Brownian excursions, stochastic integrals, and representation of Wiener functionals

Abstract

A stochastic calculus similar to Malliavin's calculus is worked out for Brownian excursions. The analogue of the Malliavin derivative in this calculus is not a differential operator, but its adjoint is (like the Skorohod integral) an extension of the Itô integral. As an application, we obtain an expression for the integrand in the stochastic integral representation of square integrable Wiener functionals; this expression is an alternative to the classical Clark-Ocone formula. Moreover, this calculus enables to construct stochastic integrals of predictable or anticipating processes (forward, backward and symmetric integrals are considered).