随机分析基础

图书标准信息

• 作者 [丹麦]麦考斯基 著
• 出版社 世界图书出版公司
• 出版时间 2009-08
• 版次 1
• ISBN 9787510005244
• 定价 28.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 212页
• 正文语种 英语

【内容简介】

　　Iknewbetter.Atthattime.staftmembersofeconomicsandmathematicsdepartmentsalreadydiscussedtheuseoftheBlackandScholesoptionpricingformula；coursesonstochasticfinancewere0fieredatleadinginstitutionssuchasETHZfirich.ColumbiaandStanford；andthereWasageneralagreementthatnotonlystudentsandstaftmembersofeconomicsandmathematicsde-partments、butalsopractitionersinfinanciaiinstitutionsshouldknowmoreaboutthisnewtopic.
　　SoonIrealizedthatthereWasnotverymuchliteraturewhichcouldbeusedforteachingstochasticcaiculusataratherelementarylevel.Ialnfullyawareofthefactthatacombinationof“elementary”and“stochasticcalculus”isacontradictioniUitselfStochasticcalculusrequiresadvancedmathematicaitechniques；thistheorycannotbefullvunderstoodifonedoesnotknowaboutthebasicsofmeasuretheory，functionalanalysisandthetheoryofstochasticprocessesHowever.Istronglybelievethataninterestedpersonwhoknowsaboutelementaryprobabilitytheoryandwhocanhandletherulesofinte-grationanddifierentiationisabletounderstandthemainideasofstochasticcalculus.ThisissupportedbymyexperiencewhichIgainedincoursesforeconomicsstatisticsandmathematicsstudentsatVUWWellingtonandtheDepartmentofMathematicsinGroningen.IgotthesameimpressionasalecturerofcrashcoursesonstochasticcalculusattheSummerSchOOl.

【目录】

ReaderGuidelines
1Preliminaries
1.1BasicConceptsflomProbabilityTheory
1.1.1RandomVariables
1.1.2RandomVectors
1.1.3IndependenceandDependence
1.2StochasticProcesses
1.3BrownianMotion
1.3.1DefiningProperties
1.3.2ProcessesDerivedfromBrownianMotion
1.3.3SimulationofBrownianSamplePaths
1.4ConditionalExpectation
1.4.1ConditionalExpectationunderDiscreteCondition
1.4.2Abouta-Fields
1.4.3TheGeneralConditionalExpectation
1.4.4RulesfortheCalculationofConditionalExpectations
1.4.5TheProjectionPropertyofConditionalExpectations
1.5Martingales
1.5.1DefiningProperties
1.5.2Examples
1.5.3TheInterpretationofaMartingaleasaFairGame
2TheStochasticIntegral
2.1TheRiemannandRiemann-StieltjesIntegrals
2.1.1TheOrdinaryRiemannIntegral
2.1.2TheRiemann-StieltjesIntegral
2.2TheItoIntegral
2.2.1AMotivatingExample
2.2.2TheItoStochasticIntegralforSimpleProcesses
2.2.3TheGeneralItoStochasticIntegral
2.3TheItoLemma
2.3.1TheClassicalChainRuleofDifferentiation
2.3.2ASimpleVersionoftheItoLemma
2.3.3ExtendedVersionsoftheItoLemma
2.4TheStratonovichandOtherIntegrals
3StochasticDifferentialEquations
3.1DeterministicDifferentialEquations
3.2ItoStochasticDifferentialEquations
3.2.1WhatisaStochasticDifferentialEquation?
3.2.2SolvingItoStochasticDifferentialEquationsbytheItoLemma
3.2.3SolvingItoDifferentialEquationsviaStratonovichCalculus
3.3TheGeneralLinearDifferentialEquation
3.3.1LinearEquationswithAdditiveNoise
3.3.2HomogeneousEquationswithMultiplicativeNoise
3.3.3TheGeneralCase
3.3.4TheExpectationandVarianceFunctionsoftheSolution
3.4NumericalSolution
3.4.1TheEulerApproximation
3.4.2TheMilsteinApproximation
4ApplicationsofStochasticCalculusinFinance
4.1TheBlack-ScholesOptionPricingFormula
4.1.1AShortExcursionintoFinance
4.1.2WhatisanOption?
4.1.3AMathematicalFormulationoftheOptionPricingProblem
4.1.4TheBlackandScholesFormula
4.2AUsefulTechnique:ChangeofMeasure
4.2.1WhatisaChangeoftheUnderlyingMeasure?
4.2.2AnInterpretationoftheBlack-ScholesFormulabyChangeofMeasure
Appendix
A1ModesofConvergence
A2Inequalities
A3Non-DifferentiabilityandUnboundedVariationofBrownianSamplePaths
A4ProofoftheExistenceoftheGeneralItoStochasticIntegral
A5TheRadon-NikodymTheorem
AoProofoftheExistenceandUniquenessoftheConditionalExpectation
Bibliography
Index
ListofAbbreviationsandSymbols