medium and smallscale analysis of financial data中小规模的金融数据分析外文翻译学位论文Word文档下载推荐.docx
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Fokker–Planckequation
1.Introduction
Oneoftheoutstandingfeaturesofthecomplexityoffinancialmarketsisthatveryoftenfinancialquantitiesdisplaynon-Gaussianstatisticsoftendenotedasheavytailedorintermittentstatistics.Tocharacterizethefluctuationsofafinancialtimeseriesx(t),mostcommonlyquantitieslikereturns,logreturnsorpriceincrementsareused.Here,weconsiderthestatisticsofthelogreturny(τ)overacertaintimescalet,whichisdefinedas
y(τ)=logx(t+τ)-logx(t),
(1)
wherex(t)denotesthepriceoftheassetattimet.Acommonproblemintheanalysisoffinancialdataisthequestionofstationarityforthediscussedstochasticquantities.Inparticularwefindinouranalysisthatthemethodsseemtoberobustagainstnonstationarityeffects.Thismaybeduetothedataselection.Notethattheuseof(conditional)returnsofscaleτcorrespondstoaspecificfilteringofthedata.Neverthelesstheparticularresultschangeslightlyfordifferentdatawindows,indicatingapossibleinfluenceofnonstationarityeffects.Inthispaperwefocusontheanalysisandreconstructionoftheprocessesforagivendatawindow(timeperiod).TheanalysispresentedismainlybasedonBayerdataforthetimespanof1993–2003.ThefinancialdatasetswereprovidedbytheKarlsruherKapitalmarktDatenbank(KKMDB).
2.Small-scaleanalysis
Oneremarkablefeatureoffinancialdataisthefactthattheprobabilitydensityfunctions(pdfs)arenotGaussian,butexhibitheavytailedshapes.Anotherremarkablefeatureisthechangeoftheshapewiththesizeofthescalevariableτ.Toanalysethechangingstatisticsofthepdfswiththescaletanon-parametricapproachischosen.Thedistancebetweenthepdfp(y(τ))onatimescaleτandapdfpT(y(T))onareferencetimescaleTiscomputed.Asareferencetimescale,T=1sischosen,whichisclosetothesmallestavailabletimescaleinourdatasetsandonwhichtherearestillsufficientevents.Inordertobeabletocomparetheshapeofthepdfsandtoexcludeeffectsduetovariationsofthemeanandvariance,allpdfsp(y(τ))havebeennormalisedtoazeromeanandastandarddeviationof1.
Asameasuretoquantifythedistancebetweenthetwodistributionsp(y(τ))andpT(y(T)),theKullback–Leiblerentropyisused.
dK(τ)=
(2)
TheevolutionofdKwithincreasingtisillustrated.Thisquantifiesthechangeoftheshapeofthepdfs.Fordifferentstockswefoundthatfortimescalessmallerthanabout1minalineargrowthofthedistancemeasureseemstobeuniversallypresent.IfanormalisedGaussiandistributionistakenasareferencedistribution,thefastdeviationfromtheGaussianshapeinthesmall-timescaleregimebecomesevident.ForlargertimescalesdKremainsapproximatelyconstant,indicatingaveryslowchangeoftheshapeofthepdfs.
3.Mediumscaleanalysis
Nextthebehaviourforlargertimescales(τ>
1min)isdiscussed.Weproceedwiththeideaofacascade.itispossibletograspthecomplexityoffinancialdatabycascadeprocessesrunninginthevariableτ.InparticularithasbeenshownthatitispossibletoestimatedirectlyfromgivendataastochasticcascadeprocessintheformofaFokker–Planckequation.Theunderlyingideaofthisapproachistoaccessstatisticsofallordersofthefinancialdatabythegeneraljointn-scaleprobabilitydensitiesp(y1,τ1;
y2,τ2;
…;
yN,τN).Hereweusetheshorthandnotationy1=y(τ1)andtakewithoutlossofgeneralityτi<
τi+1.Thesmallerlogreturnsy(τi)arenestedinsidethelargerlogreturnsy(τi+1)withcommonendpointt.
Thejointpdfscanbeexpressedaswellbythemultipleconditionalprobabilitydensitiesp(yi,ti│yi+1,ti+1;
...;
yN,tN).Thisverygeneraln-scalecharacterisationofadataset,whichcontainsthegeneraln-pointstatistics,canbesimplifiedessentiallyifthereisastochasticprocessint,whichisaMarkovprocess.Thisisthecaseiftheconditionalprobabilitydensitiesfulfilthefollowingrelations:
p(y1,τ1│y2,τ2;
y3,τ3;
yN,τN)=p(y1,τ1│y2)(3)
Consequently,
p(y1,τ1;
yN,τN)=p(y1,τ1│y2)……p(yN-1,τN-1│yN,τN)·
p(yN,τN)(4)
holds.Eq.(4)indicatestheimportanceoftheconditionalpdfforMarkovprocesses.Knowledgeofp(y,τ│y0,τ0)(forarbitraryscalesτandτ0withτ<
τ0)issufficienttogeneratetheentirestatisticsoftheincrement,encodedintheN-pointprobabilitydensityp(y1,τ1;
yN,τN).
ForMarkovprocessestheconditionalprobabilitydensitysatisfiesamasterequation,whichcanbeputintotheformofaKramers–MoyalexpansionforwhichtheKramers–MoyalcoefficientsD(K)(y,τ)aredefinedasthelimit△τ→0oftheconditionalmomentsM(K)(y,τ,△τ):
(5)
(6)
Forageneralstochasticprocess,allKramers–Moyalcoefficientsaredifferentfromzero.AccordingtoPawula’stheorem,however,theKramers–Moyalexpansionstopsafterthesecondterm,providedthatthefourthordercoefficientD(4)(y,τ)vanishes.Inthatcase,theKramers–MoyalexpansionreducestoaFokker–Planckequation(alsoknownasthebackwardsorsecondKolmogorovequation):
(7)
D
(1)isdenotedasdriftterm,D
(2)asdiffusionterm.Theprobabilitydensityp(y,τ)hastosatisfythesameequation,ascanbeshownbyasimpleintegrationofEq.(7).
4.Discussion
Theresultsindicatethatforfinancialdatatherearetwoscaleregimes.Inthesmall-scaleregimetheshapeofthepdfschangesveryfastandameasureliketheKullback–Leiblerentropyincreaseslinearly.Attimescalesofafewsecondsnotallavailableinformationmaybeincludedinthepriceandprocessesnecessaryforpriceformationtakeplace.Neverthelessthisregimeseemstoexhibitawell-definedstructure,expressedbytheverysimplefunctionalformoftheKullback–Leiblerentropywithrespecttothetimescaleτ.Theupperboundaryintimescaleforthisregimeseemstobeverysimilarfordifferentstocks.Basedonastochasticanalysiswehaveshownthatasecondtimerange,themediumscalerangeexists,wheremulti-scalejointprobabilitydensitiescanbeexpressedbyastochasticcascadeprocess.Here,theinformationonthecomprehensivemulti-scalestatisticscanbeexpressedbysimpleconditionedprobabilitydensities.Thissimplificationmaybeseeninanalogytothethermodynamicaldescriptionofagasbymeansofstatisticalmechanics.Thecomprehensivestatisticalquantityforthegasisthejointn-particleprobabilitydensity,whichdescribesthelocationandthemomentumofalltheindividualparticles.Oneessentialsimplificationforthekineticgastheoryisthesingleparticleapproximation.TheBoltzmannequationisanequationforthetimeevolutionoftheprobabilitydensityp(x;
p;
t)inone-particlephasespace,wherexandparepositionandmomentum,respectively.InanalogytothiswehaveobtainedforthefinancialdataaFokker–Planckequationforthescaletevolutionofconditionalprobabilities,p(yi,τi│yi+1,τi+1).Inourcascadepicturetheconditionalprobabilitiescannotbereducedfurthertosingleprobabilitydensities,p(yi,τi),withoutlossofinformation,asitisdoneforthekineticgastheory.
Asalastpoint,wewouldliketodrawattentiontothefactthatbasedontheinformationobtainedbytheFokker–Planckequationitispossibletogenerateartificialdatasets.Theknowledgeofconditionalprobabilitiescanbeusedtogeneratetimeseries.Oneimportantpointisthatincrementsy(τ)withcommonrightendpointsshouldbeused.Bytheknowledgeofthen-scaleconditionalprobabilitydensityofally(τi)thestochasticallycorrectnextpointcanbeselected.Wecouldshowthattimeseriesforturbulentdatageneratedbythisprocedurereproducetheconditionalprobabilitydensities,asthecentralquantityforacomprehensivemulti-scalecharacterisation.
AndreasP-Nawroth,JoachimPeinke.Carl-von-Ossietzky奥尔登堡大学,D-26111奥尔登伯格,德国[J].2008年3月30日.
中小规模的金融数据分析
摘要
财务数据随机分析已经被提出,特别是我们探讨如何统计在不同时间里记录返回的变化。
财务数据的时间规模依赖行为可分为两个区域:
第一个时间范围是被描述为普遍特征的小时就区域(范围秒)。
第二个时间范围是增加了几分钟的可以被描述为随机的级联过程的中期时间范围。
相应的Fokker-Planck方程可以从特定的数据提取,并提供了一个非平衡热力学描述的复杂的财务数据。
关键词:
经济物理学;
金融市场;
随机过程;
Fokker-Planck方程
第一章前言
复杂的金融市场的其中一个突出特点是资金数量显示非高斯统计往往被命名为重尾或间歇统计。
描述金融时间序列x(t)的波动,最常见的就是log函数或价格增量的使用。
在这里我们认为,log函数y(τ)超过一定时间t的统计,被定义为:
y(r)=logx(t+r)-logx(t)
(1)
其中x(t)是指在时间t时资产的价格。
在财务分析数据中一个常见的问题是讨论随机数量的平稳性,尤其是我们发现在我们的分析中采用什么样的方法似乎是强大的非平稳性的影响,这可能是由于数据的选择。
请注意,有条件的应用τ相当于一个特定的数据过滤。
尽管如此,特殊的结果略微改变了不同的数据窗口,显示出非平稳性影响的可能性。
在本文中,对于一个特定的数据窗口(时间段)我们侧重于分析和重建进程。
目前已有的分析主要是基于1993至2003年的拜耳数据,财务数据集是由KapitabmarktDatenbank(KKMDB)提供。
第二章小规模分析
财务数据的一个突出特点是事实上概率密度函数(pdfs)不是Gaussian,而是展览重尾形状。
另一个显著的特点是形状伴随着可变规模τ的大小而变化。
分析pdfs伴随着规模τ的变化的统计,非参数方法是一种选择。
Pdfp(y(τ))的时间T和PT(y(T))的参考时间T之间的差距是可以计算的。
作为一个参考的时间,在我们的数据集上接近最小的可用时间但仍然有足够的活动,T=1s是选择。
为了能够比较pdfs,并排除由于不同的均值和方差的影响,所有的pdfsp(y(τ))正常化为零平均,标准偏差为1。
作为衡量量化两个分布p(y(τ))和PT(y(T))之间的距离,需使用Kullback–Leibler:
dK随着t的增加而变化,量化的改变pdfs的形状。
对于不同的股票,目前我们发现时间小于1