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 $$B$$ 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