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1、Chapter1BackgroundMaterialSDEs,PDEs,andTheirInterplay1.1MathematicalFrameworkInoptimalportfoliotheory,inderivativepricingandhedgingtheory,inequityvaluationtheory,inforeignexchangeandforeignexchangederivativestheory,onehastomodelvariousunderlyingdynamicquantit
2、iestheunderlyingdynamics:pricesofstocks,interestrates,dividends,variouscashflows,andeconomicindicators.Modelingdynamicquantitiesisthecentralthemeofclassicalappliedmathemat-ics.Veryoften,itisdonebymeansof(systemsof)ordinarydifferentialequations(ODEs).Insomearea
3、s,suchasmechanicsandastronomy,thehighlevelofprecisionofphysicalmodelsandthehighlevelofprecisionofmeasureddatamakeODEsasufficientlyprecisemodelingdevicesothatthestudyoferrorsisofsecondaryimportance,ormayevenbeignored.Inmanyotherapplicationareas,evenphysicalones
4、suchasmeteorology,andinparticularthenonphysicalonessuchaseconomicsandfinance,themathematicalmodelsandthemeasureddataarebothinsufficientlyprecise.Indeed,forsuchphenomena,realityistoocomplicatedtoevenattempttomodelthemusingdeterministicmathematics.Theremainingerr
5、or,theremainingindeterminacy,isquitesignificantinsize,andthereforehastobeconsideredstatistically.ThemathematicalvehicleforsuchageneralizationofODEsisstochasticdifferentialequations(SDEs),whichonecanthinkofasrandomlyperturbedODEs.1.2WhiteNoiseandBrownianMotionT
6、ogeneratedynamicrandomnessinthemostnaturalandmostproductivefashion,oneuseswhitenoise,orequivalently(standard)Brownianmotion.Whitenoiseisadistribution-valuedstochasticprocess,whereintuitively,thedistributioncanbethoughtofasthederivativeofanondifferentiablefunc
7、tion:S.Stojanovic,NeutralandIndifferencePortfolioPricing,HedgingandInvesting:1WithApplicationsinEquityandFX,DOI10.1007/978-0-387-71418-91,©SpringerScience+BusinessMedia,LLC201121BackgroundMaterialK=1000K=10000100400300500.20.40.60.81.00.20.40.60.81.0–200–50–4
8、00Fig.1.1(Approximate)WhiteNoiseFig.1.2(Approximate)3.0BrownianMotionanditsquadraticvariationprocess2.52.01.51.00.5t0.51.01.52.02.53.0W(t)=dB(t)/dt.The(approximate)propertiesoftheprocessW
