Equations Involving Malliavin Calculus Operators: Applications and Numerical Approximation

ISBN-13: 9783319656779 / Angielski / Miękka / 2017 / 132 str.

Tijana Levajkovic; Hermann Mena

Opis

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 particular, we introduce spaces of random variables and stochastic processes, and consider processes that have finite variance on classical and fractional Gaussian white noise probability spaces. We also present processes with infinite variance, particularly Kondratiev stochastic distributions. We introduce the Wick and ordinary multiplication of the processes and state where these operations are well defined.
In Chapter 2, Generalized Operators of Malliavin Calculus, the Malliavin derivative operator D, the Skorokhod integral δ and the Ornstein-Uhlenbeck operator R 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. Moreover, we discuss fractional versions of these operators. 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. In particular, we describe the range of the operators D, δ and R.
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.

Recenzje

"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)

Spis treści

1 White Noise Analysis and Chaos Expansions: 1.1   Introduction.- 1.3  Deterministic background.- 1.2   Spaces of random variables.- 1.4 Stochastic processes.- 1.5 Operators.- References.- 2 Generalized Operators of Malliavin Calculus: 2.1 Introduction.- 2.1  The Malliavin derivative.- 2.2   The Skorokhod integral.- 2.3   The Ornstein-Uhlenbeck operator.- 2.4 Properties of the Malliavin operators.- 2.5 Fractional operators of the Malliavin calculus.- References.- 3 Equations involving Mallivin Calculus Operators: 3.1  Introduction.- 3.2 Equations with the Ornstein-Uhlenbeck operator.- 3.3  First order equation with the Malliavin derivative operator.- 3.4  Nonhomogeneous equation with the Malliavin derivative operator.- 3.5 Wick-type equations involving the Malliavin derivative.- 3.6 Integral equation.- References.- 4 Applications and Numerical Approximation: 4.1 Introduction.- 4.1  A stochastic optimal control problem.- 4.3 Operator differential algebraic equations.- 4.4 Stationary equations.- 4.5 A fractional optimal control problem.- 4.6 Numerical approximation.- References.

O autorze

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.