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The article is devoted to the developement of the method of expansion andmean-square approximation of iterated Ito stochastic integrals based ongeneralized multiple Fourier series converging in the sense of norm in thespace $L_2([t, T]^k)$ ($k$ is the multiplicity of the iterated Ito stochasticintegral). We obtain the exact and approximate expressions for the mean-squareapproximation error of iterated Ito stochastic integrals of multiplicity $k$($k\in\mathbb{N}$) from the stochastic Taylor-Ito expansion. As a result, we donot need to use redundant terms of expansions of iterated Ito stochasticintegrals that complicate the numerical methods for Ito stochastic differentialequations. Moreover, we proved the convergence with propability 1 for themethod of expansion of iterated Ito stochastic integrals based on generalizedmultiple Fourier series for the cases of multiple Fourier-Legendre series andmultiple trigonometric Fourier series. Mean-square approximation of iteratedStratonovich stochastic integrals is also considered in the article. Theresults of the article can be applied to the high-order strong numericalmethods for Ito stochastic differential equations as well as fornon-commutative semilinear stochastic partial differential equations withmultiplicative trace class noise.

Exact Calculation of the Mean-Square Error in the Method of Expansion of  Iterated Ito Stochastic integrals Based on Generalized Multiple Fourier  Series