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The basis of this book is composed on the monographs: 1. Kuznetsov D.F. Strong Approximation of Multiple Ito and Stratonovich Stochastic Integrals. Multiple Fourier series approach. St.-Petersburg Polytechnical University Press, St.-Petersburg. 2011. 282 p. (in English); 2. Dmitriy Kuznetsov. Approximation of Multiple Ito and Stratonovich Stochastic Integrals. Multiple Fourier series approach. AV Akademikerverlag, Saarbrucken. 2012, 300 p. (In English), which are the first monographs where the problem of strong (mean-square) approximation of multiple Ito and Stratonovich stochastic integrals is sistematically analyzed in the context of numerical integration of stochastic differential Ito equations. The presented book and mentioned monographs successfully use the tool of multiple and iterative Fourier series, built in the space L2 and pointwise, for the strong approximation of multiple stochastic integrals and open a new direction in researching of multiple Ito and Stratonovich stochastic integrals. We obtained a general result connected with expansion of multiple Ito stochastic integrals of any fixed multiplicity k, based on generalized multiple Fourier series converging in the space L2([t, T]x ... x [t,T]) (x .... x -- k-1 times). This result is adapted for multiple Stratonovich stochastic integrals of 1 -- 4 multiplicity for Legendre polynomial system and system of trigonometric functions, as well as for other types of multiple stochastic integrals. The theorem on expansion of multiple Stratonovich stochastic integrals with any fixed multiplicity k, based on generalized Fourier series converging pointwise is verified.
We obtained exact expressions for mean-square errors of approximation of multiple Ito stochastic integrals of 1 -- 4 multiplicity. We provided a significant practical material devoted to approximation of specific multiple Ito and Stratonovich stochastic integrals of 1 -- 5 multiplicity using the system of Legendre polynomials and the system of trigonometric functions. We compared the methods formulated in this book with existing methods. We consider some weak approximations of multiple Ito stochastic integrals. We proved the theorems about integration order replacement for multiple Ito stochastic integrals and for the multiple stochastic integrals according to martingale. We brought out two families of analytical formulas for calculation of stochastic integrals. This book will be interesting for specialists dealing with the theory of stochastic processes, applied and computational mathematics, senior students and postgraduates of technical institutes and universities, as well as for computer experts.
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