ֻ13जֻ1௹ܵ ॓ ࿐ ࿐ Б 2010୍1ᄅ JOURNALOFMANAGEMENTSCIENCESINCH ٳྙ҃ᄎ༯ቋႪЌބཨٮҦ 张卫国,肖炜麟,张惜丽(ଲ۽ն࿐۽അܵ࿐ჽ,ܼᇜ510640)ᅋေ:基于Merton的最优消费和投资组合模型,通过假设风险资产的价格变化服从几何分形布朗运动,探讨了一类具有人寿保险的最优投资消费问题.首先根据投资者在整个生命周期的消费和投保效用期望值最大的原则,利用贝尔曼动态规划原理,建立了最优投保和消费策略模型.然后在给定消费和遗赠评价效用函数的情况下,给出了最优投保和消费的闭式解,并获得了最优投资组合受模型参数变化影响的一些重要性质.最后,通过数值例子讨论了时间间隔a赫斯特指数变化时最优投保和最大期望效用的变化趋势.ܱՍ:分形布朗运动;赫斯特指数;效用函数;投资组合和消费;人寿保险ᇏٳোݼ: ໓ངѓ്:A ໓ᅣщݼ:1007-9807(2010)01-0078-070 ႄ ᅚཛଢቋႪሧൈࠏ࿊ᄴ֥႕ཙ.ၛഈ֥࣮൞ᄝڄགሧӁࡎ֥۬э߄ݖӱቋႪሧཨٮ໙ี൞ࣜ࠶ࠃᇏ၂ᇕ௴ђطڛՖࠫޅ҃ᄎ༯ࣉྛ֥.ಖط,ݓଽຓ࿐ᆀႻᇗေ֥໙ี[1].Mertonิԛਔׅ֥ࣜቋႪሧؓሧЧ൧ӆ֥նਈൌᆣ࣮іૼࣁವሧӁ֥൬ၭཨٮଆ,ѩҐႨෛࠏ॥ᇅંؓሧཨٮ໙ีੱࣜӈӯགྷԛ Ⴕொ!a ࡕڂިແ!֥ٳ҃,ѩ֥ቋႪҦࣉྛਔ࣮,۳ԛਔоൔࢳ.DuffieࣁವሧӁࡎ۬ᆭࡗթᄝሢӉ௹ཌྷܱ,ྟ২ೂ:ߛ֣֩[2]ᄝMertonଆ֥ࠎԤഈ,ॉ੮ਔሧᆀऎႵބဗཫܻ[9],ᅦົ֩[10],ဗޡਟ֩[11],ޱဇ઼ෛࠏ൬ೆ౦ঃ༯֥ቋႪཨٮބᆣಊ࿊ᄴ໙ี.֩[12][13]ࠣ[14],LoބMacKinlayLo.֩Ԯ֥[3]Jean০Ⴈෛࠏܿ߃֥ٚم,࣮ਔሧᆀሧ࿊ᄴଆၘࣜم૭ඍᆃུགྷའ,္مൡႋᄝࢹೆሧࣁऎႵჿඏ่ࡱ༯֥ቋႪཨٮაሧ໙གྷսሧᆀ֥ླေ,ࠤླྍ֥ྩᆞଆބਈ߄ٚ[4].ีYong๙ݖ᷉ٚم֤֞ਔཨٮ൝༯ࢆ౦ঃم[15].PetersބM[16]andelbrotิԛٳྙ൧ӆࡌඪ,༯ཋ௹ཋ֥ቋႪཨٮაሧ໙,ีѩԛਔоᆷԛٳྙ҃ᄎିܔޓݺֹࢳሧЧ൧ӆࡕڂൔࢳ[5].Kumarᄝቋն߄ཨٮᅼ२ིႨݦඔ֥ჰިແaӉ௹ཌྷܱྟ֩གྷའ.Hu֩[17]ᄝࡌקڄགᄵ༯,ॉ੮ઙ(છ)ᆣಊऎႵб২ࢌၞٮႨ౦ঃൈሧӁэ߄ݖӱڛՖٳྙ҃ᄎ༯,ᆣૼਔٳྙ֥ቋႪሧაཨٮҦ.ਾݚބԊڃ[6]ᄝᆣBlack Scholes൧ӆ൞၂۱ส০֥ປС൧ӆ,ѩಊ൬ၭթᄝႵࢸ҂ಒקۄಠބॉ੮ࢌၞٮႨ֥౦۳ԛቋႪሧཨٮଆ.ңޡ᪶[18]ၛᇏݓധ൧ঃ༯,֝ԛቋҵ౦ঃ༯֥ቋႪཨٮބሧҦ.ູ࣮ؓའࣉྛൌᆣ࣮,࣮іૼᇏݓᆣಊ൧ဗೋӮބਾএ߶[7]࣮ਔෛࠏ๋ᄁݖӱ༯֥ቋӆթᄝཌྷܱ,ྟіགྷູٳྙൈࡗਙ,ఃྛູڛႪཨٮაᆣಊ࿊ᄴ໙.ีᤂЏڃ֩[8]๙ݖഡ࠹ٟՖٳඔ҃ᄎ,ѩ۳ԛਔ۴ऌތථหᆷඔH֥ᆇႪ߄ෘمٳ༅ਔචᇗෛࠏ҂ಒקၹؓಸਈঔሧथҦ҄ᇧބҦ.້დބߛ֨ൢ[19,20]ҐႨൌᆣ֥ ൬۠ರ௹:2007-09-13;ྩרರ௹:2009-07-05.ࠎࣁཛଢ:ݓࡅሱಖ॓࿐ࠎࣁሧᇹཛଢ(70825005);࢝ტ҆ྍൗࡀႪྮದҌᆦӻ࠹߃ሧᇹཛଢ(06-0749);࢝ტ҆ದ໓ഠ߶॓࿐࣮ܿ߃ࠎࣁཛଢ(07JA630048).ቔᆀࡥࢺ:ᅦݓ(1963∀),ଳ,༆νूದ,Ѱൖ,࢝൱.Emai:lwgzhang@
ֻ1௹ᅦݓ֩:ٳྙ҃ᄎ༯ቋႪЌބཨٮҦ∀79∀ٚمંᆣਔᇏݓࣁವ൧ӆऎႵ؟ѓ؇ٳྙหᆘ.ႄ1 ഡBH={BH(t, ),t>0}іൕٳྙෛሢದૌളࠃඣ֥ิ,ۚᇙ؟ሧᆀष҃ᄎ BHіൕ tൈࡗଽэ߄ਈ,ᄵႵ[15]ܓઙದ൰Ќག,۲ᇕགᇕ္ᄓტطള.ູਔ۷ݺ BH= ( Ht)ֹّ႘གྷൌ౦ঃ,נԮૼބሻࢮᇏ[21]ᄝࡌഡڄགఃᇏ, ~N(0,1),H൞ތථหᆷඔ0<H<1,ሧӁࡎ֥۬э߄ݖӱڛՖࠫޅ҃ᄎ༯࣮ਔ۱ದሧᆀೂޅथקቋႪ֥ᆣಊቆކaཨٮބܓหљ֒H1=ൈ,ࣼ൞௴๙֥҃ᄎ.2ઙದ൰Ќག,ః௹ຬིႨቋն߄໙.ี҂ႿູਔٚьႨٳྙ҃ᄎ࣮ڄགሧӁࡎ۬ၛభ֥ቋႪሧaЌགࠣཨٮ໙ี࣮,Ч໓ࡌэ߄ݖӱ,൮༵ؓࣁವሧ൧ӆቓၛ༯ׄࡌഡ:ഡڄགሧӁࡎ۬э߄ݖӱڛՖٳྙ҃ᄎ,ࠎ1)൧ӆ൞ଉ҈֥,ࠧ҂ॉ੮ඥ൬aࢌၞٮႿളଁᇛ௹֥ཨٮིႨބண෮Ӂള֥၌ᄼིႨႨڄགሧӁ(ܢௐ)҂ᆦڱޣ০.௹ຬᆴݦඔቋն߄ჰᄵ,ࡹ৫ਔٳྙ҃ᄎ༯2)൧ӆᇏթᄝਆᇕሧӁॖ܂ሧᆀ࿊ᄴ.၂ᇕ֥ቋႪЌބཨٮଆ,෮֤ଆ۷ऎႵ၂Ϯྟ.ູ০ੱr(r>0)֥ڄགሧӁ(ၿྛթॻ),t(t#0)ᄝ۳קིႨݦඔ֥ࠎԤഈ,০ႨНغણܿ߃ൈख़֥ࡎ۬Ptડቀ Pr tt=ePt-Pt=r tPt+ჰ֤֞ਔཌྷႋ֥ቋႪሧaЌགࠣཨٮҦ.o( t),ࠧࣉ၂҄ࠆ֤ਔቋႪሧቆކ൳ތථหᆷඔaൈࡗdPt=rPtdt(1)ࡗۯaڄགሧӁ൬ၭੱaѯੱࠣ০ੱэ߄႕ཙ֥ਸ਼၂ᇕ൞ڄགሧӁ(ܢௐ),t(t#0)ൈख़֥၂ུᇗေྟᇉ.ቋުิ܂֥ඔᆴ২ሰඪૼਔൈࡗࡎ۬StڛՖٳྙ҃ᄎ,ఃັٳྙൔູࡗۯ taތථหᆷඔHэ߄ൈቋႪሧҦބིdSt=St(!dt+∀dBH(t))(2)Ⴈ௹ຬᆴ֥э߄൝.ఃᇏ,BH={BH(t, ),t>0}ູۀੱॢࡗ(#,FH,PH)ഈ֥ٳྙ҃ᄎ;!ູ൬ၭੱ௹ຬ,ડቀ1 ଆࡹ৫!>r;∀ູܢࡎѯੱ;HູHurstᆷඔ.3)ඵວ൙ࡱ൞၂۱৫֥ѻݖӱ[21],ࠧႮႿࣁವ༢൞၂۱ሱႮ؇ࠞն֥گᄖ༢ሧᆀᄝt_ൈख़थҦ൞ڎູtൈख़ඵວܓઙದ൰Ќག.tൈख़ໃඵວ,ᄵሧᆀڱԛЌٮ,֤֞,ሧᆀ၂ٚ૫࿙ܿхڄགҦ,ਸ਼၂ٚ૫ູਔ֤֞ۚح০طӵքڄག0;ሧᆀඵວ,ᄵሧᆀࡼࠆ֤ண֥൬ೆ,ቔ.ൈሧᆀ҂൞ᄝࢤ൳ྐ༏ൈ৫ख़ቓԛّ႘,ط൞ᄝྐ༏ղ֞၂ູ၌Ӂ۳ުದ.קਢࢸᆴൈҌቓԛथҦ.ՖطᄯӮਔ൬ၭੱ֥࠺ሧᆀᄝtൈख़֥ҍڶູWt,ཨٮݖӱູCt,ದ൰Ќགೆ֥ЌٮູQt,ணࣁحູIt.Ⴎ Ⴕொ!, ࡕڂިແ!ބ Ӊ௹ཌྷܱྟ!֩གྷའ.ڄགሧӁ(ܢௐ)ࡎ۬ྛູଆൔӈႨࠫޅ҃ᄎႿࡌഡሧᆀᄝଖൈख़ඵວ֥൙ࡱ൞ҕඔູ∃֥ࠇ๋ᄁঔݖӱটख़߂,ಖطૌمࢳࣁѻݖӱ,ѩඵວൈࡗ%൞ඵວ൙ࡱֻ֥၂ՑವሧӁ൬ၭੱٳ֥҃ࡕڂިແaӉ௹ཌྷܱ֩གྷའؿള,෮ၛሧᆀ֥ඵວ൙ࡱ൞၂۱ڛՖҕඔູ.ႮႿٳྙ҃ᄎ൞၂ᇕۚථݖӱ,ఃྟᇉᇶေ∃֥ᆷඔٳ֥҃ෛࠏэਈ.۴ऌ໓ང[21],ކЌႵࡆم҂эྟaሱཌྷරྟaިແྟa҂৵࿃ྟaӉ௹ٮა෬ணحᆭࡗ֥ܱ༢ູQt=∃It.ሧႿڄགཌྷܱྟaหᆘӉ؇aࣚ༥ࢲܒ֩,ᆃུྟᇉ൞ܢሧӁ(ܢௐ)֥б২ູ&t∃[0,1],ࠧڄགሧӁࡎྛູनऎС֥หᆘ,Ֆط֤ٳྙ҃ᄎӮSt=&tWt,ڄགሧӁPt=(1-&t)Wt,ᄵሧູख़߂ࣁವڄགሧӁࡎ۬э߄ݖӱ֥ਅݺ۽ऎᆀҍڶэ߄ݖӱູdWt=dSt+dPt-dCt-dQt,.༯૫ҐႨٳྙ҃ᄎটख़߂ڄགሧӁ(ܢௐ)ࠧڛՖ༯૫ෛࠏັٳٚӱࡎ֥۬ྛູэ߄,ଆ۷၂Ϯ߄ބ۷ିܔࣚಒdWt=(&tWt(!-r)+rWt-Ct-Qt)dt+૭ඍܢࡎྛູ֥ഈඍหᆘ &.tWt∀dBH(t)קၬႮႄ1ᆩ,dWt=(&tWt(!-r)+rWt-Ct-tൈख़ڄགሧӁ(ܢௐ)ࡎ۬St֥э߄ݖӱ࿖ٳྙ҃ݖӱ,Ⴈ Sіൕ tൈࡗଽэ߄QHt)dt+&tWt∀ (dt),Ԛൈख़t(t#0)ҍڶਈ,ᄵႵ༯૫ࢲં.Wt=W0#0.
∀80∀ܵ ॓ ࿐ ࿐ Б2010୍1ᄅሧᆀ֥ଢ֥൞ᄝಆุॖಸྸҦ∋=эਈіൕູሑэਈ֥ཌྟݦඔ,္ࣼ൞ඪᆺေ۳{(&t,Ct,It)}ᇏ࿊ᄴቋႪሧЌགބཨٮҦ,קི֥Ⴈݦඔູӈཌྷؓڄགညذྙൔ[1](ֆᄹ֥χᆜ۱ളଁᇛ௹ི֥Ⴈ௹ຬᆴݦඔቋն,ࠧݦඔ)ൈ,ሹॖၛ๙ݖրק༢ඔم֤֞(6)֥ቋႪ%ࢳ,္ࣼ൞ࠆ֤ቋႪЌބཨٮҦ.ࡌഡཨٮބ၌ J(Wt,t)=&maxEt{U(Cs)ds+B(I%+W%)}{,C,I}%tsssᄼࡎི֥ႨݦඔनູCRRAིႋݦඔ,ᄵႵ༯૫(3)ఃᇏࢲ.ં:Etіൕඔ࿐௹ຬ;%ູሧᆀ҂ಒק֥ඵວൈࡗ;U(&)ބB(&)ٳљіൕཨٮބ၌ᄼࡎིק1 ࡌഡཨٮބ၌ᄼࡎི֥ႨݦඔनູC∗Ⴈݦඔ.CRRAোt,ࠧU(Ct)=∗B(It+Wt)=۴ऌ໓ང[21]ଁี֥ࢲંᆩ,ൔ(3)ॖၛіൕູ(It+W∗t)∗,0<∗<1,ᄵଆ(6)֥ቋႪࢳູ∋J(Wt,t)=maxE(s-t)t{[U(Cs)+{&,C,I}t%-∃esss∃(!2-r)r+∃∗-∗()+H-1*C2(∗-1)∀2( 2t) ∃B(Is+Ws)]ds}(4)t=WtႮൔ(1+∃)(1-∗)(4)֤֞J(Wt,t)=maxE-∃(s-t){e[U(Cs)+∃B(Is+Ws)]}.ൈႮႿ∃∗2(!-r)-(1+∃)(1-∗)-∗(r+∃)+2H-1*2(∗-1)∀2( t)J(W+ W,t+ t)=J(W,t)+J(!-r)+ttW((&WIt=Wtttt(1+∃)(1-∗) rWt-Ct-Qt) t+JW(&tWt∀ BH+&*=!-r2t2H-1 J(t 1t+J2W)W(&tWt∀ BH)+o( t)(1-∗)∀2( t)(7)ൔᇏJ(tіൕ(J(Wt,t),J(W(іൕ(J(Wt,t)t(W,JW)Wіൕᆣૼڸ.t2(ᄝࡌഡڄགሧӁڛՖࠫޅ҃ᄎ֥ཨٮބJ(Wt,t)(W2(ၛ༯),ၹՎႵሧଆᇏ,ڄགሧӁ֥ቋႪሧб২ໃุགྷڄགሧt0=m-∃(s-t)Ӂ֥эڄག.ႮႿތථหᆷඔᄀ,ۚіൕھোڄགax{e[U(Ct)+∃B(It+Wt)]+ JW((&tWt(!-r)+rWt-Ct-Qt)+J(ሧӁ֥ᄮലᄀഒ,ऎႵ۷఼֥ӻࣲྟބ۷ౢԣ֥эt+߄൝.༯૫ษંᄝࡌഡڄགሧӁڛՖٳྙ҃ᄎ JW(&tWH-11t∀( t)+J2W)W(&tWt∀2)( 2H-1t)}֥Ќބཨٮଆᇏ,ڄགሧӁቋႪሧб২൳(5)ތථหᆷඔaൈࡗࡗۯaڄགሧӁ൬ၭੱaѯੱູࠣࡥ߄іղ,০ੱ֩ଆҕඔэ߄႕ཙ֥ᇗေྟᇉ. )-∃(s-t)(&t,Ct,It;Wt,t)=e[U(Ct)+∃B(It+ྟᇉ1 ڄགሧӁ֥ቋႪሧб২&*൞ތtWt)]+JW((&tWt(!-r)+rWt-Ct-Qt)+J(t+ථหᆷඔH֥ᄹݦඔ.JW(&tW-11ᆣૼ Ⴎࡌഡ!>rln( t)<0,۴ऌൔt∀( Ht)+JW)W(&tWt∀2)( 2H-1t)2ᄵࢲކჿඏ่ࡱॖᆩ,ቋႪЌބཨٮҦ(&* t)-2H+1t,(7),ॖ֤(&*t2(r-!)ln(=t)>0,(H(∗2( -1)∀C**t,It)Ⴎ༯૫ٚӱቆಒק)Ֆط&*t൞ތථหᆷඔH֥ᄹݦඔ.(&*t,C**t,It;Wt,t)=0ྟᇉ1ඪૼሧᆀሧႿڄགሧӁ֥б২ෛ)&(&*t,C**t,It;Wt,t)=0tሢڄགሧӁ֥҂ಒקྟ֥ࡨഒطᄹࡆ.ᆃࣼ൞ඪ(6))**C(&*t,Ct,It;Wt,t)=0ᆺေିܔࡨഒڄགሧӁ֥҂ಒקྟ,ሧᆀࡼࡆt)I(&***t,Ct,It;Wt,t)=0նؓڄགሧӁ֥ሧ.tྟᇉ2 ᄝིႨݦඔડቀ֩ྟш࠽ིႨ౦2ྙ༯,ቋႪሧቆކб২&*t൞ӈඔაሧᆀ ቋႪЌބཨٮҦࠣྟᇉ֥ҍڶႚႵਈWtܱ.၂Ϯটඪ,ࢳ(6)൞бࢠ,ିϜ॥ᇅᆣૼ ۴ऌൔ(7)ᇏ&*t֥іղൔ,ࢲં൞
ֻ1௹ᅦݓ֩:ٳྙ҃ᄎ༯ቋႪЌބཨٮҦ∀81∀ཁಖ.֥ԚҍڶW0=10ຣჭ,ཌྷؓڄགညذҕඔ∗=ྟᇉ2ඪૼᄝ֩ྟш࠽ིႨݦඔ֥౦ྙ0 4,ڄགሧӁ֥ყ௹൬ၭੱ!=,ѯੱѓሙ༯,ሧቆކ࿊ᄴაቋႪཨٮथҦܱ,Վൈ֥ҵ∀=.ູਔุགྷٳྙ҃ᄎބದ൰Ќགሧቆކ࿊ᄴ൞၂۱֥۬࣡໙ีაໃটሧࠏؓሧथҦ֥႕ཙ,༯૫Ֆᇕ౦ঃࣉྛٳ༅ษ߶.ܱᆃൌ࠽ഈ൞ིႨݦඔ֥ӈཌྷؓڄགညذ.ં൮༵ܥקތථหᆷඔHᆴ,ฐษ tэ߄ൈིหᆘ֥၂ᇕุགྷ.Ⴈ௹ຬᆴ֥э߄౦ঃ.ಖ,ުಞ tܥק,࣮ථหྟᇉ3 ڄགሧӁ֥ቋႪሧб২&*ᆷඔHэ߄ൈቋႪሧҦބིႨ௹ຬᆴ֥э߄t൞!֥ᄹݦඔ,ٳљ൞rၛࠣ∀2֥ࡨݦඔ.౦ঃ.ቋު,ٳ༅ތථหᆷඔHބ tൈэ߄֥ᆣૼ౦ྙ༯ིႨ௹ຬᆴ֥э߄൝. ۴ऌࡌഡ่ࡱ0<∗<1ބ!>r,ಸၞ֤֞ൔ(7)ಒק֥&*tડቀෘ২1 ࡌקHurstᆷඔH=.ೂݔ t=(&**t(&t(&*,০ႨMatlabೈࡱ֤&t=,Ԛཨt>0,<0,(!(r(∀2<0,.ࢲંཁಖٮਈູຣჭ,ઙದ൰Ќག֥ЌٮႨູӮ৫0 0998ຣჭ,ሧႿܢௐ֥Ԛਈູຣჭ,.ྟᇉ3ඪૼሧႿڄགሧӁ֥бᇗෛሢఃყܓઙၿྛᅏಊ֥Ԛਈູຣჭ,ᆜ۱ളଁᇛ௹൬ၭੱ֥ᄹնطᄹࡆ,ෛሢڄག০ੱބڄག௹֥ཨٮིႨ֥ቋն௹ຬᆴູ3747ჭ.ೂݔ t=ሧӁ֥ѯੱᄹնطࡨഒ.ᆃᆞ൞ሧᆀטᆜڄ0 3,০ႨMatlabೈࡱॖၛ֤ԛ&t=0 413847,ԚགሧӁሧбᇗ֥၇ऌཨٮਈູຣჭ,ದ൰Ќག֥ЌٮႨູ.ྟᇉ0 0511ຣჭ,ሧႿܢௐ֥Ԛਈູຣჭ,ܓ4 ೂݔތථหᆷඔH1<,ପહڄག2ઙၿྛᅏಊ֥Ԛਈູຣჭ,ᆜ۱ളଁᇛ௹֥ሧӁ֥ቋႪሧб২&*t൞ൈࡗࡗۯ t֥ᄹݦཨٮིႨ֥ቋն௹ຬᆴູ3735ჭ.ط๙ݖ࠹ෘॖඔᆩቋႪԚሧڄགሧӁ֥б২ၛࠣളଁᇛ௹ི֥.ೂݔތථหᆷඔH#1,ପહڄགሧӁ֥ቋ2Ⴈቋն௹ຬᆴෛሢ t֥ᄹնطࡨഒ.1۳ԛਔႪሧб২&*t൞ൈࡗࡗۯ t֥ࡨݦඔ.H=0 58 t∃[0 01,]ൈ,ളଁᇛ௹ི֥Ⴈቋᆣૼ ۴ऌൔ(7),Ⴕն௹ຬᆴ֥э߄൝.(&*t!-r-2H=)( t(1-∗)∀2(1-2H)( tႮࡌഡ่ࡱ0<∗<1ބ!>r,ೂݔH<1,2ପહ(&*t0.( >tೂݔH#1,ପહ(&*t.ࢲંཁಖӮ( ∗0৫.2tྟᇉ4ඪૼೂݔތථหᆷඔH1<,ପહڄ2གሧӁ֥ቋႪሧб২ෛሢൈࡗࡗۯ֥ᄹնطᄹࡆ;ೂݔތථหᆷඔH#1,ପહڄགሧӁ֥ቋ2Ⴊሧб২ෛሢൈࡗࡗۯ֥ᄹնطࡨഒ.1 ௹ຬིႋᆴෛൈࡗࡗۯэ߄൝3 ඔᆴෘ২ෘ২2 ࡌק t=.і1۳ԛਔHurstᆷࡌഡၿྛթॻ൬ၭੱr=,ѻٳ҃ҕඔHՖэ߄֞ൈቋႪԚሧҦၛඔ∃=,ೂݔሧᆀՖགྷᄝ(t=0)ष,ఃࠣളଁᇛ௹ཨٮིႨ֥ቋն௹ຬᆴ.
∀82∀ܵ ॓ ࿐ ࿐ Б2010୍1ᄅі1 ቋႪԚሧҦaቋնིႨބތථหᆷඔ֥ܱ༢(ֆ໊:ჭ)Table1Relationshipofoptimalinitialinvestmentstrategy,maximalutilityandHurstexponent(unit:RMB)HCI&J(W,t)HCI&J(W,t) і1іૼ,֒ၿྛ০ੱaڄགሧӁ֥ყ௹൬ၭੱބѯੱ၂קൈ,ԚቋႪཨٮਈaЌਈa4 ࢲ ંሧႿڄགሧӁ֥бᇗෛሢHurstᆷඔ֥־ᄹطᄹն.ᆃ൞ႮႿHurstᆷඔᄀն,ڄགሧӁ֥ᄮലᄀഒ,ሧᆀؓڄགሧӁ֥ሼ൝ᄀႵϜ,ՖطॖູਔିܔޓݺֹࢳሧЧ൧ӆࡕڂިແaӉၛ۴ऌሱ࠭གྷႵҍڶࡆնؓڄགሧӁ֥,ೆၛ௹ཌྷܱྟ֩གྷའ,Ч໓๙ݖࡌഡڄགሧӁ֥ࡎ۬௹ࠆ֤ࢠն֥ཨٮބЌིႨ௹ຬᆴ.ൌ࠽ഈэ߄ݖӱڛՖٳྙ҃ᄎ,ܒᄯਔࠎႿٳྙ҃,০Ⴈܿ߃ჰ,ॖၛϜၩൈख़֒ቔԚൈᄎ֥ቋႪཨٮބЌଆ.ᄝ۳קཨٮބ၌ख़,Ֆطԛၩൈख़֥ቋႪሧҦࠣཌྷႋ֥ᄼࡎི֥ႨݦඔनູCRRAݦඔ֥౦ঃ༯,ቋնཨٮЌིႨ௹ຬᆴ.ԛਔၩൈख़֥ቋႪЌބཨٮҦ.๙ݖંෘ২3 ࡌഡ tބHurstҕඔ֥э߄ٓຶູٳ༅ބඔᆴෘ২ඪૼਔތථหᆷඔaൈࡗࡗۯؓ t∃[,]H∃[,].ҐႨ܄ቋႪሧaЌགaཨٮҦބ௹ຬིႨႵሢཁᇷ֥ൔ(7)֥ቋႪࢳ,ࢲކ௹ຬིႨᆴݦඔ֥קၬ,֤ތථหᆷඔބൈࡗࡗۯൈэ߄ൈ,௹ຬིႨ႕ཙ.ّ႘ਔሧᆀᄝڄགሧӁࡎ۬эྐ༏ԉᆴ֥э߄൝ೂ༯෮ൕٳ֥౦ঃ༯,۷჻ၩሧڄགሧӁ,ၛ௹ࠆ֤ቋն.௹ຬིႨ֥གྷൌ.ڸ:ק1֥ᆣૼႮ໓ང[21]ଁี֥ࢲ,ંཨԢൔ(3)ؓൈࡗ֥၇ঠྟު္ॖၛіൕູ∋∃tV(Wt)=J(Wt,t)=maxE{[U(C{&,C,I}0%-et)+ttt∃B(It+Wt)]dt}ᄵႵ∃V(Wt)=max{U(Ct)+∃B(It+Wt)+V((Wt)(&tWt(!-r)+rWt-Ct-∃It)+12H-12 ௹ຬིႋᆴෛൈࡗࡗۯބތථหᆷඔэ߄൝V)(Wt)(&2tWt∀2)( t)}ႮቋႪྟ่ࡱॖ֤
ֻ1௹ᅦݓ֩:ٳྙ҃ᄎ༯ቋႪЌބཨٮҦ∀83∀C∗-1t-V((Wt)=0ൔ(9)ູؽࢨັٳٚӱ,Ⴈրק༢ඔمॖ∗-1ࢳ.ࡌഡٚӱ֥ࢳູV(Wt)=AW∗t,ఃᇏAրק. (It+Wt)-V((Wt)=0ᄵႵV(=A∗W∗-1t,V)=A∗(∗-1)W∗-2t.ࡼV(ބV)V((Wt)Wt(!-r)+V)(Wt)&t(Wt∀2)( 2H-1t)=0սೆൔ(9)֤(8)ՖطႵ11∃∗2(!∗-1-r)-∗(r+∃)+A=12(∗22H-1C**-1-1)∀( t)t=(V((Wt))∗,It=(V((Wt))∗-1-W((Wt)W∗∗-1[(1+∃)(1-∗)]t(r-!)t,&*Vt=22H-1V)(Wt)(Wt∀ࡼA֥ᆴսೆV(Wt)=AW∗tᇏ,ॖ֤)( t)ࡼC*ta*Itބ&*t֥ᆴս,ೆᄵႵ1V(Wt)=∗+∗2∗-1 ∃V(Wt)=(1+∃)(V((Wt))∗-11-∗(∗)+∃-r)-∗(r+∃∗(!)+2(∗-1)∀21( 2H-t)2))(!2-r)[(1+∃1W∗ t)(1-∗∗-)](r+∃)W(tV(W(V((Wtt)-2H-12V)(Wt)∀2( t)۴ऌഈൔ,֤V((Wt)ބV)(Wt),ѩսೆൔ(9)(8),Ֆطॖ֤ཌྷႋ֥ቋႪࢳൔ(7).ҕॉ໓ང:[1][J].JournalofEconomicTheory,1971,3(4):373∀413.[2]DuffieD,FlemingW,SonerM,[J].JournalofEconomicDynam icsandContro,l1997,21(4∀5):753∀782.[3][J].JournalofEconomicTheory,1997,77,402∀431.[4]ShinYH,LimBH,[J].AppliedMathematicsandComputation,2007,188(2):1801∀1811.[5] consumptionwithproportionaltransactioncosts[J].JournalofEconomicDynamicsandContro,l2007,31(4):1132∀1159.[6]ਾݚ,Ԋڃ.ࠎႿቋҵ౦ঃ֥ቋႪཨٮބሧҦ[J].ܵ॓࿐࿐Б,2001,4(6):48∀,[J].JournalofManagementSciencesinChina,2001,4(6):48∀54.(inChinese)[7]ဗೋӮ,ਾএ߶.ෛࠏ๋ᄁږ؇֥ቋႪཨٮაᆣಊ࿊ᄴҦ໙ี[J].ܵ॓࿐࿐Б,2005,8(6):83∀, range[J].Jour nalofManagementSciencesinChina,2005,8(6):83∀87.(inChinese)[8]ᤂЏڃ,ޱݱߩ,ሌۑ.චᇗෛࠏ҂ಒק่ࡱ༯֥၂Ցྟಸਈঔᅚሧ[J].ܵ॓࿐࿐Б,2007,10(3):37∀,HuHanhu,[J].JournalofManagementSciencesinChina,2007,10(3):37∀43.(inChinese)[9]ߛ֣,ဗཫܻ.ᇏݓᆣಊ൧ӆܢᆷ൬ၭٳ֥҃ൌᆣٳ༅[J].ܵ॓࿐࿐Б,2008,11(1):68∀,,ssecuritiesmarket[J].JournalofManagementSciencesinChina,2008,11(1):68∀77.(inChinese)[10]ᅦ ົ,ᅦཬม,ྦྷ ྦྷ.ഈݚܢௐ൧ӆѯ҂ؓӫྟ࣮∀∀∀GJRაVS GARCHଆ֥бࢠ[J].ඔ࠹აܵ,2005,24(6):96∀,iZhangXiaotao,∀ComparisonofGJR andVS GARCH[J].ApplicationofStatisticsandManagement,2005,24(6):96∀102.(inChinese)
∀84∀ܵ ॓ ࿐ ࿐ Б2010୍1ᄅ[11]ဗޡਟ,ӧ ൬,ჯ࠽फ.؟ѓ؇่ࡱ༯֥ૢཌྷܱྟ:ᇏݓܢ൧൬ၭੱ֥ࣜဒٳ༅[J].ܵ࿐Б,2007,4(5):618∀,ChenShou,,spower correlationpropertiesinChinaunderconditionofmultiscale[J].ChineseJournalofManagement,2007,4(5):618∀621.(inChinese)[12]ޱဇ઼,ᅦݓ,ӧࡹᇑ.ᇏݓܢ൧Ӊ࠺ၫ֥ྩᆞR/Sٳ༅[J].ඔ࠹აܵ,2006,25(1):96∀,iZhangWeiguo, termmemoryinstockmarketprices[J].Econometrica,l1991,59(5):1279∀1313.[15][J].FinancialAnalystJourna,l1989,7:434∀453.[16][M].SanFrancisco:.,1982.[17]HuY, ksendalB, ScholesmarketdrivenbyfractionalBrownianmotion[J].,2003,6(4):519∀536.[18]ңޡ᪶.ᆣಊ൧ӆگᄖྛູٳྙѓ؇ٳ༅აࠏ߶थҦ࣮[J].ࣁವ࣮,2005,295:138∀[J].JournalofFinance,2005,295:138∀145.(inChinese)[19]້ დ,ߛ֨ൢ.ࣁವ൧ӆ؟ѓ؇ٳྙགྷའࠣაڄག֥ܱܵ༢[J].ܵ॓࿐࿐Б,2003,6(1):87∀,[J].JournalofManagementSciencesinChina,2003,6(1):87∀91.(inChinese)[20]້ დ,ߛ֨ൢ.ࠎႿ؟ѓ؇ٳྙં֥ࣁವڄགҩ؇ᆷѓ࣮[J].ܵ॓࿐࿐Б,2005,8(4):50∀,[J].JournalofManagementSciencesinChina,2005,8(4):50∀59.(inChinese)[21]נԮૼ,ሻࢮᇏ.ॉ੮ದ൰Ќག֥ቋႪࣁವथҦ[J].༢۽ӱ,2003,21(5):84∀,[J].SystemsEngineering,2003,21(5):84∀87.(inChinese)Optimalstrategiesonportfolioandconsumptionwithinsuranceunderfrac tionalBrownianmotionZHANGWei guo,XIAOWei lin,ZHANGXi liSchoolofBusinessAdministration,SouthChinaUniversityofTechnology,Guangzhou510640,ChinaAbstractBasedonMerton,soptimalconsumptionandinvestmentmode,lthispaperresearchesaclassofopti malportfolioandconsumptionproblemthatcombineslifeinsurancewhentheriskyassetfollowsgeometricfrac ,sexpectedlifetimeutility,,theclosed formsolutionsfortheoptimalportfolio, more,someimportantpropertiesfortheinfluenceofparametricchangesontheoptimalportfolioarealsoob ,numericalexamplesarepresentedtodiscussthechangeovertheoptimalportfolioandinsur :fractionalBrownianmotion;Hurstexponen;tutilityfunction;portfolioandconsumption;lifeinsurance
