The covariation for Banach space valued processes and applications

Cristina Di Girolami (1, 2) , Giorgio Fabbri (3) , Francesco Russo (4, 5)

1 LMM - Laboratoire Manceau de Mathématiques
2 DEA - Dipartimento di Economia Aziendale
3 EPEE - Centre d'Etudes des Politiques Economiques
4 UMA - Unité de Mathématiques Appliquées
5 OC - Optimisation et commande

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.

Mots clés: Quadratic variation; Itô formula; Dirichlet processes; Kolmogorov equation; Calculus via regularization; Kolmogorov equation.; Infinite dimensional analysis; Tensor analysis; Clark-Ocone formula; Stochastic partial differential equations

Domaines: Probabilités [math.PR]

Cristina Di Girolami, Giorgio Fabbri, Francesco Russo. The covariation for Banach space valued processes and applications. Metrika, 2014, 77 (1), pp.51-104. ⟨10.1007/s00184-013-0472-6⟩. ⟨hal-00780430v2⟩