Malliavin calculus for product measures on R{sup}N based on chaos

摘要

Malliavin calculus is developed for each measure on R{sup}N, which is the product measure derived from an arbitrary Borel probability measure μ{sup}1 on R. Each square integrable functional on R{sup}N can be expanded into an orthogonal series of multiple integrals. The integrators are martingales, whose increments are orthogonal polynomials of F, where F is a Borel measurable bijection from R onto R such that E{sub}(μ{sup}1)e{sup}|F| < ∞, Based on this chaos decomposition result, we introduce the Malliavin derivative, the Ito integral, the Skorohod integral and prove the Clark-Ocone formula. Our approach includes Malliavin calculus on the classical Poisson space and on any abstract Wiener - Predict space over l{sub}2 Moreover, measures for which polynomials are not integrable and non-smooth measures are included.