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1、2388Ilv!arkm}Processes8.5BibliographicNotesDoob(1953,chap.V)andGihmanandSkorohod(1974,chap.II,sects.4and5)bothcom.ainexcellentdiscussionsoftransitionfunctionsandMarkovprocesses.Ageneralproofthat,givenatransitionfunctionona9StochasticDynamicProgrammingmeasurablespace.thereexistsacorrespondingst
2、ochasticprocess-theconstructionof(Z",:r-,,u."(LQ,.))inSection8.2-isduetolonescu•Tulcea.TheproofmaybefoundinShiryayev(1984,Theorem2,pp.247-249),Neveu(1965,sect.V.I),orGihmanandSkorohod(1974,Theorem3,pp.81-82).OurdiscussionofstochastickernelsinSection8.3alsodrawsheavilyonthetreatmentinthelauermo
3、sl.TheproofofTheorem8.9followstheoneinFutia(1982,Theorem5.2),whereaWiththemathematicalbackgroundinplace,wearereadytostudyslightlymoregeneralversionisgiven.dynamicprogrammingmodelsthatincorporatestochasticshocks.Westudytwospecificationsoftheproblem.Thefirstoftheseparallelsmorecloselythetreatmen
4、tofthedeterministicmodelinChapter4,butthesecondismoregeneral.Thetwoapproachescanbeillustratedbytwovariationsontheone-sectormodelofoptimalgrowth.SupposeLhaltheonegood.com,canbeconsumedorusedasseed.Outputinanyperioddependsonthequantityofseedplantedandonthequantityofrainfall.Assumethattheeffectof
5、rainfallismultiplica•tive.Rainfallisexogenouslydeterminedandstochastic.andmaybeseriallycorrelatedfromyeartoyear.Thefunctionalequationforoptimalgrowthinthiseconomyisv(k,z)=sup(U[if(k)-y)+(JLv(y,z')Q(z,dr'))..,EIO.:j"CA.IlInthisequationtheslatevariablesareh,thequantityofseedcornplantedinthesprin
6、g,andz,thequantityofrainfallduringthegrowingseason.Theonlydecisiontobemadeishowmuchseedcom,)'.Losetasideforthefollowingyear.Thisdecisionismadeinthefall,givenif(k),thequantityofcomhan'ested.Sincenofurtheruncertaint),intervenesbetweenfallandspring,yisalsothequantit)'plantedinthespring.Alternativ
7、ely,consideraneconom)'inwhichthequantityofrainfalldoesnotftucruatefromyeartoyearbutinwhichtherearemiceinthestorehousewheretheseedcornisstored.Thesizeofthemousepopula•tionisexogenouslydeterminedandstochastic,andrna)'displayserialcorrelat
