  This 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 M\'etivier 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 It\^o 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.
