Overview

Stochastic differential equations have many applications in the natural sciences. Besides, the employment of probabilistic representations together with the Monte Carlo technique allows us to reduce solution of multi-dimensional problems for partial differential equations to integration of stochastic equations. This approach leads to powerful computational mathematics that is presented in the treatise. The authors propose many new special schemes, some published here for the first time. In the second part of the book they construct numerical methods for solving complicated problems for partial differential equations occurring in practical applications, both linear and nonlinear. All the methods are presented with proofs and hence founded on rigorous reasoning, thus giving the book textbook potential. An overwhelming majority of the methods are accompanied by the corresponding numerical algorithms which are ready for implementation in practice. The book addresses researchers and graduate students in numerical analysis, physics, chemistry, and engineering as well as mathematical biology and financial mathematics.

Full Product Details

Author:   Grigori Noah Milstein ,  Michael V. Tretyakov
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   Softcover reprint of hardcover 1st ed. 2004
Dimensions:   Width: 15.50cm , Height: 3.10cm , Length: 23.50cm
Weight:   0.937kg
ISBN:  

9783642059308

ISBN 10:   3642059309
Pages:   596
Publication Date:   01 December 2010
Audience:   Professional and scholarly ,  Professional & Vocational
Replaced By:   9783030820398
Format:   Paperback
Publisher's Status:   Out of Print
Availability:   In Print  
Limited stock is available. It will be ordered for you and shipped pending supplier's limited stock.

Table of Contents

1 Mean-square approximation for stochastic differential equations.- 2 Weak approximation for stochastic differential equations.- 3 Numerical methods for SDEs with small noise.- 4 Stochastic Hamiltonian systems and Langevin-type equations.- 5 Simulation of space and space-time bounded diffusions.- 6 Random walks for linear boundary value problems.- 7 Probabilistic approach to numerical solution of the Cauchy problem for nonlinear parabolic equations.- 8 Numerical solution of the nonlinear Dirichlet and Neumann problems based on the probabilistic approach.- 9 Application of stochastic numerics to models with stochastic resonance and to Brownian ratchets.- A Appendix: Practical guidance to implementation of the stochastic numerical methods.- A.1 Mean-square methods.- A.2 Weak methods and the Monte Carlo technique.- A.3 Algorithms for bounded diffusions.- A.4 Random walks for linear boundary value problems.- A.5 Nonlinear PDEs.- A.6 Miscellaneous.- References.

Reviews

From the reviews of the first edition: Milstein and Tretyakov's book is a significant contribution to stochastic numerics. It is essential reading for anyone with serious interest in the field, either theoretical or practical. (Mathematical Reviews, 2005) The monograph presents research results of the authors concerning the numerical treatment of stochastic differential equations. ... The book is written in the style of a research exposition. It provides a rich source of mathematical theory, examples and insightful remarks and is thus essential material for those who are interested in the subject, from both a theoretical and practical viewpoint. (Evelyn Buckwar, Zentralblatt MATH, Vol. 1085, 2006)

From the reviews of the first edition: Milstein and Tretyakov's book is a significant contribution to stochastic numerics. It is essential reading for anyone with serious interest in the field, either theoretical or practical. (Mathematical Reviews, 2005) The monograph presents research results of the authors concerning the numerical treatment of stochastic differential equations. ! The book is written in the style of a research exposition. It provides a rich source of mathematical theory, examples and insightful remarks and is thus essential material for those who are interested in the subject, from both a theoretical and practical viewpoint. (Evelyn Buckwar, Zentralblatt MATH, Vol. 1085, 2006)

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