The Wiener—Itô Chaos Expansion

The celebrated Wiener-Itô chaos expansion is fundamental in stochastic analysis. In particular, it plays a crucial role in the Malliavin calculus as it is presented in the sequel. This result which concerns the representation of square integrable random variables in terms of an infinite orthogonal sum was proved in its first version by Wiener in 1938 [227]. Later, in 1951, Itô [120] showed that the expansion could be expressed in terms of iterated Itô integrals in the Wiener space setting. Before we state the theorem we introduce some useful notation and give some auxiliary results.