Classical Potential Theory And Its Probabilistic Counterpart
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Classical Potential Theory And Its Probabilistic Counterpart
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Author : J. L. Doob
language : en
Publisher: Springer Science & Business Media
Release Date : 1984-01-30
Classical Potential Theory And Its Probabilistic Counterpart written by J. L. Doob and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984-01-30 with Mathematics categories.
Potential theory and certain aspects of probability theory are intimately related, perhaps most obviously in that the transition function determining a Markov process can be used to define the Green function of a potential theory. Thus it is possible to define and develop many potential theoretic concepts probabilistically, a procedure potential theorists observe withjaun diced eyes in view of the fact that now as in the past their subject provides the motivation for much of Markov process theory. However that may be it is clear that certain concepts in potential theory correspond closely to concepts in probability theory, specifically to concepts in martingale theory. For example, superharmonic functions correspond to supermartingales. More specifically: the Fatou type boundary limit theorems in potential theory correspond to supermartingale convergence theorems; the limit properties of monotone sequences of superharmonic functions correspond surprisingly closely to limit properties of monotone sequences of super martingales; certain positive superharmonic functions [supermartingales] are called "potentials," have associated measures in their respective theories and are subject to domination principles (inequalities) involving the supports of those measures; in each theory there is a reduction operation whose properties are the same in the two theories and these reductions induce sweeping (balayage) of the measures associated with potentials, and so on.
Classical Potential Theory And Its Probabilistic Counterpart
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Author : J. L. Doob
language : en
Publisher: Springer
Release Date : 2011-09-26
Classical Potential Theory And Its Probabilistic Counterpart written by J. L. Doob and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-09-26 with Mathematics categories.
Potential theory and certain aspects of probability theory are intimately related, perhaps most obviously in that the transition function determining a Markov process can be used to define the Green function of a potential theory. Thus it is possible to define and develop many potential theoretic concepts probabilistically, a procedure potential theorists observe withjaun diced eyes in view of the fact that now as in the past their subject provides the motivation for much of Markov process theory. However that may be it is clear that certain concepts in potential theory correspond closely to concepts in probability theory, specifically to concepts in martingale theory. For example, superharmonic functions correspond to supermartingales. More specifically: the Fatou type boundary limit theorems in potential theory correspond to supermartingale convergence theorems; the limit properties of monotone sequences of superharmonic functions correspond surprisingly closely to limit properties of monotone sequences of super martingales; certain positive superharmonic functions [supermartingales] are called "potentials," have associated measures in their respective theories and are subject to domination principles (inequalities) involving the supports of those measures; in each theory there is a reduction operation whose properties are the same in the two theories and these reductions induce sweeping (balayage) of the measures associated with potentials, and so on.
Classical Potential Theory And Its Probabilistic Counterpart
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Author : Joseph L. Doob
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Classical Potential Theory And Its Probabilistic Counterpart written by Joseph L. Doob and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
From the reviews: "This huge book written in several years by one of the few mathematicians able to do it, appears as a precise and impressive study (not very easy to read) of this bothsided question that replaces, in a coherent way, without being encyclopaedic, a large library of books and papers scattered without a uniform language. Instead of summarizing the author gives his own way of exposition with original complements. This requires no preliminary knowledge. ...The purpose which the author explains in his introduction, i.e. a deep probabilistic interpretation of potential theory and a link between two great theories, appears fulfilled in a masterly manner". M. Brelot in Metrika (1986)
The Splendors And Miseries Of Martingales
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Author : Laurent Mazliak
language : en
Publisher: Springer Nature
Release Date : 2022-10-17
The Splendors And Miseries Of Martingales written by Laurent Mazliak and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-10-17 with Mathematics categories.
Over the past eighty years, martingales have become central in the mathematics of randomness. They appear in the general theory of stochastic processes, in the algorithmic theory of randomness, and in some branches of mathematical statistics. Yet little has been written about the history of this evolution. This book explores some of the territory that the history of the concept of martingales has transformed. The historian of martingales faces an immense task. We can find traces of martingale thinking at the very beginning of probability theory, because this theory was related to gambling, and the evolution of a gambler’s holdings as a result of following a particular strategy can always be understood as a martingale. More recently, in the second half of the twentieth century, martingales became important in the theory of stochastic processes at the very same time that stochastic processes were becoming increasingly important in probability, statistics and more generally in various applied situations. Moreover, a history of martingales, like a history of any other branch of mathematics, must go far beyond an account of mathematical ideas and techniques. It must explore the context in which the evolution of ideas took place: the broader intellectual milieux of the actors, the networks that already existed or were created by the research, even the social and political conditions that favored or hampered the circulation and adoption of certain ideas. This books presents a stroll through this history, in part a guided tour, in part a random walk. First, historical studies on the period from 1920 to 1950 are presented, when martingales emerged as a distinct mathematical concept. Then insights on the period from 1950 into the 1980s are offered, when the concept showed its value in stochastic processes, mathematical statistics, algorithmic randomness and various applications.
Game Theoretic Foundations For Probability And Finance
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Author : Glenn Shafer
language : en
Publisher: John Wiley & Sons
Release Date : 2019-03-21
Game Theoretic Foundations For Probability And Finance written by Glenn Shafer and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-03-21 with Business & Economics categories.
Game-theoretic probability and finance come of age Glenn Shafer and Vladimir Vovk’s Probability and Finance, published in 2001, showed that perfect-information games can be used to define mathematical probability. Based on fifteen years of further research, Game-Theoretic Foundations for Probability and Finance presents a mature view of the foundational role game theory can play. Its account of probability theory opens the way to new methods of prediction and testing and makes many statistical methods more transparent and widely usable. Its contributions to finance theory include purely game-theoretic accounts of Ito’s stochastic calculus, the capital asset pricing model, the equity premium, and portfolio theory. Game-Theoretic Foundations for Probability and Finance is a book of research. It is also a teaching resource. Each chapter is supplemented with carefully designed exercises and notes relating the new theory to its historical context. Praise from early readers “Ever since Kolmogorov's Grundbegriffe, the standard mathematical treatment of probability theory has been measure-theoretic. In this ground-breaking work, Shafer and Vovk give a game-theoretic foundation instead. While being just as rigorous, the game-theoretic approach allows for vast and useful generalizations of classical measure-theoretic results, while also giving rise to new, radical ideas for prediction, statistics and mathematical finance without stochastic assumptions. The authors set out their theory in great detail, resulting in what is definitely one of the most important books on the foundations of probability to have appeared in the last few decades.” – Peter Grünwald, CWI and University of Leiden “Shafer and Vovk have thoroughly re-written their 2001 book on the game-theoretic foundations for probability and for finance. They have included an account of the tremendous growth that has occurred since, in the game-theoretic and pathwise approaches to stochastic analysis and in their applications to continuous-time finance. This new book will undoubtedly spur a better understanding of the foundations of these very important fields, and we should all be grateful to its authors.” – Ioannis Karatzas, Columbia University
Quadrature Domains And Their Applications
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Author : Peter Ebenfelt
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-03-10
Quadrature Domains And Their Applications written by Peter Ebenfelt and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-03-10 with Mathematics categories.
Quadrature domains were singled out about 30 years ago by D. Aharonov and H.S. Shapiro in connection with an extremal problem in function theory. Since then, a series of coincidental discoveries put this class of planar domains at the center of crossroads of several quite independent mathematical theories, e.g., potential theory, Riemann surfaces, inverse problems, holomorphic partial differential equations, fluid mechanics, operator theory. The volume is devoted to recent advances in the theory of quadrature domains, illustrating well the multi-facet aspects of their nature. The book contains a large collection of open problems pertaining to the general theme of quadrature domains.
Uncertainty In Economics
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Author : Julia Köhn
language : en
Publisher: Springer
Release Date : 2017-07-04
Uncertainty In Economics written by Julia Köhn and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-07-04 with Business & Economics categories.
In this book the author develops a new approach to uncertainty in economics, which calls for a fundamental change in the methodology of economics. It provides a comprehensive overview and critical appraisal of the economic theory of uncertainty and shows that uncertainty was originally conceptualized both as an epistemic and an ontological problem. As a result of the economic professions’ attempt to become acknowledged as a science, the more problematic aspect of ontological uncertainty has been neglected and the subjective probability approach to uncertainty became dominant in economic theory. A careful analysis of ontological theories of uncertainty explains the blindness of modern economics to economic phenomena such as instability, slumps or excessive booms. Based on these findings the author develops a new approach that legitimizes a New Uncertainty Paradigm in economics.
Interaction Between Functional Analysis Harmonic Analysis And Probability
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Author : Nigel Kalton
language : en
Publisher: CRC Press
Release Date : 1995-10-12
Interaction Between Functional Analysis Harmonic Analysis And Probability written by Nigel Kalton and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995-10-12 with Mathematics categories.
Based on a conference on the interaction between functional analysis, harmonic analysis and probability theory, this work offers discussions of each distinct field, and integrates points common to each. It examines developments in Fourier analysis, interpolation theory, Banach space theory, probability, probability in Banach spaces, and more.
Functional Analysis For Probability And Stochastic Processes
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Author : Adam Bobrowski
language : en
Publisher: Cambridge University Press
Release Date : 2005-08-11
Functional Analysis For Probability And Stochastic Processes written by Adam Bobrowski and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-08-11 with Mathematics categories.
This text is designed both for students of probability and stochastic processes, and for students of functional analysis. For the reader not familiar with functional analysis a detailed introduction to necessary notions and facts is provided. However, this is not a straight textbook in functional analysis; rather, it presents some chosen parts of functional analysis that can help understand ideas from probability and stochastic processes. The subjects range from basic Hilbert and Banach spaces, through weak topologies and Banach algebras, to the theory of semigroups of bounded linear operators. Numerous standard and non-standard examples and exercises make the book suitable as a course textbook or for self-study.
Probability Statistics And Stochastic Processes For Engineers And Scientists
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Author : Aliakbar Montazer Haghighi
language : en
Publisher: CRC Press
Release Date : 2020-07-14
Probability Statistics And Stochastic Processes For Engineers And Scientists written by Aliakbar Montazer Haghighi and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-07-14 with Mathematics categories.
Featuring recent advances in the field, this new textbook presents probability and statistics, and their applications in stochastic processes. This book presents key information for understanding the essential aspects of basic probability theory and concepts of reliability as an application. The purpose of this book is to provide an option in this field that combines these areas in one book, balances both theory and practical applications, and also keeps the practitioners in mind. Features Includes numerous examples using current technologies with applications in various fields of study Offers many practical applications of probability in queueing models, all of which are related to the appropriate stochastic processes (continuous time such as waiting time, and fuzzy and discrete time like the classic Gambler’s Ruin Problem) Presents different current topics like probability distributions used in real-world applications of statistics such as climate control and pollution Different types of computer software such as MATLAB®, Minitab, MS Excel, and R as options for illustration, programing and calculation purposes and data analysis Covers reliability and its application in network queues