Uniquely combining theory, application, and computing, this book explores the spectral approach to time series analysis

The use of periodically correlated (or cyclostationary) processes has become increasingly popular in a range of research areas such as meteorology, climate, communications, economics, and machine diagnostics. Periodically Correlated Random Sequences presents the main ideas of these processes through the use of basic definitions along with motivating, insightful, and illustrative examples. Extensive coverage of key concepts is provided, including second-order theory, Hilbert spaces, Fourier theory, and the spectral theory of harmonizable sequences. The authors also provide a paradigm for nonparametric time series analysis including tests for the presence of PC structures.

Features of the book include:

An emphasis on the link between the spectral theory of unitary operators and the correlation structure of PC sequences

A discussion of the issues relating to nonparametric time series analysis for PC sequences, including estimation of the mean, correlation, and spectrum

A balanced blend of historical background with modern application-specific references to periodically correlated processes

An accompanying Web site that features additional exercises as well as data sets and programs written in MATLAB(R) for performing time series analysis on data that may have a PC structure

Periodically Correlated Random Sequences is an ideal text on time series analysis for graduate-level statistics and engineering students who have previous experience in second-order stochastic processes (Hilbert space), vector spaces, random processes, and probability. This bookalso serves as a valuable reference for research statisticians and practitioners in areas of probability and statistics such as time series analysis, stochastic processes, and prediction theory.

Applied Stochastic Processes introduces the reader to stochastic processes with a focus on the applications of the theoretical results. This text is self-contained and logically organized. It begins with a review of elementary probability, followed by an introduction to the most important subjects in the field of stochastic processes. Topics covered include Gaussian and Markovian processes, Markov Chains, Weiner and Poisson processes, Brownian motion, and queuing theory with a special highlight on diffusion processes. The reader will appreciate the clear definitions, thoroughly explained examples and interesting notes about the mathematicians referenced throughout the text. In addition, there are hundreds of advanced, multi-part problems following each chapter which enable even a novice of theoretical mathematics to master the material presented. This textbook evolved from the author's lecture notes for a graduate-level course on applied stochastic processes. It is meant for graduate-level students in electrical engineering, applied mathematics, and notably operations research.

The theory of stochastic integration, also called the Ito calculus, has a large spectrum of applications in virtually every scientific area involving random functions, but it can be a very difficult subject for people without much mathematical background. The Ito calculus was originally motivated by the construction of Markov diffusion processes from infinitesimal generators. Previously, the construction of such processes required several steps, whereas Ito constructed these diffusion processes directly in a single step as the solutions of stochastic integral equations associated with the infinitesimal generators. Moreover, the properties of these diffusion processes can be derived from the stochastic integral equations and the Ito formula. This introductory textbook on stochastic integration provides a concise introduction to the Ito calculus, and covers the following topics:

* Constructions of Brownian motion;

* Stochastic integrals for Brownian motion and martingales;

* The Ito formula;

* Multiple Wiener-Ito integrals;

* Stochastic differential equations;

* Applications to finance, filtering theory, and electric circuits.

The reader should have a background in advanced calculus and elementary probability theory, as well as a basic knowledge of measure theory and Hilbert spaces. Each chapter ends with a variety of exercises designed to help the reader further understand the material.

Hui-Hsiung Kuo is the Nicholson Professor of Mathematics at Louisiana State University. He has delivered lectures on stochastic integration at Louisiana State University, Cheng Kung University, Meijo University, and University of Rome "Tor Vergata," among others. He is also theauthor of Gaussian Measures in Banach Spaces (Springer 1975), and White Noise Distribution Theory (CRC Press 1996), and a memoir of his childhood growing up in Taiwan, An Arrow Shot into the Sun (Abridge Books 2004).

In his seminal 1982 paper, Robert F. Engle described a time series model with a time-varying volatility. Engle showed that this model, which he called ARCH (autoregressive conditionally heteroscedastic), is well-suited for the description of economic and financial price. Nowadays ARCH has been replaced by more general and more sophisticated models, such as GARCH (generalized autoregressive heteroscedastic).

This monograph concentrates on mathematical statistical problems associated with fitting conditionally heteroscedastic time series models to data. This includes the classical statistical issues of consistency and limiting distribution of estimators. Particular attention is addressed to (quasi) maximum likelihood estimation and misspecified models, along to phenomena due to heavy-tailed innovations. The used methods are based on techniques applied to the analysis of stochastic recurrence equations. Proofs and arguments are given wherever possible in full mathematical rigour. Moreover, the theory is illustrated by examples and simulation studies.

A graduate-course text, written for readers familiar with measure-theoretic probability and discrete-time processes, wishing to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed, illustrated by results concerning representations of martingales and change of measure on Wiener space, which in turn permit a presentation of recent advances in financial economics. The book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The whole is backed by a large number of problems and exercises.

This book is dedicated to the systematization and development of models, methods, and algorithms for queuing systems with correlated arrivals. After first setting up the basic tools needed for the study of queuing theory, the authors concentrate on complicated systems: multi-server systems with phase type distribution of service time or single-server queues with arbitrary distribution of service time or semi-Markovian service. They pay special attention to practically important retrial queues, tandem queues, and queues with unreliable servers. Mathematical models of networks and queuing systems are widely used for the study and optimization of various technical, physical, economic, industrial, and administrative systems, and this book will be valuable for researchers, graduate students, and practitioners in these domains.

All populations fluctuate stochastically, creating a risk of extinction that does not exist in deterministic models, with fundamental consequences for both pure and applied ecology. This book provides the most comprehensive introduction to stochastic population dynamics, combining classical background material with a variety of modern approaches, including new and previously unpublished results by the authors, illustrated with examples from bird and mammal populations, and insect communities.

2013 Reprint of 1958 Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. A series of lectures on the role of nonlinear processes in physics, mathematics, electrical engineering, physiology, and communication theory. From the preface: "For some time I have been interested in a group of phenomena depending upon random processes. One the one hand, I have recorded the random shot effect as a suitable input for testing nonlinear circuits. On the other hand, for some of the work that Professor W. A. Rosenblith and I have been doing concerning the nature of the electroencephalogram, and in particular of the alpha rhythm, it has occurred to me to use the model of a system of random nonlinear oscillators excited by a random input. . . . At the beginning we had contemplated a series of only four or five lectures. My ideas developed pari passu with the course, and by the end of the term we found ourselves with a set of fifteen lectures. The last few of these were devoted to the application of my ideas to problems in the statistical mechanics of gases. This work is both new and tentative, and I found that I had to supplement my course by the writing over of these with the help of Professer Y. W. Lee. "

This unique book combines lucid and engaging exposition, graphic and poignantly applied examples, and realistic exercises to take readers beyond the mechanics of statistical techniques. The result is a journey into the realm of practical data analysis and inference-based problem solving.

A beginner s guide to stochastic growth modeling The chief advantage of stochastic growth models over deterministic models is that they combine both deterministic and stochastic elements of dynamic behaviors, such as weather, natural disasters, market fluctuations, and epidemics. This makes stochastic modeling a powerful tool in the hands of practitioners in fields for which population growth is a critical determinant of outcomes. However, the background requirements for studying SDEs can be daunting for those who lack the rigorous course of study received by math majors. Designed to be accessible to readers who have had only a few courses in calculus and statistics, this book offers a comprehensive review of the mathematical essentials needed to understand and apply stochastic growth models. In addition, the book describes deterministic and stochastic applications of population growth models including logistic, generalized logistic, Gompertz, negative exponential, and linear. Ideal for students and professionals in an array of fields including economics, population studies, environmental sciences, epidemiology, engineering, finance, and the biological sciences, Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling: Provides precise definitions of many important terms and concepts and provides many solved example problems Highlights the interpretation of results and does not rely on a theorem-proof approach Features comprehensive chapters addressing any background deficiencies readers may have and offers a comprehensive review for those who need a mathematics refresher Emphasizes solution techniques for SDEs and their practical application to the development of stochastic population models An indispensable resource for students and practitioners with limited exposure to mathematics and statistics, Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling is an excellent fit for advanced undergraduates and beginning graduate students, as well as practitioners who need a gentle introduction to SDEs. Michael J. Panik, PhD, is Professor in the Department of Economics, Barney School of Business and Public Administration at the University of Hartford in Connecticut. He received his PhD in Economics from Boston College and is a member of the American Mathematical Society, The American Statistical Association, and The Econometric Society.

This collection of papers gives a representative cross-selectional view of recent developments in the field. After a survey paper by C. Lefevre, 17 other research papers look at stochastic modeling of epidemics, both from a theoretical and a statistical point of view. Some look more specifically at a particular disease such as AIDS, malaria, schistosomiasis and diabetes."

This is the first comprehensive book on the AIMD algorithm, the most widely used method for allocating a limited resource among competing agents without centralized control. The authors offer a new approach that is based on positive switched linear systems. It is used to develop most of the main results found in the book, and fundamental results on stochastic switched nonnegative and consensus systems are derived to obtain these results. The original and best known application of the algorithm is in the context of congestion control and resource allocation on the Internet, and readers will find details of several variants of the algorithm in order of increasing complexity, including deterministic, random, linear, and nonlinear versions. In each case, stability and convergence results are derived based on unifying principles. Basic and fundamental properties of the algorithm are described, examples are used to illustrate the richness of the resulting dynamical systems, and applications are provided to show how the algorithm can be used in the context of smart cities, intelligent transportation systems, and the smart grid.

Computational neuroscience is the theoretical study of the brain to uncover the principles and mechanisms that guide the development, organization, information processing, and mental functions of the nervous system. Although not a new area, it is only recently that enough knowledge has been gathered to establish computational neuroscience as a scientific discipline in its own right. Given the complexity of the field, and its increasing importance in progressing our understanding of how the brain works, there has long been a need for an introductory text on what is often assumed to be an impenetrable topic. The new edition of Fundamentals of Computational Neuroscience build on the success and strengths of the previous editions. It introduces the theoretical foundations of neuroscience with a focus on the nature of information processing in the brain. The book covers the introduction and motivation of simplified models of neurons that are suitable for exploring information processing in large brain-like networks. Additionally, it introduces several fundamental network architectures and discusses their relevance for information processing in the brain, giving some examples of models of higher-order cognitive functions to demonstrate the advanced insight that can be gained with such studies. Each chapter starts by introducing its topic with experimental facts and conceptual questions related to the study of brain function. An additional feature is the inclusion of simple Matlab programs that can be used to explore many of the mechanisms explained in the book. An accompanying webpage includes programs for download. The book will be the essential text for anyone in the brain sciences who wants to get to grips with this topic.

Probability theory has become a convenient language and a useful tool in many areas of modern analysis. The main purpose of this book is to explore part of this connection concerning the relations between Brownian motion on a manifold and analytical aspects of differential geometry. A dominant theme of the book is the probabilistic interpretation of the curvature of a manifold.The book begins with a brief review of stochastic differential equations on Euclidean space. After presenting the basics of stochastic analysis on manifolds, the author introduces Brownian motion on a Riemannian manifold and studies the effect of curvature on its behavior. He then applies Brownian motion to geometric problems and vice versa, using many well-known examples, e.g., short-time behavior of the heat kernel on a manifold and probabilistic proofs of the Gauss-Bonnet-Chem theorem and the Atiyah-Singer index theorem for Dirac operators. The book concludes with an introduction to stochastic analysis on the path space over a Riemannian manifold.

This text introduces engineering students to probability theory and stochastic processes. Along with thorough mathematical development of the subject, the book presents intuitive explanations of key points in order to give students the insights they need to apply math to practical engineering problems. The first seven chapters contain the core material that is essential to any introductory course. In one-semester undergraduate courses, instructors can select material from the remaining chapters to meet their individual goals. Graduate courses can cover all chapters in one semester.

The Routledge Companion to Intelligence Studies provides a broad overview of the growing field of intelligence studies. The recent growth of interest in intelligence and security studies has led to an increased demand for popular depictions of intelligence and reference works to explain the architecture and underpinnings of intelligence activity. Divided into five comprehensive sections, this Companion provides a strong survey of the cutting-edge research in the field of intelligence studies: Part I: The evolution of intelligence studies; Part II: Abstract approaches to intelligence; Part III: Historical approaches to intelligence; Part IV: Systems of intelligence; Part V: Contemporary challenges. With a broad focus on the origins, practices and nature of intelligence, the book not only addresses classical issues, but also examines topics of recent interest in security studies. The overarching aim is to reveal the rich tapestry of intelligence studies in both a sophisticated and accessible way. This Companion will be essential reading for students of intelligence studies and strategic studies, and highly recommended for students of defence studies, foreign policy, Cold War studies, diplomacy and international relations in general.

The book provides a sound mathematical base for life insurance mathematics and applies the underlying concepts to concrete examples. Moreover the models presented make it possible to model life insurance policies by means of Markov chains. Two chapters covering ALM and abstract valuation concepts on the background of Solvency II complete this volume. Numerous examples and a parallel treatment of discrete and continuous approaches help the reader to implement the theory directly in practice.

This volume in the Mastering Mathematical Finance series strikes just the right balance between mathematical rigour and practical application. Existing books on the challenging subject of stochastic interest rate models are often too advanced for Master's students or fail to include practical examples. Stochastic Interest Rates covers practical topics such as calibration, numerical implementation and model limitations in detail. The authors provide numerous exercises and carefully chosen examples to help students acquire the necessary skills to deal with interest rate modelling in a real-world setting. In addition, the book's webpage at www.cambridge.org/9781107002579 provides solutions to all of the exercises as well as the computer code (and associated spreadsheets) for all numerical work, which allows students to verify the results.

These notes are based on a course of lectures given by Professor Nelson at Princeton during the spring term of 1966. The subject of Brownian motion has long been of interest in mathematical probability. In these lectures, Professor Nelson traces the history of earlier work in Brownian motion, both the mathematical theory, and the natural phenomenon with its physical interpretations. He continues through recent dynamical theories of Brownian motion, and concludes with a discussion of the relevance of these theories to quantum field theory and quantum statistical mechanics.

This book explains the notion of Brakke's mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. The focus of study is a time-parameterized family of k-dimensional surfaces in the n-dimensional Euclidean space (1 k < n). The family is the mean curvature flow if the velocity of motion of surfaces is given by the mean curvature at each point and time. It is one of the simplest and most important geometric evolution problems with a strong connection to minimal surface theory. In fact, equilibrium of mean curvature flow corresponds precisely to minimal surface. Brakke's mean curvature flow was first introduced in 1978 as a mathematical model describing the motion of grain boundaries in an annealing pure metal. The grain boundaries move by the mean curvature flow while retaining singularities such as triple junction points. By using a notion of generalized surface called a varifold from geometric measure theory which allows the presence of singularities, Brakke successfully gave it a definition and presented its existence and regularity theories. Recently, the author provided a complete proof of Brakke's existence and regularity theorems, which form the content of the latter half of the book. The regularity theorem is also a natural generalization of Allard's regularity theorem, which is a fundamental regularity result for minimal surfaces and for surfaces with bounded mean curvature. By carefully presenting a minimal amount of mathematical tools, often only with intuitive explanation, this book serves as a good starting point for the study of this fascinating object as well as a comprehensive introduction to other important notions from geometric measure theory.

This update of the 1987 title of the same name is an examination of what is currently known about the probabilistic method, written by one of its principal developers. Based on the notes from Spencer's 1986 series of ten lectures, this new edition contains an additional lecture: The Janson Inequalities. These inequalities allow accurate approximation of extremely small probabilities. A new algorithmic approach to the Lovasz Local Lemma, attributed to Jozsef Beck, has been added to Lecture 8, as well. Throughout the monograph, Spencer retains the informal style of his original lecture notes and emphasizes the methodology, shunning the more technical "best possible" results in favour of clearer exposition. The book is not encyclopaedic - it contains only those examples that clearly display the methodology. The probabilistic method is a powerful tool in graph theory, combinatorics, and theoretical computer science. It allows one to prove the existence of objects with certain properties (e.g., colourings) by showing that an appropriately defined random object has positive probability of having those properties. Spencer retains the informal style of his original lecture notes and emphasizes the methodology, shunning the more technical "best possible" results in favor of clearer exposition. Topics include: A description via examples of the basic Probabilistic Method and its refinements; Random Graphs; The Lovasz Local Lemma and its recent algorithmic implementations; Discrepancy; Derandomization; Large Deviation Estimates; Martingales; and the recent Janson Inequalities.

This book introduces optimal control problems for large families of deterministic and stochastic systems with discrete or continuous time parameter. These families include most of the systems studied in many disciplines, including Economics, Engineering, Operations Research, and Management Science, among many others. The main objective is to give a concise, systematic, and reasonably self contained presentation of some key topics in optimal control theory. To this end, most of the analyses are based on the dynamic programming (DP) technique. This technique is applicable to almost all control problems that appear in theory and applications. They include, for instance, finite and infinite horizon control problems in which the underlying dynamic system follows either a deterministic or stochastic difference or differential equation. In the infinite horizon case, it also uses DP to study undiscounted problems, such as the ergodic or long-run average cost. After a general introduction to control problems, the book covers the topic dividing into four parts with different dynamical systems: control of discrete-time deterministic systems, discrete-time stochastic systems, ordinary differential equations, and finally a general continuous-time MCP with applications for stochastic differential equations. The first and second part should be accessible to undergraduate students with some knowledge of elementary calculus, linear algebra, and some concepts from probability theory (random variables, expectations, and so forth). Whereas the third and fourth part would be appropriate for advanced undergraduates or graduate students who have a working knowledge of mathematical analysis (derivatives, integrals, ...) and stochastic processes.

The fourth edition of this successful text provides an introduction to probability and random processes, with many practical applications. It is aimed at mathematics undergraduates and postgraduates, and has four main aims. US BL To provide a thorough but straightforward account of basic probability theory, giving the reader a natural feel for the subject unburdened by oppressive technicalities. BE BL To discuss important random processes in depth with many examples.BE BL To cover a range of topics that are significant and interesting but less routine.BE BL To impart to the beginner some flavour of advanced work.BE UE OP The book begins with the basic ideas common to most undergraduate courses in mathematics, statistics, and science. It ends with material usually found at graduate level, for example, Markov processes, (including Markov chain Monte Carlo), martingales, queues, diffusions, (including stochastic calculus with Ito's formula), renewals, stationary processes (including the ergodic theorem), and option pricing in mathematical finance using the Black-Scholes formula. Further, in this new revised fourth edition, there are sections on coupling from the past, Levy processes, self-similarity and stability, time changes, and the holding-time/jump-chain construction of continuous-time Markov chains. Finally, the number of exercises and problems has been increased by around 300 to a total of about 1300, and many of the existing exercises have been refreshed by additional parts. The solutions to these exercises and problems can be found in the companion volume, One Thousand Exercises in Probability, third edition, (OUP 2020).CP

Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance. Stochastic calculus is the mathematics of systems interacting with random noise. Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way. This fully revised edition now features a number of new topics. These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs.

Kiyosi Ito's greatest contribution to probability theory may be his introduction of stochastic differential equations to explain the Kolmogorov-Feller theory of Markov processes. Starting with the geometric ideas that guided him, this book gives an account of Ito's program.

The modern theory of Markov processes was initiated by A. N. Kolmogorov. However, Kolmogorov's approach was too analytic to reveal the probabilistic foundations on which it rests. In particular, it hides the central role played by the simplest Markov processes: those with independent, identically distributed increments. To remedy this defect, Ito interpreted Kolmogorov's famous forward equation as an equation that describes the integral curve of a vector field on the space of probability measures. Thus, in order to show how Ito's thinking leads to his theory of stochastic integral equations, Stroock begins with an account of integral curves on the space of probability measures and then arrives at stochastic integral equations when he moves to a pathspace setting. In the first half of the book, everything is done in the context of general independent increment processes and without explicit use of Ito's stochastic integral calculus. In the second half, the author provides a systematic development of Ito's theory of stochastic integration: first for Brownian motion and then for continuous martingales. The final chapter presents Stratonovich's variation on Ito's theme and ends with an application to the characterization of the paths on which a diffusion is supported.

The book should be accessible to readers who have mastered the essentials of modern probability theory and should provide such readers with a reasonably thorough introduction to continuous-time, stochastic processes."