Almost Periodic Stochastic Processes [electronic resource] /
Bezandry, Paul H.author.
Diagana, Toka.author.
Almost Periodic Stochastic Processes is among the few published books that is entirely devoted to almost periodic stochastic processes and their applications.  The topics treated range from existence, uniqueness, boundedness, and stability of solutions, to stochastic difference and differential equations. Motivated by the studies of the natural fluctuations in nature, this work aims to lay the foundations for a theory on almost periodic stochastic processes and their applications. This book is divided in to eight chapters and offers useful bibliographical notes at the end of each chapter. Highlights of this monograph include the introduction of the concept of p-th mean almost periodicity for stochastic processes and applications to various equations. The book offers some original results on the boundedness, stability, and existence of p-th mean almost periodic solutions to (non)autonomous first and/or second order stochastic differential equations, stochastic partial differential equations, stochastic functional differential equations with delay, and stochastic difference equations.  Various illustrative examples are also discussed throughout the book. The results provided in the book will be of particular use to those conducting research in the field of stochastic processing including engineers, economists, and statisticians with backgrounds in functional analysis and stochastic analysis.   Advanced graduate students with backgrounds in real analysis, measure theory, and basic probability, may also find the material in this book quite useful and engaging.
Description based upon print version of record.
Almost Periodic Stochastic Processes; Acknowledgments; Preface; Contents; Chapter 1 Banach and Hilbert Spaces; 1.1 Banach Spaces; 1.1.1 Introduction; 1.1.2 Normed Vector Spaces; 1.1.3 Examples of Banach Spaces; 1.1.4 Hölder and Lipschitz Spaces; 1.2 Hilbert Spaces; 1.2.1 Basic Definitions; 1.2.2 Orthogonality; 1.2.3 Projections; 1.3 Bibliographical Notes; Chapter 2 Bounded and Unbounded Linear Operators; 2.1 Introduction; 2.2 Linear Operators; 2.2.1 Bounded Operators; 2.2.1.1 Adjoint For Bounded Operators; 2.2.1.2 The Inverse Operator; 2.2.1.3 Compact Operators
2.2.1.4 Hilbert-Schmidt Operators2.2.2 Unbounded Linear Operators; 2.2.3 Examples of Unbounded Operators; 2.2.3.1 Closed and Closable Linear Operators; 2.2.3.2 Spectral Theory for Unbounded Linear Operators; 2.2.3.3 Symmetric and Self-Adjoint Linear Operators; 2.3 Sectorial Linear Operators; 2.3.1 Basic Definitions; 2.3.2 Examples of Sectorial Operators; 2.4 Semigroups of Linear Operators; 2.4.1 Basic Definitions; 2.4.2 Basic Properties of Semigroups; 2.4.3 Analytic Semigroups; 2.5 Intermediate Spaces; 2.5.1 Fractional Powers of Sectorial Operators; 2.5.2 The Spaces DA(α, p) and DA(α)
2.5.3 Hyperbolic Semigroups2.6 Evolution Families and Their Properties; 2.6.1 Evolution Families; 2.6.2 Estimates for U(t, s); 2.7 Bibliographical Notes; Chapter 3 An Introduction to Stochastic Differential Equations; 3.1 Fundamentals of Probability; 3.1.1 Probability and Random Variables; 3.1.2 Sequence of Events; 3.1.3 Convergence of Random Variables; 3.1.3.1 Convergence in Distribution; 3.1.3.2 Convergence in Probability; 3.1.3.3 Almost Sure Convergence; 3.1.3.4 Lp-Convergence; 3.1.4 Conditional Expectation; 3.2 Stochastic Processes; 3.2.1 Continuity; 3.2.2 Separability and Measurability
3.2.3 Stopping Times3.2.4 Gaussian Processes; 3.2.5 Martingales; 3.3 Stochastic Integrals in One Dimension; 3.3.1 Motivation; 3.3.2 Itô Integrals; 3.3.3 Itô Integrals with Stopping Time; 3.3.4 Itô Formula; 3.3.5 Diffusion Process; 3.4 Wiener Process and Stochastic Integrals in a Hilbert Space; 3.4.1 Wiener Process in a Separable Hilbert Space; 3.4.2 Stochastic Integrals in a Hilbert Space; 3.4.3 Stochastic Convolution Integrals; 3.5 Existence of Solutions of Stochastic Differential Equations in a Hilbert Space; 3.5.1 Existence and Uniqueness; 3.5.2 L2-Bounded Solutions
3.5.3 Stochastic Delay Differential Equation and Exponential Stability3.6 Bibliographical Notes; Chapter 4 p-th Mean Almost Periodic Random Functions; 4.1 Almost Periodic Functions; 4.1.1 Introduction; 4.1.2 Basic Definitions; 4.1.3 Properties of Almost Periodic Functions; 4.2 p-th Mean Almost Periodic Processes; 4.2.1 Composition of p-th Mean Almost Periodic Processes; 4.3 Bibliographical Notes; Chapter 5 Existence Results for Some Stochastic Differential Equations; 5.1 The Autonomous Case; 5.2 The Nonautonomous Case; 5.2.1 Introduction; 5.2.2 Existence of p-th Mean Almost Periodic Solutions
5.2.3 Example
Includes bibliographical references and index.
Almost Periodic Stochastic Processes is among the few published books that is entirely devoted to almost periodic stochastic processes and their applications.  The topics treated range from existence, uniqueness, boundedness, and stability of solutions, to stochastic difference and differential equations. Motivated by the studies of the natural fluctuations in nature, this work aims to lay the foundations for a theory on almost periodic stochastic processes and their applications. This book is divided in to eight chapters and offers useful bibliographical notes at the end of each chapter. Highlights of this monograph include the introduction of the concept of p-th mean almost periodicity for stochastic processes and applications to various equations. The book offers some original results on the boundedness, stability, and existence of p-th mean almost periodic solutions to (non)autonomous first and/or second order stochastic differential equations, stochastic partial differential equations, stochastic functional differential equations with delay, and stochastic difference equations.  Various illustrative examples are also discussed throughout the book. The results provided in the book will be of particular use to those conducting research in the field of stochastic processing including engineers, economists, and statisticians with backgrounds in functional analysis and stochastic analysis.   Advanced graduate students with backgrounds in real analysis, measure theory, and basic probability, may also find the material in this book quite useful and engaging.
Differential equations.
Probabilities.
Partial differential equations.
Functional analysis.
Operator theory.
Integral equations.
Ordinary Differential Equations.
Probability Theory and Stochastic Processes.
Partial Differential Equations.
Functional Analysis.
Operator Theory.
Integral Equations.