The covariation for Banach space valued processes and applications

TL;DR

This paper introduces a generalized quadratic variation concept for Banach space valued processes, extending classical notions, and applies it to stochastic calculus, Clark-Ocone formulas, and PDE representations.

Contribution

It develops a broader framework for quadratic variation in Banach spaces and introduces convolution type processes and $ar u_0$-semimartingales, extending stochastic calculus tools.

Findings

Generalized quadratic variation for Banach space processes

Application to Clark-Ocone formula for finite quadratic variation processes

Probabilistic representation of Hilbert space PDEs

Abstract

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.