The covariation for Banach space valued processes and applications.

january, 2014
Publication type:
Paper in peer-reviewed journals
Journal:
Metrika, vol. 77, pp. 51-104
arXiv:
images/icons/icon_arxiv.png 1301.5715
Keywords :
Calculus via regularization; Infinite dimensional analysis;
 Tensor analysis; Clark-Ocone formula;
Dirichlet processes; It\^o formula; Quadratic variation;
Stochastic partial differential equations; Kolmogorov equation.
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 Métivier 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 Itô 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.
BibTeX:
@article{DiG-Fab-Rus-2014,
    author={Cristina Di Girolami and Giorgio Fabbri and Francesco Russo },
    title={The covariation for Banach space valued processes and 
           applications. },
    doi={10.1007/s00184-013-0472-6 },
    journal={Metrika },
    year={2014 },
    month={1},
    volume={77 },
    pages={51--104},
}