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Collecting together selected pioneering works of the celebrated
mathematician Anatolii V. Skorokhod, this volume serves as a guide
to the theory of stochastic processes from its beginning to its
current state. It offers both an excellent bibliographic resource
and a unique opportunity for readers to gain a better understanding
of Skorokhod's original and beautiful ideas, which had a deep
impact on the development of the subject. The modern theory of
stochastic processes is a fast-growing branch of probability theory
which is now an independent science in its own right, with its own
methods and philosophy. It has many applications in various fields,
including financial mathematics, quantum physics and engineering. A
clear understanding of this theory is impossible without knowledge
of the ideas which form its base, many of which are due to
Skorokhod. The book is intended for a broad audience of researchers
and students with an interest in probability theory, stochastic
processes and their applications.
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Selected Works (Hardcover, 1st ed. 2016)
Anatolii V Skorokhod; Edited by Andrey A. Dorogovtsev, Alexey Kulik, Andrey Pilipenko, Mykola I. Portenko, …
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R3,024
Discovery Miles 30 240
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Ships in 10 - 15 working days
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Collecting together selected pioneering works of the celebrated
mathematician Anatolii V. Skorokhod, this volume serves as a guide
to the theory of stochastic processes from its beginning to its
current state. It offers both an excellent bibliographic resource
and a unique opportunity for readers to gain a better understanding
of Skorokhod's original and beautiful ideas, which had a deep
impact on the development of the subject. The modern theory of
stochastic processes is a fast-growing branch of probability theory
which is now an independent science in its own right, with its own
methods and philosophy. It has many applications in various fields,
including financial mathematics, quantum physics and engineering. A
clear understanding of this theory is impossible without knowledge
of the ideas which form its base, many of which are due to
Skorokhod. The book is intended for a broad audience of researchers
and students with an interest in probability theory, stochastic
processes and their applications.
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 x Contents 5
Stochastic processes with independent increments. Wiener and
Poisson processes. Poisson point measures...
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 x Contents 5
Stochastic processes with independent increments. Wiener and
Poisson processes. Poisson point measures...
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