  The article is devoted to the developement of the method of expansion and
mean-square approximation of iterated Ito stochastic integrals based on
generalized multiple Fourier series converging in the sense of norm in the
space $L_2([t, T]^k)$ ($k$ is the multiplicity of the iterated Ito stochastic
integral). We obtain the exact and approximate expressions for the mean-square
approximation error of iterated Ito stochastic integrals of multiplicity $k$
($k\in\mathbb{N}$) from the stochastic Taylor-Ito expansion. As a result, we do
not need to use redundant terms of expansions of iterated Ito stochastic
integrals that complicate the numerical methods for Ito stochastic differential
equations. Moreover, we proved the convergence with propability 1 for the
method of expansion of iterated Ito stochastic integrals based on generalized
multiple Fourier series for the cases of multiple Fourier-Legendre series and
multiple trigonometric Fourier series. Mean-square approximation of iterated
Stratonovich stochastic integrals is also considered in the article. The
results of the article can be applied to the high-order strong numerical
methods for Ito stochastic differential equations as well as for
non-commutative semilinear stochastic partial differential equations with
multiplicative trace class noise.
