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

### Martingale structure of Skorohod integral processes

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