Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60G99 None of the above but in this section
- 60H05 Stochastic integrals
- 60J65 Brownian motion, See also {58G32}

**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.

**Mots Clés:** *Time-space chaos ; Anticipative stochastic integration ; Malliavin operators*

**Date:** 2002-10-18

**Prépublication numéro:** *PMA-763*