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The book is devoted to the strong approximation of iterated stochasticintegrals (ISIs) in the context of numerical integration of Ito SDEs andnon-commutative semilinear SPDEs with nonlinear multiplicative trace classnoise. The monograph opens up a new direction in researching of ISIs. For thefirst time we successfully use the generalized multiple Fourier seriesconverging in the sense of norm in the space $L_2([t, T]^k)$ for the expansionand strong approximation of Ito ISIs of multiplicity $k,$ $k\in{\bf N}$(Chapter 1). This result is adapted for iterated Stratonovich stochasticintegrals of multiplicities 1 to 6 (the case of continuously differentiableweight functions and a complete orthonormal system of Legendre polynomials ortrigonometric functions in $L_2([t, T])$) and for iterated Stratonovichstochastic integrals of multiplicities 1 to 4 (the case of weight functionsidentically equal to 1 and an arbitrary complete orthonormal system offunctions in $L_2([t, T])$) (Chapter 2). The mentioned adaptation has also beencarried out for iterated Stratonovich stochastic integrals of multiplicity $k$$(k\in{\bf N})$ under the condition of convergence of trace series (Chapter 2).We derived the exact and approximate expressions for the mean-square error ofapproximation of Ito ISIs of multiplicity $k$, $k\in{\bf N}$ (Chapter 1).Furthermore, we provided a significant practical material (Chapter 5) devotedto the expansions of specific Ito and Stratonovich ISIs of multiplicities 1 to6 from the Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4) using thesystem of Legendre polynomials and the system of trigonometric functions. Themethods formulated in this book have been compared with some existing methods(Chapter 6). The results of Chapter 1 were applied (Chapter 7) to theapproximation of ISIs of multiplicity $k,$ $k\in{\bf N}$ with respect to the$Q$-Wiener process.

Strong Approximation of Iterated Ito and Stratonovich Stochastic  Integrals Based on Generalized Multiple Fourier Series. Application to  Numerical Solution of Ito SDEs and Semilinear SPDEs