Stokastik

Dieses essential bietet eine prägnante und gute verständliche Einführung in die Theorie der stochastischen partiellen Differentialgleichungen. Wir werden die dafür benötigten mathematischen Hilfsmittel wie das Bochner-Integral, das Itô-Integral und die Itô-Formel kennenlernen. Anschließend werden wir die relevanten Lösungskonzepte besprechen, Existenz- und Eindeutigkeitsresultate präsentieren und diese anhand von Anwendungsbeispielen erläutern.

This book provides a blend of quantitative and qualitative approaches to decision making, while also bridging the gap between the theory of how to make good decisions versus how people actually make decisions.  The authors present the tools and techniques of decision analysis to learn how to become a FOCCUSSED decision maker: Identify and properly Frame the decision or problem at handSpecify the goals, Objectives, and values that you are trying to achieveDevelop creative, meaningful Choices from among which you can chooseEvaluate the Consequences of selecting each alternative using your goals, objectives, and valuesThink about the key Uncertainties that could impact the decisionUnderstand the Swaps and trade-offs that you are willing to makeDevelop an approach for implementing your SolutionElicit the data you'll need from a variety of sourcesand Disseminate and communicate your decisions to others.  The authors define a decision as the choice among alternatives, based on how we value and trade-off their pros and cons, made in the face of uncertainty about what will actually happen. The decision-making process is presented as having three pillars to support the decision maker: Preferences-what we prefer, what meets our goals and objectives, and the recognition that preferences are personal to the one making the decision; Alternatives-the choices, options, or courses of action that we have, and over which we have some degree of control; and Information-what we know about the situation, what we don't know, how we connect choices to outcomes, and how we deal with uncertainty.  Key components of good decision-making include how to define your goals and objectives, how to incorporate uncertainties that we all face, and how to develop better alternatives, all of which are discussed.  Sophisticated techniques are presented in a way that is accessible to the average decision maker. Probability theory is utilized to improve decisions, and uncertainties are captured in decision trees.  Risk avoidance, risk transfer, and risk mitigation are also discussed.  Readers will gain a clear understanding of how to articulate the goals and objectives that should be the focal point of any decision.

This book is intended to provide graduate students and researchers in graph theory with an overview of the elementary methods of graph Ramsey theory. It is especially targeted towards graduate students in extremal graph theory, graph Ramsey theory, and related fields, as the included contents allow the text to be used in seminars. It is structured in thirteen chapters which are application-focused and largely independent, enabling readers to target specific topics and information to focus their study. The first chapter includes a true beginner's overview of elementary examples in graph Ramsey theory mainly using combinatorial methods. The following chapters progress through topics including the probabilistic methods, algebraic construction, regularity method, but that's not all. Many related interesting topics are also included in this book, such as the disproof for a conjecture of Borsuk on geometry, intersecting hypergraphs, Turan numbers and communication channels, etc.

Dieses Lehrbuch befasst sich leicht verständlich mit der Theorie der Kalman-Filterung. Die Autoren geben damit eine Einführung in Kalman-Filter und deren Anwendung für eingebettete Systeme. Zusätzlich wird anhand konkreter Praxisbeispiele der Kalman-Filterentwurf demonstriert ¿ Teilschritte werden im Buch ausführlich erläutert.Kalman-Filter sind die erste Wahl, um Störsignale auf den Sensorsignalen zu eliminieren. Dies ist von besonderer Bedeutung, da viele technische Systeme ihre prozessrelevanten Informationen über Sensoren gewinnen. Jeder Messwert eines Sensors weißt jedoch aufgrund verschiedener Ursachen einen Messfehler auf. Würde ein System nur auf Basis dieser ungenauen Sensorinformationen arbeiten, so wären viele Anwendungen, wie zum Beispiel ein Navigationssystem oder autonome arbeitende Systeme, nicht möglich.Das Buch ist geeignet für interessierte Bachelor- und Master-Studierende der Fachrichtungen Informatik, Maschinenbau, Elektrotechnik undMechatronik. Ebenso ist das Buch eine Hilfe für Ingenieure und Wissenschaftler, die ein Kalman-Filter z. B. für die Datenfusion oder die Schätzung unbekannter Größen in Echtzeitanwendungen einsetzen möchten.

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 avariety 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.

Stochastic elasticity is a fast developing field that combines nonlinear elasticity and stochastic theories in order to significantly improve model predictions by accounting for uncertainties in the mechanical responses of materials. However, in contrast to the tremendous development of computational methods for large-scale problems, which have been proposed and implemented extensively in recent years, at the fundamental level, there is very little understanding of the uncertainties in the behaviour of elastic materials under large strains.Based on the idea that every large-scale problem starts as a small-scale data problem, this book combines fundamental aspects of finite (large-strain) elasticity and probability theories, which are prerequisites for the quantification of uncertainties in the elastic responses of soft materials. The problems treated in this book are drawn from the analytical continuum mechanics literature and incorporate random variables as basic concepts along with mechanical stresses and strains. Such problems are interesting in their own right but they are also meant to inspire further thinking about how stochastic extensions can be formulated before they can be applied to more complex physical systems.

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 volume presents extensive research devoted to a broad spectrum of mathematics with emphasis on interdisciplinary aspects of Optimization and Probability. Chapters also emphasize applications to Data Science, a timely field with a high impact in our modern society. The discussion presents modern, state-of-the-art, research results and advances in areas including non-convex optimization, decentralized distributed convex optimization, topics on surrogate-based reduced dimension global optimization in process systems engineering, the projection of a point onto a convex set, optimal sampling for learning sparse approximations in high dimensions, the split feasibility problem, higher order embeddings, codifferentials and quasidifferentials of the expectation of nonsmooth random integrands, adjoint circuit chains associated with a random walk, analysis of the trade-off between sample size and precision in truncated ordinary least squares, spatial deep learning, efficient location-based tracking for IoT devices using compressive sensing and machine learning techniques, and nonsmooth mathematical programs with vanishing constraints in Banach spaces.The book is a valuable source for graduate students as well as researchers working on Optimization, Probability and their various interconnections with a variety of other areas.Chapter 12 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

This book extends the theory and applications of random evolutions to semi-Markov random media in discrete time, essentially focusing on semi-Markov chains as switching or driving processes. After giving the definitions of discrete-time semi-Markov chains and random evolutions, it presents the asymptotic theory in a functional setting, including weak convergence results in the series scheme, and their extensions in some additional directions, including reduced random media, controlled processes, and optimal stopping. Finally, applications of discrete-time semi-Markov random evolutions in epidemiology and financial mathematics are discussed. This book will be of interest to researchers and graduate students in applied mathematics and statistics, and other disciplines, including engineering, epidemiology, finance and economics, who are concerned with stochastic models of systems.

This book contains contributions from the participants of the international conference ¿Foundations of Modern Statistics¿ which took place at Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, during November 6¿8, 2019, and at Higher School of Economics (HSE University), Moscow, during November 30, 2019. The events were organized in honor of Professor Vladimir Spokoiny on the occasion of his 60th birthday. Vladimir Spokoiny has pioneered the field of adaptive statistical inference and contributed to a variety of its applications. His more than 30 years of research in the field of mathematical statistics had a great influence on the development of the mathematical theory of statistics to its present state. It has inspired many young researchers to start their research in this exciting field of mathematics. The papers contained in this book reflect the broad field of interests of Vladimir Spokoiny: optimal rates and non-asymptotic bounds in nonparametrics, Bayes approaches from a frequentist point of view, optimization, signal processing, and statistical theory motivated by models in applied fields. Materials prepared by famous scientists contain original scientific results, which makes the publication valuable for researchers working in these fields. The book concludes by a conversation of Vladimir Spokoiny with Markus Rei¿ and Enno Mammen. This interview gives some background on the life of Vladimir Spokoiny and his many scientific interests and motivations.

This book introduces the concept of ¿bespoke learning¿, a new mechanistic approach that makes it possible to generate values of an output variable at each designated value of an associated input variable. Here the output variable generally provides information about the system¿s behaviour/structure, and the aim is to learn the input-output relationship, even though little to no information on the output is available, as in multiple real-world problems. Once the output values have been bespoke-learnt, the originally-absent training set of input-output pairs becomes available, so that (supervised) learning of the sought inter-variable relation is then possible. Three ways of undertaking such bespoke learning are offered: by tapping into system dynamics in generic dynamical systems, to learn the function that causes the system¿s evolution; by comparing realisations of a random graph variable, given multivariate time series datasets of disparate temporal coverage; and by designing maximally information-availing likelihoods in static systems. These methodologies are applied to four different real-world problems: forecasting daily COVID-19 infection numbers; learning the gravitational mass density in a real galaxy; learning a sub-surface material density function; and predicting the risk of onset of a disease following bone marrow transplants. Primarily aimed at graduate and postgraduate students studying a field which includes facets of statistical learning, the book will also benefit experts working in a wide range of applications. The prerequisites are undergraduate level probability and stochastic processes, and preliminary ideas on Bayesian statistics.

This book offers an introduction to the field of stochastic analysis of Hermite processes. These selfsimilar stochastic processes with stationary increments live in a Wiener chaos and include the fractional Brownian motion, the only Gaussian process in this class. Using the Wiener chaos theory and multiple stochastic integrals, the book covers the main properties of Hermite processes and their multiparameter counterparts, the Hermite sheets. It delves into the probability distribution of these stochastic processes and their sample paths, while also presenting the basics of stochastic integration theory with respect to Hermite processes and sheets.The book goes beyond theory and provides a thorough analysis of physical models driven by Hermite noise, including the Hermite Ornstein-Uhlenbeck process and the solution to the stochastic heat equation driven by such a random perturbation. Moreover, it explores up-to-date topics central to current research in statistical inference for Hermite-driven models.

This book considers a broad range of areas from decision making methods applied in the contexts of Risk, Reliability and Maintenance (RRM). Intended primarily as an update of the 2015 book Multicriteria and Multiobjective Models for Risk, Reliability and Maintenance Decision Analysis, this edited work provides an integration of applied probability and decision making. Within applied probability, it primarily includes decision analysis and reliability theory, amongst other topics closely related to risk analysis and maintenance. In decision making, it includes multicriteria decision making/aiding (MCDM/A) methods and optimization models. Within MCDM, in addition to decision analysis, some of the topics related to mathematical programming areas are considered, such as multiobjective linear programming, multiobjective nonlinear programming, game theory and negotiations, and multiobjective optimization. Methods related to these topics have been applied to the context of RRM. In MCDA, several other methods are considered, such as outranking methods, rough sets and constructive approaches. The book addresses an innovative treatment of decision making in RRM, improving the integration of fundamental concepts from both areas of RRM and decision making. This is accomplished by presenting current research developments in decision making on RRM. Some pitfalls of decision models on practical applications on RRM are discussed and new approaches for overcoming those drawbacks are presented.

This book introduces the Learning-Augmented Network Optimization (LANO) paradigm, which interconnects network optimization with the emerging AI theory and algorithms and has been receiving a growing attention in network research. The authors present the topic based on a general stochastic network optimization model, and review several important theoretical tools that are widely adopted in network research, including convex optimization, the drift method, and mean-field analysis. The book then covers several popular learning-based methods, i.e., learning-augmented drift, multi-armed bandit and reinforcement learning, along with applications in networks where the techniques have been successfully applied. The authors also provide a discussion on potential future directions and challenges.

Focusing on comprehensive comparisons of the performance of stochastic optimization algorithms, this book provides an overview of the current approaches used to analyze algorithm performance in a range of common scenarios, while also addressing issues that are often overlooked. In turn, it shows how these issues can be easily avoided by applying the principles that have produced Deep Statistical Comparison and its variants. The focus is on statistical analyses performed using single-objective and multi-objective optimization data. At the end of the book, examples from a recently developed web-service-based e-learning tool (DSCTool) are presented. The tool provides users with all the functionalities needed to make robust statistical comparison analyses in various statistical scenarios.The book is intended for newcomers to the field and experienced researchers alike. For newcomers, it covers the basics of optimization and statistical analysis, familiarizing them with the subject matter before introducing the Deep Statistical Comparison approach. Experienced researchers can quickly move on to the content on new statistical approaches. The book is divided into three parts:Part I: Introduction to optimization, benchmarking, and statistical analysis - Chapters 2-4.Part II: Deep Statistical Comparison of meta-heuristic stochastic optimization algorithms - Chapters 5-7.Part III: Implementation and application of Deep Statistical Comparison - Chapter 8.

This volume collects selected papers from the Ninth High Dimensional Probability Conference, held virtually from June 15-19, 2020. These papers cover a wide range of topics and demonstrate how high-dimensional probability remains an active area of research with applications across many mathematical disciplines. Chapters are organized around four general topics: inequalities and convexity; limit theorems; stochastic processes; and high-dimensional statistics. High Dimensional Probability IX will be a valuable resource for researchers in this area.

This book characterizes the discrete event simulation and analysis using ExtendSim 10. It is a blend between theory and application leaning largely to the weight of the latter. Since the ExtendSim 8 version of the book (13 years ago) there has been significant improvements to ExtendSim, including the new Reliability library incorporated in this new, enhanced edition of the first book. There are two new chapters, one include a model simulating software reliability and inherent availability and the other is a guided project addressing the Launch Availability of a crew launch vehicle (CLV) for a limited launch window.For those unfamiliar with the first edition, there is coverage of just-enough queuing theory for building discrete event models, using the M/M/1 queuing problem involving warmup and steady-state phenomena, as well as methods for analysis and corrective adjustments. Probability distributions and their inverse transfers for random sampling are covered. The StatFit application is used for fitting and analyzing data, including goodness of fit testing. Also, there is an in-depth treatment of random number generators. A bank model is used to demonstrate hierarchical modeling and basic simulation animation. Advanced queuing processes are addressed using a circuit board production example. Detailed modeling is covered using a delivery system transfer depot handing packages for domestic delivery.

This book is an easy-to-read reference providing a link between functional analysis and diffusion processes. More precisely, the book takes readers to a mathematical crossroads of functional analysis (macroscopic approach), partial differential equations (mesoscopic approach), and probability (microscopic approach) via the mathematics needed for the hard parts of diffusion processes. This work brings these three fields of analysis together and provides a profound stochastic insight (microscopic approach) into the study of elliptic boundary value problems.The author does a massive study of diffusion processes from a broad perspective and explains mathematical matters in a more easily readable way than one usually would find. The book is amply illustrated; 14 tables and 141 figures are provided with appropriate captions in such a fashion that readers can easily understand powerful techniques of functional analysis for the study of diffusion processes in probability.The scope of the author's work has been and continues to be powerful methods of functional analysis for future research of elliptic boundary value problems and Markov processes via semigroups. A broad spectrum of readers can appreciate easily and effectively the stochastic intuition that this book conveys.  Furthermore, the book will serve as a sound basis both for researchers and for graduate students in pure and applied mathematics who are interested in a modern version of the classical potential theory and Markov processes.For advanced undergraduates working in functional analysis, partial differential equations, and probability, it provides an effective opening to these three interrelated fields of analysis. Beginning graduate students and mathematicians in the field looking for a coherent overview will find the book to be a helpful beginning. This work will be a major influence in a very broad field of study for a long time.

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 alsobe 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.