Wiener-Itô decomposition

Itô–Wiener decomposition

An orthogonal decomposition of the Hilbert space of square-integrable functions on a Gaussian space. It was first proved in 1938 by N. Wiener [a6] in terms of homogeneous chaos (cf. also Wiener chaos decomposition). In 1951, K. Itô [a1] defined multiple Wiener integrals to interpret homogeneous chaos and gave a different proof of the decomposition theorem.

Take an abstract Wiener space [a3] (cf. also Wiener space, abstract). Let be the standard Gaussian measure on . The abstract version of Wiener–Itô decomposition deals with a special orthogonal decomposition of the real Hilbert space .

Each defines a normal random variable on with mean and variance [a3]. Let . For , let be the -closure of the linear space spanned by and random variables of the form with and for . Then is an increasing sequence of closed subspaces of . Let and, for , let be the orthogonal complement of in . The elements in are called homogeneous chaos of degree . Obviously, the spaces are orthogonal. Moreover, the Hilbert space is the direct sum of for , namely, .

Fix . To describe more precisely, let be the orthogonal projection of onto the space . For , define

Then (where denotes the symmetric tensor product) and

Thus, extends by continuity to a continuous linear operator from into and is an isometric mapping (up to the constant ) from into . Actually, is surjective and so for any , there exists a unique such that and . Therefore, for any , there exists a unique sequence with such that

This is the abstract version of the Wiener–Itô decomposition theorem [a2], [a4], [a5].

Let . Define a norm on by

The Hilbert space is called the Fock space of (cf. also Fock space). The spaces and are isomorphic under the unitary operator defined by

Let be an orthonormal basis (cf. also Orthogonal basis) for . For any non-negative integers such that , define

where is the Hermite polynomial of degree (cf. also Hermite polynomials). The set is an orthonormal basis for the space of homogeneous chaos of degree . Hence the set forms an orthonormal basis for .

Consider the classical Wiener space [a3]. The Hilbert space is isomorphic to under the unitary operator , . The standard Gaussian measure on is the Wiener measure and , , is a Brownian motion. For , the random variable is exactly the Wiener integral . Let , . The random variable

is a homogeneous chaos in the space . The mapping extends by continuity to the space . For ,

where the right-hand side is a multiple Wiener integral of order as defined by Itô in [a1] and (where is the symmetrization of .) For any there exists a unique sequence of symmetric functions such that

This is the Wiener–Itô decomposition theorem in terms of multiple Wiener integrals. An orthonormal basis for is given by the set

where is an orthonormal basis for and the integrals are Wiener integrals.

References

[a1]	K. Itô, "Multiple Wiener integral" J. Math. Soc. Japan , 3 (1951) pp. 157–169
[a2]	G. Kallianpur, "Stochastic filtering theory" , Springer (1980)
[a3]	H.-H. Kuo, "Gaussian measures in Banach spaces" , Lecture Notes in Mathematics , 463 , Springer (1975)
[a4]	H.-H. Kuo, "White noise distribution theory" , CRC (1996)
[a5]	N. Obata, "White noise calculus and Fock space" , Lecture Notes in Mathematics , 1577 , Springer (1994)
[a6]	N. Wiener, "The homogeneous chaos" Amer. J. Math. , 60 (1938) pp. 897–936