随机微分方程

I.Introduction
Someproblems(1-6)wherestochasticdifferentialequationsplay
anessentialroleinthesolution
II.SomeMathematicalPreliminaries
Randomvariables,independence,stochasticprocesses
Kolmogorov'sextensiontheorem
Brownianmotion
BasicpropertiesofBrownianmotion
VersionsofprocessesandKolmogorov'scontinuitytheorem
Exercises
III.ItoIntegrals
Mathematicalinterpretationofequationsinvolving"noise"
TheItointegral
SomepropertiesoftheItointegral
Martingales
ExtensionsoftheItointegral
ComparisonbetweenItoandStratonovichintegrals
Exercises
IV.ItoProcessesandtheItoFormula
Itoprocesses(stochasticintegrals)
The1-dimensionalItoformula
Themulti-dimensionalItoformula
Themartingalerepresentationtheorem
Exercises
V.StochasticDifferentialEquations
Thepopulationgrowthmodelandotherexamples
Brownianmotionontheunitcircle
Existenceanduniquenesstheoremforstochastic
differentialequations
Weakandstrongsolutions
Exercises
VI.TheFilteringProblem
Statementofthegeneralproblem
Thelinearfilteringproblem
Step1:Z-linearandZ-measurableestimates
Step2:Theinnovationprocess
Step3:TheinnovationprocessandBrownianmotion
Step4:AnexplicitformulaforXl
Step5:ThestochasticdifferentialequationforXt
The1-dimensionalKalman-Bucyfilter
Examples
Themulti-dimensionalKalman-Bucyfilter
Exercises
VII.Diffusions:BasicProperties
DefinitionofanItodiffusion
(A)TheMarkovproperty
(B)ThestrongMarkovproperty
Hittingdistribution,harmonicmeasureandthe
meanvalueproperty
(C)Thegeneratorofadiffusion
(D)TheDynkinformula
(E)Thecharacteristicoperator
Exercises
VIII.OtherTopicsinDiffusionTheory
(A)Kolmogorov'sbackwardequation
Theresolvent
(B)TheFeynman-Kacformula.Killing
(C)Themartingaleproblem
(D)WhenisanItoprocessadiffusion?
HowtorecognizeaBrownianmotion
(E)Randomtimechange
TimechangeformulaforItointegrals
Examples:Brownianmotionontheunitsphere
Harmonicandanalyticfunctions
(F)TheGirsanovtheorem
Exercises
IX.ApplicationstoBoundaryValueProblems
(A)TheDirichletproblem
Regularpoints
ThestochasticDirichletproblem
Existenceanduniquenessofsolution
WhenisthesolutionofthestochasticDirichletproblem
alsoasolutionoftheoriginalDirichletproblem?
(B)ThePoissonproblem
Astochasticversion
Existenceofsolution
Uniquenessofsolution
ThecombinedDirichlet-Poissonproblem
TheGreenmeasure
Exercises
X.ApplicationtoOptimalStopping
Statementoftheproblem
Leastsuperharmonicmajorants
Existencetheoremforoptimalstopping
Uniquenesstheoremforoptimalstopping
Examples:SomestoppingproblemsforBrownianmotion
Rewardfunctionsassumingnegativevalues
Tiletime-inhomogeneouscase
Example:Whenistherighttimetosellthestocks?
Optimalstoppingproblemsinvolvinganintegral
Connectionwithvariationalinequalities
Exercises
XI.ApplicationtoStochasticControl
Statementoftheproblem
TheHamilton-Jacobi-Bellman(HJB)equation
AconverseoftheHJBequation
Markovcontrolsversusgeneraladaptivecontrols
Thelinearstochasticregulatorproblem
Anoptimalportfolioselectionproblem
Asimpleproblemwheretheoptimalprocessisdiscontinuous
Stochasticcontrolproblemswithterminalconditions
Exercises
AppendixA:NormalRandomVariables
AppendixB:Conditional.Expectations
AppendixC:UniformIntegrabilityandMartingale
Convergence
Solutionsandadditionalhintstosomeoftheexercises
Bibliography
ListofFrequentlyUsedNotationandSymbols
Index