ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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

Author(s):

Dmitriy Feliksovich Kuznetsov

Peter the Great Saint-Petersburg Polytechnic University
Department of Higher Mathematics
Dr. Sc., Professor

sde_kuznetsov@inbox.ru

Abstract:

The book is devoted to the strong approximation of iterated stochastic integrals in the context of numerical integration of Ito stochastic differential equations and non-commutative semilinear stochastic partial differential equations with nonlinear multiplicative trace class noise. The presented monograph open a new direction in researching of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. For the first time we successfully use the generalized multiple Fourier series (multiple Fourier-Legendre series as well as multiple trigonometric Fourier series) converging in the sense of norm in Hilbert space for the expansion and mean-square approximation of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1). The convergence with probability 1 as well as the convergence in the sense of n-th moment for the mentioned expansion have been proved (n=2, 3,...). Moreover, the expansion for iterated Ito stochastic integrals is adapted for iterated Stratonovich stochastic integrals of multiplicities 1 to 5 (Chapter 2) as well as for some other types of iterated stochastic integrals (Chapter 1). 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 integration order replacement technique for the class of iterated Ito stochastic integrals has been introduced (Chapter 3). We derived the exact and approximate expressions for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and 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) using the system of Legendre polynomials and the system 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 iterated stochastic integrals of arbitrary multiplicity k with respect to the infinite-dimensional Q-Wiener process.

Keywords

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