Stokastik

This book discusses state-of-the-art stochastic optimization algorithms for distributed machine learning and analyzes their convergence speed. The book first introduces stochastic gradient descent (SGD) and its distributed version, synchronous SGD, where the task of computing gradients is divided across several worker nodes. The author discusses several algorithms that improve the scalability and communication efficiency of synchronous SGD, such as asynchronous SGD, local-update SGD, quantized and sparsified SGD, and decentralized SGD. For each of these algorithms, the book analyzes its error versus iterations convergence, and the runtime spent per iteration. The author shows that each of these strategies to reduce communication or synchronization delays encounters a fundamental trade-off between error and runtime.

The book constitutes an introduction to stochastic calculus, stochastic differential equations and related topics such as Malliavin calculus. On the other hand it focuses on the techniques of stochastic integration and calculus via regularization initiated by the authors. The definitions relies on a smoothing procedure  of the integrator process, they generalize the usual Ito and Stratonovich integrals for Brownian motion but the integrator could also not be a semimartingale and the integrand is allowed to be anticipating. The resulting calculus requires a simple formalism: nevertheless it entails pathwise techniques even though it takes into account randomness.  It allows connecting different types of pathwise and non pathwise integrals such as Young, fractional, Skorohod integrals, enlargement of filtration and rough paths. The covariation, but also high order variations, play a fundamental role in the calculus via regularization, which can also be applied for irregular integrators. A large class of Gaussian processes, various generalizations of semimartingales such that Dirichlet and weak Dirichlet processes are revisited. Stochastic calculus via regularization has been successfully used in applications, for instance in robust finance and on modeling vortex filaments in turbulence. The book is addressed to PhD students and researchers in stochastic analysis and applications to various fields.

This book develops limit theorems for a natural class of long range random walks on finitely generated torsion free nilpotent groups. The limits in these limit theorems are Lévy processes on some simply connected nilpotent Lie groups. Both the limit Lévy process and the limit Lie group carrying this process are determined by and depend on the law of the original random walk. The book offers the first systematic study of such limit theorems involving stable-like random walks and stable limit Lévy processes in the context of (non-commutative) nilpotent groups.

This book discusses the systems of interacting particles evolving in the random media. The focus is on the study of both the finite subsystems motion and the flow, describing motion of all particles in the space. The integral characteristics of the system and mass distribution are also covered and results are illustrated with examples from turbulence theory, synchronization and DNA evolution.

This book emphasizes the use of stochastic orders as motivational tools for developing new statistical procedures. Stochastic orders have found useful applications in many disciplines, including reliability theory, survival analysis, risk theory, finance, nonparametric methods, economics and actuarial science.  Written by a statistician, this volume clarifies the connection between stochastic orders and nonparametric methods.The importance of order statistics and spacings is well recognized. Classically, they mainly focus on the case when the observations are independent and identically distributed, however, several new developments have extended the comparison of order statistics to the case of non-identically distributed or non-independent observations. In addition to giving a detailed discussion of various topics in the general area of stochastic orders, a substantial part of the book is devoted to recent research on stochastic comparisons of order statistics and spacings, including a long chapter on dependence among them. The book will be useful for graduate students and researchers in statistics, economics, actuarial science and other related disciplines.  In particular, with close to 300 references, it will be a valuable resource for reliability theorists, applied probabilists and statisticians. Readers are expected to have taken a first-year graduate level course in mathematical statistics or in applied probability. 

Basierend auf Grundkenntnissen aus der Schulzeit oder aus dem ersten Band des Gesamtwerks ¿Mathematik verstehen und anwenden¿ führt dieser zweite Band in die Vektoranalysis, in das Gebiet der Differenzialgleichungen und in die Fourier-Analysis einschließlich der Laplace-Transformation ein und beinhaltet außerdem eine Einführung in die Wahrscheinlichkeitsrechnung und Statistik. Damit er unabhängig vom ersten Band gelesen werden kann, beginnt er mit einer kurzen Zusammenfassung der wichtigsten Begriffe und Ergebnisse der Differenzial- und Integralrechnung sowie der Linearen Algebra.Zielgruppe sind Studierende der Ingenieur- und Naturwissenschaften an Fachhochschulen und Universitäten. Trotz der verständlichen Darstellung für ein Bachelor-Studium geht die mathematische Exaktheit nicht verloren. Hintergrundinformationen und Beweise ergänzen die sehr umfangreiche Stoffauswahl und bieten Anknüpfungspunkte für ein Masterstudium. Daneben erleichtern sie auch den Einstieg in Spezialvorlesungen der Mathematik wie beispielsweise die Numerik, die Funktionalanalysis und insbesondere die Fourier-Analysis.In der vierten Auflage wurden viele Anwendungsbeispiele ergänzt und der Text grundlegend überarbeitet.Stimmen zur ersten Auflage:¿Sowohl mathematisch exakt als auch äußerst anschaulich. Eine echte Bereicherung der großen Auswahl an Büchern zum Thema Ingenieurmathematik.¿Prof. Dr. Andreas Gessinger, Rheinische Fachhochschule Köln¿Der Spagat zwischen Verständlichkeit und mathematischer Tiefe ist hervorragend gelungen. Eine breite Palette von praxisorientierten Beispielen wirkt motivationsfördernd.¿Prof. Dr. Helga Tecklenburg, Hochschule für Technik, Wirtschaft und Kultur Leipzig

This proceedings book covers a wide range of topics related to uncertainty analysis and its application in various fields of engineering and science. It explores uncertainties in numerical simulations for soil liquefaction potential, the toughness properties of construction materials, experimental tests on cyclic liquefaction potential, and the estimation of geotechnical engineering properties for aerogenerator foundation design. Additionally, the book delves into uncertainties in concrete compressive strength, bio-inspired shape optimization using isogeometric analysis, stochastic damping in rotordynamics, and the hygro-thermal properties of raw earth building materials. It also addresses dynamic analysis with uncertainties in structural parameters, reliability-based design optimization of steel frames, and calibration methods for models with dependent parameters. The book further explores mechanical property characterization in 3D printing, stochastic analysis in computational simulations, probability distribution in branching processes, data assimilation in ocean circulation modeling, uncertainty quantification in climate prediction, and applications of uncertainty quantification in decision problems and disaster management. This comprehensive collection provides insights into the challenges and solutions related to uncertainty in various scientific and engineering contexts.

This book offers the reader a journey through the counterintuitive nature of Brownian motion under confinement. Diffusion is a universal phenomenon that controls a wide range of physical, chemical, and biological processes. The transport of spatially-constrained molecules and small particles is ubiquitous in nature and technology and plays an essential role in different processes. Understanding the physics of diffusion under conditions of confinement is essential for a number of biological phenomena and potential technological applications in micro- and nanofluidics, among others. Studies on diffusion under confinement are typically difficult to understand for young scientists and students because of the extensive background on diffusion processes, physics, and mathematics that is required. All of this information is provided in this book, which is essentially self-contained as a result of the authors¿ efforts to make it accessible to an audience of students from a variety of different backgrounds. The book also provides the necessary mathematical details so students can follow the technical process required to solve each problem. Readers will also find detailed explanations of the main results based on the last 30 years of research devoted to studying diffusion under confinement. The authors approach the physical problem from various angles and discuss the role of geometries and boundary conditions in diffusion. This textbook serves as a comprehensive and modern overview of Brownian motion under confinement and is intended for young scientists, graduate students, and advanced undergraduates in physics, physical chemistry, biology, chemistry, chemical engineering, biochemistry, bioengineering, and polymer and material sciences.

This textbook covers the fundamentals of reliability theory and its application for engineering processes, especially for aircraft units and systems. Reliability basis was explained for the best understanding of reliability analysis application for engineering systems in aviation industry. Several approaches for the reliability analysis and their application with examples are presented. It also introduces main trends in the modern reliability theory development.This book will be interested for university students and early-career engineers of aviation industry majors.

This book is a comprehensive exploration of the interplay between Stochastic Analysis, Geometry, and Partial Differential Equations (PDEs). It aims to investigate the influence of geometry on diffusions induced by underlying structures, such as Riemannian or sub-Riemannian geometries, and examine the implications for solving problems in PDEs, mathematical finance, and related fields. The book aims to unify the relationships between PDEs, nonholonomic geometry, and stochastic processes, focusing on a specific condition shared by these areas known as the bracket-generating condition or Hörmander's condition. The main objectives of the book are: The intended audience for this book includes researchers and practitioners in mathematics, physics, and engineering, who are interested in stochastic techniques applied to geometry and PDEs, as well as their applications in mathematical finance and electrical circuits.

This book develops alternative methods to estimate the unknown parameters in stochastic volatility models, offering a new approach to test model accuracy. While there is ample research to document stochastic differential equation models driven by Brownian motion based on discrete observations of the underlying diffusion process, these traditional methods often fail to estimate the unknown parameters in the unobserved volatility processes. This text studies the second order rate of weak convergence to normality to obtain refined inference results like confidence interval, as well as nontraditional continuous time stochastic volatility models driven by fractional Levy processes. By incorporating jumps and long memory into the volatility process, these new methods will help better predict option pricing and stock market crash risk. Some simulation algorithms for numerical experiments are provided.

This textbook presents some basic stochastic processes, mainly Markov processes. It begins with a brief introduction to the framework of stochastic processes followed by the thorough discussion on Markov chains, which is the simplest and the most important class of stochastic processes. The book then elaborates the theory of Markov chains in detail including classification of states, the first passage distribution, the concept of periodicity and the limiting behaviour of a Markov chain in terms of associated stationary and long run distributions. The book first illustrates the theory for some typical Markov chains, such as random walk, gambler's ruin problem, Ehrenfest model and Bienayme-Galton-Watson branching process; and then extends the discussion when time parameter is continuous. It presents some important examples of a continuous time Markov chain, which include Poisson process, birth process, death process, birth and death processes and their variations. These processes play a fundamental role in the theory and applications in queuing and inventory models, population growth, epidemiology and engineering systems. The book studies in detail the Poisson process, which is the most frequently applied stochastic process in a variety of fields, with its extension to a renewal process.The book also presents important basic concepts on Brownian motion process, a stochastic process of historic importance. It covers its few extensions and variations, such as Brownian bridge, geometric Brownian motion process, which have applications in finance, stock markets, inventory etc. The book is designed primarily to serve as a textbook for a one semester introductory course in stochastic processes, in a post-graduate program, such as Statistics, Mathematics, Data Science and Finance. It can also be used for relevant courses in other disciplines. Additionally, it provides sufficient background material for studying inference in stochastic processes. The book thus fulfils the need of a concise but clear and student-friendly introduction to various types of stochastic processes.

Suitable for a first undergraduate course in applied probability, this book introduces elementary probability theory and stochastic processes, and shows how probability theory can be applied fields such as engineering, computer science, management science, the physical and social sciences, and operations research.

This text presents the basic theory of random walks on infinite, finitely generated groups, along with certain background material in measure-theoretic probability. The main objective is to show how structural features of a group, such as amenability/nonamenability, affect qualitative aspects of symmetric random walks on the group, such as transience/recurrence, speed, entropy, and existence or nonexistence of nonconstant, bounded harmonic functions. The book will be suitable as a textbook for beginning graduate-level courses or independent study by graduate students and advanced undergraduate students in mathematics with a solid grounding in measure theory and a basic familiarity with the elements of group theory. The first seven chapters could also be used as the basis for a short course covering the main results regarding transience/recurrence, decay of return probabilities, and speed. The book has been organized and written so as to be accessible not only to students in probability theory, but also to students whose primary interests are in geometry, ergodic theory, or geometric group theory.

This textbook presents the basics of probability and statistical estimation, with a view to applications. The didactic presentation follows a path of increasing complexity with a constant concern for pedagogy, from the most classical formulas of probability theory to the asymptotics of independent random sequences and an introduction to inferential statistics. The necessary basics on measure theory are included to ensure the book is self-contained. Illustrations are provided from many applied fields, including information theory and reliability theory. Numerous examples and exercises in each chapter, all with solutions, add to the main content of the book.Written in an accessible yet rigorous style, the book is addressed to advanced undergraduate students in mathematics and graduate students in applied mathematics and statistics. It will also appeal to students and researchers in other disciplines, including computer science, engineering, biology, physics and economics, who are interested in a pragmatic introduction to the probability modeling of random phenomena.

This comprehensive open access book enables readers to discover the essential techniques for load forecasting in electricity networks, particularly for active distribution networks.From statistical methods to deep learning and probabilistic approaches, the book covers a wide range of techniques and includes real-world applications and a worked examples using actual electricity data (including an example implemented through shared code). Advanced topics for further research are also included, as well as a detailed appendix on where to find data and additional reading. As the smart grid and low carbon economy continue to evolve, the proper development of forecasting methods is vital. This book is a must-read for students, industry professionals, and anyone interested in forecasting for smart control applications, demand-side response, energy markets, and renewable utilization.

This book describes how reliability can be embedded into the product development using a design methodology that uses parametric accelerated lifecycle testing (ALT) .The book has these features:¿ A new reliability methodology, based on inferential statistics, that can determine whether the reliability of a mechanical/civil system is achieved.¿ A unique reliability methodology to prevent reliability disasters in new mechanical products in the field, e.g., automobiles and airplanes.¿ Robust design methodology of mechanical/civil product to withstand a variety of loads.¿ Explanation of an alternative experimental Taguchi methodology.¿ Discussion of how parametric ALT can also be used to predict product reliability¿lifetime and failure rate.¿ Detailed case studies that demonstrate parametric ALT methodology.This book will be useful for senior-level undergraduate and graduate students, professional engineers, college and university-level lecturers, researchers, and design managers in mechanical and civil engineering.

This book provides a thorough conversation on the underpinnings of Covid-19 spread modelling by using stochastics nonlocal differential and integral operators with singular and non-singular kernels. The book presents the dynamic of Covid-19 spread behaviour worldwide. It is noticed that the spread dynamic followed process with nonlocal behaviours which resemble power law, fading memory, crossover and stochastic behaviours. Fractional stochastic differential equations are therefore used to model spread behaviours in different parts of the worlds. The content coverage includes brief history of Covid-19 spread worldwide from December 2019 to September 2021, followed by statistical analysis of collected data for infected, death and recovery classes.

Sergio Albeverio gave important contributions to many fields ranging from Physics to Mathematics, while creating new research areas from their interplay. Some of them are presented in this Volume that grew out of the Random Transformations and Invariance in Stochastic Dynamics Workshop held in Verona in 2019. To understand the theory of thermo- and fluid-dynamics, statistical mechanics, quantum mechanics and quantum field theory, Albeverio and his collaborators developed stochastic theories having strong interplays with operator theory and functional analysis. His contribution to the theory of (non Gaussian)-SPDEs, the related theory of (pseudo-)differential operators, and ergodic theory had several impacts to solve problems related, among other topics, to thermo- and fluid dynamics. His scientific works in the theory of interacting particles and its extension to configuration spaces lead, e.g., to the solution of open problems in statistical mechanics and quantum field theory. Together with Raphael Hoegh Krohn he introduced the theory of infinite dimensional Dirichlet forms, which nowadays is used in many different contexts, and new methods in the theory of Feynman path integration. He did not fear to further develop different methods in Mathematics, like, e.g., the theory of non-standard analysis and p-adic numbers.

"The Art of Randomness teaches readers to harness the power of randomness (and Python code) to solve real-world problems in programming, science, and art through hands-on experiments-from simulating evolution to encrypting messages to making machine-learning algorithms. Each chapter describes how randomness plays into the given topic area, then proceeds to demonstrate its problem-solving role with hands-on experiments to work through using Python code"--

The main subject of the book is stochastic analysis and its various applications to mathematical finance and statistics of random processes. The main purpose of the book is to present, in a short and sufficiently self-contained form, the methods and results of the contemporary theory of stochastic analysis and to show how these methods and results work in mathematical finance and statistics of random processes. The book can be considered as a textbook for both senior undergraduate and graduate courses on this subject. The book can be helpful for undergraduate and graduate students, instructors and specialists on stochastic analysis and its applications.