Queues and L vy Fluctuation Theory

Author: Krzysztof Dębicki
Publisher: Springer
ISBN: 3319206931
Format: PDF
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The book provides an extensive introduction to queueing models driven by Lévy-processes as well as a systematic account of the literature on Lévy-driven queues. The objective is to make the reader familiar with the wide set of probabilistic techniques that have been developed over the past decades, including transform-based techniques, martingales, rate-conservation arguments, change-of-measure, importance sampling, and large deviations. On the application side, it demonstrates how Lévy traffic models arise when modelling current queueing-type systems (as communication networks) and includes applications to finance. Queues and Lévy Fluctuation Theory will appeal to postgraduate students and researchers in mathematics, computer science, and electrical engineering. Basic prerequisites are probability theory and stochastic processes.

Fluctuations of L vy Processes with Applications

Author: Andreas Kyprianou
Publisher: Springer Science & Business Media
ISBN: 3642376320
Format: PDF, Docs
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Lévy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their application appears in the theory of many areas of classical and modern stochastic processes including storage models, renewal processes, insurance risk models, optimal stopping problems, mathematical finance, continuous-state branching processes and positive self-similar Markov processes. This textbook is based on a series of graduate courses concerning the theory and application of Lévy processes from the perspective of their path fluctuations. Central to the presentation is the decomposition of paths in terms of excursions from the running maximum as well as an understanding of short- and long-term behaviour. The book aims to be mathematically rigorous while still providing an intuitive feel for underlying principles. The results and applications often focus on the case of Lévy processes with jumps in only one direction, for which recent theoretical advances have yielded a higher degree of mathematical tractability. The second edition additionally addresses recent developments in the potential analysis of subordinators, Wiener-Hopf theory, the theory of scale functions and their application to ruin theory, as well as including an extensive overview of the classical and modern theory of positive self-similar Markov processes. Each chapter has a comprehensive set of exercises.

Random Processes by Example

Author: Mikhail Lifshits
Publisher: World Scientific
ISBN: 9814522309
Format: PDF, ePub, Mobi
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This volume first introduces the mathematical tools necessary for understanding and working with a broad class of applied stochastic models. The toolbox includes Gaussian processes, independently scattered measures such as Gaussian white noise and Poisson random measures, stochastic integrals, compound Poisson, infinitely divisible and stable distributions and processes. Next, it illustrates general concepts by handling a transparent but rich example of a “teletraffic model”. A minor tuning of a few parameters of the model leads to different workload regimes, including Wiener process, fractional Brownian motion and stable Lévy process. The simplicity of the dependence mechanism used in the model enables us to get a clear understanding of long and short range dependence phenomena. The model also shows how light or heavy distribution tails lead to continuous Gaussian processes or to processes with jumps in the limiting regime. Finally, in this volume, readers will find discussions on the multivariate extensions that admit a variety of completely different applied interpretations. The reader will quickly become familiar with key concepts that form a language for many major probabilistic models of real world phenomena but are often neglected in more traditional courses of stochastic processes. Contents:Preliminaries:Random Variables: A SummaryFrom Poisson to Stable VariablesLimit Theorems for Sums and Domains of AttractionRandom VectorsRandom Processes:Random Processes: Main ClassesExamples of Gaussian Random ProcessesRandom Measures and Stochastic IntegralsLimit Theorems for Poisson IntegralsLévy ProcessesSpectral RepresentationsConvergence of Random ProcessesTeletraffic Models:A Model of Service SystemLimit Theorems for the WorkloadMicropulse ModelSpacial Extensions Readership: Graduate students and researchers in probability & statistics. Keywords:Fractional Brownian Motion;Gaussian Process;Independently Scattered Measure;Lévy Process;Limit Theorem;Long Range Dependence;Micropulse Model;Poisson Random Measure;Random Process;Stable Process;Stochastic Process;Teletraffic Model;White Noise;Wiener ProcessKey Features:A thorough choice of self-contained material packed in a small volume enabling the reader to focus on really important issues and reach the frontline of research in a pretty short timeMain examples explaining the theory originating from the modern research fieldHandling full scale examples in an in-depth manner (unusual for a textbook) brings in a touch of research work in an otherwise routine learning methodReviews: “This is a nicely written book on stochastic processes from a very special perspective on the topic, inspired by the limiting behavior of a teletraffic model. The richness of this model needs to introduce many concepts of stochastic process theory which are not mainstream in the existing literature. Thus the book appears as a fresh and appealing addition. Interested researchers in pure and applied mathematics can find a comprehensive presentation of the topic for the first time in book format.” Mathematical Reviews Clippings

Risk Management and Simulation

Author: Aparna Gupta
Publisher: CRC Press
ISBN: 1439835950
Format: PDF
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The challenges of the current financial environment have revealed the need for a new generation of professionals who combine training in traditional finance disciplines with an understanding of sophisticated quantitative and analytical tools. Risk Management and Simulation shows how simulation modeling and analysis can help you solve risk management problems related to market, credit, operational, business, and strategic risk. Simulation models and methodologies offer an effective way to address many of these problems and are easy for finance professionals to understand and use. Drawing on the author’s extensive teaching experience, this accessible book walks you through the concepts, models, and computational techniques. How Simulation Models Can Help You Manage Risk More Effectively Organized into four parts, the book begins with the concepts and framework for risk management. It then introduces the modeling and computational techniques for solving risk management problems, from model development, verification, and validation to designing simulation experiments and conducting appropriate output analysis. The third part of the book delves into specific issues of risk management in a range of risk types. These include market risk, equity risk, interest rate risk, commodity risk, currency risk, credit risk, liquidity risk, and strategic, business, and operational risks. The author also examines insurance as a mechanism for risk management and risk transfer. The final part of the book explores advanced concepts and techniques. The book contains extensive review questions and detailed quantitative or computational exercises in all chapters. Use of MATLAB® mathematical software is encouraged and suggestions for MATLAB functions are provided throughout. Learn Step by Step, from Basic Concepts to More Complex Models Packed with applied examples and exercises, this book builds from elementary models for risk to more sophisticated, dynamic models for risks that evolve over time. A comprehensive introduction to simulation modeling and analysis for risk management, it gives you the tools to better assess and manage the impact of risk in your organizations. The book can also serve as a support reference for readers preparing for CFA exams, GARP FRM exams, PRMIA PRM exams, and actuarial exams.

Stochastic Multi Stage Optimization

Author: Pierre Carpentier
Publisher: Springer
ISBN: 3319181386
Format: PDF
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The focus of the present volume is stochastic optimization of dynamical systems in discrete time where - by concentrating on the role of information regarding optimization problems - it discusses the related discretization issues. There is a growing need to tackle uncertainty in applications of optimization. For example the massive introduction of renewable energies in power systems challenges traditional ways to manage them. This book lays out basic and advanced tools to handle and numerically solve such problems and thereby is building a bridge between Stochastic Programming and Stochastic Control. It is intended for graduates readers and scholars in optimization or stochastic control, as well as engineers with a background in applied mathematics.

Levy Processes in Credit Risk

Author: Wim Schoutens
Publisher: John Wiley & Sons
ISBN: 0470685069
Format: PDF, ePub, Mobi
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This book is an introductory guide to using Lévy processes for credit risk modelling. It covers all types of credit derivatives: from the single name vanillas such as Credit Default Swaps (CDSs) right through to structured credit risk products such as Collateralized Debt Obligations (CDOs), Constant Proportion Portfolio Insurances (CPPIs) and Constant Proportion Debt Obligations (CPDOs) as well as new advanced rating models for Asset Backed Securities (ABSs). Jumps and extreme events are crucial stylized features, essential in the modelling of the very volatile credit markets - the recent turmoil in the credit markets has once again illustrated the need for more refined models. Readers will learn how the classical models (driven by Brownian motions and Black-Scholes settings) can be significantly improved by using the more flexible class of Lévy processes. By doing this, extreme event and jumps can be introduced into the models to give more reliable pricing and a better assessment of the risks. The book brings in high-tech financial engineering models for the detailed modelling of credit risk instruments, setting up the theoretical framework behind the application of Lévy Processes to Credit Risk Modelling before moving on to the practical implementation. Complex credit derivatives structures such as CDOs, ABSs, CPPIs, CPDOs are analysed and illustrated with market data.

Module Theory Extending Modules and Generalizations

Author: Adnan Tercan
Publisher: Birkhäuser
ISBN: 3034809522
Format: PDF, Docs
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The main focus of this monograph is to offer a comprehensive presentation of known and new results on various generalizations of CS-modules and CS-rings. Extending (or CS) modules are generalizations of injective (and also semisimple or uniform) modules. While the theory of CS-modules is well documented in monographs and textbooks, results on generalized forms of the CS property as well as dual notions are far less present in the literature. With their work the authors provide a solid background to module theory, accessible to anyone familiar with basic abstract algebra. The focus of the book is on direct sums of CS-modules and classes of modules related to CS-modules, such as relative (injective) ejective modules, (quasi) continuous modules, and lifting modules. In particular, matrix CS-rings are studied and clear proofs of fundamental decomposition results on CS-modules over commutative domains are given, thus complementing existing monographs in this area. Open problems round out the work and establish the basis for further developments in the field. The main text is complemented by a wealth of examples and exercises.

Metastability

Author: Anton Bovier
Publisher: Springer
ISBN: 3319247778
Format: PDF, ePub, Docs
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This monograph provides a concise presentation of a mathematical approach to metastability, a wide-spread phenomenon in the dynamics of non-linear systems - physical, chemical, biological or economic - subject to the action of temporal random forces typically referred to as noise, based on potential theory of reversible Markov processes. The authors shed new light on the metastability phenomenon as a sequence of visits of the path of the process to different metastable sets, and focuses on the precise analysis of the respective hitting probabilities and hitting times of these sets. The theory is illustrated with many examples, ranging from finite-state Markov chains, finite-dimensional diffusions and stochastic partial differential equations, via mean-field dynamics with and without disorder, to stochastic spin-flip and particle-hop dynamics and probabilistic cellular automata, unveiling the common universal features of these systems with respect to their metastable behaviour. The monograph will serve both as comprehensive introduction and as reference for graduate students and researchers interested in metastability.

Quantum Lie Theory

Author: Vladislav Kharchenko
Publisher: Springer
ISBN: 3319227041
Format: PDF, ePub, Docs
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This is an introduction to the mathematics behind the phrase “quantum Lie algebra”. The numerous attempts over the last 15-20 years to define a quantum Lie algebra as an elegant algebraic object with a binary “quantum” Lie bracket have not been widely accepted. In this book, an alternative approach is developed that includes multivariable operations. Among the problems discussed are the following: a PBW-type theorem; quantum deformations of Kac--Moody algebras; generic and symmetric quantum Lie operations; the Nichols algebras; the Gurevich--Manin Lie algebras; and Shestakov--Umirbaev operations for the Lie theory of nonassociative products. Opening with an introduction for beginners and continuing as a textbook for graduate students in physics and mathematics, the book can also be used as a reference by more advanced readers. With the exception of the introductory chapter, the content of this monograph has not previously appeared in book form.