the Wiener-Ito integral
/ VEE-ner EE-toh /
The Wiener-Ito integral is the concrete realization of each Wiener chaos: it integrates a deterministic kernel against n copies of white noise to produce a random variable sitting exactly in the n-th chaos. For n = 1 it is the ordinary Wiener integral integral f dB of a deterministic function; for general n it is the multiple stochastic integral that builds the higher layers of the chaos decomposition. It is how you write a chaos element down as an integral instead of an abstract Hermite combination.
Single (n = 1) integral: for f in L^2(R_+), define I_1(f) = integral f(t) dB_t, a centered Gaussian with E[I_1(f)^2] = integral f(t)^2 dt (the Ito/Wiener isometry); this is the first chaos. Multiple (n-fold) integral: for a symmetric kernel f in L^2((R_+)^n), I_n(f) = n! times the iterated integral integral_(t_1 < t_2 < ... < t_n) f dB_(t_1) ... dB_(t_n) over the increasing simplex, equivalently the integral of f over all of (R_+)^n with the diagonals removed. The key facts are the isometry E[I_n(f) I_n(g)] = n! < f_sym, g_sym >_(L^2) and orthogonality across orders, E[I_m(f) I_n(g)] = 0 for m != n. The bridge to Hermite polynomials: if ||h|| = 1 then I_n(h tensor ... tensor h) = H_n(I_1(h)), so multiple integrals of tensor powers are exactly Hermite polynomials of first-chaos variables. One must exclude the diagonals {t_i = t_j}; integrating over them would reintroduce the (dt) terms and break the isometry — this is the multiple-integral shadow of Ito's correction.
Why it matters: the map f -> I_n(f) is a (rescaled) isometry from symmetric L^2 kernels onto the n-th chaos, so the whole chaos decomposition becomes 'F = sum I_n(f_n) for a unique sequence of symmetric kernels f_n.' This is the representation behind Malliavin calculus, the martingale representation theorem (every L^2 Brownian functional is a single Ito integral of a predictable integrand — its first-order Clark-Ocone term), and quantitative CLTs for functionals living in a fixed chaos. Caveat: it is genuinely n-LINEAR and the diagonals must be removed; naively raising integral f dB to a power does NOT give I_n.
The double integral I_2(f tensor f) with ||f|| = 1 equals (integral f dB)^2 - integral f^2 dt = Z^2 - 1 = H_2(Z), where Z = integral f dB is standard normal. The subtraction of integral f^2 dt is exactly the removal of the diagonal {t_1 = t_2}; without it you would get Z^2, which is not in the 2nd chaos because it has a nonzero mean.
Removing the diagonal turns Z^2 into Z^2 - 1 = H_2(Z): the multiple Wiener integral lands precisely in the second chaos.
Multiple Wiener integrals require deleting the diagonals; integrating over them reintroduces Ito's (dt) correction, so a power of integral f dB is NOT the same as the n-fold integral.