cristina Girolami (), Giorgio Fabbri and Francesco Russo ()

Metrika: International Journal for Theoretical and Applied Statistics, 2014, vol. 77, issue 1, 104 pages

Abstract: This article focuses on a recent concept of covariation for processes taking values in a separable Banach space $$B$$ 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 $$\chi $$ χ of the dual of the projective tensor product of $$B$$ 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 $$\bar{\nu }_0$$ ν ¯ 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. Copyright Springer-Verlag Berlin Heidelberg 2014

Keywords: Calculus via regularization; Infinite dimensional analysis; Clark–Ocone formula; Itô formula; Quadratic variation; Stochastic partial differential equations; Kolmogorov equation; 60G22; 60H05; 60H07; 60H15; 60H30; 26E20; 35K90; 46G05 (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (3)

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Working Paper: The covariation for Banach space valued processes and applications (2013)
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DOI: 10.1007/s00184-013-0472-6

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