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

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Anticipative stochastic integration based on time-space chaos


Code(s) de Classification MSC:

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