stochaLsticCalculusofVariations(orMalliavinCalculus)consists,inbrief,inconstructingandexploitingnaturaldifferentiablestructuresonabstractDrobabilityspaces;inotherwords,StochasticCalculusofVariationsproceedsfromamergingofdifferentialcalculusandprobabilitytheory.AsoptimizationunderarandomenvironmentiSattheheartofmathemat’icalfinance,andasdifferentialcalculusiSofparamountimportanceforthesearchofextrema,itisnotsurprisingthatStochasticCalculusofVariationsappearsinmathematicalfinance.Thecomputationofpricesensitivities(orGreekslobviouslybelongstotherealmofdifferentialcalculus.
Nevertheless,StochasticCalculusofVariationsWasintroducedrelativelylateinthemathematicalfinanceliterature:firstin1991withtheOcone-Karatzashedgingformula,andsoonafterthat,manyotherapplicationsalDearedinvariousotherbranchesofmathematicalfinance;in1999anewirapetuscamefromtheworksofP.L.Lionsandhisassociates.