Theory and Statistical Applications of Stochastic Processes
Yuliya Mishura Georgiy Shevchenko
First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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ISBN 978-1-78630-050-8
Chapter1.StochasticProcesses.GeneralProperties. Trajectories,Finite-dimensionalDistributions ..............3
1.1.Definitionofastochasticprocess.....................3
1.2.Trajectoriesofastochasticprocess.Someexamples ofstochasticprocesses..............................5
1.2.1.Definitionoftrajectoryandsomeexamples..............5
1.2.2.Trajectoryofastochasticprocessas arandomelement................................8
1.3.Finite-dimensionaldistributionsofstochastic processes:consistencyconditions........................10
1.3.1.Definitionandpropertiesoffinite-dimensional distributions...................................10
1.3.2.Consistencyconditions.........................11
1.3.3.Cylindersetsandgenerated σ -algebra.................13
1.3.4.Kolmogorovtheoremontheconstructionofastochastic processbythefamilyofprobabilitydistributions..............15
1.4.Propertiesof σ -algebrageneratedbycylindersets. Thenotionof σ -algebrageneratedbyastochasticprocess..........19
Chapter2.StochasticProcesseswithIndependent Increments .....................................21
2.1.Existenceofprocesseswithindependentincrements intermsofincrementalcharacteristicfunctions................21
2.2.Wienerprocess................................24
2.2.1.One-dimensionalWienerprocess...................24
2.2.2.Independentstochasticprocesses.Multidimensional Wienerprocess.................................24
2.3.Poissonprocess...............................27
2.3.1.Poissonprocessdefinedviatheexistencetheorem..........27
2.3.2.Poissonprocessdefinedviathedistributions oftheincrements................................28
2.3.3.Poissonprocessasarenewalprocess.................30
2.4.CompoundPoissonprocess.........................33
2.5.Lévyprocesses................................34
2.5.1.Wienerprocesswithadrift.......................36
2.5.2.CompoundPoissonprocessasaLévyprocess............36
2.5.3.SumofaWienerprocesswithadriftand aPoissonprocess................................36 2.5.4.Gammaprocess.............................37
2.5.5.StableLévymotion...........................37
2.5.6.StableLévysubordinatorwithstability parameter
Chapter3.GaussianProcesses.IntegrationwithRespectto GaussianProcesses
3.1.Gaussianvectors...............................39
3.2.TheoremofGaussianrepresentation(theoremon normalcorrelation)................................42
3.3.Gaussianprocesses.............................44
3.4.ExamplesofGaussianprocesses......................46
3.4.1.WienerprocessasanexampleofaGaussianprocess........46
3.4.2.FractionalBrownianmotion......................48
3.4.3.Sub-fractionalandbi-fractionalBrownianmotion..........50
3.4.4.Brownianbridge.............................50
3.4.5.Ornstein–Uhlenbeckprocess......................51
3.5.Integrationofnon-randomfunctionswithrespect toGaussianprocesses..............................52
3.5.1.Generalapproach............................52
3.5.2.Integrationofnon-randomfunctionswithrespect totheWienerprocess..............................54
3.5.3.Integrationw.r.t.thefractionalBrownianmotion...........57
3.6.Two-sidedWienerprocessandfractionalBrownian motion:Mandelbrot–vanNessrepresentationoffractional Brownianmotion.................................60
3.7.RepresentationoffractionalBrownianmotionasthe Wienerintegralonthecompactintegral....................63
Chapter4.Construction,PropertiesandSomeFunctionalsofthe WienerProcessandFractionalBrownianMotion ............67
4.1.ConstructionofaWienerprocessontheinterval [0, 1] .........67
4.2.ConstructionofaWienerprocesson R+ .................72
4.3.Nowheredifferentiabilityofthetrajectoriesof aWienerprocess.................................74
4.4.PowervariationoftheWienerprocessandofthe fractionalBrownianmotion...........................77
4.4.1.Ergodictheoremforpowervariations.................77
4.5.Self-similarstochasticprocesses......................79
4.5.1.Definitionofself-similarityandsomeexamples...........79
4.5.2.Powervariationsofself-similarprocesses onfiniteintervals................................80
Chapter5.MartingalesandRelatedProcesses ..............85
5.1.Notionofstochasticbasiswithfiltration.................85
5.2.Notionof(sub-,super-)martingale:elementary properties.....................................86
5.3.Examplesof(sub-,super-)martingales..................87
5.4.Markovmomentsandstoppingtimes...................90
5.5.Martingalesandrelatedprocesseswithdiscretetime..........96
5.5.1.Upcrossingsoftheintervalandexistence ofthelimitofsubmartingale..........................96
5.5.2.Examplesofmartingaleshavingalimitandof uniformlyandnon-uniformlyintegrablemartingales............102
5.5.3.Lévyconvergencetheorem.......................104
5.5.4.Optionalstopping............................105
5.5.5.Maximalinequalitiesfor(sub-,super-) martingales...................................108
5.5.6.Doobdecompositionfortheintegrableprocesses withdiscretetime................................111
5.5.7.Quadraticvariationandquadraticcharacteristics: Burkholder–Davis–Gundyinequalities....................113
5.5.8.ChangeofprobabilitymeasureandGirsanov theoremfordiscrete-timeprocesses......................116
5.5.9.Stronglawoflargenumbersformartingales withdiscretetime................................120
5.6.Lévymartingalestopped..........................126
5.7.Martingaleswithcontinuoustime.....................127
Chapter6.RegularityofTrajectoriesofStochastic Processes
......................................131
6.1.Continuityinprobabilityandin L2 (Ω, F , P) ...............131
6.2.Modificationofstochasticprocesses:stochastically equivalentandindistinguishableprocesses...................133
6.3.Separablestochasticprocesses:existenceof separablemodification..............................135
6.4.Conditionsof D -regularityandabsenceofthe discontinuitiesofthesecondkindforstochasticprocesses..........138
6.4.1.Skorokhodconditionsof D -regularityinterms ofthree-dimensionaldistributions.......................138
6.4.2.Conditionsofabsenceofthediscontinuities ofthesecondkindformulatedintermsofconditional probabilitiesoflargeincrements.......................144
6.5.Conditionsofcontinuityoftrajectoriesof stochasticprocesses...............................148
6.5.1.Kolmogorovconditionsofcontinuityinterms oftwo-dimensionaldistributions.......................148
6.5.2.Höldercontinuityofstochasticprocesses: asufficientcondition..............................152
6.5.3.Conditionsofcontinuityintermsof conditionalprobabilities............................154
Chapter7.MarkovandDiffusionProcesses
................157
7.1.Markovproperty...............................157
7.2.ExamplesofMarkovprocesses.......................163
7.2.1.Discrete-timeMarkovchain......................163
7.2.2.Continuous-timeMarkovchain....................165
7.2.3.Processwithindependentincrements.................168
7.3.Semigroupresolventoperatorandgeneratorrelated tothehomogeneousMarkovprocess......................168
7.3.1.SemigrouprelatedtoMarkovprocess.................168
7.3.2.Resolventoperatorandresolventequation..............169
7.3.3.Generatorofasemigroup........................171
7.4.Definitionandbasicpropertiesofdiffusionprocess...........175
7.5.Homogeneousdiffusionprocess.Wienerprocess asadiffusionprocess...............................179
7.6.Kolmogorovequationsfordiffusions...................181
Chapter8.StochasticIntegration
8.1.Motivation..................................187
8.2.DefinitionofItôintegral..........................189
8.2.1.ItôintegralofWienerprocess.....................195
8.3.ContinuityofItôintegral..........................197
8.4.ExtendedItôintegral............................199
8.5.ItôprocessesandItôformula........................203
8.6.Multivariatestochasticcalculus......................212
8.7.MaximalinequalitiesforItômartingales.................215
8.7.1.StronglawoflargenumbersforItô localmartingales................................218
8.8.LévymartingalecharacterizationofWienerprocess...........220
8.9.Girsanovtheorem..............................223
8.10.Itôrepresentation..............................228
Chapter9.StochasticDifferentialEquations
9.1.Definition,solvabilityconditions,examples...............233
9.1.1.Existenceanduniquenessofsolution.................234
9.1.2.Somespecialstochasticdifferentialequations............238
9.2.Propertiesofsolutionstostochasticdifferential equations......................................241
9.3.Continuousdependenceofsolutionsoncoefficients...........245
9.4.Weaksolutionstostochasticdifferentialequations............247
9.5.SolutionstoSDEsasdiffusionprocesses.................249
9.6.Viability,comparisonandpositivityofsolutionsto stochasticdifferentialequations.........................252
9.6.1.Comparisontheoremforone-dimensionalprojectionsof stochasticdifferentialequations........................257
9.6.2.Non-negativityofsolutionstostochastic differentialequations..............................258
9.7.Feynman–Kacformula...........................258
9.8.Diffusionmodeloffinancialmarkets...................260
9.8.1.Admissibleportfolios,arbitrageandequivalent martingalemeasure...............................263
9.8.2.Contingentclaims,pricingandhedging................266
Part2.StatisticsofStochasticProcesses
10.1.Driftanddiffusionparameterestimationinthelinear regressionmodelwithdiscretetime.......................273
10.1.1.Driftestimationinthelinearregressionmodel withdiscretetimeinthecasewhentheinitialvalueisknown.......274
xTheoryandStatisticalApplicationsofStochasticProcesses
10.1.2.Driftestimationinthecasewhentheinitialvalue isunknown...................................277
10.2.Estimationofthediffusioncoefficientinalinear regressionmodelwithdiscretetime.......................277
10.3.Driftanddiffusionparameterestimationinthelinear modelwithcontinuoustimeandtheWienernoise..............278
10.3.1.Driftparameterestimation......................279
10.3.2.Diffusionparameterestimation...................280
10.4.Parameterestimationinlinearmodelswithfractional Brownianmotion.................................281
10.4.1.EstimationofHurstindex.......................281
10.4.2.Estimationofthediffusionparameter................283
10.5.Driftparameterestimation.........................284
10.6.Driftparameterestimationinthesimplest autoregressivemodel...............................285
10.7.Driftparametersestimationinthehomogeneous diffusionmodel..................................289
Chapter11.FilteringProblem.Kalman-BucyFilter
11.1.Generalsetting...............................293
11.2.Auxiliarypropertiesofthenon-observableprocess...........294
11.3.Whatisanoptimalfilter..........................295
11.4.Representationofanoptimalfilterviaanintegral equationwithrespecttoanobservableprocess................296
11.5.IntegralWiener-Hopfequation......................299
Preface
Thisbookisconcernedwithbothmathematicaltheoryofstochasticprocessesand sometheoreticalaspectsofstatisticsforstochasticprocesses.Ourgeneralideawasto combineclassictopicsofthetheoryofstochasticprocesses–measure-theoretic issuesofexistence,processeswithindependentincrements,Gaussianprocesses, martingales,continuityandrelatedpropertiesoftrajectoriesandMarkovproperties–withcontemporarysubjects–stochasticanalysis,stochasticdifferentialequations, fractionalBrownianmotionandparameterestimationindiffusionmodels.Amore detailedexpositionofthecontentsofthebookisgivenintheIntroduction.
Weaimedtomakethepresentationofmaterialasself-containedaspossible.With thisinmind,wehaveincludedseveralcompleteproofs,whichareofteneither omittedfromtextbooksonstochasticprocessesorreplacedbysomeinformalor heuristicarguments.Forthisreason,wehavealsoincludedsomeauxiliarymaterials, mainlyrelatedtodifferentsubjectsofrealanalysisandprobabilitytheory,inthe comprehensiveappendix.However,wecouldnotcoverthefullscopeofthetopic,so asubstantialbackgroundincalculus,measuretheoryandprobabilitytheoryis required.
Thebookisbasedonlecturecourses, Theoryofstochasticprocesses, Statisticsof stochasticprocesses, Stochasticanalysis, Stochasticdifferentialequations, Theoryof Markovprocesses, GeneralizedprocessesoffractionalBrownianmotionand Diffusionprocesses,taughtregularlyintheMechanicsandMathematicsFacultyof TarasShevchenkoNationalUniversityofKyivand Stochasticdifferentialequations lecturecoursestaughtattheUniversityofVeronainSpring2016; Fractional Brownianmotionandrelatedprocesses:stochasticcalculus,statisticalapplications andmodeling taughtinSchoolinBedlewoinMarch2015; FractionalBrownian motionandrelatedprocesses taughtatUlmUniversityinJune2015;anda FractionalBrownianmotioninanutshell mini-coursegivenatthe7thJagna InternationalConferencein2014.
Thebookistargetedatthewidestaudience:studentsofmathematicalandrelated programs,postgraduatestudents,postdoctoralresearchers,lecturers,researchers, practitionersinthefieldsconcernedwiththeapplicationofstochasticprocesses,etc. Thebookwouldbemostusefulwhenaccompaniedbyaprobleminstochastic processes;werecommend[GUS10]asitmatchesourtopicsbest.
Wewouldliketoexpressourgratitudetoeveryonewhomadethecreationofthis bookpossible.Inparticular,wewouldliketothankŁukaszStettner,Professoratthe DepartmentofProbabilityTheoryandMathematicsofFinance,Instituteof Mathematics,PolishAcademyofSciences;LucaDiPersio,AssistantProfessoratthe DepartmentofComputerScienceattheUniversityofVerona;EvgenySpodarev, ProfessorandDirectoroftheInstituteofStochasticsatUlmUniversity,fortheir hospitalitywhilehostingYuliyaMishuraduringlecturecourses.Wewouldalsolike tothankAlexanderKukush,ProfessorattheDepartmentofMathematicalAnalysis ofTarasShevchenkoNationalUniversityofKyiv,forproofreadingthestatisticalpart ofthemanuscript,andEvgeniyaMunchak,PhDstudentattheDepartmentof Probability,Statistics,andActuarialMathematicsofTarasShevchenkoNational UniversityofKyiv,forherhelpintypesettingthemanuscript.
YuliyaM ISHURA GeorgiyS HEVCHENKO
September2017
Intheworldthatsurroundsus,alotofeventshavearandom(nondeterministic) structure.Atmolecularandsubatomiclevels,allnaturalphenomenaarerandom. Movementofparticlesinthesurroundingenvironmentisaccidental.Numerical characteristicsofcosmicradiationandtheresultsofmonitoringtheeffectofionizing radiationarerandom.Themajorityofeconomicfactorssurroundingassetpriceson financialmarketsvaryrandomly.Despiteeffortstomitigateriskandrandomness, theycannotbecompletelyeliminated.Moreover,incomplexsystems,itisoften easiertoreachanequilibriumstatewhentheyarenottootightlycontrolled. Summing-up,chancemanifestsitselfinalmosteverythingthatsurroundsus,and thesemanifestationsvaryovertime.Anyonecansimulatetime-varyingrandomness bytossingacoinorrollingadicerepeatedlyandrecordingtheresultsofsuccessive experiments.(Ifaphysicalrandomnumberisunavailable,oneofthenumerous computeralgorithmstogeneraterandomnumberscanbeused.)Inviewofthis ubiquityofrandomness,thetheoryofprobabilityandstochasticprocesseshasalong history,despitethefactthattherigorousmathematicalnotionofprobabilitywas introducedlessthanacenturyago.Letusspeakmoreonthishistory.
Peoplehaveperceivedrandomnesssinceancienttimes,forexample,gambling alreadyexistedinancientEgyptbefore3000BC.Itisdifficulttotellexactlywhen systematicattemptstounderstandrandomnessbegan.Probably,themostnotable werethosemadebytheprominentancientGreekphilosopherEpicurus(341–270 BC).AlthoughhisviewswereheavilyinfluencedbyDemocritus,heattacked Democritus’materialism,whichwasfullydeterministic.Epicurusinsistedthatall atomsexperiencesomerandomperturbationsintheirdynamics.Althoughmodern physicsconfirmstheseideas,Epicurushimselfattributedtherandomnesstothefree willofatoms.Thephenomenonofrandomdetoursofatomswascalled clinamen (cognatetoinclination)bytheRomanpoetLucretius,whohadbrilliantlyexposed Epicurus’philosophyinhispoem OntheNatureofThings
Movingclosertopresenttimes,letusspeakofthetimeswheretherewasno theoryofstochasticprocesses,physicswasalreadyawell-developedsubject,but therewasn’tanyequipmentsuitabletostudyobjectsinsufficientlysmallmicroscopic detail.In1825,botanistRobertBrownfirstobservedaphenomenon,latercalled Brownianmotion,whichconsistedofachaoticmovementofapollenparticleina vessel.Hecouldnotcomeupwithamodelofthissystem,sojuststatedthatthe behaviorisrandom.
Asuitablemodelforthephenomenonaroseonlyseveraldecadeslater,inavery differentproblem,concernedwiththepricingoffinancialassetstradedonastock exchange.AFrenchmathematicianLouisBachelier(1870–1946),whoaimedtofind amathematicaldescriptionofstochasticfluctuationsofstockprices,provideda mathematicalmodelinhisthesis“Théoriedelaspéculation”[BAC95],whichwas defendedattheUniversityofParisin1900.Themodelis,inmodernterms,a stochasticprocess,whichischaracterizedbythefactthatitsincrementsintime,ina certainstatisticalsense,areproportionaltothesquarerootofthetimechange;this “squareroot”phenomenonhadalsobeobservedearlierinphysics;Bachelierwasthe firsttoprovideamodelforit.Looselyspeaking,accordingtoBachelier,theasset price St attime t ismodeledby
St = at + b√tξ,
where a,b areconstantcoefficients,and ξ isarandomvariablehavingGaussian distribution.
TheworkofBachelierwasundervalued,probablyduetothefactthatapplied mathematicswasvirtuallyabsentatthetime,aswellasconciseprobabilitytheory. BachelierspenthisfurtherlifeteachingindifferentuniversitiesinFranceandnever returnedtothetopicofhisthesis.Itwasonlybroughttothespotlight50yearsafter itspublication,afterthedeathofBachelier.Now,Bachelierisconsideredaprecursor ofmathematicalfinance,andtheprincipalorganizationinthissubjectbearshisname: BachelierFinanceSociety.
OtherworkswhichfurtheredunderstandingtowardsBrownianmotionweremade byprominentphysicists,AlbertEinstein(1879–1955)andMarianSmoluchowski (1872–1917).Theirarticles[EIN05]and[VON06]explainedthephenomenonof Brownianmotionbythermalmotionofatomsandmolecules.Accordingtothis theory,themoleculesofagasareconstantlymovingwithdifferentspeedsin differentdirections.Ifweputaparticle,sayofpollenwhichhasasmallsurfacearea, insidethegas,thentheforcesfromimpactswithdifferentmoleculesdonot compensateeachother.Asaresult,this Brownian particlewillexperienceachaotic movementwithvelocityanddirectionchangingapproximately 1014 timesper second.Thisgaveaphysicalexplanationtothephenomenonobservedbythe botanist.Italsoturnedoutthatakinetictheoryofthermalmotionrequireda
stochasticprocess Bt .EinsteinandSmoluchowskinotonlydescribedthisstochastic process,butalsofounditsimportantprobabilisticcharacteristics.
Onlyaquarterofacenturylater,in1931,AndreyKolmogorov(1903–1987)laid thegroundworkforprobabilitytheoryinhispioneeringworks AbouttheAnalytical MethodsofProbabilityTheory and FoundationsoftheTheoryofProbability [KOL31, KOL77].ThisallowedhisfellowresearcherAleksandrKhinchin(1894–1859)togive adefinitionofstochasticprocessinhisarticle[KHI34].
ThereisananecdoterelatedtotheroleofKhinchinindefiningastochastic processandtheoriginsofthe“stochastic”asasynonymforrandomness(theoriginal Greekwordmeans“guessing”and“predicting”).TheysaythatwhenKhinchin definedtheterm“randomprocess”,itdidnotgowellwiththeSovietauthorities.The reasonisthatthenotionofrandomprocessusedbyKhinchincontradicteddialectical materialism(diamat).Indiamat,similarlytoDemocritus’materialism,allprocesses innaturearecharacterizedbytotallydeterministicdevelopment,transformation,etc., sothephrase“randomprocess”itselfsoundedparadoxical.Asaresult,toavoiddire consequences(werecallthat1934wastheapogeeofStalin’sGreatTerror),Khinchin hadtochangethename.Aftersomeresearch,hecameupwiththeterm“stochastic”, from στoχαστικ ´ ητ ´ εχνη ,theGreektitleof ArsConjectandi,acelebratedbookby JacobBernoulli(1655–1705)publishedin1713,whichcontainsmanyclassicresults. BeingpopularizedlaterbyWilliamFeller[FEL49]andJosephDoob[DOO53],this becameastandardnotioninEnglishandGermanliterature.Perhapsparadoxically,in Russianliterature,theterm“stochasticprocesses”didnotliveforlong.The1956 RussiantranslationofDoob’smonograph[DOO53]ofthisnamewasentitled Probabilisticprocesses,andnowthestandardnameis randomprocess
Analternativeexplanation,given,forexample,in[DEL17],attributestheterm “stochastic”toLadislausWładysławBortkiewicz(1868–1931),Russianeconomist andstatistician,whoinhispaper, DieIterationen [BOR17],definedtheterm “stochastic”as“theinvestigationofempiricalvarieties,whichisbasedonprobability theory,and,therefore,onthelawoflargenumbers.Butstochasticisnotsimply probabilitytheory,butaboveallprobabilitytheoryandapplications”.Thismeaning correlateswiththeonegivenin ArsConjectandi byJacobBernoulli,sothetrue originofthetermprobablyissomewherebetweenthesetwostories.Itisalsoworth mentioningthatBortkiewiczisknownforprovingthe Poissonapproximation theorem abouttheconvergenceofbinomialdistributionswithsmallparameterstothe Poissondistribution,whichhecalled thelawofsmallnumbers.
ThishistoricaldiscussionwouldbeincompletewithoutmentioningPaulLévy (1886–1971),aFrenchmathematicianwhomademanyimportantcontributionsto thetheoryofstochasticprocesses.Manyobjectsandtheoremsnowbearhisname: Lévyprocesses, Lévy-Khinchinrepresentation, Lévyrepresentation,etc.Among
otherthings,hewrotethefirstextensivemonographonthe(mathematicalmodelof) Brownianmotion[LÉV65].
FurtherimportantprogressinprobabilitytheoryisrelatedtoNorbertWiener (1894–1964).Hewasajackofalltrades:aphilosopher,ajournalist,butthemost importantlegacythatheleftwasasamathematician.Inmathematics,hisinterestwas verybroad,fromnumbertheoryandrealanalysis,toprobabilitytheoryandstatistics. Besidesmanyotherimportantcontributions,hedefinedanintegral(ofadeterministic function)withrespecttothemathematicalmodelofBrownianmotion,whichnow bearshisname:a Wienerprocess (andthecorrespondingintegraliscalleda Wiener integral).
TheideasofWienerweredevelopedbyKiyoshiItô(1915–2008),whointroduced anintegralofrandomfunctionswithrespecttotheWienerprocessin[ITÔ44].This leadtotheemergenceofabroadfieldof stochasticanalysis,aprobabilistic counterparttorealintegro-differentialcalculus.Inparticular,hedefined stochastic differentialequations (thenameisself-explanatory),whichallowedustostudy diffusionprocesses,whicharenaturalgeneralizationsoftheWienerprocess.Aswith Lévy,manyobjectsinstochasticanalysisarenamedafterItô: Itôintegral, Itô process, Itôrepresentation, Wiener-Itôdecomposition,etc.
Animportantcontributiontothetheoryofstochasticprocessesandstochastic differentialequationswasmadebyUkrainianmathematiciansIosifGihman (1918–1985)andmorenotablybyAnatoliySkorokhod(1930–2011).Theirbooks [GIH72,GIK04a,GIK04b,GIK07]arenowclassicalmonographs.Therearemany thingsinstochasticanalysisnamedafterSkorokhod: Skorokhodintegral, Skorokhod space, Skorokhodrepresentation,etc.
Ourbook,ofcourse,isnotthefirstbookonstochasticprocesses.Theyare describedinmanyothertexts,fromsomeofwhichwehaveborrowedmanyideas presentedhere,andwearegratefultotheirauthorsforthetexts.Itisimpossibleto mentioneverysinglebookhere,soweciteonlyfewtextsofourselection.We apologizetotheauthorsofmanyotherwonderfultextswhichwearenotabletocite here.
Theextensivetreatmentofprobabilitytheorywithallnecessarycontextis availableinthebooksofP.Billingsley[BIL95],K.-L.Chung[CHU79], O.Kallenberg[KAL02],L.KoralovandY.Sinai[KOR07],M.Loève [LOÈ77,LOÈ78],D.Williams[WIL91].Itisalsoworthmentioningtheclassic monographofP.Billingsley[BIL99]concernedwithdifferentkindsofconvergence conceptsinprobabilitytheory.
Forbookswhichdescribethetheoryofstochasticprocessesingeneral,we recommendthatthereaderlooksatthemonographbyJ.Doob[DOO53],the extensivethree-volumemonographbyI.GikhmanandA.Skorokhod
[GIK04a,GIK04b,GIK07],thetextbooksofZ.BrzezniakandT.Zastawniak [BRZ99],K.-L.Chung[CHU79],G.Lawler[LAW06],S.Resnick[RES92], S.Ross[ROS96],R.SchillingandL.Partzsch[SCH14],A.Skorokhod[SKO65], J.Zabczyk[ZAB04].ItisalsoworthmentioningthebookbyA.Bulinskiyand A.Shiryaev[BUL05],fromwhichweborrowedmanyideas;unfortunately,itisonly availableinRussian.MartingaletheoryiswellpresentedinthebooksofR.Liptser andA.Shiryaev[LIP89],J.JacodandA.Shiryaev[JAC03],L.Rogersand D.Williams[ROG00a],andtheclassicmonographofD.RevuzandM.Yor [REV99].TherearemanyexcellenttextsrelatedtodifferentaspectsofLévy processes,includingthebooksofD.Applebaum[APP09],K.Sato[SAT13], W.Schoutens[SCH03],andthecollection[BAR01].
Stochasticanalysisnowstandsasanindependentsubject,sotherearemany bookscoveringdifferentaspectsofit.ThebooksofK.-L.ChungandD.Williams [CHU90],I.KaratzasandS.Shreve[KAR91],H.McKean[MCK69],J.-F.LeGall [LEG16],L.RogersandD.Williams[ROG00b]coverstochasticanalysisin general,andthemonographofP.Protter[PRO04]goesmuchdeeperintointegration issues.Stochasticdifferentialequationsanddiffusionprocessesarethesubjectofthe best-sellingtextbookofB.Øksendal[ØKS03],andthemonographsofN.Ikedaand S.Watanabe[IKE89],K.ItôandH.McKean[ITÔ74],A.Skorokhod[SKO65],and D.StrookandS.Varadhan[STR06].TheultimateguidetoMalliavincalculusis givenbyD.Nualart[NUA06].Concerningfinancialapplications,stochasticanalysis ispresentedinthebooksofT.Björk[BJÖ04],M.Jeanblanc,M.Yor,and M.Chesney[JEA09],A.Shiryaev[SHI99],andS.Shreve[SHR04].
Differentaspectsofstatisticalmethodsforstochasticprocessesarecoveredbythe booksofP.BrockwellandR.Davis[BRO06],C.Heyde[HEY97],Y.Kutoyants [KUT04],G.SeberandA.Lee[SEB03].
Finally,fractionalBrownianmotion,oneofthemainresearchinterestsofthe authorsofthisbook,iscoveredbythebooksofF.Biagini etal. [BIA08],Y.Mishura [MIS08],I.Nourdin[NOU12],D.Nualart[NUA06],andbylecturenotesof G.Shevchenko[SHE15].
Ourbookconsistsoftwoparts:thefirstisconcernedwiththetheoryofstochastic processesandthesecondwithstatisticalaspects.
Inthefirstchapter,wedefinethemainsubjects:stochasticprocess,trajectoryand finite-dimensionaldistributions.Wediscussthefundamentalissues:existenceand constructionofastochasticprocess,measurabilityandotheressentialproperties,and sigma-algebrasgeneratedbystochasticprocesses.
Thesecondchapterisdevotedtostochasticprocesseswithindependent increments.Adefinitionisgivenandsimplecriteriawhichprovidetheexistenceare
discussed.Wealsoprovidenumerousimportantexamplesofprocesseswith independentincrements,includingLévyprocesses,andstudytheirproperties.
Thethirdchapterisconcernedwithasubclassofstochasticprocesses,arguably themostimportantforapplications:Gaussianprocesses.First,wediscussGaussian randomvariablesandvectors,andthenwegiveadefinitionofGaussianprocesses. Furthermore,wegiveseveralimportantexamplesofGaussianprocessesanddiscuss theirproperties.Then,wediscussintegrationwithrespecttoGaussianprocessesand relatedtopics.ParticularattentionisgiventofractionalBrownianmotionandWiener processes,withdiscussionofseveralintegralrepresentationsoffractionalBrownian motion.
ThefourthchapterfocusesonsomedelicatepropertiesoftwoGaussian processes,whichareofparticularinterestforapplications:theWienerprocessand fractionalBrownianmotion.Inparticular,anexplicitconstructionoftheWiener processisprovidedandnowheredifferentiabilityofitstrajectoriesisshown.Having inmindthequestionofparameterestimationforstochasticprocesses,wealso discusstheasymptoticbehaviorofpowervariationsfortheWienerprocessand fractionalBrownianmotioninthischapter.
Inthefifthchapter,weattemptedtocoverthemaintopicsinthemartingale theory.Themainfocusisonthediscretetimecase;however,wealsogiveseveral resultsforstochasticprocesses.Inparticular,wediscussthenotionsofstochastic basiswithfiltrationandstoppingtimes,limitbehaviorofmartingales,optional stoppingtheorem,Doobdecomposition,quadraticvariations,maximalinequalities byDoobandBurkholder-Davis-Gundy,andthestronglawoflargenumbers.
Thesixthchapterisdevotedtopropertiesoftrajectoriesofastochasticprocess. Weintroducedifferentnotionsofcontinuityaswellasimportantconceptsof separability,indistinguishabilityandstochasticequivalence,andestablishseveral sufficientconditionsforcontinuityoftrajectoriesandforabsenceofdiscontinuities ofthesecondkind.Tothebestofourknowledge,thisisthefirsttimethatthe differentaspectsofregularityandcontinuityarecomprehensivelydiscussedand compared.
TheseventhchapterdiscussesMarkovprocesses.Thedefinition,togetherwith severalimportantexamples,isfollowedbyanalyticaltheoryofMarkovsemigroups. Thechapterisconcludedbytheinvestigationofdiffusionprocesses,whichservesas abridgetostochasticanalysisdiscussedinthefollowingchapters.Weprovidea definitionandestablishimportantcriteriaandcharacterizationofdiffusionprocesses. WepayparticularattentiontotheforwardandbackwardKolmogorovequations, whichareofgreatimportanceforapplications.
Intheeighthchapter,wegivetheclassicalintroductiontostochasticintegration theory,whichincludesthedefinitionandpropertiesofItôintegral,Itôformula,
multivariatestochasticcalculus,maximalinequalitiesforstochasticintegrals, GirsanovtheoremandItôrepresentation.
Theninthchapter,whichclosesthetheoreticalpartofthebook,isconcernedwith stochasticdifferentialequations.Wegiveadefinitionofstochasticdifferential equationsandestablishtheexistenceanduniquenessofitssolution.Several propertiesofthesolutionareestablished,includingintegrability,continuous dependenceofthesolutionontheinitialdataandonthecoefficientsoftheequation. Furthermore,weprovethatsolutionstostochasticdifferentialequationsarediffusion processesandprovidealinktopartialdifferentialequations,theFeynman-Kac formula.Finally,wediscussthediffusionmodelofafinancialmarket,givingnotions ofarbitrage,equivalentmartingalemeasure,pricingandhedgingofcontingent claims.
Thetenthchapteropensthesecondpartofthebook,whichisdevotedtostatistical aspects.Itstudiestheestimationofparametersofstochasticprocessesindifferent scenarios:inalinearregressionmodelwithdiscretetime,inacontinuoustimelinear modeldrivenbyWienerprocess,inmodelswithfractionalBrownianmotions,ina linearautoregressivemodelandinhomogeneousdiffusionmodels.
Intheeleventhchapter,theclassicproblemofoptimalfilteringisstudied.A statisticalsettingisdescribed,thenarepresentationofoptimalfilterisgivenasan integralwithrespecttoanobservableprocess.Finally,theintegralWiener-Hopf equationisderived,alinearstochasticdifferentialequationfortheoptimalfilteris derived,andtheerroroftheoptimalfilterisidentifiedintermsofsolutionofthe Riccatiequation.Inthecaseofconstantcoefficients,theexplicitsolutionsofthese equationsarefound.
Auxiliaryresults,whicharereferredtointhebook,arecollectedinAppendices 1and2.InAppendix1,wegiveessentialfactsfromcalculus,measuretheoryand theoryofoperators.Appendix2containsimportantfactsfromprobabilitytheory.
TheoryofStochasticProcesses
Theory and Statistical Applications of Stochastic Processes, First Edition. Yuliya Mishura and Georgiy Shevchenko. © ISTE Ltd 2017. Published by ISTE Ltd and John Wiley & Sons, Inc.
1.1.Definitionofastochasticprocess
Let (Ω, F , P) beaprobabilityspace.Here, Ω isasamplespace,i.e.acollectionof allpossibleoutcomesorresultsoftheexperiment,and F isa σ -field;inotherwords, (Ω, F ) isameasurablespace,and P isaprobabilitymeasureon F .Let (S , Σ) be anothermeasurablespacewith σ -field Σ,andletusconsiderthefunctionsdefined onthespace (Ω, F ) andtakingtheirvaluesin (S , Σ).Recallthenotionofrandom variable.
D EFINITION 1.1.– A randomvariable ontheprobabilityspace (Ω, F ) withthevalues inthemeasurablespace (S , Σ) isameasurablemap Ω ξ →S ,i.e.amapforwhichthe followingconditionholds:thepre-image ξ 1 (B ) ofanyset B ∈ Σ belongsto F Equivalentformsofthisdefinitionare:forany B ∈ Σ,wehavethat
ξ 1 (B ) ∈F , or,forany B ∈ Σ,wehavethat
{ω : ξ (ω ) ∈ B }∈F
Considerexamplesofrandomvariables.
1)Thenumbershownbyrollingafairdie.Here,
Theory and Statistical Applications of Stochastic Processes, First Edition. Yuliya Mishura and Georgiy Shevchenko. © ISTE Ltd 2017. Published by ISTE Ltd and John Wiley & Sons, Inc.
2)Thepriceofcertainassetsonafinancialmarket.Here, (Ω, F ) candependon themodelofthemarket,andthespace S ,asarule,coincideswith R+ =[0, +∞).
3)Coordinatesofamovingairplaneatsomemomentoftime.Peopleuse differentcoordinatesystemstodeterminethecoordinatesoftheairplanethathas threecoordinatesatanytime.Thecoordinatesaretimedependentandrandom,to someextent,becausetheyareundertheinfluenceofmanyfactors,someofwhichare random.Here, S = R3 fortheCartesiansystem,or S = R2 × [0, 2π ] forthe cylindricalsystem,or S = R × [0,π ] × [0, 2π ] forthesphericalsystem.
Now,weformalizethenotionofastochastic(random)process,definedon (Ω, F , P).Wewilltreatarandomprocessasasetofrandomvariables.Thatsaid, introducetheparameterset T withelements t : t ∈ T.
D EFINITION 1.2.– Stochasticprocessontheprobabilityspace (Ω, F , P), parameterizedbytheset T andtakingvaluesinthemeasurablespace (S , Σ),isaset ofrandomvariablesoftheform
Xt = {Xt (ω ),t ∈ T,ω ∈ Ω}, where Xt (ω ): T × Ω → S .
Thus,eachparametervalue t ∈ T isassociatedwiththerandomvariable Xt taking itsvaluein S .Sometimes,wecall S aphasespace.Theoriginofthetermcomes fromthephysicalapplicationsofstochasticprocesses,ratherthanfromthephysical problemswhichstimulatedthedevelopmentofthetheoryofstochasticprocessestoa largeextent.
Hereareothercommondesignationsofstochasticprocesses:
X (t),ξ (t),ξt ,X = {Xt ,t ∈ T}
Thelastdesignationisthebestinthesensethatitdescribestheentireprocessas asetoftherandomvariables.Thedefinitionofarandomprocesscanberewrittenas follows:forany t ∈ T andanyset B ∈ Σ
X 1 t (B ) ∈F .
Anotherform:forany t ∈ T andanyset B ∈ Σ
{ω : Xt (ω ) ∈ B }∈F
Ingeneral,thespace S candependonthevalueof t, S = St ,but,inthisbook, space S willbefixedforanyfixedstochasticprocess X = {Xt ,t ∈ T}.If S = R, thentheprocessiscalledrealorreal-valued.Additionally,weassumeinthiscase that Σ= B (R),i.e. (S , Σ)=(R, B (R)),where B (S ) isaBorel σ -fieldon S .If
S = C,theprocessiscalledcomplexorcomplex-valued,andif S = Rd ,d> 1, theprocessiscalledvectororvector-valued.Inthiscase, (S , Σ)=(C, B (C)) and (S , Σ)=(Rd , B (Rd )),respectively.
Concerningtheparameterset T,asarule,itisinterpretedasatimeset.Ifthetime parameteriscontinuous,thenusuallyeither T =[a,b],or [a, +∞) or R.Ifthetime parameterisdiscrete,thenusuallyeither T = N =1, 2, 3,...,or T = Z+ = N ∪ 0 or T = Z.
Theparametersetcanbemultidimensional,e.g. T = Rm ,m> 1.Inthiscase, wecalltheprocessarandomfield.Theparametersetcanalsobemixed,thesocalledtime–spaceset,becausewecanconsidertheprocessesoftheform X (t,x)= X (t,x,ω ),where (t,x) ∈ R+ × Rd .Inthiscase,weinterpret t astimeand x ∈ Rd asthecoordinateinthespace Rd
Therecanbemoreinvolvedcases,e.g.itispossibletoconsiderrandommeasures μ(t,A,ω ),where t ∈ R+ ,A ∈B (Rd ),orrandomprocessesdefinedonthegroups, whoseorigincomesfromphysics.Wewillnotconsiderindetailthetheoryofsuch processes.
Inwhatfollows,weconsiderthereal-valuedparameter,i.e. T ⊂ R,sothatwecan regardtheparameterastime,asdescribedabove.
1.2.Trajectoriesofastochasticprocess.Someexamplesofstochastic processes
1.2.1. Definitionoftrajectoryandsomeexamples
Astochasticprocess X = {Xt (ω ),t ∈ T,ω ∈ Ω} isafunctionoftwovariables, oneofthembeingatimevariable t ∈ T andtheotheroneasamplepoint(elementary event) ω ∈ Ω.Asmentionedearlier,fixing t ∈ T,wegetarandomvariable Xt (·).In contrast,fixing ω ∈ Ω andfollowingthevaluesthat X (ω ) takesasthefunctionof parameter t ∈ T,wegetatrajectory(path,samplepath)ofthestochasticprocess.The trajectoryisafunctionof t ∈ T and,forany t,ittakesitsvaluein S .Changingthe valueof ω ,wegetasetofpaths.TheyareschematicallydepictedinFigure1.1.
Letusconsidersomeexamplesofrandomprocessesanddrawtheirtrajectories. First,werecalltheconceptofindependenceofrandomvariables.
D EFINITION 1.3.– Randomvariables {ξα ,α ∈A},where A issomeparameterset, arecalled mutuallyindependent ifforanyfinitesubsetofindices {α1 ,...,αk }⊂A and,foranymeasurablesets A1 ,...,Ak ,wehavethat
ω1
ω2 t
ω3
Figure1.1. Trajectoriesofastochasticprocess.Foracolorversionof thefigure,seewww.iste.co.uk/mishura/stochasticprocesses.zip
1.2.1.1. Randomwalks
A randomwalk isaprocesswithdiscretetime,e.g.wecanput T = Z+ .Let {ξn ,n ∈ Z+ } beafamilyofrandomvariablestakingvaluesin Rd ,d ≥ 1.Put Xn = n i=0 ξi .Stochasticprocess X = {Xn ,n ∈ Z+ } iscalledarandomwalkin Rd .Inthecasewhere d =1,wehavearandomwalkintherealline.Ingeneral,the randomvariables ξi canhavearbitrarydependencebetweenthem,butthemost developedtheoryisinthecaseofrandomwalkswithmutuallyindependentand identicallydistributedvariables {ξn ,n ∈ Z+ }.If,additionally,anyrandomvariable ξn takesonlytwovalues a and b withrespectiveprobabilities P{ξn = a} = p and P{ξn = b} = q =1 p ∈ (0, 1),thenwehaveaBernoullirandomwalk.If a = b and p = q = 1 2 ,thenwehaveasymmetricBernoullirandomwalk.Thetrajectoryof therandomwalkconsistsofindividualpoints,andisshowninFigure1.2.
1.2.1.2. Renewalprocess
Let {ξn ,n ∈ Z+ } beafamilyofrandomvariablestakingpositivevalueswith probability1.Stochasticprocess N = {Nt ,t ≥ 0} canbedefinedbythefollowing formula: Nt = 0,t<ξ1 ; sup{n ≥ 1: n i=1 ξi ≤ t},t ≥ ξ1
Stochasticprocess N = {Nt ,t ≥ 0} iscalleda renewalprocess.Trajectoriesofa renewalprocessarestep-wisewithstep1.Theexampleofthetrajectoryisrepresented inFigure1.3.
Figure1.2. Trajectoriesofarandomwalk
Figure1.3. Trajectoriesofarenewalprocess
Randomvariables T1 = ξ1 ,T2 = ξ1 + ξ2 ,... arecalledjumptimes,arrivaltimes orrenewaltimesoftherenewalprocess.Thelatternamecomesfromthefactthat therenewalprocesseswereconsideredinappliedproblemsrelatedtomomentsof failureandreplacementofequipment.Intervals [0,T1 ] and [Tn ,Tn+1 ],n ≥ 1 are calledrenewalintervals.
1.2.1.3. Stochasticprocesseswithindependentvaluesandthosewith independentincrements
D EFINITION 1.4.– Astochasticprocess X = {Xt ,t ≥ 0} iscalleda processwith independentvalues iftherandomvariables {Xt ,t ≥ 0} aremutuallyindependent.
Itwillbeshownlater,inExample6.1,thatthetrajectoriesofprocesseswith independentvaluesarequiteirregularand,forthisreason,theprocesseswith independentvaluesarerelativelyrarelyusedtomodelphenomenainnature, economics,technics,society,etc.
D EFINITION 1.5.– Astochasticprocess X = {Xt ,t ≥ 0} iscalleda processwith independentincrements,if,foranysetofpoints 0 ≤ t1 <t2 <...<tn ,therandom variables Xt1 ,Xt2 Xt1 ,...,Xtn Xtn 1 aremutuallyindependent.
Hereisanexampleofarandomprocesswithdiscretetimeandindependent increments.
Let X = {Xn ,n ∈ Z+ } bearandomwalk, Xn = n i=0 ξi ,andtherandom variables {ξi ,i ≥ 0} bemutuallyindependent.Evidently,forany 0 ≤ n1 <n2 < ...<nk ,therandomvariables
aremutuallyindependent;therefore, X isaprocesswithdiscretetimeandindependent increments.Randomprocesseswithcontinuoustimeandindependentincrementsare consideredindetailinChapter2.
1.2.2. Trajectoryofastochasticprocessasarandomelement
Let {Xt ,t ∈ T} beastochasticprocesswiththevaluesinsomeset S .Introduce thenotation S T = {y = y (t),t ∈ T} forthefamilyofallfunctionsdefinedon T andtakingvaluesin S .Anothernotationcanbe S T = ×t∈T St ,withall St = S or simply S T = ×t∈T S ,whichemphasizesthatanyelementfrom S T iscreatedinsuch awaythatwetakeallpointsfrom T,assigningapointfrom S toeachofthem.For example,wecanconsider S [0,∞) or S [0,T ] forany T> 0.Now,thetrajectoriesofa randomprocess X belongtotheset S T .Thus,consideringthetrajectoriesaselements oftheset S T ,wegetthemapping X :Ω →S T ,thattransformsanyelementof Ω intosomeelementof S T .Wewouldliketoaddressthequestionofthemeasurability ofthismapping.Tothisend,weneedtofinda σ -field ΣT ofsubsetsof S T suchthat themapping X is F -ΣT -measurable,andthis σ -fieldshouldbethesmallestpossible. First,letusproveanauxiliarylemma.
L EMMA 1.1.– Let Q and R betwospaces.Assumethat Q isequippedwith σ -field F ,and R isequippedwith σ -field G ,where G isgeneratedbysomeclass K ,i.e. G = σ (K ).Then,themapping f : Q→R is F -G -measurableifandonlyifitis F -K -measurable,i.e.forany A ∈ K ,thepre-imageis f 1 (A) ∈F .
P ROOF.–Necessityisevident.Toprovesufficiency,weshouldcheckthat,inthecase wherethepre-imagesofallsetsfrom K undermapping f belongto F ,thepre-images ofallsetsfrom G undermapping f belongto F aswell.Introducethefamilyofsets
K1 = {B ∈G : f 1 (B ) ∈F}
Thepropertiesofpre-imagesimplythat K1 isa σ -field.Indeed,
f 1 ∞ n=1 Bn = ∞ n=1
f 1 (Bn ) ∈F ,
if f 1 (Bn ) ∈F ,
f 1 (C2 \ C1 )= f 1 (C2 ) \ f 1 (C1 ) ∈F , if f 1 (C1 ) ∈F ,i =1, 2,and f 1 (R)= Q∈F .Itmeansthat K1 ⊃ σ (K )= G , whencetheprooffollows.
Therefore,tocharacterizethemeasurabilityofthetrajectories,wemustfinda “reasonable”subclassofsetsof S T ,theinverseimagesofwhichbelongto F .
D EFINITION 1.6.– Letthepoint t0 ∈ T andtheset A ⊂S ,A ∈ Σ befixed. Elementarycylinder withbase A overpoint t0 isthefollowingsetfrom S T :
C (t0 ,A)= {y = y (t) ∈S T : y (t0 ) ∈ A}.
If S = R and A issomeinterval,then C (t0 ,A) isrepresentedschematicallyin Figure1.4.Elementarycylinderconsistsofthefunctionswhosevaluesatpoint t0 belongtotheset A.
Let Kel betheclassofelementarycylinders,and Kel = σ (Kel ),withthe σ -field beinggeneratedbytheelementarycylinders.
T HEOREM 1.1.– Foranystochasticprocess X = {Xt ,t ∈ T},themapping X : Ω → S T ,whichassignstoanyelement ω ∈ Ω thecorrespondingtrajectory X ( ,ω ), is F –Kel -measurable.
P ROOF.–Accordingtolemma1.1,itissufficienttocheckthatthemapping X is F –Kel -measurable.Lettheset C (t0 ,A) ∈ Kel .Then,thepre-image X 1 (C (t0 ,A))= {ω ∈ Ω: X (t0 ,ω ) ∈ A}∈F ,andthetheoremisproved.
C OROLLARY 1.1.– The σ -field Kel ,generatedbytheelementarycylinders,isthe smallest σ -field ΣT suchthatforanystochasticprocess X ,themapping ω →{Xt (Ω),t ∈ T} is F -ΣT -measurable.
Figure1.4. Trajectoriesthatbelongtoelementarycylinder. Foracolorversionofthefigure,see www.iste.co.uk/mishura/stochasticprocesses.zip
1.3.Finite-dimensionaldistributionsofstochasticprocesses: consistencyconditions
Therearetwomainapproachestocharacterizingastochasticprocess:bythe propertiesofitstrajectoriesandbysomenumber-valuedcharacteristics,e.g.by finite-dimensionaldistributionsofthevaluesoftheprocess.Ofcourse,these approachesarecloselyrelated;however,anyofthemhasitsownspecifics.Nowwe shallconsiderfinite-dimensionaldistributions.
1.3.1. Definitionandpropertiesoffinite-dimensionaldistributions
Let X = {Xt ,t ∈ T} beastochasticprocesstakingitsvaluesinthemeasurable space (S , Σ).Forany k ≥ 0,considerthespace S (k ) ,thatis,aCartesianproductof S : S (k ) = S×S× ... ×S k = ×k i=1 S .
Letthe σ -field Σ(k ) ofmeasurablesetson S (k ) begeneratedbyallproductsof measurablesetsfrom Σ
D EFINITION 1.7.– Finite-dimensionaldistributionsoftheprocess X isafamilyof probabilitiesoftheform P = {P{(Xt1 ,Xt2 ,...,Xtk ) ∈ A(k ) },k ≥ 1,ti ∈ T, 1 ≤ i ≤ k,A(k ) ∈ Σ(k ) }.
R EMARK 1.1.–Often,especiallyinappliedproblems, finite-dimensional distributions aredefinedasthefollowingprobabilities:
P1 = {P{Xt1 ∈ A1 ,...,Xtk ∈ Ak },k ≥ 1,ti ∈ T,Ai ∈ Σ, 1 ≤ i ≤ k }.
Sincewecanwrite
P{Xt1 ∈ A1 ,...,Xtk ∈ Ak } =P (Xt1 ,...,Xtk ) ∈×k i=1 Ai , and ×k i=1 Ai ∈ Σ(k ) ,thefollowinginclusionisevident: P1 ⊂ P.Theinclusion isstrict,becausethesetsoftheform ×k i=1 A(i) donotexhaust Σ(k ) unless k =1 However,belowwegivearesultwherecheckingsomepropertiesfor P isequivalent tocheckingthemfor P1
1.3.2. Consistencyconditions
Let π = {l1 ,...,lk } beapermutationofthecoordinates {1,...,k },i.e. li are distinctindicesfrom1to k .Denotefor A(k ) ∈ Σ(k ) by π (A(k ) ) thesetobtainedfrom A(k ) bythecorrespondingpermutationofcoordinates,e.g.
π ×k i=1 Ai = ×k i=1 Ali .
Denotealso π (Xt1 ,...,Xtk )=(Xti1 ,...,Xtik ) therespectivepermutationof vectorcoordinates (Xt1 ,...,Xtk ).Considerseveralconsistencyconditionswhich finite-dimensionaldistributionsofrandomprocessesandthecorresponding characteristicfunctionssatisfy.
Consistencyconditions (A):
1)Forany 1 ≤ k ≤ l ,anypoints ti ∈ T, 1 ≤ i ≤ l ,andanyset A(k ) ∈ Σ(k )
P{(Xt1 ,...,Xtk ,Xtk+1 ,...,Xtl ) ∈ A(k ) ×S (l k ) } =P{(Xt1 ,...,Xtk ) ∈ A(k ) }.
2)Foranypermutation π
P{π (Xt1 ,...,Xtk ) ∈ π (A(k ) )} =P{(Xt1 ,...,Xtk ) ∈ A(k ) }. [1.1]
R EMARK 1.2.–Assumenowthat S = R andconsiderthecharacteristicfunctionsthat correspondtothefinite-dimensionaldistributionsofstochasticprocess X .Denote
ψ (λ1 ,...,λk ; t1 ,...,tk )=Eexp ⎧ ⎨ ⎩ i k j =1 λj Xtj ⎫ ⎬ ⎭ ,
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In Line, from muzzle to muzzle 15 yards.
In Column of Route 177 ” ” Sub-divisions 87 ” ” Divisions 87 ” ” Half battery 72 ”
A Gun, or Waggon, with 4 Horses covers 11 yards of ground, from front to rear.
For every additional pair of Horses 4 yards should be added.
A Battery of 6 Guns, when limbered up at full intervals, occupies from
Right to left 78 yards.
Front to rear 26 yards.
On each flank, 22 additional yards should be allowed.
A battery of 6 Guns, when unlimbered for Action, at full intervals, occupies from
Right to left 78 yards.
Front to rear 37 yards.
The space required for reversing a Gun with 4 Horses is 9 yards, and for a Waggon about 8 yards.
NAMES OF THE PRINCIPAL PARTS OF A FIELD GUN CARRIAGE.
A Block, or Trail. J Tire, or Streak. g Portfire clipper q Handspike shoe.
B Cheeks, or Brackets. K Rivets. h Locking plate. r Handspike pin.
C Axletree. L Tire, or Streak bolts. i Trail plate bolt. s Handspike ring.
D Ogee. a Eye, or Capsquare bolts. k Trail plate. t Axletree arms.
E Trunnion holes. b Capsquares. l Trail plate eye. u Dragwashers.
F Wheel. c Axletree bands. m Chain eye bolt. v Nave hoops.
G Felly. d Bracket bolts. n Locking chain. w Elevating screw.
H Spokes. e Transom bolts. o Breast, or advancing chain. x Handles of elevating screw.
I Nave. f Trunnion plates. p Trail handles. y Elevating screw box.
P 9 PR B F C .
Section 5. Method of performing the duties of serving ordnance.
Section 6. Ranges.
Section 7. Method of laying a piece of ordnance.
Section 8. Limbering up.
F ( , , ) | Halt: Limber up.
Section 9. Unlimbering, or coming into action.
To the front, rear, right or left. The reverse of limbering up.
Section 10. Moving with the Prolonge.
P .
T .
Nos. 1
Right about face.
H . Front.
U P .
P .
Nos. 3
A Drive on.
Section 11. Mounting field ordnance, with the materials belonging to the battery.
Section 12. Dismounting field ordnance, with the materials belonging to the battery.
Section 13. Shifting shafts.
Section 14. Disengaging a shaft horse, when he falls, or is disabled in action.
Section 15. Changing wheels, when the lifting jack is not at hand.
Section 16. Shifting the medium 12 pounder.
Section 17. To remove disabled field artillery.
Section 18. Exercise with Drag-ropes.
1. A light 6 pounder with its limber requires 15 men, six of whom are told off entirely for the drag-ropes, the other men at the gun also assisting in manning them: No. 9 is always in the shafts, and No. 8 at the point of the shaft, near side. A 9 pounder requires additional men, and a double set of drag-ropes.
2. The drag-rope men are numbered off from 10 upward. Nos. 10, 12, 14, are with the left drag-rope; 11, 13, 15, with the right; 10, 11, carry the drag-ropes.
3. The gun being limbered up, and the detachment and drag-rope men in the order of march, at the word “H ,” Nos. 8 and 9 get into their places; 10 and 11 move outside the gun detachments to the rear, and hook on to the gun drag-washers, passing the end of the drag-ropes at once to the front.
The Nos. then man the drag-ropes as follows:—Nos. 10 and 11 outside, close to the drag-washer; 2, 12, 3, and 13 the centre of the drag-rope; 4, 14, 5, and 15 the front; 6 and 7 the ends The gun detachments inside, and the drag-rope men outside. No. 1 at the point of the shafts, off-side. At the word “U ,” Nos. 10 and 11
unhook, coil up the drag-ropes; and the whole then form the order of march.
4. At the word “A ,” whether to the “ ,” “ ,” “ ,” or “ ,” the drag-ropes are at once quitted; Nos. 10 and 11 unhook, and coil them up; and the whole of the drag-rope men retire with the limber, forming in front of it two deep, as they were numbered off. In limbering up, the drag-rope men form the order of march, and wait for the word to hook on.
FORMATION OF A BATTERY.
A battery of Artillery is generally composed of six pieces of ordnance, to which a Company of Artillery is attached. The number of ammunition, forge, and store waggons varies according to the nature of the ordnance.
Section 19. Fitting of saddles, bridles, harness, &c.
Section 20. Harnessing.
Section 21. Carrying forage.
Section 22. Instruction for Drivers.
Section 23. Parade, and inspection.
The Battery, limbered up, is told off by sub-divisions, divisions, and half batteries.
One gun and its waggon constitute a sub-division.
Two sub-divisions ” a division.
Three sub-divisions ” a half battery.
The battery is numbered from right to left by sub-divisions. It is then told off into three divisions. No. 1 the right; No. 2 the centre; No. 3 the left. Sub-divisions Nos 1, 3, and 5 are also distinguished as right sub-divisions of divisions; and Nos. 2, 4, and 6 the left; the two centre sub-divisions are also to be named. It is also told off into half batteries, and these are distinguished by right, centre, and left subdivisions of half batteries. The gun of direction should always be
named. A flank gun is generally named with a battery of four guns, and the right centre gun with a battery of six guns.
Spare carriages, with the battery, form a third, and, if necessary, a fourth line, in rear. The forge and store-waggon always in the centre, and the ammunition waggons on the flanks, covering those in the front line.
Section 24. Posts, and duties of Officers, and mounted Noncommissioned officers, &c., at exercise.
Second Captain.
In line, limbered up.—One horse’s length in rear of the centre.
In column.—Two horses’ length from the centre on the reverse flank.
In action.—He assists the Captain in general superintendence.
He dresses all points of formation, gives the word “Steady,” when they have been correctly taken up, and the formation completed. When required he commands a division.
Subalterns.
In line, limbered up.—The senior on the right of the right division; the second on the left of the left division; the junior on the right of the centre division.
In column of route.—On the pivot flanks of their leading subdivisions.
In column of divisions.—On the pivot flanks of their respective divisions.
In column of half-batteries.—The subaltern of the centre division, on the pivot flank of the leading half battery. The others continue on the same flank of their sub-divisions as when in line.
In action.—Between the guns of their divisions, a little in rear. They command the divisions to which they are attached, dressing in line with, and close to the leaders, and always with the guns.
In shifting from one flank to the other —It is always along the front, and at a canter; and in joining the new sub-division, the officer always turns his horse’s head inwards.
Staff Serjeants.
In line, limbered up.—The senior on the right of the marker of the right division. The junior on the left of the marker of the left division.
In column of route.—One on the reverse flank of the leading gun; the other on the pivot flank of the rear carriage.
In column of divisions.—One between the guns of the leading, the other between the guns of the rear division.
In column of half-batteries.—On the reverse flank of the waggons of each half battery.
They take up points in changes of position. In line formations, 10 yards from the flank sub-divisions; in column formations, 10 yards in front and rear. They dress the markers when there is no staff officer; and the limbers and waggons in action.
Markers.
In line, limbered up.—In line with the leaders of the waggons, and covering their officers.
In column of route.—With their leading waggons covering their officers. (Without waggons, covering their officers, and in line with the centre horses of the gun.)
In column of divisions, and half-batteries.—They cover their officers. (Without waggons, on the reverse flank of their divisions.)
In action.—Those of the right, and centre divisions on the right of the leaders of the limbers of their right sub-divisions. The marker of the left division, on the left of the leaders of the limber of the left subdivision.
They take up points in changes of position; in line formations, for the sub-division nearest the one of formation; in column formations, for the pivot sub-division.
Farriers, and Artificers.
The farrier is generally attached to the forge; but when the battery is limbered up he is in the centre, in rear of the second captain. The other artificers are told off in the gun detachments when not mounted.
Trumpeters.
In line, limbered up.—On the right of the battery, in line with it, one horse’s length distant.
In column.—One horse’s length in front.
During manœuvres.—One with the commander; the other in rear of the battery.
MANŒUVRES OF A BATTERY OF SIX PIECES
Section 25. B .
1. To advance. Commanding officer’s word of command repeated by officers.
[16]T —M .
The officer, and marker of the subdivision of direction take up points.
2. To retire.
R ( ) —M .
3. To come into action.
A .
Senior staff-serjeants.—Left reverse.
4. To diminish (or increase) intervals on the march.
To diminish.
H , , — -
Nos. 1. Right (or left) half turn—Trot —Front turn—Trot—(except No. 1 of
the named sub-division).
To increase.
F - .
5. To take ground to a flank.
R ( ) — M .
The officers shift to the pivot flank of what will become their leading subdivisions.
6. To make a half turn on the march.
R ( ) —M
7. To form Column of divisions in rear of a flank.
F .
Centre division—Right reverse— March—By the left—Left take ground— Halt—Dress.
M . Left division—Right reverse—March —By the left—Left half turn—Left take ground—Halt—Dress.
F .
Centre division—Left reverse—March
—By the right—Right take ground— Right take ground—Halt—Dress.
M . Right division—Left reverse—March
—By the right—Right half turn—Right take ground—Halt—Dress.
8. To form Column of divisions in front of a flank.
F —
Centre division—Forward—March— Left take ground—Waggons close intervals—Right take ground—Halt— Dress.
M
F —
M .
Right division—Forward—March— Left half turn—Left half turn—Waggons close intervals—Right take ground— Halt—Dress.
Centre division—Forward—March— Right take ground—Waggons close intervals—Left take ground—Halt— Dress.
Left division—Forward—March— Right half turn—Right half turn— Waggons close intervals—Left take ground—Halt—Dress.
9. To form Column of divisions on the centre division. This manœuvre is a combination of Nos. 7 and 8.
10. To change front to the rear. First method. By a countermarch.
T —G , W —M —
R F —H D .
The officers shift to the pivot flank of their leading guns—viz., to the left of 1, 3, and 5 guns.
Second method. On the centre.
C —M .
Centre division—Sub-divisions inwards about wheel—March—Halt— Dress.
Right division—March—Left wheel— Left wheel—Halt—Dress.
Left division—March—Right wheel— Right wheel—Halt—Dress.
Third method. When at diminished intervals; on the march.
T ; . L — —H —F .
11. To change front to a flank.
First method. Right (or left) back, on a flank sub-division. Nos. 1.
C N . 6. M .
C N . 1. M .
6. Left wheel—Left about wheel—Halt —Dress.
5. } Right reverse.
4. } Left shoulders.
3. } Right reverse.
2. } Halt—Dress.
1. } Nos. 1.
1. Right wheel—Right about wheel— Halt—Dress.
2. } Left reverse.
3. } Right shoulders.
4. } Left reverse.
5. } Halt.
6. } Dress.
For Action.
C , N . 6, . M (
The named sub-division comes into action in the new direction; the others proceed as before, and come into action to the rear.
N . 1, . M ).
Second method. Right (or left) forward, on a flank sub-division. Nos. 1.
C
N . 6. M .
C
N . 1— M .
6. Right wheel—Right about wheel— Halt—Dress.
5. } 4. } 3. } Right shoulders. 2. } Halt—Dress.
1. } Nos. 1.
1. Left wheel—Left about wheel—Halt —Dress.
2. } 3. } Left shoulders.
4. } Halt.
5. } Dress. 6. }
For Action.
C
N . 1 — M —(
C
N . 6, —M ).
The named sub-division comes into action in the new direction; the others proceed as before, and come into action to the front.
Third method. To the right (or left) on a central sub-division, one flank thrown forward, the other back. This is a combination of the First and Second methods.
Note. A battery may change its front, Half right, or Half left, on the same principle as already detailed. The commanding officer’s
word would be “C , H ( H ) S - .”
These manœuvres can be executed on the same principle, by divisions, or half batteries.
A Battery can also change front on a moveable pivot by a simple wheel.
12. To advance from a flank, in column.
A , —M .
A , —M .
Right division. Forward by the left. March.
Centre, and left divisions
Nos. 1.
Right take ground. March—Left take ground.
2. } Waggon right. Waggon rear.
4. }
Left division
Centre, and right divisions
Nos. 1.
Forward by the right—March.
Left take ground— March. Right take ground.
5.} Waggon left. Waggon rear.
3.}
13. To advance from the centre, in double column of subdivisions.
A ,
Centre division—Forward.—Trot— March.
—M .
Right division—Left take ground— March.
Left division—Right take ground— March.
Nos. 1.
2. } Right take ground—Trot.
1. } Right take ground.
5. } Left take ground—Trot.
6. } Left take ground.
14. To move from a flank along the front in a column of divisions.
M , , .
M .
Forward—March. Left wheel.
Note. To advance from the left, along the front, is done in the same manner.
15. To advance from a flank, in echellon of sub-divisions.
A , - M .
Advancing from the Left is done on the same principle.
Note. A Battery in echellon of Sub-divisions, if required to change its front when in action, can do so at the word “A ” ( ), by merely throwing the trails round, and bringing the guns into the new direction, the limbers and waggons forming in rear of their guns.
16. To advance from a flank in echellon of divisions.
A ,
The right division advances, the centre moves off in succession, when
—M at its wheeling distance, in rear of the leading one.
The left division follows in the same manner.
Advancing from the Left is done on the same principle.
Note. A Battery in echellon of Divisions, if required to change its front when in action, does so as follows:—
C
,
M .
Nos. 1.
2. }
4. } Action left.
6. }
1. } Front limber up.
3. } Left wheel—Halt.
5. } Action front.
An echellon of Half batteries is formed in the same manner as that of divisions; the rear half battery must, however, keep its wheeling distance from the leading one. When in action, if the front is to be changed, it is better to do it on a centre gun.
Retirements in echellon, are done on the same principle as the advance.
17. To retire from a flank in column.
First Method.
R —M .
Right division—Sub-divisions inwards about wheel—March.
Centre and left divisions—Forward— Right wheel—March—Right wheel.
To retire with the Left division is done on the same principle.
Second method.
R , —M .
Right division—Right reverse— March.
Centre, and left divisions—Right take
ground—March—Right take ground.
Nos. 1.
2. } Gun left.
4. } Gun rear.
18. To retire from the centre in a double column of sub-divisions.
In order to perform this manœuvre, the battery should be reversed, and then (with waggons leading) it is performed in the same manner as the advance from the centre, in a double column.
19. To retire from a flank by alternate Half batteries, in action.
When a battery in line, in action, is ordered to retire from a flank by alternate Half batteries, the named half battery at once limbers up to the rear, retires to its distance in echellon, and comes into action. As soon as this half battery is in action, the other limbers up to the rear, retires, passes the half battery in action, and so on. The senior officer of each half battery gives the word of command.
Note.—This manœuvre would generally be practised with the prolonge.
20. To break into column to a flank.
B . M .
Nos. 1.
1. } Right take ground.
3. } Guns front.
5. }
2. }
4. } Right wheel.
6. }
Breaking into Column to the Left can be done on the same principle.
A Column of Half batteries can be formed in the same manner; the pivot sub-divisions wheeling as before, but the others after taking ground, must incline away to gain their required intervals.
Note.—This movement would generally be employed in breaking into column from line, to march past with other troops; and with half batteries it would be done at reduced intervals.
21. To increase, and diminish the front.
First method.
From Column of route, to form Column of divisions, on the march.
Column, right in front.
F .
Right division—Forward by the left.
Centre and left divisions.—Forward by the left. Trot—Walk. Nos. 1.
2. }
4. } Left half turn—Trot—Front turn.
6. }
Column, left in front.
F .
Left division—Forward by the right.
Centre and right divisions.—Forward by the right—Trot—Walk. Nos. 1.
5. }
3. } Right half turn—Trot.
1. } Front turn.
Second method.
From Column of route, to form Column of divisions, in succession.
Column, right in front.
I Nos. 1.
Column, left in front.
I .
1. }
3. } Halt.
5. }
2. } Left half turn.
4. } Front turn.
6. } Halt—Dress.
Nos. 1.
6. } 4. } Halt.
2. }
5. } Right half turn.
3. } Front turn.
1. } Halt—Dress.
Third method.
From Column of divisions, to form Column of route, on the march.
Column right in front.
F .
Centre and left divisions—Halt.
Nos. 1.
1. Forward.
3. } Forward—March.
5. }
2. } Halt—Right half turn—March.
4. } Front turn.
6. }
Column left in front.
F .
Centre and right divisions—Halt.
Nos. 1.
6. Forward.
4. } Forward—March.
2. }
5. } Halt—Left half turn.
3. } March—Front turn.
1. }
22. To bring the Rear to the front, in succession, on the march. First method. In Column of route.
R -
Nos. 1.
6. } 5. }
4. } In succession { Right half turn.
3. } { Front turn.
2. }
Second method. In Column of divisions.
R .
Centre and rear divisions in succession.
Inwards close—Forward—Full intervals.
23. To form Line on the leading division. Divisions right in front.
L —M .
Centre division—Left take ground— March—Right take ground—Halt— Dress.
Left division—Left take ground—
Divisions left in front.
R —M .
March—Right half turn—Right half turn —Halt—Dress.
Centre division—Right take ground— March—Left take ground—Halt— Dress.
Right division—Right take ground— March—Left half turn—Left half turn— Halt—Dress.
On the March.
The centre and rear divisions make a half turn towards the intended line, and come up at an increased pace.
For Action.
R ( ) —M .
Halt—Action front.
24. To form Line on the rear division.
Divisions right in front.
R —M .
Divisions left in front.
L —M
Centre division—Right take ground— March—Right take ground—Right reverse—Halt—Dress.
Right division—Right take ground— March—Right half turn—Right half turn —Right reverse—Halt—Dress.
Centre division—Left take ground— March—Left take ground—Left reverse —Halt—Dress.
Left division—Left take ground— March—Left half turn—Left half turn— Left reverse—Halt—Dress.
For Action.
