Discrete Stochastic Processes and Applications

1 Discrete time, countable space Summary. Throughout this chapter, S is a
countable space, called the state space; a discrete-time stochastic process on S
is a collection X of S−valued random variables (Xn) neN indexed by time n. We
shall ...

Author: Jean-François Collet

Publisher: Springer

ISBN: 3319740180

Category: Mathematics

Page: 220

View: 237

This unique text for beginning graduate students gives a self-contained introduction to the mathematical properties of stochastics and presents their applications to Markov processes, coding theory, population dynamics, and search engine design. The book is ideal for a newly designed course in an introduction to probability and information theory. Prerequisites include working knowledge of linear algebra, calculus, and probability theory. The first part of the text focuses on the rigorous theory of Markov processes on countable spaces (Markov chains) and provides the basis to developing solid probabilistic intuition without the need for a course in measure theory. The approach taken is gradual beginning with the case of discrete time and moving on to that of continuous time. The second part of this text is more applied; its core introduces various uses of convexity in probability and presents a nice treatment of entropy.

Stochastic Processes with Applications to Finance

Stochastic Processes with Applications to Finance shows that this is not necessarily so.

Author: Masaaki Kijima

Publisher: CRC Press

ISBN: 9781584882244

Category: Mathematics

Page: 288

View: 785

In recent years, modeling financial uncertainty using stochastic processes has become increasingly important, but it is commonly perceived as requiring a deep mathematical background. Stochastic Processes with Applications to Finance shows that this is not necessarily so. It presents the theory of discrete stochastic processes and their applications in finance in an accessible treatment that strikes a balance between the abstract and the practical. Using an approach that views sophisticated stochastic calculus as based on a simple class of discrete processes-"random walks"-the author first provides an elementary introduction to the relevant areas of real analysis and probability. He then uses random walks to explain the change of measure formula, the reflection principle, and the Kolmogorov backward equation. The Black-Scholes formula is derived as a limit of binomial model, and applications to the pricing of derivative securities are presented. Another primary focus of the book is the pricing of corporate bonds and credit derivatives, which the author explains in terms of discrete default models. By presenting important results in discrete processes and showing how to transfer those results to their continuous counterparts, Stochastic Processes with Applications to Finance imparts an intuitive and practical understanding of the subject. This unique treatment is ideal both as a text for a graduate-level class and as a reference for researchers and practitioners in financial engineering, operations research, and mathematical and statistical finance.

Discrete Stochastic Processes

Discrete stochastic processes find wide and diverse applications in operations
research, communication, control, computer systems, management science, etc.
Paradoxically, we shall spend relatively little of our time discussing these
particular ...

Author: Robert G. Gallager

Publisher: Springer Science & Business Media

ISBN: 146152329X

Category: Technology & Engineering

Page: 271

View: 571

Stochastic processes are found in probabilistic systems that evolve with time. Discrete stochastic processes change by only integer time steps (for some time scale), or are characterized by discrete occurrences at arbitrary times. Discrete Stochastic Processes helps the reader develop the understanding and intuition necessary to apply stochastic process theory in engineering, science and operations research. The book approaches the subject via many simple examples which build insight into the structure of stochastic processes and the general effect of these phenomena in real systems. The book presents mathematical ideas without recourse to measure theory, using only minimal mathematical analysis. In the proofs and explanations, clarity is favored over formal rigor, and simplicity over generality. Numerous examples are given to show how results fail to hold when all the conditions are not satisfied. Audience: An excellent textbook for a graduate level course in engineering and operations research. Also an invaluable reference for all those requiring a deeper understanding of the subject.

Introduction to Probability and Stochastic Processes with Applications

The book is also a valuable reference for researchers and practitioners in the fields of engineering, operations research, and computer science who conduct data analysis to make decisions in their everyday work.

Author: Liliana Blanco Castañeda

Publisher: John Wiley & Sons

ISBN: 1118344960

Category: Mathematics

Page: 614

View: 313

An easily accessible, real-world approach to probability andstochastic processes Introduction to Probability and Stochastic Processes withApplications presents a clear, easy-to-understand treatment ofprobability and stochastic processes, providing readers with asolid foundation they can build upon throughout their careers. Withan emphasis on applications in engineering, applied sciences,business and finance, statistics, mathematics, and operationsresearch, the book features numerous real-world examples thatillustrate how random phenomena occur in nature and how to useprobabilistic techniques to accurately model these phenomena. The authors discuss a broad range of topics, from the basicconcepts of probability to advanced topics for further study,including Itô integrals, martingales, and sigma algebras.Additional topical coverage includes: Distributions of discrete and continuous random variablesfrequently used in applications Random vectors, conditional probability, expectation, andmultivariate normal distributions The laws of large numbers, limit theorems, and convergence ofsequences of random variables Stochastic processes and related applications, particularly inqueueing systems Financial mathematics, including pricing methods such asrisk-neutral valuation and the Black-Scholes formula Extensive appendices containing a review of the requisitemathematics and tables of standard distributions for use inapplications are provided, and plentiful exercises, problems, andsolutions are found throughout. Also, a related website featuresadditional exercises with solutions and supplementary material forclassroom use. Introduction to Probability and StochasticProcesses with Applications is an ideal book for probabilitycourses at the upper-undergraduate level. The book is also avaluable reference for researchers and practitioners in the fieldsof engineering, operations research, and computer science whoconduct data analysis to make decisions in their everyday work.

Fundamentals of Probability and Stochastic Processes with Applications to Communications

X tn Discrete Stochastic Process A discrete stochastic process is defined as a
collection of RVs for discrete time points as follows: Xt1 { 2R } (6.7) {(),X (t2 ),...,X (
ti ),...X (tn) }= X(ti),t i Continuous Stochastic Process A continuous stochastic ...

Author: Kun Il Park

Publisher: Springer

ISBN: 3319680757

Category: Technology & Engineering

Page: 275

View: 456

This book provides engineers with focused treatment of the mathematics needed to understand probability, random variables, and stochastic processes, which are essential mathematical disciplines used in communications engineering. The author explains the basic concepts of these topics as plainly as possible so that people with no in-depth knowledge of these mathematical topics can better appreciate their applications in real problems. Applications examples are drawn from various areas of communications. If a reader is interested in understanding probability and stochastic processes that are specifically important for communications networks and systems, this book serves his/her need.

Stochastic Processes and Their Applications

However , the terminology stochastic processes is more common and will be
used throughout the book . In order to ... Conversely , every sequence of random
variables can be interpreted as stochastic process in discrete time . If T is an
interval ...

Author: Frank Beichelt

Publisher: CRC Press

ISBN: 9780415272322

Category: Mathematics

Page: 338

View: 321

This book introduces stochastic processes and their applications for students in engineering, industrial statistics, science, operations research, business, and finance. It provides the theoretical foundations for modeling time-dependent random phenomena encountered in these disciplines. Through numerous science and engineering-based examples and exercises, the author presents the subject in a comprehensible, practically oriented way, but he also includes some important proofs and theoretically challenging examples and exercises that will appeal to more mathematically minded readers. Solutions to most of the exercises are included either in an appendix or within the text.

An Introduction to Stochastic Processes with Applications to Biology

x F(x) 1 1 Sometimes we shall use the term probability density function to include
both the p.d.f. of a continuous random variable and the p.m.f. of a discrete
random variable. In addition, sometimes the notation Prob{·} is used in place of P(
·) or ...

Author: Linda J. S. Allen

Publisher: CRC Press

ISBN: 143989468X

Category: Mathematics

Page: 496

View: 441

An Introduction to Stochastic Processes with Applications to Biology, Second Edition presents the basic theory of stochastic processes necessary in understanding and applying stochastic methods to biological problems in areas such as population growth and extinction, drug kinetics, two-species competition and predation, the spread of epidemics, and the genetics of inbreeding. Because of their rich structure, the text focuses on discrete and continuous time Markov chains and continuous time and state Markov processes. New to the Second Edition A new chapter on stochastic differential equations that extends the basic theory to multivariate processes, including multivariate forward and backward Kolmogorov differential equations and the multivariate Itô’s formula The inclusion of examples and exercises from cellular and molecular biology Double the number of exercises and MATLAB® programs at the end of each chapter Answers and hints to selected exercises in the appendix Additional references from the literature This edition continues to provide an excellent introduction to the fundamental theory of stochastic processes, along with a wide range of applications from the biological sciences. To better visualize the dynamics of stochastic processes, MATLAB programs are provided in the chapter appendices.

Stochastic Processes

Written by one of the world's leading information theorists, evolving over twenty years of graduate classroom teaching and enriched by over 300 exercises, this is an exceptional resource for anyone looking to develop their understanding of ...

Author: Robert G. Gallager

Publisher: Cambridge University Press

ISBN: 1107435315

Category: Technology & Engineering

Page: 568

View: 179

This definitive textbook provides a solid introduction to discrete and continuous stochastic processes, tackling a complex field in a way that instils a deep understanding of the relevant mathematical principles, and develops an intuitive grasp of the way these principles can be applied to modelling real-world systems. It includes a careful review of elementary probability and detailed coverage of Poisson, Gaussian and Markov processes with richly varied queuing applications. The theory and applications of inference, hypothesis testing, estimation, random walks, large deviations, martingales and investments are developed. Written by one of the world's leading information theorists, evolving over twenty years of graduate classroom teaching and enriched by over 300 exercises, this is an exceptional resource for anyone looking to develop their understanding of stochastic processes.

Stochastic Processes with Applications to Finance Second Edition

This section demonstrates how to obtain results in Brownian motions from the
discrete counterparts. As the first example, consider a standard Brownian motion
{z(t)}, and let ]P'*(A) I 1E 11.. Y(T)], Y(t) I QWHZt/Q. (1230) It is readily seen that lP'
* ...

Author: Masaaki Kijima

Publisher: CRC Press

ISBN: 143988482X

Category: Business & Economics

Page: 343

View: 912

Financial engineering has been proven to be a useful tool for risk management, but using the theory in practice requires a thorough understanding of the risks and ethical standards involved. Stochastic Processes with Applications to Finance, Second Edition presents the mathematical theory of financial engineering using only basic mathematical tools that are easy to understand even for those with little mathematical expertise. This second edition covers several important developments in the financial industry. New to the Second Edition A chapter on the change of measures and pricing of insurance products Many examples of the change of measure technique, including its use in asset pricing theory A section on the use of copulas, especially in the pricing of CDOs Two chapters that offer more coverage of interest rate derivatives and credit derivatives Exploring the merge of actuarial science and financial engineering, this edition examines how the pricing of insurance products, such as equity-linked annuities, requires knowledge of asset pricing theory since the equity index can be traded in the market. The book looks at the development of many probability transforms for pricing insurance risks, including the Esscher transform. It also describes how the copula model is used to model the joint distribution of underlying assets. By presenting significant results in discrete processes and showing how to transfer the results to their continuous counterparts, this text imparts an accessible, practical understanding of the subject. It helps readers not only grasp the theory of financial engineering, but also implement the theory in business.

Basics of Applied Stochastic Processes

This volume gives an in-depth description of the structure and basic properties of these stochastic processes.

Author: Richard Serfozo

Publisher: Springer Science & Business Media

ISBN: 3540893326

Category: Mathematics

Page: 443

View: 852

Stochastic processes are mathematical models of random phenomena that evolve according to prescribed dynamics. Processes commonly used in applications are Markov chains in discrete and continuous time, renewal and regenerative processes, Poisson processes, and Brownian motion. This volume gives an in-depth description of the structure and basic properties of these stochastic processes. A main focus is on equilibrium distributions, strong laws of large numbers, and ordinary and functional central limit theorems for cost and performance parameters. Although these results differ for various processes, they have a common trait of being limit theorems for processes with regenerative increments. Extensive examples and exercises show how to formulate stochastic models of systems as functions of a system’s data and dynamics, and how to represent and analyze cost and performance measures. Topics include stochastic networks, spatial and space-time Poisson processes, queueing, reversible processes, simulation, Brownian approximations, and varied Markovian models. The technical level of the volume is between that of introductory texts that focus on highlights of applied stochastic processes, and advanced texts that focus on theoretical aspects of processes.

The Elements of Stochastic Processes with Applications to the Natural Sciences

CHAPTER 15 Approximations to Stochastic Processes 15.1 Introduction It is clear
from the discussions of stochastic processes ... 15.2 Continuous approximations
to discrete processes We have already seen in Chapter 14 how in some ...

Author: Norman T. J. Bailey

Publisher: John Wiley & Sons

ISBN: 9780471523680

Category: Mathematics

Page: 264

View: 225

Develops an introductory and relatively simple account of the theory and application of the evolutionary type of stochastic process. Professor Bailey adopts the heuristic approach of applied mathematics and develops both theoretical principles and applied techniques simultaneously.

Stochastic Processes with Applications

This book develops systematically and rigorously, yet in an expository and lively manner, the evolution of general random processes and their large time properties such as transience, recurrence, and convergence to steady states.

Author: Rabi N. Bhattacharya

Publisher: SIAM

ISBN: 0898716896

Category: Mathematics

Page: 184

View: 561

This book develops systematically and rigorously, yet in an expository and lively manner, the evolution of general random processes and their large time properties such as transience, recurrence, and convergence to steady states. The emphasis is on the most important classes of these processes from the viewpoint of theory as well as applications, namely, Markov processes. The book features very broad coverage of the most applicable aspects of stochastic processes, including sufficient material for self-contained courses on random walks in one and multiple dimensions; Markov chains in discrete and continuous times, including birth-death processes; Brownian motion and diffusions; stochastic optimization; and stochastic differential equations. This book is for graduate students in mathematics, statistics, science and engineering, and it may also be used as a reference by professionals in diverse fields whose work involves the application of probability.

Approximation and Weak Convergence Methods for Random Processes with Applications to Stochastic Systems Theory

The versatility of the method is illustrated by the simple examples in the sequel
and the practical applications in chapters 8-10. ... The general idea is most easily
understood when dealing with an averaging problem, where the limit process
satisfies an ODE (ordinary differential ... Sections 4 and 5, cover the discrete and
continuous parameter cases, respectively, when the noise is "state dependent.

Author: Harold Joseph Kushner

Publisher: MIT Press

ISBN: 9780262110907

Category: Computers

Page: 269

View: 776

Control and communications engineers, physicists, and probability theorists, among others, will find this book unique. It contains a detailed development of approximation and limit theorems and methods for random processes and applies them to numerous problems of practical importance. In particular, it develops usable and broad conditions and techniques for showing that a sequence of processes converges to a Markov diffusion or jump process. This is useful when the natural physical model is quite complex, in which case a simpler approximation la diffusion process, for example) is usually made. The book simplifies and extends some important older methods and develops some powerful new ones applicable to a wide variety of limit and approximation problems. The theory of weak convergence of probability measures is introduced along with general and usable methods (for example, perturbed test function, martingale, and direct averaging) for proving tightness and weak convergence. Kushner's study begins with a systematic development of the method. It then treats dynamical system models that have state-dependent noise or nonsmooth dynamics. Perturbed Liapunov function methods are developed for stability studies of nonMarkovian problems and for the study of asymptotic distributions of non-Markovian systems. Three chapters are devoted to applications in control and communication theory (for example, phase-locked loops and adoptive filters). Smallnoise problems and an introduction to the theory of large deviations and applications conclude the book. Harold J. Kushner is Professor of Applied Mathematics and Engineering at Brown University and is one of the leading researchers in the area of stochastic processes concerned with analysis and synthesis in control and communications theory. This book is the sixth in The MIT Press Series in Signal Processing, Optimization, and Control, edited by Alan S. Willsky.

Inertial Navigation Systems with Geodetic Applications

6.3 Stochastic Processes A stochastic (or, random) process is a collection,
discrete or continuous, of random variables associated with a deterministic
parameter, usually a time or space coordinate. At each point in time or in space,
the process ...

Author: Christopher Jekeli

Publisher: Walter de Gruyter

ISBN: 3110800233

Category: Science

Page: 365

View: 953

This book covers all aspects of inertial navigation systems (INS), including the sensor technology and the estimation of instrument errors, as well as their integration with the Global Positioning System (GPS) for geodetic applications. Complete mathematical derivations are given. Both stabilized and strapdown mechanizations are treated in detail. Derived algorithms to process sensor data and a comprehensive explanation of the error dynamics provide not only an analytical understanding but also a practical implementation of the concepts. A self-contained description of GPS, with emphasis on kinematic applications, is one of the highlights in this book. The text is of interest to geodesists, including surveyors, mappers, and photogrammetrists; to engineers in aviation, navigation, guidance, transportation, and robotics; and to scientists involved in aerogeophysics and remote sensing.

Modelling and Application of Stochastic Processes

Motivated by his work , model reduction techniques for discrete - time stochastic
systems were developed by Fujishige et al ( 6 ] who used the principal
component analysis of stochastic processes , and by Baram ( 7 ) , Larimore ( 8 ) ,
Desai et ...

Author: Uday B. Desai

Publisher: Springer Science & Business Media

ISBN: 9780898381771

Category: Science

Page: 288

View: 819

The subject of modelling and application of stochastic processes is too vast to be exhausted in a single volume. In this book, attention is focused on a small subset of this vast subject. The primary emphasis is on realization and approximation of stochastic systems. Recently there has been considerable interest in the stochastic realization problem, and hence, an attempt has been made here to collect in one place some of the more recent approaches and algorithms for solving the stochastic realiza tion problem. Various different approaches for realizing linear minimum-phase systems, linear nonminimum-phase systems, and bilinear systems are presented. These approaches range from time-domain methods to spectral-domain methods. An overview of the chapter contents briefly describes these approaches. Also, in most of these chapters special attention is given to the problem of developing numerically ef ficient algorithms for obtaining reduced-order (approximate) stochastic realizations. On the application side, chapters on use of Markov random fields for modelling and analyzing image signals, use of complementary models for the smoothing problem with missing data, and nonlinear estimation are included. Chapter 1 by Klein and Dickinson develops the nested orthogonal state space realization for ARMA processes. As suggested by the name, nested orthogonal realizations possess two key properties; (i) the state variables are orthogonal, and (ii) the system matrices for the (n + l)st order realization contain as their "upper" n-th order blocks the system matrices from the n-th order realization (nesting property).

Discrete Time Series Processes and Applications in Finance

In a given state i, the price follows a random walk (with Gaussian or Student
residues) with constant volatility σi. This model is structurally close to a stochastic
volatility process, as the dynamics of the volatility is independent of the prices.

Author: Gilles Zumbach

Publisher: Springer Science & Business Media

ISBN: 3642317413

Category: Business & Economics

Page: 319

View: 883

This book surveys empirical properties of financial time series, discusses their mathematical basis, and describes uses in risk evaluation, option pricing or portfolio construction. The author introduces and assesses a range of processes against the benchmark.

Essentials of Stochastic Processes

... a stochastic process . For applications , t refers to time , and xt ( W ) represents
the value of a quantity occurring by chance at time t . ... In case T is discrete , the
stochastic process is usually called a random sequence . A stochastic process xt

Author: Kiyosi Itō

Publisher: American Mathematical Soc.

ISBN: 9780821838983

Category: Mathematics

Page: 171

View: 356

This book is an English translation of Kiyosi Ito's monograph published in Japanese in 1957. It gives a unified and comprehensive account of additive processes (or Levy processes), stationary processes, and Markov processes, which constitute the three most important classes of stochastic processes. Written by one of the leading experts in the field, this volume presents to the reader lucid explanations of the fundamental concepts and basic results in each of these three major areas of the theory of stochastic processes. With the requirements limited to an introductory graduate course on analysis (especially measure theory) and basic probability theory, this book is an excellent text for any graduate course on stochastic processes. Kiyosi Ito is famous throughout the world for his work on stochastic integrals (including the Ito formula), but he has made substantial contributions to other areas of probability theory as well, such as additive processes, stationary processes, and Markov processes (especially diffusion processes), which are topics covered in this book. For his contributions and achievements, he has received, among others, the Wolf Prize, the Japan Academy Prize, and the Kyoto Prize.

Stochastic Processes

Written in a simple and accessible manner, this book addresses that inadequacy and provides guidelines and tools to study the applications.

Author: Narahari Umanath Prabhu

Publisher: World Scientific

ISBN: 9812706267

Category: Mathematics

Page: 341

View: 678

Most introductory textbooks on stochastic processes which cover standard topics such as Poisson process, Brownian motion, renewal theory and random walks deal inadequately with their applications. Written in a simple and accessible manner, this book addresses that inadequacy and provides guidelines and tools to study the applications. The coverage includes research developments in Markov property, martingales, regenerative phenomena and Tauberian theorems, and covers measure theory at an elementary level.

Stochastic Processes

This textbook was developed from a course in stochastic processes given by the
authors over many years to ... and this is reflected in the book by including
applications and examples that students can quickly understand and relate to. ...
Some of the easier material on discrete random processes is included in
Chapters ...

Author: Peter Watts Jones

Publisher: CRC Press

ISBN: 1420099809

Category: Mathematics

Page: 232

View: 867

Based on a highly popular, well-established course taught by the authors, Stochastic Processes: An Introduction, Second Edition discusses the modeling and analysis of random experiments using the theory of probability. It focuses on the way in which the results or outcomes of experiments vary and evolve over time. The text begins with a review of relevant fundamental probability. It then covers several basic gambling problems, random walks, and Markov chains. The authors go on to develop random processes continuous in time, including Poisson, birth and death processes, and general population models. While focusing on queues, they present an extended discussion on the analysis of associated stationary processes. The book also explores reliability and other random processes, such as branching processes, martingales, and a simple epidemic. The appendix contains key mathematical results for reference. Ideal for a one-semester course on stochastic processes, this concise, updated textbook makes the material accessible to students by avoiding specialized applications and instead highlighting simple applications and examples. The associated website contains Mathematica® and R programs that offer flexibility in creating graphs and performing computations.

Applications of Discrete time Markov Chains and Poisson Processes to Air Pollution Modeling and Studies

We consider the following forms of stochastic processes: discrete-time Markov
chains, Poisson processes (a type of continuous-time Markov chain), and also
some more general forms of counting processes (of which the Poisson process is

Author: Eliane Regina Rodrigues

Publisher: Springer Science & Business Media

ISBN: 1461446457

Category: Mathematics

Page: 107

View: 193

​In this brief we consider some stochastic models that may be used to study problems related to environmental matters, in particular, air pollution. The impact of exposure to air pollutants on people's health is a very clear and well documented subject. Therefore, it is very important to obtain ways to predict or explain the behaviour of pollutants in general. Depending on the type of question that one is interested in answering, there are several of ways studying that problem. Among them we may quote, analysis of the time series of the pollutants' measurements, analysis of the information obtained directly from the data, for instance, daily, weekly or monthly averages and standard deviations. Another way to study the behaviour of pollutants in general is through mathematical models. In the mathematical framework we may have for instance deterministic or stochastic models. The type of models that we are going to consider in this brief are the stochastic ones.​