Stokastik

This volume presents a selection of texts that reflects the current research streams in probability, with an interest toward topics such as filtrations, Markov processes and Markov chains as well as large deviations, Stochastic Partial Differential equations, rough paths theory, quantum probabilities and percolation on graphs.The featured contributors are R. L. Karandikar and B. V. Rao, C. Leuridan, M. Vidmar, L. Miclo and P. Patie, A. Bernou, M.-E. Caballero and A. Rouault, J. Dedecker, F. Merlevède and E. Rio, F. Brosset, T. Klein, A. Lagnoux and P. Petit, C. Marinelli and L. Scarpa, C. Castaing, N. Marie and P. Raynaud de Fitte, S. Attal, J. Deschamps and C. Pellegrini, and N. Eisenbaum.

This book provides a captivating journey through the realms of classical and quantum systems as it unravels the profound influence that noise may have on their static and dynamic properties. The first part of the book offers succinct yet enlightening discussions on foundational topics related to noise. The second part focuses on a variety of applications, where a diverse spectrum of noise effects in physical systems comes to life, meticulously presented and thoughtfully analyzed. Whether you are a curious student or a dedicated researcher, this book is your key to gaining invaluable insights into noise effects in physical systems. ¿The book has the merit of presenting several topics scattered in the literature and could become a very useful reference.¿ Giovanni Jona-Lasinio, Sapienza ¿ Università di Roma, Italy

Unsupervised domain adaptation (UDA) is a challenging problem in machine learning where the model is trained on a source domain with labeled data and tested on a target domain with unlabeled data. In recent years, UDA has received significant attention from the research community due to its applicability in various real-world scenarios. This book provides a comprehensive review of state-of-the-art UDA methods and explores new variants of UDA that have the potential to advance the field.The book begins with a clear introduction to the UDA problem and is mainly organized into four technical sections, each focused on a specific piece of UDA research. The first section covers criterion optimization-based UDA, which aims to learn domain-invariant representations by minimizing the discrepancy between source and target domains. The second section discusses bi-classifier adversarial learning-based UDA, which creatively leverages adversarial learning by conducting a minimax game between the feature extractor and two task classifiers. The third section introduces source-free UDA, a novel UDA setting that does not require any raw data from the source domain. The fourth section presents active learning for UDA, which combines domain adaptation and active learning to reduce the amount of labeled data needed for adaptation.This book is suitable for researchers, graduate students, and practitioners who are interested in UDA and its applications in various fields, primarily in computer vision. The chapters are authored by leading experts in the field and provide a comprehensive and in-depth analysis of the current UDA methods and new directions for future research. With its broad coverage and cutting-edge research, this book is a valuable resource for anyone looking to advance their knowledge of UDA.

This book publishes select papers presented at the 4th International Conference on Frontiers in Industrial and Applied Mathematics (FIAM-2021), held at the Sant Longowal Institute of Engineering and Technology, Longowal, Punjab, India, from 21¿22 December 2021. Most of the papers deal with mathematical theory embedded with its applications to engineering and sciences. This book illustrates numerical simulation of scientific problems and the state-of-the-art research in industrial and applied mathematics, including various computational and modeling techniques with case studies and concrete examples. Graduate students and researchers, who are interested in real applications of mathematics in the areas of computational and theoretical fluid dynamics, solid mechanics, optimization and operations research, numerical analysis, bio-mathematics, fuzzy, control and systems theory, dynamical systems and nonlinear analysis, algebra and approximation theory, will find the book useful.

Petri nets model concurrent and distributed systems where active components communicate through the production and absorption of various kinds of resources. Although the dynamic properties of such systems may be very complex, they may sometimes be connected to the static structure of a Petri net. Many properties are decidable, but their complexity may be huge. It is often opportune to restrict oneself to classes of systems, to partial algorithms, and to similar but simpler properties. Instead of analysing a given system, it is also possible to search for a system satisfying some desired properties by construction. This comprehensive textbook/reference presents and discusses these issues in-depth in the context of one of the most fundamental Petri net models, called place/transition nets. The presentation is fortified by means of many examples and worked exercises. Among topics addressed: ¿ In which order may actions may be generated and scheduled? ¿ What states and configurations may be reached in a concurrent system? ¿ Which interesting classes of systems can be analysed relatively efficiently? ¿ Is it possible to synthesise a system of some class from its behaviour? ¿ How can systems be represented algebraically, compositionally, and concisely? This unique text, based on introductory as well as on advanced courses on distributed systems, will serve as an invaluable guide for students and (future) researchers interested in theoretical¿as well as in practical¿aspects of Petri nets and related system models. Eike Best has been a full professor (now retired) affiliated to Carl von Ossietzky Universität Oldenburg, Germany. Raymond Devillers has been a full professor (now retired) affiliated to Université Libre de Bruxelles, Belgium. The authors have a long record as collaborators in the fields of Petri nets and the semantics of concurrency.

Statistik und Wahrscheinlichkeit ¿ leicht gemachtist ein Lehr- und Nachschlagewerkfür Schule, Studium und Beruf.Das Buch ist zum Selbststudium geeignet.Alle wichtigen Begriffe, Regeln und Methodenwerden in verständlicher Form dargestellt,in Handlungsanweisungen übertragen undan Beispielen erläutertJedes Kapitel derDatenanalyseStatistikRegressionKombinatorikWahrscheinlichkeitDichte- und VerteilungsfunktionenTestkonzeptionHypothesentestsBayes-Statistikwird durch Übungsaufgaben mitvollständigem Lösungsweg komplettiert.

Kolmogorov equations are a fundamental bridge between the theory of partial differential equations and that of stochastic differential equations that arise in several research fields.This volume collects a selection of the talks given at the Cortona meeting by experts in both fields, who presented the most recent developments of the theory. Particular emphasis has been given to degenerate partial differential equations, Itô processes, applications to kinetic theory and to finance.

The book collects papers on several topics in probability and stochastic processes. These papers have been presented at a conference organised in honour of Professor Rajeeva L. Karandikar who turned 65 in 2021. He was a distinguished researcher and a teacher at the Indian Statistical Institute (ISI), Delhi Centre, for many years. He has been a multi-faceted academician, interacting with the Government of India and the industry. He has left an indelible mark in every endeavour of his and in his several different avatars¿be it in the ISI, in the industry or as Director of Chennai Mathematical Institute. This book will be useful to senior undergraduate and graduate students, as well as researchers in probability, statistics and related fields.

This book provides a systematic presentation of the major results in the field of the theory of k-out-of-n systems obtained in recent years and their applications for the reliability assessment of high-altitude unmanned platforms. Mathematical models, methods, and algorithms, presented in the book, will make a significant contribution to the development of reliability theory and the theoretical foundations of unmanned UAV-based aerial communications networks in the framework of the concept of creating the 5G and beyond networks. The book gives a description of new mathematical methods and approaches (based on decomposable semi-regenerative processes, simulation and machine learning methods, and inventory models) to the study of the complex k-out-of-n systems, which makes it possible to carry out numerical calculations of reliability indicators. Organized into five chapters, each chapter begins with a summary of the main definitions andresults contained in the chapter. The content of this book is based on the original results developed by the authors, many of which appear for the first time in book form.

Large numbers of studies of the KdV equation have appeared since the pioneering paper by Gardner, Greene, Kruskal, and Miura in 1967. Most of those works have employed the inverse spectral method for 1D Schrödinger operators or an advanced Fourier analysis. Although algebraic approaches have been discovered by HirotäSato and Marchenko independently, those have not been fully investigated and analyzed. The present book offers a new approach to the study of the KdV equation, which treats decaying initial data and oscillating data in a unified manner. The author¿s method is to represent the tau functions introduced by HirotäSato and developed by Segal¿Wilson later in terms of the Weyl¿Titchmarsh functions (WT functions, in short) for the underlying Schrödinger operators. The main result is stated by a class of WT functions satisfying some of the asymptotic behavior along a curve approaching the spectrum of the Schrödinger operators at +¿ in an order of -(n-1/2)for the nth KdV equation. This class contains many oscillating potentials (initial data) as well as decaying ones. Especially bounded smooth ergodic potentials are included, and under certain conditions on the potentials, the associated Schrödinger operators have dense point spectrum. This provides a mathematical foundation for the study of the soliton turbulence problem initiated by Zakharov, which was the author¿s motivation for extending the class of initial data in this book. A large class of almost periodic potentials is also included in these ergodic potentials. P. Deift has conjectured that any solutions to the KdV equation starting from nearly periodic initial data are almost periodic in time. Therefore, our result yields a foundation for this conjecture. For the reader¿s benefit, the author has included here (1) a basic knowledge of direct and inverse spectral problem for 1D Schrödinger operators, including the notion of the WT functions; (2)Satös Grassmann manifold method revised by Segal¿Wilson; and (3) basic results of ergodic Schrödinger operators.

This book introduces readers to the financial markets, derivatives, structured products and how the products are modelled and implemented by practitioners. In addition, it equips readers with the necessary knowledge of financial markets needed in order to work as product structurers, traders, sales or risk managers.This second edition substantially extends, updates and clarifies the previous edition. New materials and enhanced contents include, but not limited to, the role of central counterparties for derivatives transactions, the reference rates to replace LIBOR, risk-neutral modelling for futures and forward, discussions and analysis on risk-neutral framework and numéraires, discrete dividend modelling, variance reduction techniques for Monte Carlo method, finite difference method analysis, tree method, FX modelling, multi-name credit derivatives modelling, local volatility model, forward variance model and local-stochastic volatility model to reflect market practice.As the book seeks to unify the derivatives modelling and the financial engineering practice in the market, it will be of interest to financial practitioners and academic researchers alike. The book can also be used as a textbook for the following courses:¿ Financial Mathematics (undergraduate level)¿ Stochastic Modelling in Finance (postgraduate level)¿ Financial Markets and Derivatives (undergraduate level)¿ Structured Products and Solutions (undergraduate/postgraduate level)

This textbook invites readers to explore mathematical thinking by finding the beauty in the subject. With an accessible tone and stimulating puzzles, the author will convince curious non-mathematicians to continue their studies in the area. It has an expansive scope, covering everything from probability and graph theory to infinities and Newton¿s method. Many examples of proofs appear as well, offering readers the opportunity to explore these topics with the amount of rigor that suits them. Programming exercises in Python are also included to show how math behaves in action.Mathematical Thinking is an ideal textbook for transition courses aimed at undergraduates moving from lower level to more advanced topics, as well as for math recruitment and invitational courses at the freshman or sophomore level. It may also be of interest in computer science departments and can be used as a supplemental text for courses in discrete mathematics and graph theory.

Vector Quantization, a pioneering discretization method based on nearest neighbor search, emerged in the 1950s primarily in signal processing, electrical engineering, and information theory. Later in the 1960s, it evolved into an automatic classification technique for generating prototypes of extensive datasets. In modern terms, it can be recognized as a seminal contribution to unsupervised learning through the k-means clustering algorithm in data science.In contrast, Functional Quantization, a more recent area of study dating back to the early 2000s, focuses on the quantization of continuous-time stochastic processes viewed as random vectors in Banach function spaces. This book distinguishes itself by delving into the quantization of random vectors with values in a Banach space¿a unique feature of its content. Its main objectives are twofold: first, to offer a comprehensive and cohesive overview of the latest developments as well as several new results in optimal quantization theory, spanning both finite and infinite dimensions, building upon the advancements detailed in Graf and Luschgy's Lecture Notes volume. Secondly, it serves to demonstrate how optimal quantization can be employed as a space discretization method within probability theory and numerical probability, particularly in fields like quantitative finance. The main applications to numerical probability are the controlled approximation of regular and conditional expectations by quantization-based cubature formulas, with applications to time-space discretization of Markov processes, typically Brownian diffusions, by quantization trees.While primarily catering to mathematicians specializing in probability theory and numerical probability, this monograph also holds relevance for data scientists, electrical engineers involved in data transmission, and professionals in economics and logistics who are intrigued by optimal allocation problems.

This revised book provides an accessible presentation of concepts from probability theory, statistical methods, the design of experiments, and statistical quality control. It is shaped by the experience of the two teachers teaching statistical methods and concepts to engineering students. Practical examples and end-of-chapter exercises are the highlights of the text, as they are purposely selected from different fields. Statistical principles discussed in the book have a great relevance in several disciplines like economics, commerce, engineering, medicine, health care, agriculture, biochemistry, and textiles to mention a few.Organised into 16 chapters, the revised book discusses four major topics¿probability theory, statistical methods, the design of experiments, and statistical quality control. A large number of students with varied disciplinary backgrounds need a course in basics of statistics, the design of experiments and statistical quality control at an introductory level to pursue their discipline of interest. No previous knowledge of probability or statistics is assumed, but an understanding of calculus is a prerequisite. The whole book also serves as a master level introductory course in all the three topics, as required in textile engineering or industrial engineering.

The volume includes a collection of peer-reviewed contributions from among those presented at the FNE, the main conference organized every two years by the Mexican Statistical Society (AME), and the 2020 AME Virtual Meeting. Statistical research in Latin America is prolific and research networks span both within and outside the region. As much of the work is typically carried out and published in Spanish, a large portion of the interested public is denied access to interesting findings, and the goal of this volume is therefore to provide access to selected works from Mexican collaborators and their international research networks to a wider audience. It may be especially attractive to academics interested in the latest methodological advances, while professionals from other disciplines may also find value in these new tools for data analysis. In 2021, the conference broadly focused on the interdisciplinary aspects of Statistics.

This book presents a selection of peer-reviewed contributions to the fifth Bayesian Young Statisticians Meeting, BaYSM 2021, held virtually due to the COVID-19 pandemic on 1-3 September 2021. Despite all the challenges of an online conference, the meeting provided a valuable opportunity for early career researchers, including MSc students, PhD students, and postdocs to connect with the broader Bayesian community.The proceedings highlight many different topics in Bayesian statistics, presenting promising methodological approaches to address important challenges in a variety of applications. The book is intended for a broad audience of people interested in statistics, and provides a series of stimulating contributions on theoretical, methodological, and computational aspects of Bayesian statistics.

This book explains the basic theory of Hilbert C*-module in detail, covering a wide range of applications from generalized index to module framework. At the center of the book, the Beurling-Deny criterion is characterized between operator valued Dirichlet forms and quantum Markov semigroups, hence opening a new field of quantum probability research. The general scope of the book includes: basic theory of Hilbert C*-modules; generalized indices and module frames; operator valued Dirichlet forms; and quantum Markov semigroups.This book will be of value to scholars and graduate students in the fields of operator algebra, quantum probability and quantum information.

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.

Arbeitsbuch Stochastik (Einführung und Grundzüge der Maßtheorie)Dieses Arbeitsbuch enthält die Aufgaben, Hinweise, Lösungen und Lösungswege der Kapitel 2 bis 8 des Lehrbuchs Stochastik: Eine Einführung mit Grundzügen der Maßtheorie. Durch die Offenlegung der Lösungswege und der Lösungen ist das Werk bestens geeignet zum Selbststudium, zur Vorlesungsbegleitung und als Prüfungsvorbereitung.Das Werk umfasst mehr als 330 Übungsaufgaben, die in Verständnisfragen, Beweisaufgaben und Rechenaufgaben gegliedert sind, und ist in der vorliegenden zweiten Auflage auf die zweite Auflage des Hauptwerks angepasst.

This book is devoted to the study of multivariate discrete q-distributions, which is greatly facilitated by existing multivariate q-sequences and q-functions. Classical multivariate discrete distributions are defined on a sequence of independent and identically distributed Bernoulli trials, with either being a success of a certain rank (level) or a failure. The author relaxes the assumption that the probability of success of a trial is constant by assuming that it varies geometrically with the number of trials and/or the number of successes. The latter is advantageous in the sense that it permits incorporating the experience gained from the previous trials and/or successes, which leads to multivariate discrete q-distributions. Furthermore, q-multinomial and negative q-multinomial formulae are obtained. Next, the book addresses q-multinomial and negative q-multinomial distributions of the first and second kind. The author also examines multiple q-Polya urn model, multivariate q-Polya and inverse q-Polya distributions. Presents definitions and theorems that highlight key concepts and worked examples to illustrate the various applicationsContains numerous exercises at varying levels of difficulty that consolidate the presented concepts and resultsIncludes hints and answers to all exercises via the appendix and is supplemented with an Instructor's Solution Manual

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 graduate textbook provides an alternative to discrete event simulation. It describes how to formulate discrete event systems, how to convert them into Markov chains, and how to calculate their transient and equilibrium probabilities. The most appropriate methods for finding these probabilities are described in some detail, and templates for efficient algorithms are provided. These algorithms can be executed on any laptop, even in cases where the Markov chain has hundreds of thousands of states. This book features the probabilistic interpretation of Gaussian elimination, a concept that unifies many of the topics covered, such as embedded Markov chains and matrix analytic methods.The material provided should aid practitioners significantly to solve their problems. This book also provides an interesting approach to teaching courses of stochastic processes.