title: Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)
metadata.name: dc.contributor.author
metadata.value: Kuznetsov, Dmitriy Feliksovich
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2023-04-11T11:16:18Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2023-04-11T11:16:18Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2023-03
metadata.lang: -
metadata.name: dc.identifier.other
metadata.value: http://doi.org/10.21638/11701/spbu35.2023.110
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/41283
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: This is the third edition of the monograph (first edition 2020, second edition 2021) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the first time we use the generalized multiple Fourier series converging in the sense of norm in Hilbert space for the expansion of iterated Ito stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process (Chapter 1). Sections 1.11-1.13 (Chapter 1) are new and generalize the results of Chapter 1 obtained earlier by the author and are also closely related to the multiple Wiener stochastic integral introduced by Ito in 1951. The convergence with probability 1 as well as the convergence in the sense of n-th (n=2, 3,...) moment for the expansion of iterated Ito stochastic integrals have been proved (Chapter 1). Moreover, the rate of both types of convergence has been established. The main difference between the third and second editions of the book is that the third edition includes original material (Chapter 2, Sections 2.10-2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process. The above approach allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. In addition, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. These results (Chapter 2) adapt the results of Chapter 1 for iterated Stratonovich stochastic integrals. Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). These approximations were obtained using Legendre polynomials and trigonometric functions. The methods constructed in the book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity k) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: en
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Differential Equations and Control Processes;Issue 1
metadata.lang: -
metadata.name: dc.subject
metadata.value: approximation in the sense of n-th moment
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: approximation with probability 1
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: convergence in the sense of norm in Hilbert space
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: expansion
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: exponential Milstein scheme
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: exponential Wagner-Platen scheme
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: generalized iterated Fourier series
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: generalized multiple Fourier series
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: high-order strong numerical method
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: Hilbert-Schmidt operator
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: infinite-dimensional Q-Wiener process
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: iterated Ito stochastic integral
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: iterated Stratonovich stochastic integral
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: Ito stochastic differential equation
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: Legendre polynomials
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: mean-square approximation
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: multi-dimensional Wiener process
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: multiple Fourier-Legendre series
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: multiple trigonometric Fourier series
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: non-commutative semilinear stochastic partial differential equation
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: nonlinear multiplicative trace class noise
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: Parseval equality
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: stochastic Ito-Taylor expansion
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: stochastic Stratonovich-Taylor expansion
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: trace class operator
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Book
metadata.lang: en_GB
resources.filename: 151-1097.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/41283/1/151-1097.pdf
resources.format: Adobe PDF
resources.size: 5,33 MB