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

CNRS U.M.R. 7599
| ||

``Probabilités et Modèles Aléatoires''
| ||

**Auteur(s): **

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

- 60H05 Stochastic integrals
- 60H07 Stochastic calculus of variations and the Malliavin calculus

**Résumé:** Let the process $\left\{ Y_{t}, t\in[0,1] \right\}$, have the form $Y_{t}=\delta \left( u%
\mathbf{1}_{\left[ 0,t\right] }\right) $, where $\delta $ stands for a
Skorohod integral with respect to Brownian motion, and $u$ is a measurable
process verifying some suitable regularity conditions. We use a recent
result by Tudor (2004), to prove that $Y_{t}$ can be represented as the
limit of linear combinations of processes that are products of forward and
backward Brownian martingales. Such a result is a further step towards the
connection between the theory of continuous-time (semi)martingales, and that
of anticipating stochastic integration. We establish an explicit link
between our results and the classic characterization, due to Duc and Nualart
(1990), of the chaotic decomposition of Skorohod integral processes. We also
explore the case of Skorohod integral processes that are time-reversed
Brownian martingales, and provide an \textquotedblleft
anticipating\textquotedblright\ counterpart to the classic Optional Sampling
Theorem for It\^{o} stochastic integrals.

**Mots Clés:** *Malliavin calculus ; Anticipating stochastic integration ; Martingale theory ; Stopping times*

**Date:** 2004-06-09

**Prépublication numéro:** *PMA-919*