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)
arXiv:
images/icons/icon_arxiv.png 1105.4419
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) },
}