ࠏྏބࠏஞଐ࿒∗``ݢ৫(൮ࣜ࠶સၞն࿐ࣜ࠶࿐ჽ)ᅋေ`Ԯ֥ሧӁקࡎં҂षྟࣜ࠶ದࡌק,HarrisonandKreps(1978)[12]іૼ,٢ఙھࡌק߶֝ᇁሧӁࡎ۬༢ֹۚႿఃࠎԤࡎ.ခሢᆃ۱නਫ਼,ScheinkanandXiong[19]ิԛ၂۱Ⴕॖࢳ༅ྟ֥৵࿃ൈࡗଆ,ᆷԛொྐ߶֝ᇁݖ؇ࠏࢌၞ,ႮՎႄؿࠏஞଐ.Ч໓ٳ༅ਔࠏྏᄝࠏࢌၞᇏ֥ቔႨ,ࡼࠏྏᆷѓႄೆ֞ScheinkanandXiong[19]֥ଆ,ษંਔࠏྏؓࠏஞଐ֥႕ཙ.ܱሳ`ࣜ࠶ದᇶܴࡎ,ொྐ,ࠎԤࡎ,ࠏ௹ಃ,ࠏྏ,ࠏஞଐ.``ሧЧሧӁקࡎ࣮ਵთ֥ཁᇷหᆘ൞,ંંߎ൞ൌᆣ,҂ಒקྟϲဆሢᇗေ࢘.ޅࣁವଆ,ቔູԛؿׄࡌקሧᆀ૫ؓ҂ಒקྟ.ޅࣁವଆ,ఃЧᇉଽಸടࠣሧᆀ҂ಒקྟྛູ֥Ԋࠌ,طᆃུԊࠌቋᇔႻ߶ّ႘֞൧ӆࡎ۬ᇏಀ.ሧЧሧӁקࡎ࣮൞ೂޅԩᆃུ໙ี֥?``࣍20୍ট֥ሧЧሧӁקࡎ࣮,ᄝ၂۱Ф“ޓݺֹ”ܒࡹ֥ॿࡏᇏࣉྛ.ھॿࡏࡌק൧ӆส০,఼טሧЧሧӁ߭Б֥ࢲܒ.ॖၛᆣૼ,ส০ࡌק༯ෛࠏᅼགྷၹሰ(SDF)сಖթᄝ,طSDFႻିϜ۲ᇕሧЧሧӁ֥ࠆ০ః൧ӆࡎ۬৳༢ఏট,ᆃФ൪ູ൞Arrow-Debreu၂Ϯनޙંᄝࣁವ࣮ᇏႋႨ֥ࢲݔ.Arrow-Debreu၂Ϯनޙંಪູ,ॴ௹ଆᇏૄ၂௹ཌྷؓႿૄ۱ሱಖሑթᄝሑࡎ۬,ሧЧሧӁࡎ۬ॖіགྷູھሧӁໃট۲ᇕॖି൬ၭᄝሑࡎ۬ࡆಃ༯֥ࡆಃބ.``ᆃ۱ॿࡏФಪູ“ޓݺ”,൞ၹູॖ๙ݖᄝఃᇏሔࡆ၂ུཌྷܱࡌקࠆ֤ఃޓ؟Ⴕၩၬ֥ࢲݔ.২ೂ,ሔࡆ൧ӆປСྟࡌק,ॖၛᆣૼSDF҂թᄝطື၂;ሔࡆSDF܄܋ԊࠌႵཌྟܱ༢ࡌק,ॖၛ֝ԛሧЧሧӁ߭Б֥ཌྟଆ—CAPM;ሔࡆޓݺקၬਔིႨݦඔ֥ࡆሹࣜ࠶ದթᄝ֥ࡌק,ॖࡹ৫SDFࡆሹࣜ࠶ದш࠽ིႨᆭࡗ֥ܱ༢.ቋ֥࣍၂ུ࣮മᇀಪູ,ࠧь൞ቋ࣍ྍؿᅚఏট֥ྛູࣁವ࿐္ିᄝᆃ۱ॿࡏᇏࠆ֤ࢳ.ᆃ۱ॿࡏೂՎᆭ“ປૅ”,ၛᇀႿDuffie[10]ಪູ,“1969-1979୍൞ሧӁקࡎ࣮֥ߛࣁ୍ս...୍࣍ট,ࠇᆀඪ1979୍ၛট,෮Ⴕ۽ቔݸ২ຓ൞ೡແྟ֥”.∗[ቔᆀࡥࢺ]࿒,ଳ,1964୍ള,൮ࣜ࠶સၞն࿐ྐ༏࿐ჽڬ࢝൱,Ѱൖ.ݢ৫,ଳ,1955୍ള,Кࣘݴॢݴ฿ն࿐ࣜ࠶ܵ࿐ჽ࢝൱,Ѱൖള֝ഽ,൮ࣜ࠶સၞն࿐࡙1
``Campbell[8]ؓՎӻ҂ܴׄ,“҂ڎಪھਵთᄪ௹࣮Ӯݔᆭሜᄀ,ᆺ༐ຬіૼ1979-1999୍ᆭࡗ္֥࣮٤ӈႵࡎᆴ”.࣐ܵCampbell[8]္ಪູ,“ෛࠏᅼགྷၹሰթᄝ่֥ࡱ٤ӈ၂Ϯ,ၛᇀႿॖၛࠫެݸཋᇅֹႨ֞෮Ⴕ֥ࣁವඔऌᇏಀ”.മᇀಪູ,ࣂުሧЧሧӁקࡎ࣮֥ᇶေज़ี൞“ಪ്थקSDF֥ࣜ࠶৯ਈ”.൞ѩ҂ڎಪఃᇏթᄝ֥໙ี,ᆷԛႨᆃ۱ॿࡏᇏࣉྛקࡎ࣮߶მ֞ޓ؟.২ೂ,Banz[2]ؿགྷ֥“ཬ܄ඳིႋ”,Basu[3]֩ؿགྷ֥“ࡎᆴིႋ”,JegadeeshandSheridan[14]ؿགྷ֥“Ԋਈིႋ”֩ࢳᆭ(puzzle).ѩಪູ“”߶“ࠗؿྍ֥ંӁള”.``ഈඍၩၬགྷའ֥ӁളіૼԮ֥ሧӁקࡎં൞҂ປૅ֥.ః҂ປૅྟ҆ٳჷႿ֥ࠎЧࡌഡ.ھંࡌק:(1)൧ӆྐ༏ؓ෮Ⴕࣜ࠶ದ൞܄ष֥;(2)ࣜ࠶ದࢥྟ,ࠧࣜ࠶ದᇶܴྐཌྷሔིႨቋն߄.ࣜ࠶ದི֥Ⴈݦඔܿٓ֩ඌྟ่ࡱӮ৫,ᄵॖၛᆣૼ൧ӆഈ֥ಒթᄝनޙࡎ۬,ݺႨ֥SDFקࡎॿࡏ္൞ીႵ໙ี֥.൞,ᆃུࡌק่ࡱ֤҂֞Ќᆣ,২ೂࣜ࠶ದ֥ᇶܴྐ҂,ᄵनޙࡎ֥۬թᄝໃсିܔФЌᆣ.২ೂ,[23]֥ٳ༅іૼ,ਆ۱ࣜ࠶ದ(Иܴބুܴࣜ࠶ದ)֥ਆ௹ଆᇏ,ᇶܴۀੱྐ֥҂Ⴕॖି֝ᇁ൧ӆनޙࡎ۬҂թᄝ.࠻ಖ൧ӆनޙࡎ۬թᄝ่֥ࡱໃсିܔЌᆣ,नޙࡎ۬ߎିࢳ൧ӆಆ҆(ࠇᆀն҆ٳ)ࡎ۬གྷའગ?ScheinkanandXiong[19]πൕ,ႋھࠎႿࣜ࠶ದؓሧЧሧӁ֥ᇶܴࡎটࢳ൧ӆࡎ۬,ᆃဢࠇྸିܔࢳ۷؟֥གྷའ.``HarrisonandKreps(1978)[12]ಪູ,ࣜ࠶ದ֥ᇶܴྐ҂൧ӆ࣌ᆸછॢ,ᄵሧӁ֥ࡎ۬ေۚႿఃࠎԤࡎ.ခሢᆃ۱නਫ਼,ScheinkanandXiong[19]ิԛਔॖປಆࢳ༅֥৵࿃ൈࡗଆ(ࡥӫູS-Xଆ),ษંਔࣜ࠶ದொྐ֝ᇁ֥ࠏஞଐ໙ี,ٳ༅ਔૅݓܢ൧ஞଐ(২ೂବථղक़ຩܢஞଐ)ࢨ؍֥၂ུఅၳགྷའ.ఃᇏ,ࣜ࠶ದ֥ொྐ֝ᇁఃؓሧЧሧӁ“ٳޣ෮֤”֥ᇶܴࡎ(္ФӫູࠎԤࡎ)֥҂၂ᇁ,طᆃᇕ҂၂ᇁႻَࢌҭ֥ᇶܴࡎ֤ࣜ࠶ದᆭࡗ֥ࢌၞَؿള,ჷႿ֮ઙۚછ֥“ሧЧ෮֤”(္Фӫູࠏ௹ಃࡎ,ࠇᆀஞଐ)္ၹՎؿള.൞,ScheinkanandXiong[19]ѩໃษંࠏྏؓࠏஞଐ֥႕ཙ.Ч໓ࠎႿS-Xଆ,ٳ༅ਔࠏྏᄝࠏࢌၞᇏ֥ቔႨ,ิԛਔࠏྏᆷѓ,ѩࡼః༢ֹ֝ೆ֞ଆᇏ.ᄝՎࠎԤഈ,Ч໓ษંਔࠏྏؓࠏஞଐ֥႕ཙ.``Ч໓Ⴎ5ࢫܒӮ,ֻ1ࢫࢺകS-Xଆ.ֻ2ࢫษંࠏྏᄝࠏࢌၞᇏఏ֥ቔႨ,ิԛࠏྏᆷѓ,ѩࡼః֝ೆ֞S-Xଆᇏ.ֻ3ࢫษંႵࠏྏᆷѓ֥S-Xଆଆ֥ࢳ༅໙ี,ѩ۳ԛਔඔᆴٳ༅֥ࠎЧනਫ਼.ֻ4ࢫษંਔࠏྏؓࠏஞଐ֥႕ཙ,ࣉྛཌྷܱ֥ඔᆴٳ༅ބࣜ࠶ٳ༅.ֻ5ࢫ൞ࢲં.2
1S-Xଆொྐބࣁವ൧ӆ``ྏ࿐࣮іૼ,ದႵொྐ֥ౠཟ.AlpertandRaiffa[1]ބBrenner,Koehler,Liber-man,andTversky[7]ؿགྷ,ದૌӈӈౠཟႿࢠֹۚࡎሱ࠭ႚႵᆩ്֥ࣚಒ؇.ᆃུܴׄᄝHirshleifer[13]ބBarberandOdean[4]ᇏ֤֞ड़ק.ᄝࣁವ࿐ਵთ,࣮ᆀૌၘࣜषؿԛཌྷႋ֥ଆٳ༅ொྐؓࣁವ൧ӆᄯӮ֥႕ཙ.KyleandWang[15]ބDaniel,HirshleiferandSubrahmanyam[9]ႨொྐࢳࣁವሧӁ߭Б֥ॖყҩྟ.Odean[16]іૼ,ொྐॖି֝ᇁݖ؇ࢌၞ.BernardoandWelch[5]ᄵษંਔఒြᇶ๙ݖࢆ֮ࠢᇏౠཟطࠆ֤֥ொྐ൬ၭ.ഈඍ࣮൞ࡼொྐଆ߄ູؓྐ༏ࣚಒ؇֥ݖۚܙ࠹.ScheinkanandXiong[19]္ҐႨᆃᇕࡹଆٚم.``ொྐ֥ሧᆀಪູ,ሱ࠭ؓሧЧሧӁࡎᆴ֥ࡎ۷ູࣚಒ.ᆃဢ,ொྐࣼ߶֝ᇁ٤ఊՑྐࠇಪ്ҵၳ.০Ⴈनᆴ߭گঔݖӱ֥ྟᇉॖၛᆣૼ,ૄ၂۱ொྐ֥ሧᆀಕุ߶ಪູ,ఃಕุሧᆀ֥ྐ߶ຶಡሢሱ֥࠭ྐෛࠏѯ.Varian[21]ބHarrisandRaviv[11]࣮ਔ٤ఊՑྐဟ֝ᇁ֥ࠏࢌၞ.Odean[17]ބBarberandOdean[4]֥ൌᆣ࣮іૼ,ࢌၞᆀ֥ݖ؇ࢌၞॖିݝఃڞ০,ఃᇏݖ؇ࢌၞಯಖჷႿொྐ.ொྐࠏଆ``ഡ൧ӆഈᆺթᄝ၂۱ڄགሧӁ,ఃටൈࡎᆴ࠺ູ{ft},ӫᆭູࠎԤݖӱ.ࡌഡࣜ࠶ದ҂ିᆰࢤܴҳ֞ࠎԤэਈЧദᆺᆩ֡ఃડቀ༯૫֥ෛࠏັٳٚӱൔ:dft=−λ(ft−f¯)dt+σfdZft,()ఃᇏf¯ູෛࠏݖӱ{ft}֥ൈࡗन,{Zft}ູѓሙ҃ᄎ.ࡌഡھڄགሧӁ֥܂۳Ⴕཋ.ડቀ܄ൔ()֥ෛࠏݖӱൌ࠽ഈࣼ൞෮໌ύضථฌ-Нक़ݖӱ֥नᆴ߭گݖӱ,ఃनᆴ߭گ؇֥նཬ౼थႿҕඔλ֥նཬ,λᄀնनᆴ߭گ؇ᄀॹ.ॖၛᆣૼ,ෛࠏັٳٚӱൔ()Ⴕ݅֡ື၂఼֥ࢳ,ѩႵ∫()tft=f¯+exp{−λt(f0−f¯)}+exp{−λ(t−s)}σfdZfs.()0ູษંഈ֥ٚь,ࡌഡԚሑf0ڛՖᆞٳ҃.``ࡌഡڄགሧӁ֥ٳޣݖӱ{Dt},౼थႿႻ҂ປಆ౼थႿࠎԤݖӱ{ft},іགྷູࠎԤݖӱथק҆ٳބ٤ࠎԤݖӱथק҆ٳ(ᄮၻ)֥ןࡆ,ᆃၩሢડቀෛࠏັٳٚӱൔdDt=ftdt+σddZDt,()3
ఃᇏ{ZDt}ູބ{Zft}ཌྷ৫֥ѓሙ҃ᄎ.``ࡌഡթᄝ{sAt}ބ{sBt}ਆ۱ྐݼݖӱ,ٳљડቀ༯૫֥ෛࠏັٳٚӱൔdsAt=ftdt+σsdZAt,()dsBt=ftdt+σsdZBtఃᇏ,{ft}a{Dt}a{sAt}ބ{sAt}൞ཌྷ৫֥ѓሙ҃ᄎ.൙ൌഈ,{Dt}္൞ྐݼݖӱ,҂֥൞ᄝ༯૫֥ഡקᇏ,ھྐݼؓ{ft}Ԯղሙಒ؇൞Ф෮Ⴕࣜ࠶ದಪॖ֥,ط{sAt}ބ{sAt}ᄵ҂ಖ.``ࡌഡ෮Ⴕ֥ࣜ࠶ದ൞ڄགᇏ৫֥.ᆃဢഡקᆺ൞མಞ࠹ෘэ֤۷ࡥֆ၂ུ,ѩ֤ྐބྐ༏ᄝሧЧሧӁࡎ۬थק֥ݖӱᇏ֥ቔႨіགྷ֥۷ູౢ༉.ႮႿࣜ࠶ದࢥڄགᇏ৫,෮ၛؓሧЧሧӁ൬ೆੀࣉྛࡎൈႨᅼགྷၹሰ(1+r)−1ࠧॖ,ఃᇏ֥rູڄག০ੱ.ಖط,ႮႿٳޣݖӱᇏթᄝᄮၻ,෮ၛࣜ࠶ದؓࠎԤݖӱࣉྛ؎ൈߎေҕॉྐݼݖӱ,ࣜ࠶ದؓࠎԤݖӱ֥؎ᆰࢤ႕ཙ֞ఃؓሧЧሧӁ֥ࠎԤࡎ.ࣜ࠶ದؓ܄ൔ()ᇏ֥ྐݼAބྐݼB֥ࣚಒ؇Ⴕཌྷ֥ಪॖӱ؇,ᄵӫࣜ࠶ದૌؓྐݼ֥ಪॖ൞ྟ֥,ᆃൈૌؓࠎԤݖӱ{ft}ቔԛ֥ܙࡎ{fˆRt}ડቀ༯૫֥ෛࠏັٳٚӱൔ:dfˆRRt=−λ(fˆRt−f¯)dt+γRdW()σA(t)+γRdWRt)+γRdWRsσB(sσd(t),dఃᇏ√()λ2+σ21fσ+2D2σ2−λsγR=1,σ2+2Dσ2sdWRA(t)=1σ(dsAst−fˆRtdt)dWRB(t)=1σ(dsBst−fˆ)dWRD(t)=1σ(dDRst−fˆtdt)ॖၛᆣૼWRA(t)aWRA(t)ބWRA(t)൞ཌྷ৫֥ѓሙ҃ᄎ.ѩ,ޓཁಖ{fˆRt}൞ൈࡗनູf¯֥नᆴ߭گ֥ෛࠏݖӱ(ऎุ֥࠹ෘބᆣૼҕRogersandWilliams[18].ෛࠏݖӱ{fˆRt}іൕྟࣜ࠶ದؓࠎԤݖӱ֥ܙ࠹,ܣФӫູྟࣜ࠶ದ֥ྐ.ਸ਼ຓ,ߎॖၛ۴ऌ၈ฬ܄ൔ࠹ෘԛೂ༯ࢲݔ:ؓႿၩ֥t≥0Ⴕ[]fˆRRt=f¯+exp{−λt}(fˆ0−f¯)+MRt,()ఃᇏ∫MRγRtt=exp{−λ(t−s)}dWRσD(s)D∫0t+γRexp{−λ(t−s)}dWRσA(s)+γR∫texp{−λ(t−s)}dZRs0σB(s)s0ູ᷉,ѩડቀ่ࡱE0MRt=
ྐҵၳ``ѓሙ࢝॓,২ೂKyleandWang[15]ބOdean[16],ࡼ҂ࣜ࠶ದಕุ֥ྐ,หљ൞ᄝྐݼࣚಒ؇ഈ֥ҵၳο༯ඍٚൔଆ߄:``ಕุAބBྐݼAބB֥ुم҂,భᆀಪູ{sAt}۷ॖौ၂ུ,ުᆀཌྷّ.భᆀ۴ऌdDt=ftdt+σDdZDt,.7)dsAt=ftdt+σsφdZAt,φ≥1,(1dsBt=ftdt+σsdZBt,ิ܂֥ྐ༏FAt,ԛ{ft}֥ੲѯ{fAt}ఃᇏfˆAt=E[ft|FAt],ӫᆭູಕุAؓ{ft}֥ྐ.ުᆀ۴ऌdDt=ftdt+σDdZDt,dsAt=ftdt+σsdZA()t,dsBt=ftdt+σsφdZBt,φ≥1,ิ܂֥ྐ༏FBt,ԛ{ft}֥ੲѯ{fBt},ఃᇏfBt=E[ft|FBt],ӫູಕุBؓ{ft}֥ྐ.``۴ऌRogersandWilliams[18]֥ษંॖၛᆣૼ:fˆAtބfˆBt൞ൈࡗनູf¯֥नᆴ߭گݖӱ,ડቀ༯ඍෛࠏັٳٚӱൔ:dfˆAt=−λ(fˆAt−f¯)dt+φγσdWAsA(t)+γσdWAsB(t)+γσDdWAD(t),()dfˆBt=−λ(fˆBt−f¯)dt+φγσdWBdWBsA(t)+γσdWBsB(t)+γσDD(t)ఃᇏ√()λ2+σ21fσD2+1+φ2σ2−λsγ=1σD2+1+φ2,σ2sφdWAA(t)=σ(dsAst−fˆAtdt),dWAB(t)=1Bσ(dsst−fˆAtdt)dWAD(t)=1σD(dDt−fˆAtdt)dWAA(t)=φσ(dsAst−fˆAtdt)dWAB(t)=1σ(dsBst−fˆAtdt),dWAD(t)=1σD(dDt−fˆAtdt)5
ߎॖၛᆣૼ,(1)ᄝಕุ֥ࣜ࠶ದAငᇏ,{WA∗(t)},∗=A,B,Dູཌྷ৫֥ѓሙ҃ᄎ;(2)ᄝಕุ֥ࣜ࠶ದBငᇏ,{WB∗(t)},∗=A,B,Dູཌྷ৫֥ѓሙ҃ᄎ.ط,০Ⴈ၈ฬ܄ൔࣉྛࡥֆֹ࠹ෘ҂ؿགྷ:[]fˆAt=f¯+exp{−λt}(fˆA0−f¯)+MAt[],()fˆBt=f¯+exp{−λt}(fˆB0−f¯)+MBtఃᇏ,{MAt}ᄝಕุA֥ࣜ࠶ದငᇏ൞नᆴູ0֥᷉,{MBt}ᄝಕุB֥ࣜ࠶ದငᇏ္൞नᆴູ0֥᷉,ఃඔ࿐іղൔູ∫tMAt=γexp{−λ(t−s)}dWAσD(s)D0∫t+γR∫exp{−λ(t−s)}dWAγRtxp{−λ(t−s)}dZAσA(s)+eB(s),s∫0σs0tMBt=γRexp{−λ(t−s)}dWBσD(s)D0t+γR∫texp{−λ(t−s)}dWBσA(s)+γR∫exp{−λ(t−s)}dZBs0σB(s).s0``༯૫ॉ੮ሧᆀ֥ྐҵၳ໙ี.קၬӫ{gAt},ఃᇏgAt=fˆBt−fˆAt,ູಕุBཌྷؓႿಕุA֥ྐҵၳ;ӫ{gBt},ఃᇏgBt=fˆAt−fˆBtູಕุBཌྷؓႿಕุA֥ྐҵၳ.``ၹູಕุA֥ࣜ࠶ದಪູ,ྐݼsAtбྐݼsBt۷ູࣚಒ,ѩ္ᆩ֡ಕุB֥ࣜ࠶ದ“հ༂ֹ”ಪູ,ྐݼsBtбྐݼsAt۷ູࣚಒ.ၹՎ,ಕุA֥ࣜ࠶ದಪູྐݼ{sAt}ᄝໃটࡼ۷؟ֹّ႘ٳޣ֥ൌ࠽౦ঃ,ᆃၩሢ,ಕุB֥ࣜ࠶ದ֥ྐfˆBtࡼཟЧಕุྐfˆAt߭گ.ၹՎgAtࡼ൞ཟൈࡗनᆴ0߭گ֥ෛࠏݖӱ.,ಕุB္߶Ⴕোර֥ुم.ਸ਼ຓ,܄ൔ()ၩሢ,۲ಕุ෮ொݺ֥ྐݼᄝః۲ሱငᇏႵཌྷ֥ঔσs/φ,۲ಕุ෮҂ொݺ֥ྐݼᄝః۲ሱ֥ငᇏ္Ⴕཌྷ֥ঔσs.෮ၛ,ᄝૌ۲ሱငᇏ{gAt}ބ{gBt}൞नᆴ߭گ֥,طनᆴ߭گڛՖ֥ۀੱܿᄵ္ႋھ၂ဢ.ᆃུᆰܴٳ༅ଽಸФۀওᄝ༯ඍଁีᇏ.ଁี[19]ρ=λ+γ√(1+φ2))σ2+γsσD2>0(,σg=1+1φ2(φ2−1)γσsᄵႵ༯૫֥ࢲંӮ৫:1.ᄝಕุA֥ࣜ࠶ದငᇏdgAt=−ρgAtdt+σgdWAg(t).()ѩ,ᄝಕุA֥ࣜ࠶ದငᇏ,WAg(t)ູѓሙ҃ᄎ,৫ႿfˆAt֥ྍ༏(innovation);6
2.ᄝಕุB֥ࣜ࠶ದငᇏdgBt=−ρgBtdt+σgdWBg(t).()ѩ,ᄝಕุB֥ࣜ࠶ದငᇏ,WBg(t)ູѓሙ҃ᄎ,৫ႿfˆBt֥ྍ༏(innovation).ᇿؓྐݼொݺ(φ>1)֥թᄝ֤σg>0,طφᄀնρᄀն,ᆃၩሢgtཟनᆴ߭گ֥؇ᄀॹ.2ࣜ࠶ದ֥ࠏྏ٤ӻႵᆀࠏྏ``܄ൔ()іૼ҂ಕุ֥ࣜ࠶ದؓሧЧሧӁටൈࡎᆴfˆAtބfˆBt֥ܙ࠹൞҂࣐ཌྷ֥,ࣜӈ൞ّ֮֮ۚۚگࢌҭ,ᆃၩሢૌؓሧЧሧӁ֥ࠎԤࡎᆴݖӱ(ٳޣᅼགྷ֥ࡆሹ)֥ܙ࠹–ࠎԤࡎ{v∗t}္൞҂࣐,ఃᇏ∗=A,B.༯૫֥ଁีіղਔᆃ۱නམ.ଁี҂ಕุࣜ࠶ದؓሧЧሧӁ֥ࠎԤࡎೂ༯:[22](∫+∞)v∗t=E∗texp{−rs}dDs=f¯+fˆ∗t−f¯∗=A,B.()trr+,λ``ଁีіૼਔࣜ࠶ದࠎԤࡎܱ༢ൔ.ࣜ࠶ದࢥྟ,ࠧૌؓྐݼAބB֥ुمປಆཌྷ,ᄵૌؓሧЧሧӁ֥ࡎປಆ၂ᇁ,ၹՎીႵۚછ֮ઙ֥౦ঃؿള,္ࣼીႵࠏส০֥ࠏ߶.ᆃൈ֥ࠎЧࡎᆴЧᇉഈࣼ൞ሧЧሧӁקࡎં෮ษં֥नޙࡎ۬.ಖط,ᆃ֥ษંၛࣜ࠶ದؓྐݼ֥ुم҂၂ᇁູభิ,ܣส০ࠏ߶ॖିԛགྷ(҂ࣜ࠶ದؓሧЧሧӁටൈࡎᆴ֥ܙ࠹҂࣐ཌྷ,ࠇۚࠇّ֮گࢌҭ,ଁีᄵၩሢૌؓሧЧሧӁ֥ࡎ္߶ೂՎ).``࠻ಖࠏࠏ߶թᄝ,ପહෛᆭطট֥໙ีࣼ൞ࣜ࠶ದ߶ᄝ؟նӱ؇ഈ০Ⴈᆃུࠏ߶,ᆃടࠣࠏྏ໙ี.Ч࣮ބScheinkanandXiong[19]֥၂۱҂ׄᆞᄝႿՎ,ުᆀ҂ॉ੮ࠏྏ,ܣႵᙣӈ.൙ൌഈ,Ч࣮οᅶ[19]֥ଆุ༢֝ԛᄝࢌၞׄഈ,ࢌၞࡎູ۬છ֥ٚࡎᆴࡎࡆࢌၞٮႨ,ൈ္൞ઙ֥ٚࡎᆴࡎ.ᆃ۱ࢲંᆰܴࢳཌྷິМ,ၹູၩሢ,ሧЧሧӁ֥ઙೆᆀ০ႨఃܓઙሧӁᄝໃটࠆ֤֥൬ၭ,ېݺିܔַཧܓೆൈᆦڱ֥ӮЧ,ପહູહေઙೆሧӁ?൞ູਔ࿙၂ᇕҍڶ֥ಒקྟગ?൞,ଆᇏ֥ࣜ࠶ದॖ൞ڄགᇏ৫֥.൙ൌഈ,ؓႿܓઙᆀটඪఃྛູᆺିડቀӻႵᆀ֥ေބ൧ӆܵᆀ֥ေ–ࢌၞٮႨቓ܊ང.ಖط,ൗࢸഈીႵ૧ٮ֥Ҕ,۷ીႵದ჻ၩ૧ٮູದิ܂Ҕ.7
``ପહ,ࠏྏᄝଆᇏႋھೂޅ૭ඍ?ᆺေܴҳࣜ࠶ದ(ሧӁӻႵᆀބ٤ӻႵᆀ)൞ೂޅቓԛ࿊ᄴ֥,ࣼ҂ؿགྷࢲં.൮༵ॉҳሧӁӻႵᆀ.൙ൌഈ,෮ႵӻႵᆀૄൈૄख़૫ؓਆᇕ࿊ᄴ,၂൞છԛሧӁ,ਸ਼၂۱൞࿃ӻႵሧӁ.൞๙ݖછԛሧӁࠆ֤֥൬ၭ֮ႿӻႵᆀ֥ࠎԤࡎ,ᄵӻႵᆀ߶࿊ᄴ࿃ӻႵሧӁ;ۚႿࠎԤࡎ,ᄵሧӁӻႵᆀႵॖି࿊ᄴછԛሧӁ.ሧᆀಪູ֒భ൞ቋࡄࢌၞൈख़,ᄵ߶છԛఃӻႵ֥ሧӁ.ᆃु҂֞ࣜ࠶ದ֥ࠏྏ,ၹູӻႵᆀၘࣜᅧਔ࿊ᄴࠏ߶,෮૫ਢ֥໙ี൞ೂޅ০Ⴈᆃ۱࿊ᄴࠏ߶.ၹՎ,ᆇᆞथקࠏ֥ྏת༆թᄝႿ٤ሧӁӻႵᆀ၂ٚ.``ࢤ༯টॉҳ٤ሧӁӻႵᆀ.൙ൌഈ,෮Ⴕ٤ሧӁӻႵᆀૄൈૄख़္߶૫ؓਆᇕ࿊ᄴ,၂൞ઙೆሧӁ–๙ݖሧࠆ֤࿊ᄴࠏ߶,ਸ਼၂۱൞҂ઙೆሧӁ.ᆃൈ,ሧᆀေॉ੮၂֊ሱ࠭ӻႵਔሧӁᄵ߶ᄝໃটࠆ֤ᄸဢ֥൬ၭ໙ี,ࠧቔູӻႵᆀ֥ࡎᆴࡎ໙ี.ᇀႿӻႵᆀ֥ࡎᆴࡎ൞હ,ૌࡼᄝཬࢫษં,ᆃ҂ٞࡌഡӻႵᆀሹ൞ିܔቓԛࡎᆴࡎ.٤ӻႵᆀಪູ,ሱ࠭ӮູӻႵᆀުؓሧӁቔԛ֥ࡎᆴࡎ֮Ⴟ൧ӆࢌၞࡎ۬,ᄵॖၛड़קھ٤ሧӁӻႵᆀ҂߶ઙೆሧӁӮູӻႵᆀ;٤ӻႵᆀಪູ,ሱ࠭ӮູӻႵᆀުؓሧӁቔԛ֥ࡎᆴࡎۚႿ൧ӆࢌၞࡎ۬,ᄵھ٤ሧӁӻႵᆀႵॖିઙೆሧӁӮູӻႵᆀ,ᇀႿઙೆߎ൞҂ઙೆᄵ౼थႿఃࡎᆴࡎބࢌၞࡎ۬ࡎҵ֥նཬ.``൙ൌഈ,ࠏྏࣼթᄝႿഈඍࡎҵᇏ.ࡎҵݖཬ,ᄵ҂ቀၛ་ႄ٤ሧӁӻႵᆀઙೆሧӁࣉྛࠏ;ࡎҵࢠն,ᄵॖၛՏࠗ٤ሧӁሧᆀઙೆሧӁࣉྛሧ.ࡌഡ෮Ⴕࣜ࠶ದႵࠏਢࢸѓሙ.෮໌ࠏਢࢸѓሙ൞ᆷ,ࡎҵӑݖਢࢸѓሙᄵ٤ӻႵᆀࣼ߶ઙࣉሧӁࣉྛࠏ,ڎᄵ҂ࣉྛࠏ.ᆃ۱ਢࢸѓሙ൞၂۱ᇶܴ֥ת༆,ၹࣜ࠶ದ֥҂ط҂,ູษંഈ֥ٚьࡌק෮Ⴕࣜ࠶ದ֥ࠏਢࢸѓሙ൞ཌྷ,ࢥູb.ھӈඔbൌ࠽ഈॖၛख़߂ࣜ࠶ದૌ௴ђ֥ࠏྏ,bཬᄵၩሢࣜ࠶ದࠏྏն,ࠧႵࠏࠏ߶ࣼࠏ,ّᆭbնᄵၩሢࠏྏཬ,ࠧીႵն֥ࠏࠏ߶ࣼ҂ࣉྛࠏ.ၹՎ,ࢌၞିܔؿള֥сေ่ࡱіགྷູ༯ඍ҂֩ൔ:po¯t−pt≥b,()ఃᇏptіൕࢌၞࡎ۬,po¯tіൕઙೆᆀഡམ֥ሱ࠭ӮູӻႵᆀުؓሧЧሧӁ֥ࡎࡎᆴ,ఃᇏo¯∈{A,B}іൕ٤ሧЧሧӁӻႵᆀ.ႮႿଆࡌקࣜ࠶ದದඔᇙ؟,෮ၛᇙ؟ܓઙᆀᆭࡗ֥ࣩᆚ߶ഈඍ҂֩ൔ()эӮ֩ൔ,ࠧpo¯t−pt=b.()ӻႵᆀࡎᆴࡎ``భ૫ษંਔ٤ሧЧሧӁӻႵᆀ֥࿊ᄴᇏ֥ࠏྏ,ఃᇏടࠣ၂֊ሱ࠭ӮູሧӁӻႵᆀೂޅࣉྛሧӁ֥ࡎᆴࡎ໙ี,ࠧሧӁӻႵᆀ֥ࡎᆴࡎ໙ี,ᆃࡼࣉ၂҄ษંᆃ۱8
໙ี.ഡpotіൕൈख़tሧЧሧӁ֥ӻႵᆀؓሱ࠭ӻႵ֥ҍӁ֥ࡎᆴࡎ,ఃᇏo=A,BіൕሧЧሧӁ֥ӻႵᆀ.``൮༵ॉ੮ࢌၞٮႨ໙ี.ӻႵᆀછԛܢௐ֥ൈީ҂ॖх૧ֹ֥ေڱԛࢌၞӮЧ,ᆃུӮЧॖၛ൞ࢌၞٮႨࠇඥڮ֩.ࡌഡૄછԛ၂۱ֆ໊֥ᆣಊေڱԛcჭ֥ӮЧ.ၹՎ,ೂݔӻႵᆀ࿊ᄴᄝt+τൈख़છԛఃሧӁ,ࠆ֤֥൬ၭႋھູpt+τ−c,ఃᇏpt+τіൕt+τൈख़֥ࢌၞࡎ۬.ॉ੮֞ࢌၞؿള֥сေ่ࡱ(),ӻႵᆀછԛఃሧӁࠆ֤֥൬ೆॖၛіൕູpo¯t+τ−(b+c),ఃᇏpo¯t+τ,o¯∈{A,B}іൕઙೆᆀᄝt+τൈख़ؓሧЧሧӁ֥ࡎࡎᆴ.ࣂުູіඍഈ֥ࡥࢱ,ࡼഈඍb+c၂ֹіൕູc.ᆃၩሢࣂު֥cऎႵࢌၞٮႨބࠏྏචᇗݣၬ.ࡌקࡌഡࠏྏѩ٤౫ն.ᆃၩሢ,ંႵࢌၞٮႨሹႵc>0.ၹູcູࢌၞٮႨބࠏྏ֥ןࡆ.``ႮႿሧӁӻႵᆀؓఃሧӁ֥؇൞,ࢌၞ෮֤֮ႿఃࠎԤࡎᄵ࿊ᄴ࿃ӻႵሧӁ,ۚႿఃࠎԤࡎᄵ࿊ᄴఃಪູቋఞ֥֒ൈࠏછԛሧӁ.tൈख़֥ӻႵᆀॉ੮ቋࡄࢌၞൈख़ູt+τ∗,ఃᇏτ∗≥0.ᆃൈ֥છԛ෮֤po¯t+τ∗−cႋູvot+τ∗+[(po¯t+τ∗−vot+τ∗)−c]+.Ⴟ൞ӻႵᆀؓఃӻႵሧӁ֥֥ࡎᆴࡎູ{+[(po¯t+τ∗−vot+τ∗)−c]+e−}rτ,τ≥0ࠧvot+{[(po¯t+τ−vot+τ)−c]+e−rτ}.τ≥0ఃᇏEotіൕሧӁӻႵᆀ֥ᄝൈख़t่֥ࡱყ௹.ഈൔ֥Ⴗ؊Ⴎਆ҆ٳܒӮ.భᆀູӻႵᆀ֥ࠎԤࡎ,ުᆀᄵোරႿᆳྛࡎູ۬(po¯t+τ−vot+τ−c)+֥ૅൔᆣಊіགྷਔӻႵᆀؓ൭ᇏሧӁࠏࠏ߶֥ࡎᆴࡎ.קၬഡpot=vot+{[(po¯t+τ−vot+τ)−c]+e−rτ},()τ≥0ఃᇏEotіൕሧӁӻႵᆀ֥ᄝൈख़t่֥ࡱყ௹,τіൕઔغॖڏ๔ᆸൈࡗ.ӫpotູሧӁӻႵᆀ֥ఃӻႵሧӁ֥ᇶܴࡎ.ӫqot=pot−votູሧӁӻႵᆀ֥ఃӻႵሧӁ֥ࠏ௹ಃࡎ.9
``܄ൔ()֥Ⴗ؊Ⴎਆ҆ٳܒӮ.భᆀӫູӻႵᆀ֥ࠎԤࡎ,ުᆀӫູሧЧ෮֤ࡎ.ܱႿሧЧ෮֤ࡎqot,ॖၛᆣૼ༯૫֥ଁี.ଁี[22]ࠏ௹ಃqotડቀܱ༢{(o)}t+τqot=¯t+τ+ge−rτ,()τ≥0r+−cλ+ఃᇏ,qo¯t+τіൕt+τൈ٤ሧӁӻႵᆀo¯ܓೆሧӁުࠆ֤ሧӁ֥ࠏ௹ಃ.ᇿᆃ֥ษંބScheinkanandXiong[19]҂.ުᆀ֥ษં҂ടࠣࠇ҂ᆰࢤടࠣࠏࠏ߶০Ⴈٚൔބࠏྏ֩໙ี,ၹՎ࣐ܵିܔ֝ԛሧЧሧӁ֥ࡎᆴࡎູࠎԤࡎބࠏ௹ಃ֥ןࡆ,࣐ܵఃࠎԤࡎ֥ބЧ္࣮֥ປಆཌྷ,൞ః௹ಃ෮ડቀ֥ܱ༢ൔބૌ֥܄ൔ()҂,ູ{()}qot=¯t+τ+got+τce−rτ.()τ≥0r+−λ൞,ᆞ൞ႮႿ҂թᄝ+,෮ၛഈඍіൕ҂ିФ൪ູ௹ಃ֥ࡎ.``۷ູᇗေ֥൞,[19]֥c္҂ݣႵޅࠏྏഈ֥ݣၬ.ॖၛᆣૼ,ᄝࢌၞׄ∗tࢌၞࡎ۬pt∗ઙડቀܱ༢:pt∗=pot∗+c=po¯t∗.()܄ൔ()֥ᆣૼೂ༯:܄ൔ()ၩሢᄝࢌၞׄ∗tႵ༯૫֥ܱ༢qot∗=qo¯t∗+got∗cr+−λӮ৫.Ⴎ܄ൔ()ބקၬᆩpot∗=po¯t∗−c.ᆃၩሢ,܄ൔ()Ӯ৫.܄ൔ()֥ࣜ࠶ࢳ൞,ሧЧሧӁ֥ઙೆᆀ০ႨఃܓઙሧӁᄝໃটࠆ֤֥൬ၭ,ېݺିܔַཧܓೆൈᆦڱ֥ӮЧ,ପહູહေઙೆሧӁ?൞ູਔ࿙၂ᇕҍڶ֥҂ಒקગ?൞,ଆᇏ֥ࣜ࠶ದ൞ڄགᇏ৫֥.ၹՎ,ંೂޅ҂ିࢳܓઙᆀઙೆሧӁ֥ࠏޅᄝ.ಖط,Ч࣮֥ଆഡקᄵିक़ڛ[19]֥ᆃུ໙ี.10
3ଆࢳ༅–ࠏ௹ಃࢳ``۴ऌభ૫֥ษં,ሧЧሧӁӻႵᆀؓఃሧӁ֥ࡎᆴࡎpotॖၛіൕູࠎԤࡎvotބࠏࠏ߶ࡎ–ࠏ௹ಃqotᆭބ,ᆃུӮٳ֥ඔ࿐іൕೂ༯:pot=vot+qot,()−f¯vot=f¯+fˆot()rr+,λ{()}qooτt=¯t+τ+got+−ce−rτ.()τ≥0r+λ+ఃᇏ,܄ൔ()ᇏࠎԤࡎ֥ඔ࿐ࢳ༅൞ౢԣ֥,ط܄ൔ()ࣇࣇ൞၂۱ඔ࿐ٚӱ.ೂݔିܔࢳᆃ۱ٚӱ,ᄵ܄ൔ()֥ඔ࿐ࢳ္ࣼౢԣਔ.Чࢫษંٚӱ()֥ࢳ໙ี.ࠏ௹ಃࢲܒ``οᅶ܄ൔ,ࠏ௹ಃ൞ᆳྛࡎູ۬(po¯t+τ−vot+τ−c)+,֥ૅൔઙಃ֥ࡎᆴݖӱ.ھࠏ௹ಃіགྷ֥൞ӻႵᆀؓఃሧӁࠏࠏ߶֥ࡎᆴࡎ,ପહӻႵᆀ൞ೂޅ০Ⴈᆃᇕࠏ߶?ࢤ༯টૌࡼษંᆃ۱໙ี.``܄ൔ()ᇏ֥votіൕӻႵᆀؓఃሧӁ֥ࠎЧࡎ,Ⴎਆ҆ٳܒӮ.၂҆ٳ൞ሧӁൈࡗनࡎᆴ֥ᅼགྷ,ఃᅼགྷູr−1;ਸ਼၂҆ٳ൞ӻႵᆀಪູ֥ሧЧሧӁටൈࡎᆴؓఃൈࡗनࡎᆴМ֥ᅼགྷ,ႮႿᆃᇕМіགྷ֥҂໗ק,෮ၛఃᅼགྷੱေն၂ུ,ູ(r+λ)−1.ၹՎ,ࠎԤࡎିܔّ႘ӻႵᆀؓఃሧӁࡎᆴЧദ֥ಪ്,ഉໃّ႘ઙછචٚಪ്֥ҵၳ෮थק֥ଽಸ.܄ൔ()ᇏ֥qot,іགྷਔӻႵᆀؓᄝໃট֥ଖ۱ൈख़છԛሧӁࠆ֤֥ӑԛఃࠎԤࡎ҆ٳ֥ࡎ,ᆃ၂҆ٳࡎᆴႋھބӻႵᆀЧದؓሧӁࠎЧࡎᆴ֥ಪ്ܱ,ބઙછචٚؓڄགሧӁࠎЧࡎᆴಪ്֥ҵၳႵܱ.ၹՎ,qotॖၛФіགྷູgo=fˆo¯t−fˆot֥ݦඔ.ਸ਼၂ٚ૫,ႮႿૌᆃॉ੮֥ൈࡗཋࢸູཋ,෮ၛᆃ۱ݦඔ҂ၛൈࡗູэਈ.ၹՎ,ॖၛಪູթᄝଖ۱֤qot=q(got).()Ӯ৫֥ݦඔq(·)ᇿૅൔ௹ಃקࡎᇏ္ॉ੮োරഡק,ൌ࠽ഈྟൗࢸᇏषؿԛ֥၂ུࠎЧٚم,ᄝ٤ྟ࣮ᇏ္٤ӈᆴ֤ࢹ.``ࢤ༯টॉ੮ႋھڮჍݦඔq(·)ޅᇕྟᇉ.భ૫֥ٳ༅іૼ,ࠏ௹ಃ֥ࡎᆴ౼थႿචٚؓሧЧሧӁටൈࡎᆴಪ്֥ҵၳ,ᆃᇕҵၳ֝ᇁਔචٚؓሧӁࠎЧࡎಪ്֥ҵၳ,11
ࢠն֥ҵၳ֤ࣜ࠶ದᆭࡗ֥ࢌၞିܔؿള.၂Ϯটඪᆃᇕҵၳᄀն,֩րࢌၞղӮ֥ൈࡗτࣼ߶ᄀཬ,ّᆭᄵᄀն.܄ൔ()ᇏ֥ᅼགྷሰe−rτ֥թᄝіૼ,q(·)ႋھ൞၂۱ֆט־ᄹ֥ݦඔ.ႮႿᆃሢᇗॉ੮֥൞ࠏྏቀܔնطࢌၞٮႨ҂൞֥ۚ౦ঃ,෮ၛթᄝࠏࢌၞؿള֥ॖିྟ.ၹՎؓҕඔcॖၛೂ༯ࡌק.ࡌקࢌၞٮႨ҂նѩࠏྏ҂ཬ,ࠧૌ֥ןࡆc҂ն,֤ࠏࢌၞିܔࣜӈؿള,ࠧӻႵᆀ֩րࢌၞ֥ൈࡗӉ؇Ⴕཋ,҂߶ཋᆸֹ֩ր༯ಀ.ᇿ෮໌“c҂ն”္ၩሢᄝࢌၞׄ∗tӮࢌࡎсྶۚႿܓઙ֥ٚࠎԤࡎ,ࠧpo¯t∗−c>vo¯t∗.()൙ൌഈ,ૌ༯૫֥ษં൞ᄝ“c҂ն”֥భิ༯ᅚष֥.ࠎႿmathlab֥ඔᆴ൫ဒіૼ,cбࢠն֥ൈީ,۴ऌૌ֥ࡎ܄ൔ֝ԛ֥ஞଐູڵඔ.ᆃѩ҂ၩሢᆞ֥Ⴕ෮໌ڵஞଐթᄝ,ط൞ၩሢࣜ࠶ದ֥ؓሧЧሧӁ֥ಪ്ҵၳ҂ቀၛՏࠗࠏࠃؿള.``ᆃၩሢࠏ௹ಃ֥ࡎᆴႋھູᆞ,ࠧݦඔq(·)۬ᆞ.ਸ਼ຓ,ႮႿݦඔq(·)҂ܻ߁ൈඔ࿐ٳ༅֥൭م҂ିႨ,ᆃߎࡌקq(·)ܻ߁.ഈඍٳ༅ॖ݂ବӮ༯૫֥ࡌק.ࡌקթᄝ၂۱־ᄹ֥a۬ᆞ֥aܻ߁ݦඔq(·),֤ࠏ௹ಃູq∗t=q(g∗t),∗=o,o¯.``Ⴟ൞,ٚӱ()ॖၛФіൕູ{()}q(got)=(go¯t+τ)+got+τe−rτ,τ≥0{(r+−cλ+}=essτ.supEotq(−got+τ)+go)t+τ−ce−r.()τ≥0r+λ+ѩॖၛᆣૼ,ଁี[22]ࠏ௹ಃ֥ᅼགྷݖӱ{e−rtq(got)}ູഈ᷉,ѩડቀӈັٳ҂֩ൔ:1σ22gq′′(x)−ρxq′(x)−rq(x)≤0.()``ഈඍٳ༅іૼ,ӻႵᆀؓఃሧӁ֥ࡎᆴࡎ౼थႿࠏ௹ಃ֥ࡎ,ࠏ௹ಃ֥ࡎᄵ݂ࢲູັٳ҂֩ൔ()֥ࢳ.ັٳ҂֩ൔ֥ࢳႋھၛັٳٚӱ֥ࢳູభิ,ᆃၩሢߎေؓ{e−rtq(got)}ࣉྛധೆษં(ఃབྷ༥ֻࢫ).12
ັٳٚӱ֥ࢳ``ᆃษંັٳٚӱ1σ22gq′′(x)−ρxq′(x)−rq(x)=0()֥ࢳ໙ี,ູࠏ௹ಃࡎቓሙС.ູख़߂()֥ࢳ,ླေ༯૫֥ݦඔh(x).()Ur1ρ22ρ,2,σ2xifx≤0hg(x)=()(),()2piMr1ρ2ρ,Γ(1ρ,2,σ2x−Ur1σ2x2ifx>02+r2ρ)Γ(12)2g2ρ,2gఃᇏM:R3→RބU:R3→RູKummerݦඔ,ఃࢳ༅іղൔٳљູM(a,b,y)=2y21+ay+(a)+···+(a)nyn+···,b(b)22!(b)nn!{}piM(a,b,y)M(1+a−b,2−b,y)U(a,b,y)=2sinpibΓ(1+a−−y1−bb)Γ(b)Γ(a)Γ(2−,b)(a)n=a(a+1)(a+2)···(a+n−1),(a)0=1.ଁี[19]ݦඔh(x)ູӈັٳٚӱൔ()֥ࢳ,ѩડቀ༯૫֥ྟᇉ:(x)֥3ࢨၛଽ֥֝ඔթᄝ,ѩડቀ่ࡱh>0,h′>0,h′′>0,h′′′>0,limx→−∞h(x)=0ބlimx→−∞h′(x)=0.ѩᄝx=0ׄႵh(0)=pi()(),()Γ1122+r2ρ√ρh′(0)=pi()().()Γ1rσ22Γ2ρ2.ഡu(x)ᄝࡗ(−∞,0]۬ᆞֆט־ᄹ.u(x)ູັٳٚӱ()֥ࢳ,ᄵсקթᄝ၂۱ӈඔβ>0֤u(x)=βh(x).ࠏ௹ಃ֥קࡎ``ᆃॉ੮ࠏ௹ಃקࡎ໙ี,ູՎߎႋھؓࠏ௹ಃ֥උྟቓധೆษં.13
ࢌၞൈࡗ֥ׄ࿊ᄴބࠏ௹ಃקࡎ``൮༵,ॉҳࢌၞׄ∗t.ࢌၞׄ∗tഈ,Ӯࢌࡎ۬pt∗ડቀ่ࡱ൞pt∗=po¯t−c≥pot.Ⴎ܄ൔ()-()ᆩq(got)≤gotr++q(−goλt)−c.ٟᅶૅൔ௹ಃקࡎ֥ٚم,קၬࠏ௹ಃ֥࿃ྛބӻႵೂ༯.קၬٳљӫCބSູ࿃ྛთބ๔ᆸთ,ఃᇏ{∣}∣C=x∣q(x)>x+q(−x)−c,∣{r+λ∣}∣xS=x∣q(x)≤(−x)−c.∣r++qλᆃၩሢ,got∈S֥ӻႵᆀ֥ቋࢌၞҦ൞છԛఃӻႵ֥ሧӁ;got∈C֥ӻႵᆀ֥ቋࢌၞҦ൞࿃ӻႵఃሧӁ.``ఃՑ,ؓ܄ൔ()ࣉྛܴҳ.ཁطၞ,ࠏ௹ಃোරႿᆳྛࡎູ۬(po¯t+τ−vot+τ−c)+֥ૅൔઙಃ.ࢌၞൈख़∗t,ӻႵᆀ๙ݖછԛఃሧӁࠆ֤֥ሧЧ෮ູ֤q(−got∗)+got∗()r+−c.λھ෮֤Ⴎ3۱҆ٳܒӮ,ֻؽ۱҆ٳіགྷਔӻႵᆀିܔࠆ֤֥ࠎԤࡎࡎҵࡎ,ֻ۱҆ٳіൕՖ௹ಃᆳྛ෮֤ᇏัԢ҂݂છԛᆀ෮Ⴕ֥҆ٳ(ູࢌၞٮႨބࠏྏָཌ֥ןࡆ).ֻ၂۱҆ٳᄵ൞၂۱௹ಃ,ఃࣜ࠶ࢳೂ༯:ႮႿܓઙᆀӻႵҍӁު,҂ିࠆ֤ਔҍӁఃު֥ٳޣߎିࠆ֤ರުᄝൡ֥֒ൈࠏઙӻሧӁࠆ֤ሧЧ෮֤֥ࠏ߶,ܓઙᆀؓھࠏ߶֥ࡎᆴࡎܒӮਔ܄ൔ()Ⴗ؊ֻ֥၂҆ٳ.ഈඍٳ༅іૼ,છԛᆀሔ֥ᇶေ҆ٳႮࠎԤࡎࡎҵܒӮ,طҵࡎᄀն,ӻႵᆀછԛሧӁࠆ֤֥ሧЧ෮֤ࣼႋھᄀն.ູՎቔԛೂ༯ሱಖ֥ࡌഡ.ࡌק.ࠎԤࡎ֥ࡎҵູڵ.ࠧgot<0֥ൈީ,ӻႵᆀ҂߶ࣉྛࢌၞ.ᆃၩሢ,ݦඔq(x)ᄝx<0֥ൈႋھડቀ่ࡱq(x)>q(−x)++−λ2.ӻႵᆀ֥ሧЧ෮֤,ෛࠎԤࡎࡎҵ֥ᄹնطᄹն.ᆃၩሢ,q(−x)+xr+λ−cູֆט־ᄹݦඔ.14
ଁี[22]ᄝࡌקބ༯ॖၛᆣૼ,թᄝk∗>(r+λ)c֤C=(−∞,k∗).ᆃၩሢ,S=[k∗,+∞).``ഈඍଁีіૼᄝൈࡗࡗ[0,τ∗],ఃᇏτ∗=inf{t|got=k∗},࿃ӻႵࠏ௹൞ӻႵᆀ֥ቋႪ࿊ᄴ.ٟᅶૅൔ௹ಃקࡎં֥ׅษંᆩ,ෛࠏݖӱ{e−rtq(got)ູ᷉.Ⴎଁีᆣૼᆩ,ෛࠏັٳٚӱൔ[dq(got)e−]rt[]=−rq(got)−ρgotq′(got)+1σ2e−rtdt+σgq′(go2gq′′(got)t)e−rtdWot,Ӯ৫.ᆃၩሢ1σ22gq′′(x)−ρxq′(x)−rq(x)=0,−∞<x≤k∗.ၹູq(x)ᄝࡗ(−∞,0]۬aֆט־ᄹѩડቀொັٳٚӱൔ(),෮ၛႮଁีᆩ,թᄝβ>0֤q(x)=βh(x),x∈(∞,k∗).ᄝՎࠎԤഈ,༯૫֥ק۳ԛࠏ௹ಃࡎᆴݦඔ֥ऎุྙൔ.ק[22]ࠏ௹ಃ֥ࡎᆴݦඔq(x)ೂ༯:βh(x),x<k∗q(x)=x,()+βh(−x)−c,x≥k∗r+λఃᇏ,β=1,(h′(k∗)+h′(−k∗))(r+λ)k∗ູٚӱ[x−c(r+λ)][h′(x)−h′(−x)]−h(x)+h(−x)=0֥ື၂ࢳ.ᄜંࠏ௹ಃ֥קࡎ``༯૫۳ԛࠏ௹ಃקࡎ֥ਸ਼ຓ၂۱іൕྙൔ.ഡu(x,k)=Eo[e−rτ|g0t=x],τ=inf{s|got+s>k},x≤k,іൕໃটሧᆀ֥ྐဟҵ֞ղk֥ൈ֥གྷࣁཟགྷൈख़tᅼགྷ֥ᅼགྷၹሰ,ఃᇏxіൕሧᆀ֒భ֥ྐဟҵ.Standardᄝ(KarlinandTaylor[20],page243)ᇏіૼuູ٤ڵ֥ܱႿx۬ֆט־ᄹ֥ݦඔ,ѩડቀັٳٚӱൔ1σ22guxx−ρxux=ru,u(k,k)=1.()Ⴎଁีіૼ,թᄝβ1>0֤u(x,k)=β1h(x).Ⴛၹູu(x,k)|x=k=βh(k)=1,෮ၛႵu(x,k)=h(x).h(k)Ⴈᅼགྷၹሰu(x,k),ॖၛڮჍࠏ௹ಃࡎ܄ൔ()ၛྍ֥ࢳ,ऎุࢳೂ༯:15
ࠏઙಃ֥ࡎݦඔq(x)ॖၛіൕೂ༯:[()]kq(x)=sup−k)−cu(x,k).()k≥0r++q(λ܄ൔ()іૼሧᆀ֥ሧӁӻႵᆀ֥ቋႪࢌၞҦູ,ᄝෛྐဟҵk֥ᄹࡆط҂؎ᄹࡆ֥kr+λ+q(−k)−cބෛሢ֩րൈࡗࡆնطࡨഒ֥ᅼགྷu(x,k)ᆭࡗ࿙၂۱ކ֥ޙׄk∗.ྐဟҵ൮Ցղ֞k∗֥ൈࡗູࢌၞؿളൈࡗ.ᄝھൈࡗׄഈؿളࢌၞࠆ֤֥ሧЧ෮֤֥ყ௹,ູࠧӻႵᆀ֒భྐဟҵູxൈؓఃࠏ௹ಃቓ֥ࡎ.ࠏ௹ಃ֥ඔᆴ࠹ෘٚم``ູਔିܔࠎႿmatlabؓࠏ௹ಃ֥ࡎᆴݦඔq(x)aᆳྛࡎ۬ݦඔe(x)=g(−x)+x/(r+λ)−cބ᷉ࡎݦඔm(x)=βh(x)ࣉྛਔඔᆴ࠹ෘ.ᆃࡼࠏ௹ಃ֥ࡎᆴݦඔіགྷູβh(x),x<k∗q(x)=;()βh(x)+xc,x≥k∗r+−λᆳྛࡎ۬ݦඔэྙູ)q(x)=(0,x<0;()βh(x)+x,x≥0r+−cλ+᷉ࡎݦඔβh(x)ᇏ֥ࠎԤݦඔh(x)ູӈັٳٚӱൔ1σ22gq′′(x)−ρxq′(x)−rq(x)=0.()֥ࢳ,ఃԚ่ࡱh(0)=pi()()Γ12γ1(2+r√2ρ)piρ.()h′(0)=()()Γ1σ2g2Γr2ρ4ࠏྏބࠏஞଐࠏஞଐ``ᄝ၂ՑܱႿଖཛሧЧሧӁ֥ࢌၞᇏ,ઙٚ჻ၩၛۚႿఃࠎЧࡎ֥ࡎ۬Ӯࢌ,ᄵӫھࢌၞࡎ۬թᄝஞଐ(ཌྷؓႿઙٚ).ھሧӁᄝଖ؍ൈࡗଽ֥ࢌၞᇏሹ൞ೂՎ,ᄵӫھሧ16
Ӂᄝᆃ۱ൈࡗ؍ଽթᄝஞଐ.ᆃၩሢ,ॴ௹ଆᇏஞଐ൞၂۱౼ᆴ٤ڵ֥ෛࠏݖӱ.ೂݔࣜ࠶ದؿགྷ൧ӆ֥ࢲܒथקਔࠏࠃିܔൌགྷ,ࠧႵࠏࠏ߶ॖၛ০Ⴈ,ᄵఃؓሱ࠭ӻႵ֥ሧЧሧӁ֥ࡎߎေЇওؓః๙ݖࠏࠃطିࠆ֤֥ሧЧ෮֤֥ࡎᆴࡎ.ႮՎӁള֥ஞଐӫູࠏஞଐ.``ሧӁࢌၞׄ∗tഈ,છٚؓఃሧӁ֥ࡎᆴࡎpot∗,ေ֮Ⴟࢌၞࡎ۬po¯t∗−c,(ҕཬࢫ“ӻႵᆀࡎᆴࡎ”ᇏ֥ษં).ᆃၩሢ,ӻႵᆀ֥ࠎԤࡎvot∗္ေ֮Ⴟ֥ࢌၞࡎ۬,ఃҵᄵႋھնႿࢌၞׄഈࠏ௹ಃ֥ࡎᆴq(got∗),ࠧ(po¯t−c)−vo∗t∗≥q(got∗).()ਸ਼၂ٚ૫,ܓઙᆀ֥ࡎᆴࡎູpo¯t∗=vo¯t∗+q(−got∗),෮ၛఃࠎԤࡎބӮࢌࡎ֥۬ࡎҵູq(−got∗)−c.()ႮႿgot∗>0,෮ၛ(q(−got∗)−c)<[(po¯t−co)−v∗t∗].()ܣॖקၬொྐࠏஞଐೂ༯:קၬ(ொྐࠏஞଐ)ӫӮࢌࡎ۬ބܓઙᆀ֥ؓሧЧሧӁ֥ࠎԤࡎᆭࡎ۬ҵᆭ༯ཋູࠏஞଐ.ႮႿࠏஞଐބࠏྏႵᆰࢤ֥ܱ༢,ܣࡼః࠺ᆭູ,b(c).ࠏྏؓࠏஞଐ֥႕ཙ``۴ऌק,b(c)ඔ࿐іղൔູ:{}b(c)=h(−k∗)−c.()(r+λ)[h′(k∗)+h′(−k∗)]+ᆃ֥+൞ᄝ܄ൔഈؓࢌၞ҂ॖିؿള֥౦ঃࣉྛཋᇅ,ၹູᆳྛࡎູ۬ڵൈࠏ௹ಃ҂ିФྛ,ࠧࢌၞ҂ିؿള.ᇿᆃ֥ஞଐ܄ൔ()ބ[19]֥ࠏஞଐ܄ൔႵ෮ҵၳ.[19]֥܄ൔູh(−k∗),()(r+λ)[h′(k∗)+h′(−k∗)]طЧ໓֥܄ൔ,бᆭ؟ਔӈඔcބ౼ᆞژݼ{·}+.ᆃၩሢ:1.ᆃ֥ࠏஞଐࣇࣇᄝࢌၞٮႨູ0طࠏྏ౫ն֥ൈީҌބ[19]֥܄ൔᄝඔᆴ࠹ෘࢲݔഈཌྷ;2.֒ࢌၞᆀ֥ࠏྏޓཬࠇࢌၞٮႨޓն֥ൈީ,Їওࠏࢌၞᄝଽ֥ޓ؟ࢌၞ߶๔ᆸ,ᆃൈ֥ࠏஞଐ࣍ެູ
ඔᆴٳ༅c from 0 to 2``Чඔᆴ২ᇏ֥5Ⴈ0֥ҕඔ౼ሱ[19],ھ໓ངࠎႿ၂༢ਙൌᆣࢲݔؓཌྷܱҕඔࣉྛਔབྷ༥ඪૼ.ఃᇏ,ڄ45ག০ੱູr=,ڄགሧӁनᆴ߭گ؇ູλ=1,ሧᆀொྐ؇ູφ=,ڄགሧ4Ӂ0֥ঔ؇ູ√σf=100,ྐݼ֥ঔ؇σsބٳޣ֥ঔ؇σdູ√10.ႮႿЧ࣮ᇏ֥cऎႵࢌၞٮႨބࠏྏචᇗݣၬ,෮ၛູਔ࣮ࠏྏؓ35ࠏஞଐ֥႕ཙ,Ч໓ࡼҕඔcၛູࡗۯ࠹౼ᆴ100Ց,ႮՎॖၛࠆ֤“ࠏྏބࢌၞٮႨؓஞଐ֥႕ཙ30”:ࠏྏބࢌၞٮႨؓஞଐ֥႕ཙ``ඔᆴ൫ဒࢲݔ(ҕ1)іૼਔࠏྏބࢌၞٮႨ֥ࡆሹcؓࠏஞଐb(c)֥႕ཙ,ऎุࢲݔೂ༯:(1)֒cࢠཬൈ,b(c)ؓఃэ٤ӈૹۋ.ෛሢc֥ᄹࡆ,ᆃᇕૹۋӱ؇҂؎༯ࢆ;(2)֒cᄹࡆ֞၂קӱ؇(ӑݖ)ࠏஞଐཨാູ0.(3)b(c)ູc֥ֆט־ࡨݦඔ,ѩॖၛᆣૼ:ଁี[22]()√0)Γrσ2limc)=h(=2ρ()g.()c→b(02(r+λ)h′(0)2(r+λ1)Γρ2+r2ρᇿ[19]֥ษંॖၛಪູ൞ᆃษંࢲݔ֥ห൹౦ঃ–ࢌၞٮႨູ0ࠏྏཋն֥౦ঃ,෮ၛ[19]္Ⴕ܄ൔ().ᆃ۱܄ൔᄝᄝษંࠏஞଐഈཋ֥ൈީ൞Ⴕၭ֥.18b
ࣜ࠶ٳ༅``ԮሧӁקࡎં֥Ӯ৫,၇ঠႿఃЇওྐ༏ؓӫބࣜ࠶ದࢥྟ֩ࠎЧࡌק֥Ӯ৫.ᆃဢ֥ଆીႵࠏᆀളթ֥ॢࡗ.ಖط၂֊٢ॺఃᇏ֥ଖུࡌק.౦ঃࣼ߶Ⴕޓն֥҂,ؓႿሧӁࡎᆴ֥ؿགྷ,ࠏႵൈఏࠒࠞቔႨႻ൞ఃཨࠞቔႨ.২ೂ,ྐ༏҂ؓӫ൧ӆഈթᄝࠏ,ᄝሧӁࡎ۬ିܔႵّི႘൧ӆྐ༏–“Ⴕི൧ӆ”่֥ࡱ༯,ᆃᇕྛູؓႿሧӁࡎᆴؿགྷႵࠒࠞቔႨ.ऎႵொྐሧᆀ֥X-Sଆෙಖ္թᄝࠏ,൞ᆃ֥ࠏᆺିՏࠗஞଐ֥ؿള,ಏᇹႿሧӁЧദࡎᆴ֥ؿགྷ.ᆃඪૼ,ࠏྏ֥ቔႨႵൈࠒࠞႵൈཨࠞ,҂ି၂ۀطં.``ࠏྏّ႘֥ਔࣜ࠶ದ֥ࠏಣ౦,ࠏྏնၩሢࠏࠃَ,ᆃ๙ӈ߶֝ᇁࠏஞଐ֥ᄹն.ᆃႵᇹႿඪૼູહીႵહཞဢ֥ႮሧӁ֥ࡎ۬ߎ߶ശ,മᇀߎ߶ശ֞ଖᇕຂࠎЧࣜ࠶ࢳ֥ࡎ໊,ᆃဢ၂ᇕܯགྷའ.ಖط,ᆃ֥ࠏྏ֥ൌགྷ൞ၛࠏࠏ߶֥ग़ܴթᄝູభิ֥,S-Xଆᇏࠏࠏ߶ჷႿࣜ࠶ದ֥ொྐ.၂Ϯটඪ,ᆃᇕࠏ߶թᄝႿࣜ࠶ದؓໃট҂ಒקྟ֥ಪ്Чദ҂ಒק,ᆃᇕ҂ಒקྟ္Фӫູ“ห҂ಒקྟ”,ӈӈ֝ᇁࣜ࠶ದؓሧӁᇶܴࡎ֥٤ఊՑྟ.ૌಪູ,ᇶܴࡎ֥٤ఊՑྟ൞֝ᇁࠏࠏ߶ؿള֥ࠎԤ.``ೂݔ൧ӆഈ֥ۚӮЧྐݼ–২ೂٳޣྐݼݖ,֮ӮЧྐݼݖ఼–২ೂ“ყ௹”a”ีҋ”മᇀ“ۀ”֩֩,ᄵఃࣜ࠶ದޓॖି߶ऎႵห҂ಒקྟ,ᆃൈࠏஞଐ໙ีႋھФᇗ൪.ܴҳഈᆣᆷඔࠎЧཌᇏ֥1୍ཌ(2),ॖၛؿགྷ၂۱ၩන֥གྷའ:2004୍9-10ᄅࡗഈᆣᆷඔϴෛሢޓն֥ࢌၞਈശ,طᆃᇕശᄜࢌਈղ֞ቋն֥ൈީ֨ഈਔפڂ.୍03ᄅ30ರ18ൈ33ٳ,ჷሱКࣘӤБ֥ีູ“ᇏݓܢ൧ğ1500ၡჭಣఫ4ᄅ30ರభࣙྏնӜ”ᇏ֥၂؍໓ሳරެູᆃᇕགྷའቓਔଖᇕᇿ:``࣍ࠫ฿,Aܢ৵࿃նי,ᄝ1200֥ׄྏܱ१ഈ༯ّگ.ᆇᆞ֥ჰၹ൞હ?൞oਆ߶pყ௹ઋॢ,൞ࣜ࠶ඔऌ҂ুܴಞޡܴט॥ყ௹ᄜ؇௧ࠎࣁيॢ,ߎ൞Ќག܄ඳၹູѰᒮطҐ౼֥ඁ߭ྛູࡗࢤಞࠎࣁᇗҝܢඵו?ࠇྸᆃМުߎႵ۷പ֥৯ਈ.``ཨ༏ದൖರཟ࠺ᆀ,ᆇᆞ֥ᇶ࢘ఃൌ൞Ֆ؟ٚࣥࣉೆAܢ൧ӆದлശᆴ֥1500ၡჭݚຓႳሧ,ᆞ൞ૌ֥څॾa҂࠹ӮЧֹ൲,Ҍ֝ᇁਔଢభೂՎҙਛ֥ܢ൧अ૫.ቋॖஃ֥൞đoቋሐܴ֥೪יpഉໃট֞,طᆇᆞ֥նཋࡼᄝ4ᄅ30ರቐႷࢆਢ,ପࡼ൞oಣఫpࣙྏնӜ֥ൈख़.``ႮႿᆃ؍໓ሳቔԛ֥ყބଢభܢ൧֥౦ঃ٤ӈ၂ᇁ,෮ၛᄝଖᇕӱ؇ഈॖၛཌྷྐఃᇏ֥“ಣఫ”ඪ.ၹՎॖၛࣉ၂҄ሔ໙,ᆃུಣఫ൞ޅൈࣉೆܢ൧֥,ఃؓܢ൧֥ࣉೆႵ19
Figure2:ഈᆣᆷඔࠎЧཌ–1୍ཌ၂۱ཌྷؓࠢᇏ֥ൈ௹.ೂݔ၂్උൌ,ପહ2ᇏ֥9-10ᄅགྷའॖၛູS-Xଆ֥ࢳ৯ิ܂ଖᇕӱ؇֥ᆦӻ.ᆴ֤ᇿၩ֥൞ಣఫႤೆު,ૌ҂߶Ф০Ⴈ۲ᇕࠏࠏ߶,္߶ᇶᇅᄯࠏࠏ߶,ᆃ൞S-Xଆ෮҂ିࢳ֥.5ࢲં``Ч໓࣮ਔ[19]ิԛ֥၂۱Ⴕॖࢳ༅ྟ֥৵࿃ൈࡗஞଐଆ(S-Xଆ),ᆃ֥ஞଐჷႿ٤ఊՑྐࣜ࠶ದ֥ݖ؇ࠏࢌၞྛູ,ఃᇏ֥٤ఊՑྐႻ൞Ⴎࣜ࠶ದ֥ொྐᄯӮ֥.൙ൌഈ,ࠏࠏ߶൞ࠏྛູթᄝ֥сေ่ࡱط҂൞ԉٳ่ࡱ,ࠏྏ֥նཬဢ္߶႕ཙࠏྛູ֥նཬ.ಖط,[19]ѩໃॉ੮ᆃ۱໙ี.Ч໓ࡼࠏᆀ֥ྏָཌcቔູޙਈࠏྏնཬ֥ᆷѓ֝ೆ֞S-Mଆ,ঔնਔଆ֥ࢳٓຶ,ᄹ఼ਔఃࢳି৯.෮໌ࠏྏָཌc൞ᆷࣜ࠶ದॉ੮ൌീࠏྛູ֥ቋ֮ࡎҵ,ӻႵᆀࣇ֒൧ӆࡎ۬ۚႿఃᇶܴࡎc۱ֆ໊֥ൈީҌ߶ॉ੮છԛ,ܓઙᆀࣇ֒൧ӆࡎ֮۬Ⴟఃᇶܴࡎc۱ֆ໊֥ൈީҌ߶ॉ੮ઙೆ.ѩ๙ݖඔᆴٳ༅іૼ,ࠏٮႨᄀཬ,ࠏྏᄀն,ࠏஞଐᄀն.``ᆴ֤ᇿၩ֥൞,S-Xଆᇏ֥ࠏࠏ߶൞ग़ܴթᄝ֥,ࣜ࠶ದ֥ࠏྛູᆺ൞ࠎႿؓᆃᇕࠏ߶֥Ф০Ⴈ.ದૌ๙ӈܴҳ֥֞ஞଐӈӈϴෛሢն“ಣఫ”ཟ൧ӆ֥Ⴄೆ,ᆃུಣఫ҂Ф০ႨၘႵࠏࠏ߶,ߎ߶ᇶԷᄯჰЧીႵ֥ྍ֥ࠏࠏ߶.ؓႿᆃᇕགྷའ,S-XଆႨ൞൳ཋ֥.ູࣉ၂҄ঔնS-Xଆ֥ൡႨٓຶ,රެႋھധೆਔࢳଆࠎЧٚӱМު֥ັܴࢲܒ,ᆃ္൞ࣂުླေࣉ၂࣮֥҄໙ี.20
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