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

Abstract

The book is devoted to the strong approximation of iterated stochastic integrals (ISIs) in the context of numerical integration of Ito SDEs and non-commutative semilinear SPDEs with nonlinear multiplicative trace class noise. The monograph opens up a new direction in researching of ISIs. For the first time we successfully use the generalized multiple Fourier series converging in the sense of norm in $L_2([t,T{]}^k)$ for the expansion and strong approximation of Ito ISIs of multiplicity $k$, $k\in N$ (Chapter $1$). This result has been adapted for Stratonovich ISIs of multiplicities $1$ to $8$ (Chapter $2$). We derived the exact and approximate expressions for the mean-square error of approximation of Ito ISIs of multiplicity $k,$ $k\in N$ (Chapter $1$). We provided a significant practical material (Chapter $5$) devoted to the expansions of specific Ito and Stratonovich ISIs of multiplicities $1$ to $6$ from the Taylor-Ito and Taylor-Stratonovich expansions (Chapter $4$) using the CONS of Legendre polynomials and the CONS of trigonometric functions. The methods formulated in this book have been compared with some existing methods (Chapter $6$). The results of Chapter $1$ were applied (Chapter $7$) to the approximation of ISIs of multiplicity $k$, $k\in N$ with respect to the Q-Wiener process. This book has many overlaps with the monograph: Dmitriy F. Kuznetsov, Mikhail D. Kuznetsov, Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB. 2026, 1599 pp., arXiv:2003.14184v80. Thus, both monographs are placed within a single submission, and their Internet links will differ only by the version numbers within arXiv:2003.14184.

1248 pp, title and running titles were corrected

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