The covariation for Banach space valued processes and applications

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 χ 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 ν̄0-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