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1、FunctionalItocalculusandstochasticintegralrepresentationofmartingalesRamaContDavid-AntoineFourni´eFirstdraft:June2009.Thisversion:June2010.∗AbstractWedevelopanon-anticipativecalculusforfunctionalsofacontinuoussemimartingale,usinganotionofpathwisefunctionalderiv
2、ative.AfunctionalextensionoftheItoformulaisderivedandusedtoobtainaconstructivemartingalerepresentationtheoremforaclassofcontinuousmartingalesverifyingaregularityproperty.BycontrastwiththeClark-Haussmann-Oconefor-mula,thisrepresentationinvolvesnon-anticipativequ
3、antitieswhichcanbecomputedpathwise.Theseresultsareusedtoconstructaweakderivativeactingonsquare-integrablemartingales,whichisshowntobetheinverseoftheItointegral,andderiveanintegrationbypartsformulaforItostochasticintegrals.Weshowthatthisweakderivativemaybeviewed
4、asanon-anticipative“lifting”oftheMalliavinderivative.RegularfunctionalsofanItomartingalewhichhavethelocalmartingalepropertyarechar-acterizedassolutionsofafunctionaldifferentialequation,forwhichauniquenessresultisgiven.Keywords:stochasticcalculus,functionalcalcul
5、us,Itoformula,integrationbyparts,Malliavinderivative,martingalerepresentation,semimartingale,Wienerfunctionals,functionalFeynman-Kacformula,Kolmogorovequation,Clark-Oconeformula.arXiv:1002.2446v3[math.PR]8Jun2010∗WethankBrunoDupireforsharinghisoriginalideaswith
6、us.R.ContthanksHansF¨ollmer,JeanJacodandthelatePaulMalliavinforhelpfulcommentsanddiscussions.1Contents1Introduction32Functionalsrepresentationofnon-anticipativeprocesses42.1Horizontalandverticalperturbationofapath......................62.2Regularityfornon-antic
7、ipativefunctionals........................62.3Measurabilityproperties..................................73Pathwisederivativesofnon-anticipativefunctionals83.1Horizontalandverticalderivatives............................83.2Obstructionstoregularity................
8、.................113.3Uniquenessresultsforverticalderivatives........................114FunctionalItocalculus164.1FunctionalItoformula...................................164.
