on inference for partially observed nonlinear diffusion models using the metropolis–hastings algorithm

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1、Biometrika(2001),88,3,pp.603–621©2001BiometrikaTrustPrintedinGreatBritainOninferenceforpartiallyobservednonlineardiffusionmodelsusingtheMetropolis–HastingsalgorithmBG.O.ROBERTSDepartmentofMathematicsandStatistics,LancasterUniversity,Lancaster,LA14YF,U.K.g.o.roberts@lancaster.ac.uk

2、O.STRAMERDepartmentofStatisticsandActuarialScience,UniversityofIowa,IowaCity,Iowa52242,U.S.A.stramer@stat.uiowa.eduSInthispaper,weintroduceanewMarkovchainMonteCarloapproachtoBayesiananalysisofdiscretelyobserveddiffusionprocesses.Wetreatthepathsbetweenanytwodatapointsasmiss

3、ingdata.Assuch,weshowthat,becauseoffulldependencebetweenthemissingpathsandthevolatilityofthediffusion,therateofconvergenceofbasicalgorithmscanbearbitrarilyslowiftheamountoftheaugmentationislarge.Weofferatransformationofthediffusionwhichbreaksdowndependencybetweenthetransformedmissing

4、pathsandthevolatilityofthediffusion.WethenproposetwoefficientMarkovchainMonteCarloalgorithmstosamplefromtheposterior-distributionofthetrans-formedmissingobservationsandtheparametersofthediffusion.WeapplyourresultstoexamplesinvolvingsimulateddataandalsotoEurodollarshort-ratedata.Someke

5、ywords:Diffusionprocess;Independencesampler;MarkovchainMonteCarlo.1.IDiffusionprocessesarenaturalstatisticalmodelsformanynaturalphenomena,includ-ingfinancialtimeseries.However,inferenceiscomplicatedbythefactthatcompletedatadescribingthediffusionsamplepathisofnecessitynotava

6、ilable.Infactdatawillbeadiscretesetofobservationsoftheprocessoversomefinitetimeperiod,andthemarginallikelihoodgiventhisdiscretesetofobservationswillalmostalwaysbeunavailable.ThiscomplicatesbothclassicalandBayesianinference,thoughherewewillconcentrateonaBayesiantreatmentoftheproblem

7、.SupposeforinstancethatXisadiffusionprocesssatisfyingdX=sdB+b(t,X,h)dt,(1)tttforsomeRd-valuedparameterhandBrownianmotionB,observedatdiscretetimepointst=(t,t,...,t).Occasionally,tmaybesufficientlyfinetoallowtheEulerapproximating01NMarkovchainX~N{X+(t−t)b(t,X,h),s2(t−t)}(2)ttii−1i−1tii−

8、1ii−1i−1604G.O.RO.S