Infinite Divisibility Of Probability Distributions On The Real Line
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Infinite Divisibility Of Probability Distributions On The Real Line
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Author : Fred W. Steutel
language : en
Publisher: CRC Press
Release Date : 2003-10-03
Infinite Divisibility Of Probability Distributions On The Real Line written by Fred W. Steutel and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-10-03 with Mathematics categories.
Infinite Divisibility of Probability Distributions on the Real Line reassesses classical theory and presents new developments, while focusing on divisibility with respect to convolution or addition of independent random variables. This definitive, example-rich text supplies approximately 100 examples to correspond with all major chapter topics and reviews infinite divisibility in light of the central limit problem. It contrasts infinite divisibility with finite divisibility, discusses the preservation of infinite divisibility under mixing for many classes of distributions, and investigates self-decomposability and stability on the nonnegative reals, nonnegative integers, and the reals.
Advances In Heavy Tailed Risk Modeling
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Author : Gareth W. Peters
language : en
Publisher: John Wiley & Sons
Release Date : 2015-05-21
Advances In Heavy Tailed Risk Modeling written by Gareth W. Peters 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 2015-05-21 with Mathematics categories.
ADVANCES IN HEAVY TAILED RISK MODELING A cutting-edge guide for the theories, applications, and statistical methodologies essential to heavy tailed risk modeling Focusing on the quantitative aspects of heavy tailed loss processes in operational risk and relevant insurance analytics, Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk presents comprehensive coverage of the latest research on the theories and applications in risk measurement and modeling techniques. Featuring a unique balance of mathematical and statistical perspectives, the handbook begins by introducing the motivation for heavy tailed risk processes. A companion with Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk, the handbook provides a complete framework for all aspects of operational risk management and includes: Clear coverage on advanced topics such as splice loss models, extreme value theory, heavy tailed closed form loss distribution approach models, flexible heavy tailed risk models, risk measures, and higher order asymptotic approximations of risk measures for capital estimation An exploration of the characterization and estimation of risk and insurance modeling, which includes sub-exponential models, alpha-stable models, and tempered alpha stable models An extended discussion of the core concepts of risk measurement and capital estimation as well as the details on numerical approaches to evaluation of heavy tailed loss process model capital estimates Numerous detailed examples of real-world methods and practices of operational risk modeling used by both financial and non-financial institutions Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk is an excellent reference for risk management practitioners, quantitative analysts, financial engineers, and risk managers. The handbook is also useful for graduate-level courses on heavy tailed processes, advanced risk management, and actuarial science.
Topics In Infinitely Divisible Distributions And L Vy Processes Revised Edition
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Author : Alfonso Rocha-Arteaga
language : en
Publisher: Springer Nature
Release Date : 2019-11-02
Topics In Infinitely Divisible Distributions And L Vy Processes Revised Edition written by Alfonso Rocha-Arteaga and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-11-02 with Mathematics categories.
This book deals with topics in the area of Lévy processes and infinitely divisible distributions such as Ornstein-Uhlenbeck type processes, selfsimilar additive processes and multivariate subordination. These topics are developed around a decreasing chain of classes of distributions Lm, m = 0,1,...,∞, from the class L0 of selfdecomposable distributions to the class L∞ generated by stable distributions through convolution and convergence. The book is divided into five chapters. Chapter 1 studies basic properties of Lm classes needed for the subsequent chapters. Chapter 2 introduces Ornstein-Uhlenbeck type processes generated by a Lévy process through stochastic integrals based on Lévy processes. Necessary and sufficient conditions are given for a generating Lévy process so that the OU type process has a limit distribution of Lm class. Chapter 3 establishes the correspondence between selfsimilar additive processes and selfdecomposable distributions and makes a close inspection of the Lamperti transformation, which transforms selfsimilar additive processes and stationary type OU processes to each other. Chapter 4 studies multivariate subordination of a cone-parameter Lévy process by a cone-valued Lévy process. Finally, Chapter 5 studies strictly stable and Lm properties inherited by the subordinated process in multivariate subordination. In this revised edition, new material is included on advances in these topics. It is rewritten as self-contained as possible. Theorems, lemmas, propositions, examples and remarks were reorganized; some were deleted and others were newly added. The historical notes at the end of each chapter were enlarged. This book is addressed to graduate students and researchers in probability and mathematical statistics who are interested in learning more on Lévy processes and infinitely divisible distributions.
Counterexamples In Probability
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Author : Jordan M. Stoyanov
language : en
Publisher: Courier Corporation
Release Date : 2014-01-15
Counterexamples In Probability written by Jordan M. Stoyanov and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with Mathematics categories.
"While most mathematical examples illustrate the truth of a statement, counterexamples demonstrate a statement's falsity. Enjoyable topics of study, counterexamples are valuable tools for teaching and learning. The definitive book on the subject in regards to probability, this third edition features the author's revisions and corrections plus a substantial new appendix. 2013 edition"--
Jump Sdes And The Study Of Their Densities
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Author : Arturo Kohatsu-Higa
language : en
Publisher: Springer
Release Date : 2019-08-13
Jump Sdes And The Study Of Their Densities written by Arturo Kohatsu-Higa and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-08-13 with Mathematics categories.
The present book deals with a streamlined presentation of Lévy processes and their densities. It is directed at advanced undergraduates who have already completed a basic probability course. Poisson random variables, exponential random variables, and the introduction of Poisson processes are presented first, followed by the introduction of Poisson random measures in a simple case. With these tools the reader proceeds gradually to compound Poisson processes, finite variation Lévy processes and finally one-dimensional stable cases. This step-by-step progression guides the reader into the construction and study of the properties of general Lévy processes with no Brownian component. In particular, in each case the corresponding Poisson random measure, the corresponding stochastic integral, and the corresponding stochastic differential equations (SDEs) are provided. The second part of the book introduces the tools of the integration by parts formula for jump processes in basic settings and first gradually provides the integration by parts formula in finite-dimensional spaces and gives a formula in infinite dimensions. These are then applied to stochastic differential equations in order to determine the existence and some properties of their densities. As examples, instances of the calculations of the Greeks in financial models with jumps are shown. The final chapter is devoted to the Boltzmann equation.
Fundamental Aspects Of Operational Risk And Insurance Analytics
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Author : Marcelo G. Cruz
language : en
Publisher: John Wiley & Sons
Release Date : 2015-01-20
Fundamental Aspects Of Operational Risk And Insurance Analytics written by Marcelo G. Cruz 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 2015-01-20 with Mathematics categories.
A one-stop guide for the theories, applications, and statistical methodologies essential to operational risk Providing a complete overview of operational risk modeling and relevant insurance analytics, Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk offers a systematic approach that covers the wide range of topics in this area. Written by a team of leading experts in the field, the handbook presents detailed coverage of the theories, applications, and models inherent in any discussion of the fundamentals of operational risk, with a primary focus on Basel II/III regulation, modeling dependence, estimation of risk models, and modeling the data elements. Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk begins with coverage on the four data elements used in operational risk framework as well as processing risk taxonomy. The book then goes further in-depth into the key topics in operational risk measurement and insurance, for example diverse methods to estimate frequency and severity models. Finally, the book ends with sections on specific topics, such as scenario analysis; multifactor modeling; and dependence modeling. A unique companion with Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk, the handbook also features: Discussions on internal loss data and key risk indicators, which are both fundamental for developing a risk-sensitive framework Guidelines for how operational risk can be inserted into a firm’s strategic decisions A model for stress tests of operational risk under the United States Comprehensive Capital Analysis and Review (CCAR) program A valuable reference for financial engineers, quantitative analysts, risk managers, and large-scale consultancy groups advising banks on their internal systems, the handbook is also useful for academics teaching postgraduate courses on the methodology of operational risk.
Loss Models
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Author : Stuart A. Klugman
language : en
Publisher: John Wiley & Sons
Release Date : 2013-08-05
Loss Models written by Stuart A. Klugman 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 2013-08-05 with Business & Economics categories.
An essential resource for constructing and analyzing advanced actuarial models Loss Models: Further Topics presents extended coverage of modeling through the use of tools related to risk theory, loss distributions, and survival models. The book uses these methods to construct and evaluate actuarial models in the fields of insurance and business. Providing an advanced study of actuarial methods, the book features extended discussions of risk modeling and risk measures, including Tail-Value-at-Risk. Loss Models: Further Topics contains additional material to accompany the Fourth Edition of Loss Models: From Data to Decisions, such as: Extreme value distributions Coxian and related distributions Mixed Erlang distributions Computational and analytical methods for aggregate claim models Counting processes Compound distributions with time-dependent claim amounts Copula models Continuous time ruin models Interpolation and smoothing The book is an essential reference for practicing actuaries and actuarial researchers who want to go beyond the material required for actuarial qualification. Loss Models: Further Topics is also an excellent resource for graduate students in the actuarial field.
Exercises In Statistical Reasoning
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Author : Michael R. Schwob
language : en
Publisher: CRC Press
Release Date : 2025-04-07
Exercises In Statistical Reasoning written by Michael R. Schwob and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-04-07 with Mathematics categories.
Students cultivate learning techniques in school that emphasize procedural problem solving and rote memorization. This leads to efficient problem solving for familiar problems. However, conducting novel research is an exercise in creative problem solving that is at odds with a procedural approach; it requires thinking deeply about the topic and crafting solutions to unique problems. It is not easy to move from a topic-based, carefully curated curriculum to the daunting world of independent research, where solutions are unknown and may not even exist. In developing this book, we considered our experiences as graduate students that faced this transition. Exercises in Statistical Reasoning is a collection of exercises designed to strengthen creative problem-solving skills. The exercises are designed to encourage readers to understand the key points of a problem while seeking knowledge, rather than separating out these two activities. To complete the exercises, readers may need to reference the literature, which is how research-based knowledge is often acquired. Features of the Exercises The exercises are self-contained, though several build upon concepts from previous problems. Each exercise opens with a brief introduction that emphasizes the relevance of the content. Then, the problem statement is presented as a series of intermediate questions. For each exercise, we suggest one possible solution, though many may exist. Following each solution, we discuss the historical background of the content and points of interest. For many exercises, a brief demonstration is provided that illustrates relevant concepts. There is an abundance of high-quality textbooks that cover a vast range of statistical topics. However, there is also a lack of texts that focus on the development of problem-solving techniques that are required for conducting novel statistical research. We believe that this book helps fill the gap. Any reader familiar with graduate-level classical and Bayesian statistics may use this book. The goal is to provide a resource that such students can use to ease their transition to conducting novel research.
Wiener Chaos Moments Cumulants And Diagrams
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Author : Giovanni Peccati
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-04-06
Wiener Chaos Moments Cumulants And Diagrams written by Giovanni Peccati 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 2011-04-06 with Mathematics categories.
The concept of Wiener chaos generalizes to an infinite-dimensional setting the properties of orthogonal polynomials associated with probability distributions on the real line. It plays a crucial role in modern probability theory, with applications ranging from Malliavin calculus to stochastic differential equations and from probabilistic approximations to mathematical finance. This book is concerned with combinatorial structures arising from the study of chaotic random variables related to infinitely divisible random measures. The combinatorial structures involved are those of partitions of finite sets, over which Möbius functions and related inversion formulae are defined. This combinatorial standpoint (which is originally due to Rota and Wallstrom) provides an ideal framework for diagrams, which are graphical devices used to compute moments and cumulants of random variables. Several applications are described, in particular, recent limit theorems for chaotic random variables. An Appendix presents a computer implementation in MATHEMATICA for many of the formulae.
L Vy Matters V
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Author : Lars Nørvang Andersen
language : en
Publisher: Springer
Release Date : 2015-10-24
L Vy Matters V written by Lars Nørvang Andersen and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-10-24 with Mathematics categories.
This three-chapter volume concerns the distributions of certain functionals of Lévy processes. The first chapter, by Makoto Maejima, surveys representations of the main sub-classes of infinitesimal distributions in terms of mappings of certain Lévy processes via stochastic integration. The second chapter, by Lars Nørvang Andersen, Søren Asmussen, Peter W. Glynn and Mats Pihlsgård, concerns Lévy processes reflected at two barriers, where reflection is formulated à la Skorokhod. These processes can be used to model systems with a finite capacity, which is crucial in many real life situations, a most important quantity being the overflow or the loss occurring at the upper barrier. If a process is killed when crossing the boundary, a natural question concerns its lifetime. Deep formulas from fluctuation theory are the key to many classical results, which are reviewed in the third chapter by Frank Aurzada and Thomas Simon. The main part, however, discusses recent advances and developments in the setting where the process is given either by the partial sum of a random walk or the integral of a Lévy process.