SpringerBriefs in Mathematics

Equations Involving Malliavin Calculus Operators

Applications and Numerical Approximation

Authors: Levajković, Tijana, Mena, Hermann

  • Friendly approach for solving stochastic equations with singular data
  • Novel applications of operators of the Malliavin calculus
  • From theoretical to numerical results of SPDEs with singular data
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eBook $39.99
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  • ISBN 978-3-319-65678-6
  • Digitally watermarked, DRM-free
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Softcover $54.99
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About this book

This book provides a comprehensive and unified introduction to stochastic differential equations and related optimal control problems. The material is new and the presentation is reader-friendly. A major contribution of the book is the development of generalized Malliavin calculus in the framework of white noise analysis, based on chaos expansion representation of stochastic processes and its application for solving several classes of stochastic differential equations with singular data involving the main operators of Malliavin calculus. In addition, applications in optimal control and numerical approximations are discussed. 

The book is divided into four chapters. The first, entitled White Noise Analysis and Chaos Expansions, includes notation and provides the reader with the theoretical background needed to understand the subsequent chapters. 

In Chapter 2, Generalized Operators of Malliavin Calculus, the Malliavin derivative operator, the Skorokhod integral and the Ornstein-Uhlenbeck operator are introduced in terms of chaos expansions. The main properties of the operators, which are known in the literature for the square integrable processes, are proven using the chaos expansion approach and extended for generalized and test stochastic processes. 

Chapter 3, Equations involving Malliavin Calculus operators, is devoted to the study of several types of stochastic differential equations that involve the operators of Malliavin calculus, introduced in the previous chapter. Fractional versions of these operators are also discussed.

Finally, in Chapter 4, Applications and Numerical Approximations are discussed. Specifically, we consider the stochastic linear quadratic optimal control problem with different forms of noise disturbances, operator differential algebraic equations arising in fluid dynamics, stationary equations and fractional versions of the equations studied – applications never covered in the extant literature. Moreover, numerical validations of the method are provided for specific problems."


About the authors

Tijana Levajković is currently a postdoctoral researcher at the at the Department of Mathematics, University of Innsbruck. Her main research interests are in the fields of functional and stochastic analysis, particularly in infinite dimensional stochastic analysis, white noise analysis, Maliavin calculus, generalized stochastic processes, stochastic partial differential equations, algebras of generalized functions and optimal control.

Hermann Mena is professor at Yachay Tech, Ecuador. He also has an affiliation at the Department of Mathematics of Univeristy of Innsbruck, Austria. His research interests include applied mathematics, numerical analysis and optimal control. Particularly, deterministic and stochastic optimal control theory, numerical methods for optimal control problems and uncertainty quantification.

Table of contents (4 chapters)

  • White Noise Analysis and Chaos Expansions

    Levajković, Tijana (et al.)

    Pages 1-41

  • Generalized Operators of Malliavin Calculus

    Levajković, Tijana (et al.)

    Pages 43-73

  • Equations Involving Mallivin Calculus Operators

    Levajković, Tijana (et al.)

    Pages 75-96

  • Applications and Numerical Approximation

    Levajković, Tijana (et al.)

    Pages 97-132

Buy this book

eBook $39.99
price for USA (gross)
  • ISBN 978-3-319-65678-6
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $54.99
price for USA
  • ISBN 978-3-319-65677-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Equations Involving Malliavin Calculus Operators
Book Subtitle
Applications and Numerical Approximation
Authors
Series Title
SpringerBriefs in Mathematics
Copyright
2017
Publisher
Springer International Publishing
Copyright Holder
The Author(s)
eBook ISBN
978-3-319-65678-6
DOI
10.1007/978-3-319-65678-6
Softcover ISBN
978-3-319-65677-9
Series ISSN
2191-8198
Edition Number
1
Number of Pages
X, 132
Number of Illustrations and Tables
1 b/w illustrations, 6 illustrations in colour
Topics