扩散马尔可夫过程和鞅第1卷_PDF下载[32MB-百度云]L.C.G.Rogers,D.Williams著

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Longago（orsoitseemstoday）,Chungwroteonpage196ofhisbook[1]:’Onewondersifthepresenttheoryofstochasticprocessesisnotstilltoodifficultforapplications.’Advancesinthetheorysincethattimehavebeenphenomenal,butthesehavebeenaccompaniedbyanincreaseinthetechnicaldifficultyofthesubjectsobewilderingastogiveaquaintcharmtoChung’suseoftheword’still’.Meyerwritesintheprefacetohisdefinitiveaccountofstochasticintegraltheory:’…ilfaut…uncoursdesixmoissurlesdefinitions.Quepeutonyfaire?’IhavethoughtupasintuitiveapictureofthesubjectasIcan,writtenitdownatspeed,andrefusedtobeluredbackbypiety（orevenbywit!）tocancelhalfaline.’First’intuition,whichiswhatyouneedwhenyouarelearningthesubject,israw,roughandready;and,asyouhaveguessed,Imaketheexcusethatitdemandsacompatiblestyleandlackofpolish.NotethatIwrote’firstintuition’.Consideranexample.Meyer’sconceptofarightprocessisexactlyrightforMarkovprocesstheory,buttheconceptistheresultofalongevolution.Tounderstanditproperly,youneedahighlydevelopedintuition,andthattakestimetoacquire.Thedifficultywiththebestadvancedliteratureisthatitsauthorshavetoomuchintuition;nevermakethemistakeofthinkingotherwise.

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SomeFrequentlyUsedNotationCHAPTERⅠ.BROWNIANMOTION1.INTRODUCTION　1.WhatisBrownianmotion,andwhystudyit　2.Brownianmotionasamartingale　3.BrownianmotionasaGaussianprocess　4.BrownianmotionasaMarkovprocess　5.Brownianmotionasadiffusionandmartingale2.BASICSABOUTBROWNIANMOTION　6.ExistenceanduniquenessofBrownianmotion　7.Skorokhodembedding　8.Donsker”sInvariancePrinciple　9.Exponentialmartingalesandfirst-passagedistributions　10.Somesample-pathproperties　11.Quadraticvariation　12.ThestrongMarkovproperty　13.Reflection　14.ReflectingBrownianmotionandlocaltime　15.Kolmogorov”stest　16.BrownianexponentialmartingalesandtheLawoftheIteratedLogarithm3.BROWNIANMOTIONINHIGHERDIMENSIONS　17.SomemartingalesforBrownianmotion　18.Recurrenceandtransienceinhigherdimensions　19.SomeapplicationsofBrownianmotiontocomplexanalysis　20.WindingsofplanarBrownianmotion　21.Multiplepoints,conepoints,cutpoints　22.PotentialtheoryofBrownianmotioninRdd≥3　23.Brownianmotionandphysicaldiffusion4.GAUSSIANPROCESSESANDLEVYPROCESSES　Gaussianprocesses　　24.ExistenceresultsforGaussianprocesses　　25.Continuityresults　　26.Isotropicrandomflows　　27.Dynkin”sIsomorphismTheorem　Levyprocesses　　28.Levyprocesses　　29.FluctuationtheoryandWiener-Hopffactorisation　　30.LocaltimeofLevyprocessesCHAPTERⅡ.SOMECLASSICALTHEORY　1.BASICMEASURETHEORY　　Measurabilityandmeasure　　　1.Measurablespaces;a-algebras;n-systems;d-systems　　　2.Measurablefunctions　　　3.Monotone-ClassTheorems　　　4.Measures;theuniquenesslemma;almosteverywhere;a.e.u,∑　　　5.Caratheodory”sExtensionTheorem　　　6.Innerandouteru-measures;completion　　Integration　　　7.Definitionoftheintegralfdu　　　8.Convergencetheorems　　　9.TheRadon-NikodymTheorem;absolutecontinuity;<　　　10.Inequalities;andspacesp≥1　　Productstructures　　　11.Producta-algebras　　　12.Productmeasure;Fubini''sTheorem　　　13.Exercises　2.BASICPROBABILITYTHEORY　　Probabilityandexpectation　　　14.Probabilitytriple;almostsurelya.s.;a.s.P,a.s.P,F　　　15.limsupEn:FirstBorel-CantelliLemma　　　16.Lawofrandomvariable;distributionfunction:jointlaw　　　17.Expectation:EX;F　　　18.Inequalities:Markov,Jensen,Schwarz,Tchebychev　　　19.Modesofconvergenceofrandomvariables　　UniformintegrabilityandL1convergence　　　20.Uniformintegrability　　　21.L1convergence　　Independence　　　22.Independenceofa-algebrasandofrandomvariables　　　23.Existenceoffamiliesofindependentvariables　　　24.Exercises3.STOCHASTICPROCESSES4.DISCRETE-PARAMETERMARTINGALETHEORY5.CONTINUOUS-PARAMETERSUPERMARTINGALESCHAPTERⅢ.MARKOVPROCESSES1.TRANSITIONFUNCTIONSANDRESOLVENTS2.FELLER-DYNKINPROCESSES3.ADDITIVEFUNCTIONALS4.APPROACHTORAYPROCESSES:5.RAYPROCESSES6.APPLICATIONSReferencesforVolumes1and2IndextoVolumes1and2