Abstract
Malliavin calculus is developed for each measure on ℝℕ, which is the product measure derived from an arbitrary Borel probability measure μ1 on ℝ. Each square integrable functional on ℝℕ can be expanded into an orthogonal series of multiple integrals. The integrators are martingales, whose increments are orthogonal polynomials of E, where E is a Borel measurable bijection from ℝ onto ℝ such that . Based on this chaos decomposition result, we introduce the Malliavin derivative, the Itô 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–Fréchet space over l 2. Moreover, measures for which polynomials are not integrable and non-smooth measures are included.
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