  The article is devoted to a new proof of the expansion for iterated Ito
stochastic integrals with respect to the components of a multidimensional
Wiener process. The above expansion is based on Hermite polynomials and
generalized multiple Fourier series in arbitrary complete orthonormal systems
of functions in a Hilbert space. In 2006, the author obtained a similar
expansion, but with a lesser degree of generality. Namely, for the case of
continuous or piecewise continuous complete orthonornal systems of functions in
a Hilbert space. In this article, the author generalizes the expansion of
iterated Ito stochastic integrals obtained by him in 2006 to the case of an
arbitrary complete orthonormal systems of functions in a Hilbert space using a
new approach based on the Ito formula. The obtained expansion of iterated Ito
stochastic integrals is useful for constructing of high-order strong numerical
methods for systems of Ito stochastic differential equations with
multidimensional non-commutative noise.
