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1、Chapter11Themultifractalmarkets11.1Multifractalityasanewstylisedfact11.1.1Themultifractalscalingbehaviouroftimeseries11.1.1.1AnalysingcomplexsignalsTheuseoffractionalBrownianmotion(fBm)describedinSection(10.1.2),whichinvolvesarestrictiontoacertainHolderconti
2、nuityHofthepathsforalltimesoftheprocess,istoorestrictiveformanyapplications,andvariabletime-dependentHoldercontinuitiesofthepathsareneeded.Thatis,inthecasewheretheHurstexponent,H,changesovertime,thefractionalBrownianmotionbecomesasomewhatrestrictivemodel,una
3、bletocapturemorefullythecomplexdynamicsoftheseries.AsexplainedbyKantelhardtetal.[2002],manyseriesdonotexhibitasimplemonofractalscalingbehaviourdescribedwithasinglescalingexponent.Theremayexistcrossover(time-)scalesnseparatingregimeswithdifferentscalingexpon
4、ents,suchaslong-rangecorrelationsonsmallscales,n<>n.Alternatively,thescalingbehaviourmaybemorecomplicated,sothatdifferentscalingexponentsarerequiredfordifferentpartoftheseries.Forinstanc
5、e,thescalingbehaviourinthefirsthalfoftheseriesdifferfromthatinthesecondhalf.Incomplexsystem,suchdifferentscalingbehaviourcanbeobservedformanyinterwovenfractalsubsetsofthetimeseries,inwhichcaseamultitudeofscalingexponentsisrequiredforafulldescriptionofthescali
6、ngbehaviour,andamultifractalanalysismustbeapplied.Whilethetermfractalwasmainlyassociatedwithdeterministicchaos,originatingfromasmallnumberofgeneratingequations,inthe80sanewclassofstochasticfractalsdevelopedtermedmultifractals,withprocessesoriginatinginhigh-d
7、imensionalsystems(seeSchertzeretal.[1991]).Ingeneral,twodifferenttypesofmultifractalityintimeseriescanbedistinguished1.Multifractalityduetoabroadprobabilitydensityfunction(pdf)forthevaluesofthetimeseries.Onecannotremovethemultifractalitybyshufflingtheseries.2
8、.Multifractalityduetodifferentlong-range(time-)correlationsofthesmallandlargefluctuations.Inthiscase,thepdfofthevaluescanbearegulardistributionwithfinitemoments.Thecorrespondingshuffledserieswillex
