Equations Involving Malliavin Calculus Operators

1. Front Matter

   Pages i-x

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2. White Noise Analysis and Chaos Expansions

   • Tijana Levajković, Hermann Mena

   Pages 1-41

3. Generalized Operators of Malliavin Calculus

   • Tijana Levajković, Hermann Mena

   Pages 43-73

4. Equations Involving Mallivin Calculus Operators

   • Tijana Levajković, Hermann Mena

   Pages 75-96

5. Applications and Numerical Approximation

   • Tijana Levajković, Hermann Mena

   Pages 97-132

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 integraland 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."

• stochastic processes
• generalized stochastic processes
• stochastic differential equations
• Malliavin operators
• white noise analysis
• chaos expansion
• stochastic optimal control problems
• operator differential algebraic equations
• partial differential equations

“This book aims to give a unified introduction to stochasatic differential equations involving Malliavin calculus operators. The intended audience is that of researchers and graduate students interested in stochastic partial differential equations and related fields.” (John Masson Noble, Mathematical Reviews, June, 2018)

“The intended audience are researchers and graduate students interested in stochastic partial differential equations and related fields. This book is self-contained for readers familiar with white noise analysis and Malliavin calculus.” (Yuliya S. Mishura, zbMATH 1388.60007, 2018)

• Department of Mathematics, University of Innsbruck, Innsbruck, Austria

  Tijana Levajković, Hermann Mena

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.

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