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
Download:
HAL:
arXiv:
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}, }