随机分析及应用

1PreliminariesFromCalculus

1.1FunctionsinCalculus

1.2VariationofaFunction

1.3RiemannIntegralandStieltjesIntegral

1.4LebesguesMethodofIntegration

1.5DifferentialsandIntegrals

1.6TaylorsFormulaandOtherResults

2ConceptsofProbabilityTheory

2.1DiscreteProbabilityModel

2.2ContinuousProbabilityModel

2.3ExpectationandLebesgueIntegral

2.4TransformsandConvergence

2.5IndependenceandCovariance

2.6Normal(Gaussian)Distributions

2.7ConditionalExpectation

2.8StochasticProcessesinContinuousTime

3BasicStochasticProcesses

3.1BrownianMotion

3.2PropertiesofBrownianMotionPaths

3.3ThreeMartingalesofBrownianMotion

3.4MarkovPropertyofBrownianMotion

3.5HittingTimesandExitTimes

3.6MaximumandMinimumofBrownianMotion

3.7DistributionofHittingTimes

3.8ReflectionPrincipleandJointDistributions

3.9ZerosofBrownianMotion.ArcsineLaw

3.10SizeofIncrementsofBrownianMotion

3.11BrownianMotioninHigherDimensions

3.12RandomWalk

3.13StochasticIntegralinDiscreteTime

3.14PoissonProcess

3.15Exercises

BrownianMotionCalculus

4.1DefinitionofIt6Integral

4.2ItoIntegralProcess

4.3ItoIntegralandGaussianProcesses

4.4ItosFormulaforBrownianMotion

4.5ItoProcessesandStochasticDifferentials

4.6ItosFormulaforIt6Processes

4.7ItoProcessesinHigherDimensions

4.8Exercises

StochasticDifferentialEquations

5.1DefinitionofStochasticDifferentialEquations

5.2StochasticExponentialandLogarithm

5.3SolutionstoLinearSDEs

5.4ExistenceandUniquenessofStrongSolutions

5.5MarkovPropertyofSolutions

5.6WeakSolutionstoSDEs

5.7ConstructionofWeakSolutions

5.8BackwardandForwardEquations

5.9StratanovichStochasticCalculus

5.10Exercises

6DiffusionProcesses

6.1MartingalesandDynkinsFormula

6.2CalculationofExpectationsandPDEs

6.3TimeHomogeneousDiffusions

6.4ExitTimesfromanInterval

6.5RepresentationofSolutionsofODEs

6.6Explosion

6.7RecurrenceandTransience

6.8DiffusiononanInterval

6.9StationaryDistributions

6.10Multi-DimensionalSDEs

6.11Exercises

7Martingales

7.1Definitions

7.2UniformIntegrability

7.3MartingaleConvergence

7.4OptionalStopping

7.5LocalizationandLocalMartingales

7.6QuadraticVariationofMartingales

7.7MartingaleInequalities

7.8ContinuousMartingales.ChangeofTime

7.9Exercises

8CalculusForSemimartingales

8.1Semimartingales

8.2PredictableProcesses

8.3Doob-MeyerDecomposition

8.4IntegralswithrespecttoSemimartingales

8.5QuadraticVariationandCovariation

8.6ItSsFormulaforContinuousSemimartingales

8.7LocalTimes

8.8StochasticExponential

8.9CompensatorsandSharpBracketProcess

8.10ItSsFormulaforSemimartingales

8.11StochasticExponentialandLogarithm

8.12Martingale(Predictable)Representations

8.13ElementsoftheGeneralTheory

8.14RandomMeasuresandCanonicalDecomposition

8.15Exercises

9PureJumpProcesses

9.1Definitions

9.2PureJumpProcessFiltration

9.3ItSsFormulaforProcessesofFiniteVariation

9.4CountingProcesses

9.5MarkovJumpProcesses

9.6StochasticEquationforJumpProcesses

9.7ExplosionsinMarkovJumpProcesses

9.8Exercises

10ChangeofProbabilityMeasure

10.1ChangeofMeasureforRandomVariables

10.2ChangeofMeasureonaGeneralSpace

10.3ChangeofMeasureforProcesses

10.4ChangeofWienerMeasure

10.5ChangeofMeasureforPointProcesses

10.6LikelihoodFunctions

10.7Exercises

11ApplicationsinFinance:StockandFXOptions

11.1FinancialDeriwtivesandArbitrage

11.2AFiniteMarketModel

11.3SemimartingaleMarketModel

11.4DiffusionandtheBlack-ScholesModel

11.5ChangeofNumeraire

11.6Currency(FX)Options

11.7Asian,LookbackandBarrierOptions

11.8Exercises

12ApplicationsinFinance:Bonds,RatesandOption

12.1BondsandtheYieldCurve

12.2ModelsAdaptedtoBrownianMotion

12.3ModelsBasedontheSpotRate

12.4MertonsModelandVasiceksModel

12.5Heath-Jarrow-Morton(HJM)Model

12.6ForwardMeasures.BondasaNumeraire

12.7Options,CapsandFloors

12.8Brace-Gatarek-Musiela(BGM)Model

12.9SwapsandSwaptions

12.10Exercises

13ApplicationsinBiology

13.1FellersBranchingDiffusion

13.2Wright-FisherDiffusion

13.3Birth-DeathProcesses

13.4BranchingProcesses

13.5StochasticLotka-VolterraModel

13.6Exercises

14ApplicationsinEngineeringandPhysics

14.1Filtering

14.2RandomOscillators

14.3Exercises

SolutionstoSelectedExercises

References

Index

　　本书是随机分析方面的名著之一。以主题广泛丰富，论述简洁易懂而又不失严密著称。书中阐述了各领域的典型应用，包括数理金融、生物学、工程学中的模型。还提供了很多示例和习题，并附有解答。第2版增加了讲述证券，利率及其期权的一章，并在全书增加了许多新内容，以反映随机分析研究和应用的最新成果。本书可作为高年级本科生和研究生的随机分析和金融数学的教材，也非常适合各领域专业人士自学。　　本书介绍了随机分析的理论和应用两方面的知识。内容涉及积分和概率论的基础知识、基本的随机过程，布朗运动和伊藤过程的积分、随机微分方程、半鞅积分、纯离散过程，以及随机分析在金融、生物、工程和物理等方面的应用。书中有大量的例题和习题，并附有答案，便于读者进行深层次的学习。　　本书非常适合初学者阅读，可作为高等院校经管、理工和社科类各专业高年级本科生和研究生随机分析和金融数学的教材，也可供相关领域的科研人员参考。【作者简介】　　FimaCKlebaner，澳夫利亚Monash(莫纳什)大学教授，IMS(国际数理统计学会)会士，著名数理统计和金融数学家。主要研究领域有：随饥过程、概率应用、随机分析、金融数学、动态系统的随机扰动等。