随机分析应用导论：英文版

　　During the past twenty years， there has been an increasing demand for tools and methods of Stochastic Calculus in various disciplines. One of the greatest demands appears to have come from the growing area of Mathematical Finance where Stochastic Calculus is used for pricing and hedging of financial derivatives， such as options. In Engineering， most popular applications of Stochastic Calculus are in filtering and control theory. In Physics， Stochastic Calculus is used to study the effects of random excitations on various physical phenomena. In Biology， Stochastic Calculus is used to model the effects Of stochastic variability in reproduction and environment on populations.

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1PreliminariesFromCalculus
1.1ContinuousandDifferentiableFunctions
1.2RightandLeft-ContinuousFunctions
1.3VariationofaFunction
1.4RiemannIntegral
1.5StieltjesIntegral
1.6DifferentialsandIntegrals
1.7Taylor'sFormulaandotherresults
2ConceptsofProbabilityTheory
2.1DiscreteProbabilityModel
2.2ContinuousProbabilityModel
2.3ExpectationandLebesgueIntegral
2.4TransformsandConvergence
2.5IndependenceandConditioning
2.6StochasticProcessesinContinuousTime
3BasicStochasticProcesses
3.1BrownianMotion
3.2BrownianMotionasaGaussianProcess
3.3PropertiesofBrownianMotionPaths
3.4ThreeMartingalesofBrownianMotion
3.5MarkovPropertyofBrownianMotion
3.6ExitTimesandHittingTimes
3.7MaximumandMinimumofBrownianMotion
3.8DistributionofHittingTimes
3.9ReflectionPrincipleandJointDistributions
3.10ZerosofBrownianMotion.ArcsineLaw
3.11SizeofIncrementsofBrownianMotion
3.12BrownianMotioninHigherDimensions
3.13RandomWalk
3.14StochasticIntegralinDiscreteTime
3.15PoissonProcess
3.16Exercises
4BrownianMotionCalculus
4.1DefinitionofIt5Integral
4.2Itointegralprocess
4.3Ito'sFormulaforBrownianmotion
4.4StochasticDifferentialsandIt8Processes
4.5Ito'sformulaforfunctionsoftwovariables
4.6StochasticExponential
4.7ItoProcessesinHigherDimensions
4.8Exercises
5StochasticDifferentialEquations
5.1DefinitionofStochasticDifferentialEquations
5.2StrongSolutionstoSDE's
5.3SolutionstoLinearSDE's
5.4ExistenceandUniquenessofStrongSolutions
5.5MarkovPropertyofSolutions
5.6WeakSolutionstoSDE's
5.7ExistenceandUniquenessofWeakSolutions
5.8BackwardandForwardEquations.
5.9Exercises
6DiffusionProcesses
6.1MartingalesandDynkin'sformula
6.2CalculationofExpectationsandPDE's
6.3HomogeneousDiffusions
6.4ExitTimesFromanInterval
6.5RepresentationofSolutionsofPDE's
6.6Explosion
6.7RecurrenceandTransience
6.8DiffusiononanInterval
6.9StationaryDistributions
6.10MultidimensionalSDE's
6.11Exercises
7Martingales
7.1Definitions
7.2UniformIntegrability
7.3MartingaleConvergence
7.4OptionalStopping
7.5Localization.LocalMartingales
7.6QuadraticVariationofMartingales
7.7MartingaleInequalities
7.8Continuousmartingales
7.9ChangeofTimeinSDE's
7.10MartingaleRepresentations
7.11Exercises
8CalculusForSemimartingales
8.1Semimartingales
8.2QuadraticVariationandCovariation
8.3PredictableProcesses
8.4Doob-MeyerDecomposition
8.5DefinitionofStochasticIntegral
8.6PropertiesofStochasticIntegrals
8.7Ito'sFormula:continuouscase
8.8LocalTimes
8.9StochasticExponential
8.10CompensatorsandSharpBracketProcess
8.11Ito'sFormula:generalcase
8.12ElementsoftheGeneralTheory
8.13Exercises
9PureJumpProcesses
9.1Definitions
9.2PureJumpProcessFiltration
9.3Ito'sFormulaforProcessesofFiniteVariation
9.4CountingProcesses
9.5MarkovJumpProcesses
9.6StochasticequationforMarkovJumpProcesses
9.7ExplosionsinMarkovJumpProcesses
9.8Exercises
10ChangeofProbabilityMeasure
10.1ChangeofMeasureforRandomVariables
10.2EquivalentProbabilityMeasures
10.3ChangeofMeasureforProcesses.
10.4ChangeofDriftinDiffusion
10.5ChangeofWienerMeasure
10.6ChangeofMeasureforPointProcesses
10.7LikelihoodRatios
10.8Exercises
11ApplicationsinFinance
11.1FinancialDerivativesandArbitrage
11.2AFiniteMarketModel
11.3SemimartingaleMarketModel
11.4DiffusionandBlack-ScholesModel
11.5InterestRatesModels
11.6Options,Caps,Floors,SwapsandSwaptions
11.7Exercises
12ApplicationsinBiology
12.1BranchingDiffusion
12.2Wright-FisherDiffusion
12.3Birth-DeathProcesses
12.4Exercises
13ApplicationsinEngineeringandPhysics
13.1Filtering
13.2StratanovichCalculus
13.3RandomOscillators
13.4Exercises
References