Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Anticipative stochastic integration based on time-space chaos

Auteur(s):

Code(s) de Classification MSC:

• 60G99 None of the above but in this section
• 60H05 Stochastic integrals
Résumé: We use the concept of time-space chaos (see Peccati (2001a,b and 2002a,b)) to write an orthogonal decomposition of the space of square integrable functionals of a standard Brownian motion $X$ on $\left[ 0,1\right]$, say $% L^{2}\left( X\right)$, yielding an isomorphism between $L^{2}\left( X\right)$ and a semi symmetric'' Fock space over a class of deterministic functions. This allows to define a derivative operator on $L^{2}\left( X\right)$, whose adjoint is an anticipative stochastic integral with respect to $X$, that we name \textit{time-space Skorohod integral}. We show that the domain of such an integral operator contains the class of progressively measurable stochastic processes, and that time-space Skorohod integrals coincide with It\^{o} integrals on this set. We show that there exist stochastic processes for which a time-space Skorohod integral is well defined, even if they are not integrable in the usual Skorohod sense (see Skorohod (1976)). Several examples are discussed in detail.