Publisher Description

Thiscollectionofproblemsisplannedasatextbookforuniversitycoursesinthe theoryofstochasticprocessesandrelatedspecialcourses. Theproblemsinthebook haveawidespectrumofthelevelofdif cultyandcanbeusefulforreaderswith variouslevelsofmasteringinthetheoryofstochasticprocesses. Togetherwithte- nicalandillustrativeproblemsintendedforbeginners,thebookcontainsanumber ofproblemsoftheoreticalnaturethatcanbeusefulforstudentsandundergraduate studentsthatpursueadvancedstudiesinthetheoryofstochasticprocessesandits- plications. Amongothers,theimportantaimofthebookistoprovideateachingstaff anef cienttoolforpreparingseminarstudies,tests,andexamsconcerninguniversity coursesinthetheoryofstochasticprocessesandrelatedtopics. Whilecomposingthe book,theauthorshavepartiallyusedthecollectionsofproblemsinprobabilityt- ory[16,65,75,83]. Also,someexercisesandproblemsfromthemonographsand textbooks[4,9,19,22,82]wereused. Atthesametime,alargepartofourproblem bookcontainsoriginalmaterial. Thebookisorganizedasfollows. Theproblemsarecollectedintochapters,each chapterbeingdevotedtoacertaintopic.Atthebeginningofeachchapter,theth- reticalgroundsforthecorrespondingtopicaregivenbrie ytogetherwiththelistof bibliography,whichthereadercanuseinordertostudythistopicinmoredetail. For themostoftheproblems,eitherhintsorcompletesolutions(oranswers)aregiven, andsomeoftheproblemsareprovidedwithbothhintsandsolutions(answers). H- ever,theauthorsdonotrecommendthatareaderusethehintssystematically,because solvingaproblemwithoutassistanceismuchmoreusefulthanusingaready-made idea. Somestatementsthathaveaparticulartheoreticalinterestareformulatedon theoreticalgrounds,andtheirproofsareformulatedasproblemsforthereader. Such problemsaresuppliedwitheithercompletesolutionsordetailedhints. Inordertoworkwiththeproblembookef ciently,areadershouldbeacquainted withprobabilitytheory,calculus,andmeasuretheorywithinthescopeofresp- tiveuniversity courses. Standard notions, suchas random variable, measurability, independence, Lebesgue measure and integral, and so on are used without ad- tionaldiscussion. Allthenewnotionsandstatementsrequiredforsolvingthepr- lemsaregiveneitherontheoreticalgroundsorintheformulationsoftheproblems vii viii Preface straightforwardly.However,sometimesanotionisusedinthetextbeforeitsformal de nition. Forinstance,theWienerandPoissonprocessesareprocesseswithin- pendentincrementsandthusareformallyintroducedinaTheoreticalgroundsfor Chapter5,buttheseprocessesareusedwidelyintheproblemsofChapters2to4. Theauthorsrecommendthatareaderwhocomestoanunknownnotionorobject usetheIndexinorderto ndthecorrespondingformalde nition. Thesamerec- mendationconcernssomestandardabbreviationsandsymbolslistedattheendofthe book. Someproblemsinthebookformcycles:solutionstooneofthemaregrounded onstatementsofothersoronauxiliaryconstructionsdescribedinsomepreceding solutions. Sometimes,onthecontrary,itisproposedtoprovethesamestatement withindifferentproblemsusingessentiallydifferenttechniques. Theauthorsrec- mendareaderpayspeci cattentiontothesefruitfulinternallinksbetweenvarious topicsofthetheoryofstochasticprocesses. Everypartofthebookwascomposedsubstantiallybyoneauthor. Chapters1-6, and16arecomposedbyA. Kulik,Chapters7,12-15,18,and19byYu. Mishura, Chapters 8-10 by A. Pilipenko, Chapter 17 by A. Kukush, and Chapter 20 by D. Gusak. Chapter11waspreparedjointlybyD. GusakandA. Pilipenko.Atthe sametime,everyauthorhasmadeacontributiontootherpartsofthebookbyprop- ingseparateproblemsorcyclesofproblems,improvingpreliminaryversionsoft- oreticalgrounds,andeditingthe naltext. The authors would like to express their deep gratitude to M. Portenko and A. Ivanovfortheircarefulreadingofapreliminaryversionofthebookandva- ablecommentsthatledtosigni cantimprovementofthetext. Theauthorsarealso gratefultoT. Yakovenko,G. Shevchenko,O. Soloveyko, Yu. Kartashov, Yu. K- menko,A. Malenko,andN. Ryabovafortheirassistanceintranslation,preparing lesandpictures,andcomposingthesubjectindexandreferences. Thetheoryofstochasticprocessesisanextendeddiscipline,andtheauthors- derstandthattheproblembookinitscurrentformmaycausecriticalremarksfrom readers,concerningeitherthestructureofthebookorthecontentofseparatech- ters. Whilepublishingtheproblembookinitscurrentform,theauthorsareopenfor remarks,comments,andpropositions,andexpressinadvancetheirgratitudetoall theircorrespondents. Kyiv DmytroGusak December2008 AlexanderKukush AlexeyKulik YuliyaMishura AndreyPilipenko Contents 1 De?nition of stochastic process. Cylinder?-algebra, ?nite-dimensional distributions, the Kolmogorov theorem...1Theoreticalgrounds ...1 Bibliography...3 Problems...3 Hints...7 AnswersandSolutions...9 2 Characteristics of a stochastic process. Mean and covariance functions. Characteristic functions...11 Theoreticalgrounds ...11 Bibliography...13 Problems...13 Hints...16 AnswersandSolutions...17 3 Trajectories. Modi?cations. Filtrations...21 Theoreticalgrounds ...21 Bibliography...24 Problems...24 Hints...29 AnswersandSolutions...31 4 Continuity. Differentiability. Integrability...33 Theoreticalgrounds ...33 Bibliography...34 Problems...34 Hints...38 AnswersandSolutions...40 ix Contents 5 Stochastic processes with independent increments. Wiener and Poisson processes. Poisson point measures...

Notes

Contains over 1000 high quality exercises on stochastic processes

Presents a modern approach to topics such as sample paths and optimal stopping

Ideal for professors who need exercises for exams, and graduate students wishing to learn about stochastic processes

Back Cover

This book is a collection of exercises covering all the main topics in the modern theory of stochastic processes and its applications, including finance, actuarial mathematics, queuing theory, and risk theory. The aim of this book is to provide the reader with the theoretical and practical material necessary for deeper understanding of the main topics in the theory of stochastic processes and its related fields. The book is divided into chapters according to the various topics. Each chapter contains problems, hints, solutions, as well as a self-contained theoretical part which gives all the necessary material for solving the problems. References to the literature are also given. The exercises have various levels of complexity and vary from simple ones, useful for students studying basic notions and technique, to very advanced ones that reveal some important theoretical facts and constructions. This book is one of the largest collections of problems in the theory of stochastic processes and its applications. The problems in this book can be useful for undergraduate and graduate students, as well as for specialists in the theory of stochastic processes.

Table of Contents

Definition of stochastic process. Cylinder #x03C3;-algebra, finite-dimensional distributions, the Kolmogorov theorem.- Characteristics of a stochastic process. Mean and covariance functions. Characteristic functions.- Trajectories. Modifications. Filtrations.- Continuity. Differentiability. Integrability.- Stochastic processes with independent increments. Wiener and Poisson processes. Poisson point measures.- Gaussian processes.- Martingales and related processes in discrete and continuous time. Stopping times.- Stationary discrete- and continuous-time processes. Stochastic integral over measure with orthogonal values.- Prediction and interpolation.- Markov chains: Discrete and continuous time.- Renewal theory. Queueing theory.- Markov and diffusion processes.- It#x00F4; stochastic integral. It#x00F4; formula. Tanaka formula.- Stochastic differential equations.- Optimal stopping of random sequences and processes.- Measures in a functional spaces. Weak convergence, probability metrics.Functional limit theorems.- Statistics of stochastic processes.- Stochastic processes in financial mathematics (discrete time).- Stochastic processes in financial mathematics (continuous time).- Basic functionals of the risk theory.

Review

From the reviews:"Chapter deals with the statistics of stochastic processes, mainly hypotheses testing, a relatively uncommon subject. … The major strength of this problem book is the breadth and depth of coverage that five experts in their respective subfields condensed in only 375 pages. … the book is a valuable addition to the literature on stochastic processes. … any course in stochastics at the advanced undergraduate or beginning to intermediate graduate level is almost sure to interest its table of contents substantially." (Giuseppe Castellacci, Mathematical Reviews, Issue 2011 f)"Advanced undergraduates and postgraduates in mathematics, and teaching staff at these levels. This is a book in the Springer series on Problem Books in Mathematics, presenting a series of problems … . Each of the 20 chapters in this book has a condensed outline of the topic being considered, a bibliography, the problems, and then hints or solutions to most of the problems." (David J. Hand, International Statistical Review, Vol. 78 (3), 2010)"This book provides a collection of more than 800 problems for the theory of stochastic processes. It is divided into 20 chapters that cover different aspects of this theory. … this compilation is new in its broadness and completeness for the theory of stochastic processes and is well suited for students in their self-studies as well as lecturers to prepare their classes in this field of probability theory." (Claudia Hein, Zentralblatt MATH, Vol. 1189, 2010)"Each chapter consists of a brief review of theory followed by … a list of problems, hints (for the solution of) pertaining to most of the problems in the chapter, and a section giving 'Answers and Solutions' for many but not necessarily all problems. … It might also be used in seminars or in advanced topics courses. … There is also a set of graphical representations of various stochastic processes. … an excellent contribution and anyone who works through the problems will be well rewarded." (Donald E. Myers, Technometrics, Vol. 53 (3), August, 2011)

Long Description

Thiscollectionofproblemsisplannedasatextbookforuniversitycoursesinthe theoryofstochasticprocessesandrelatedspecialcourses. Theproblemsinthebook haveawidespectrumofthelevelofdif cultyandcanbeusefulforreaderswith variouslevelsofmasteringinthetheoryofstochasticprocesses. Togetherwithte- nicalandillustrativeproblemsintendedforbeginners,thebookcontainsanumber ofproblemsoftheoreticalnaturethatcanbeusefulforstudentsandundergraduate studentsthatpursueadvancedstudiesinthetheoryofstochasticprocessesandits- plications. Amongothers,theimportantaimofthebookistoprovideateachingstaff anef cienttoolforpreparingseminarstudies,tests,andexamsconcerninguniversity coursesinthetheoryofstochasticprocessesandrelatedtopics. Whilecomposingthe book,theauthorshavepartiallyusedthecollectionsofproblemsinprobabilityt- ory[16,65,75,83]. Also,someexercisesandproblemsfromthemonographsand textbooks[4,9,19,22,82]wereused. Atthesametime,alargepartofourproblem bookcontainsoriginalmaterial. Thebookisorganizedasfollows. Theproblemsarecollectedintochapters,each chapterbeingdevotedtoacertaintopic. Atthebeginningofeachchapter,theth- reticalgroundsforthecorrespondingtopicaregivenbrie ytogetherwiththelistof bibliography,whichthereadercanuseinordertostudythistopicinmoredetail. For themostoftheproblems,eitherhintsorcompletesolutions(oranswers)aregiven, andsomeoftheproblemsareprovidedwithbothhintsandsolutions(answers). H- ever,theauthorsdonotrecommendthatareaderusethehintssystematically,because solvingaproblemwithoutassistanceismuchmoreusefulthanusingaready-made idea. Somestatementsthathaveaparticulartheoreticalinterestareformulatedon theoreticalgrounds,andtheirproofsareformulatedasproblemsforthereader. Such problemsaresuppliedwitheithercompletesolutionsordetailedhints. Inordertoworkwiththeproblembookef ciently,areadershouldbeacquainted withprobabilitytheory,calculus,andmeasuretheorywithinthescopeofresp- tiveuniversity courses. Standard notions, suchas random variable, measurability, independence, Lebesgue measure and integral, and so on are used without ad- tionaldiscussion. Allthenewnotionsandstatementsrequiredforsolvingthepr- lemsaregiveneitherontheoreticalgroundsorintheformulationsoftheproblems vii viii Preface straightforwardly. However,sometimesanotionisusedinthetextbeforeitsformal de nition. Forinstance,theWienerandPoissonprocessesareprocesseswithin- pendentincrementsandthusareformallyintroducedinaTheoreticalgroundsfor Chapter5,buttheseprocessesareusedwidelyintheproblemsofChapters2to4. Theauthorsrecommendthatareaderwhocomestoanunknownnotionorobject usetheIndexinorderto ndthecorrespondingformalde nition. Thesamerec- mendationconcernssomestandardabbreviationsandsymbolslistedattheendofthe book. Someproblemsinthebookformcycles:solutionstooneofthemaregrounded onstatementsofothersoronauxiliaryconstructionsdescribedinsomepreceding solutions. Sometimes,onthecontrary,itisproposedtoprovethesamestatement withindifferentproblemsusingessentiallydifferenttechniques. Theauthorsrec- mendareaderpayspeci cattentiontothesefruitfulinternallinksbetweenvarious topicsofthetheoryofstochasticprocesses. Everypartofthebookwascomposedsubstantiallybyoneauthor. Chapters1-6, and16arecomposedbyA. Kulik,Chapters7,12-15,18,and19byYu. Mishura, Chapters 8-10 by A. Pilipenko, Chapter 17 by A. Kukush, and Chapter 20 by D. Gusak. Chapter11waspreparedjointlybyD. GusakandA. Pilipenko. Atthe sametime,everyauthorhasmadeacontributiontootherpartsofthebookbyprop- ingseparateproblemsorcyclesofproblems,improvingpreliminaryversionsoft- oreticalgrounds,andeditingthe naltext. The authors would like to express their deep gratitude to M. Portenko and A. Ivanovfortheircarefulreadingofapreliminaryversionofthebookandva- ablecommentsthatledtosigni cantimprovementofthetext. Theauthorsarealso gratefultoT. Yakovenko,G. Shevchenko,O. Soloveyko, Yu. Kartashov, Yu. K- menko,A. Malenko,andN. Ryabovafortheirassistanceintranslation,preparing lesandpictures,andcomposingthesubjectindexandreferences. Thetheoryofstochasticprocessesisanextendeddiscipline,andtheauthors- derstandthattheproblembookinitscurrentformmaycausecriticalremarksfrom readers,concerningeitherthestructureofthebookorthecontentofseparatech- ters. Whilepublishingtheproblembookinitscurrentform,theauthorsareopenfor remarks,comments,andpropositions,andexpressinadvancetheirgratitudetoall theircorrespondents. Kyiv DmytroGusak December2008 AlexanderKukush AlexeyKulik YuliyaMishura AndreyPilipenko Contents 1 De'nition of stochastic process. Cylinder?-algebra, ?nite-dimensional distributions, the Kolmogorov theorem. . . . . . . . . . 1 Theoreticalgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Hints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 AnswersandSolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Characteristics of a stochastic process. Mean and covariance functions. Characteristic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Theoreticalgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Hints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 AnswersandSolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Trajectories. Modi'cations. Filtrations. . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Theoreticalgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Hints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 AnswersandSolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4 Continuity. Differentiability. Integrability. . . . . . . . . . . . . . . . . . . . . . . . . 33 Theoreticalgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Hints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 AnswersandSolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ix Contents 5 Stochastic processes with independent increments. Wiener and Poisson processes. Poisson point measures. . . . . . . . . . . . . . . . . . . . .

Review Quote

From the reviews: "Chapter deals with the statistics of stochastic processes, mainly hypotheses testing, a relatively uncommon subject. ... The major strength of this problem book is the breadth and depth of coverage that five experts in their respective subfields condensed in only 375 pages. ... the book is a valuable addition to the literature on stochastic processes. ... any course in stochastics at the advanced undergraduate or beginning to intermediate graduate level is almost sure to interest its table of contents substantially." (Giuseppe Castellacci, Mathematical Reviews, Issue 2011 f) "Advanced undergraduates and postgraduates in mathematics, and teaching staff at these levels. This is a book in the Springer series on Problem Books in Mathematics, presenting a series of problems ... . Each of the 20 chapters in this book has a condensed outline of the topic being considered, a bibliography, the problems, and then hints or solutions to most of the problems." (David J. Hand, International Statistical Review, Vol. 78 (3), 2010) "This book provides a collection of more than 800 problems for the theory of stochastic processes. It is divided into 20 chapters that cover different aspects of this theory. ... this compilation is new in its broadness and completeness for the theory of stochastic processes and is well suited for students in their self-studies as well as lecturers to prepare their classes in this field of probability theory." (Claudia Hein, Zentralblatt MATH, Vol. 1189, 2010) "Each chapter consists of a brief review of theory followed by ... a list of problems, hints (for the solution of) pertaining to most of the problems in the chapter, and a section giving 'Answers and Solutions' for many but not necessarily all problems. ... It might also be used in seminars or in advanced topics courses. ... There is also a set of graphical representations of various stochastic processes. ... an excellent contribution and anyone who works through the problems will be well rewarded." (Donald E. Myers, Technometrics, Vol. 53 (3), August, 2011)

Feature

Contains over 1000 high quality exercises on stochastic processes Presents a modern approach to topics such as sample paths and optimal stopping Ideal for professors who need exercises for exams, and graduate students wishing to learn about stochastic processes

Details