Trotter-Kato Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications...

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T. E. Govindan

Trotter-Kato Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications

T. E. Govindan

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English

Springer International Publishing AG
02 July 2024

Cybernetics & systems theory; Calculus & mathematical analysis; Differential calculus & equations; Probability & statistics; Stochastics

This is the first comprehensive book on Trotter-Kato approximations of stochastic differential equations (SDEs) in infinite dimensions and applications. This research monograph brings together the varied literature on this topic since 1985 when such a study was initiated. The author provides a clear and systematic introduction to the theory of Trotter-Kato approximations of SDEs and also presents its applications to practical topics such as stochastic stability and stochastic optimal control. The theory assimilated here is developed slowly and methodically in digestive pieces.

The book begins with a motivational chapter introducing several different models that highlight the importance of the theory on abstract SDEs that will be considered in the subsequent chapters. The author next introduces the necessary mathematical background and then leads the reader into the main discussion of the monograph, namely, the Trotter-Kato approximations of many classes of SDEs in Hilbert spaces, Trotter-Kato approximations of SDEs in UMD Banach spaces and some of their applications. Most of the results presented in the main chapters appear for the first time in a book form. The monograph also contains many illustrative examples on stochastic partial differential equations and one in finance as an application of the Trotter-Kato formula. The key steps are included in all proofs which will help the reader to get a real insight into the theory of Trotter-Kato approximations and its use.

This book is intended for researchers and graduate students in mathematics specializing in probability theory. It will also be useful to numerical analysts, engineers, physicists and practitioners who are interested in applying the theory of stochastic evolution equations. Since the approach is based mainly in semigroup theory, it is accessible to a wider audience including non-specialists in stochastic processes.

By: T. E. Govindan
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Edition: 2024 ed.
Dimensions: Height: 235mm, Width: 155mm,
ISBN: 9783031427909
ISBN 10: 3031427904
Pages: 311
Publication Date: 02 July 2024
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

1 Introduction and Motivating Examples.- 2 Mathematical Machinery.- 3 Trotter-Kato Approximations of Stochastic Differential Equations.- 4 Trotter-Kato Approximations of Stochastic Differential Equations in UMD Banach Spaces.- 6 Applications to Stochastic Optimal Control.- Appendix A: Nuclear and Hilbert-Schmidt Operators.- Appendix B: Convergence of Analytic Semigroups.- Appendix C: The Pettis Measurability Theorem.- Appendix D: R-Boundedness and γ-Boundedness.- Appendix E: The Feynman-Kac Formula.- Bibliographical Notes and Remarks.- Bibliography.

T. E. Govindan earned a Master’s degree in Statistics from the Indian Institute of Technology, Kanpur, India, followed by a Ph.D. in Mathematics in 1991 from the Indian Institute of Technology, Bombay, India. After serving the University of Bombay, India for over eight years, he moved to Mexico in 1999. Currently, he is a Professor of Mathematics at the National Polytechnic Institute in Mexico. He specializes in stochastic analysis, particularly on stochastic differential equations. He is the author of the book on Yosida approximations of stochastic differential equations in infinite dimensions and applications, Probability Theory and Stochastic Modelling, Vol. 79, Springer, 2016.