Exact Calculation of the MeanSquare Error in the Method of Expansion of Iterated Ito Stochastic integrals Based on Generalized Multiple Fourier Series
Abstract
The article is devoted to the developement of the method of expansion and meansquare 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 meansquare error of approximation of iterated Ito stochastic integrals of multiplicity $k$ ($k\in\mathbb{N}$) from the stochastic TaylorIto expansion in the framework of the mentioned method. 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 of the method of expansion of iterated Ito stochastic integrals based on generalized multiple Fourier series for the cases of multiple FourierLegendre series and multiple trigonometric Fourier series. Meansquare approximation of iterated Stratonovich stochastic integrals is also considered in the article. The results of the article can be applied to the highorder strong numerical methods for Ito stochastic differential equations as well as noncommutative semilinear stochastic partial differential equations with multiplicative trace class noise (in accordance with the meansquare criterion of convergence).
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 arXiv:
 arXiv:1801.01079
 Bibcode:
 2018arXiv180101079K
 Keywords:

 Mathematics  Probability
 EPrint:
 57 pages. Minor changes. arXiv admin note: substantial text overlap with arXiv:1712.08991, arXiv:1712.09746, arXiv:1801.00231