# Generalized covariation and extended Fukushima decompositions for Banach space valued processes. Application to windows of Dirichlet processes.

june, 2012

Publication type:

Paper in peer-reviewed journals

Journal:

Infinite Dimensional Analysis, Quantum Probability and Related Topics (IDA-QP)., vol. 15(2)

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Keywords :

Covariation and quadratic variation; calculus via regularization; infinite-dimensional analysis; tensor analysis; Dirichlet processes; representation of Path-dependent random variables; Malliavin calculus; generalized Fukushima decomposition.

Abstract:

This paper concerns a class of Banach valued processes which
have finite quadratic variation. The notion introduced
here generalizes the classical one, of Métivier and
Pellaumail which is quite restrictive.
We make use of the notion
of $\chi$-covariation which is a generalized notion of covariation
for processes with values in two Banach spaces $B_{1}$ and $B_{2}$.
$\chi$ refers to a suitable subspace of the dual of the projective
tensor product of $B_{1}$ and $B_{2}$.
We investigate some $C^{1}$ type transformations for various classes of
stochastic processes admitting a $\chi$-quadratic variation and
related properties. If $\X^1$ and $\X^2$ admit a $\chi$-covariation,
$F^i: B_i \rightarrow \R$, $i = 1, 2$ are of class
$C^1$ with some supplementary assumptions then the covariation of
the real processes
$F^1(\X^1)$ and $F^2(\X^2)$ exist. \\
A detailed analysis will be devoted
to the so-called window processes.
Let $X$ be a real continuous process;
the $C([-\tau,0])$-valued process $X(\cdot)$ defined by
$X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$, is called {\it window} process.
Special attention is given to transformations of
window processes associated with
Dirichlet and weak Dirichlet processes.
This will constitute a significant Fukushima decomposition
for functionals of windows of (weak) Dirichlet processes.
As applications, we give a new technique for representing
path-dependent random variables.

BibTeX:

@article{DiG-Rus-2012, author={Cristina Di Girolami and Francesco Russo }, title={Generalized covariation and extended Fukushima decompositions for Banach space valued processes. Application to windows of Dirichlet processes. }, doi={10.1142/S0219025712500075 }, journal={Infinite Dimensional Analysis, Quantum Probability and Related Topics (IDA-QP). }, year={2012 }, month={6}, volume={15(2) }, }