Anticipative stochastic integration based on time-space chaos

Abstract

We use the concept of time-space chaos (see Peccati (Ann. Inst. Poincaré 37(5) (2001) 607; Prépublication n. 648 du Laboratoire de Probabilités et Modèles Aléatoires de l'Université Paris VI; Chaos Brownien d'espace-temps, décompositions de Hoeffding et problèmes de convergence associés, Ph.D. Thesis, Université Paris IV, 2002; Bernoulli 9(1) (2003) 25)) to write an orthogonal decomposition of the space of square integrable functionals of a standard Brownian motion X on [0,1], say L 2( X), yielding an isomorphism between L 2( X) and a “semi-symmetric” Fock space over a class of deterministic functions. This allows to define a derivative operator on L 2( X), whose adjoint is an anticipative stochastic integral with respect to X, that we name 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 Itô 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 (Theory Probab. Appl. 20 (1975) 219)). Several examples are discussed in detail.