ඔ࿐ᄖᆽ(2011).(PRC)3๋ঔݖӱ༯֥Ќགഅӊᅏੱଆ࣮༱֨ڂ,ٮູၿ,ਾޡࡹ(νߪ۽ӱն࿐ඔ࿐ჽ,νߪ241000)ᅋေ:Ч໓࣮ਔᄝႵࣁವӮЧ֥౦ঃ༯,ջႵ๋ঔݖӱ֥Ќགഅӊᅏੱ(SR)ଆ֥໙ีē০ႨGirsanovקࣉྛҩ؇эߐ֥ٚمၛ๋ࠣঔݖӱ༯֥ुᅨ௹ಃקࡎ܄ൔ,ࠆ֤ਔЌགഅᇔ௹൬ၭ֥གྷᆴ֥ࢲݔēܼਔ҂ջ๋ঔݖӱ֥Ќགഅӊᅏੱଆ֥ࢲݔ.ܱՍ:๋ঔݖӱĠӊᅏੱĠGirsanovקĠࣁವӮЧMR(2010)ᇶีٳোݼ:60H10;91B30ᇏٳোݼ:;O232໓ངѓ്:A໓ᅣщݼ:0255-7797(2011)03-0554-051ႄ໓ང[1–4]ษંਔЌགഅࣜࣁವ֥ڄགܵࠏē໓ང[5]࣮ਔջႵଉ҈ӮЧ֥ӊᅏੱֆ௹ଆ,ᄝՎଆ,္ࡹ৫ਔႮ҃ᄎ౺֥ӊᅏੱଆēᇙ෮ᇛᆩ,ࣁವ൧ӆᇏॖି߶ԛགྷࡗ؎o๋֥pēၹՎ,Ч໓ଆ൫ᄝ໓ང[5]֥ࠎԤഈ,ࡹ৫Ⴎ҃ᄎބѻݖӱ౺֥ӊᅏੱଆēҐႨGirsanovקࣉྛҩ؇эߐ,০Ⴈ๋ঔݖӱ༯֥ुᅨ௹ಃקࡎ܄ൔ,࠹ෘԛਔЌགഅᇔ௹൬ၭ֥གྷᆴ,ھଆܼਔགྷႵࢲંē2ѻ๋༯֥ӊᅏੱ(SR)ଆႮႿ֒܄ඳᄎቔᄝଉ҈ӮЧ֥ߌ༯܄ඳࣜႏ֥ӮЧ൞υ֥,ၹՎЌགഅڄགܵҦ၇ঠႿଉ҈ӮЧ֥ྟᇉē၂Ϯটඪ,ଉ҈ӮЧᇶေႵᇕྙൔğඥ൬֥҂ؓӫaࣁವӮЧބսӮЧēᄝЌགਵთ,Ќག܄ඳສສࣜਆᇕحຓ֥ڄགӮЧğචᇗඥ൬ӮЧ[6]aࡓܵӮЧēᄝՎ,ૌࣇࣇॉ੮ଉ҈ӮЧྙൔູࣁವӮЧ,ࡹ৫ཌྷႋ֥ڵᅏބӊᅏੱଆ,ٳ༅Ќགഅ෮ҐႨ֥ڄགܵҦ.ࡌഡ෮Ⴕ֥ᅏༀᄝ௹ଌᆦڱ.ഡ௹ଌൈख़T֥Ќག܄ඳሧӁູAT,ڵᅏູLT,ૌӫΛT=AT/LTູ܄ඳ֥ӊᅏੱ.JູЌࡓअ൙༵ܿק֥ࣁವᅰθ.ೂݔ௹ଌൈख़T܄ඳ֥ӊᅏੱΛT>J,ᄵӫЌག܄ඳԩႿࣁವࡲूሑēّᆭ,֒ΛT≤Jൈ,ᄵЌག܄ඳԩႿࣁವሑ;ᄝھሑ༯,Ќག܄ඳႵ၂۱حຓ֥ാ,ھാაሧӁATӮб২,б২༢ඔູ(1−ω),ω∈[0,1).ࣉ၂ֹ҄,ॉ੮ࣁವӮЧ֥࣪ሧӁ֥ᇔᆴωATӑݖ௹ଌൈख़T܄ඳ֥ᅏༀLT,ࠧωAT>LTࠇᆀΛT>1/ω,ᄵӫЌག܄ඳԩႿࣁವႵӊᅏି৯ሑ;ّᆭ,ωJ<1ࠇᆀJ<1/ω,ᄵӫЌག܄ඳԩႿӊᅏି৯ሑ.ሸഈ෮ඍ,Ќག܄ඳࣜ۱҂֥ࣜ࠶ሑ,ࠧࣁವࡲूaࣁವႵӊᅏ৯aӊᅏି৯.∗൬۠ರ௹:2009-11-04ࢤ൬ರ௹:2010-04-14ࠎࣁཛଢğݓࡅሱಖ॓࿐ࠎࣁ(10826098);ݓࡅo973pཛଢ(2007CB814901);νߪസሱಖ॓࿐ࠎࣁ(090416225);νߪസۚሱಖ॓࿐ࠎࣁ(KJ2010A037;KJ2010B026)ēቔᆀࡥࢺ:༱֨ڂ(1979–),ଳ,νߪ,ණൖ,ࢃഽ,ᇶေ࣮ٚཟ:ࣁವඔ࿐აࣁವ۽ӱ.
༱֨ڂ֩:๋ঔݖӱ༯֥Ќགഅӊᅏੱଆ࣮555Λtᄝڄགᇏྟۀੱҩ؇Q༯֥ݖӱູdΛt=µΛΛtdt+σΛΛtdWΛt+ΛtdQt,t∈[0,T],ఃᇏµΛູ၍༢ඔ,σΛູঔ༢ඔ,(WΛt,0≤t≤T)ᄝջੀۀੱॢࡗ(Ω,F,(Ft),Q)༯൞ѓሙ҃ᄎ.ഡN1(t),···,NM(t)൞఼؇ٳљູλ1,···,λM֥ཌྷ৫֥ѻݖӱ,Ⴟ൞MN(t)=∑Nm(t)൞఼؇ູMλ=∑λm֥ѻݖӱ.ഡY1,Y2,···൞၂ਙ৫ٳ֥҃m=1m=1N(ෛࠏэਈQ{Yi=ym}=p(ym)=λmλ,β=EYi.קၬگކѻݖӱູQt=∑t)Yiēi=1ሸഈ෮ඍ,ᄵЌགഅᇔ௹൬ၭ֥གྷᆴູVSR0=EQ{e−rT[(ωAT−LT)+1{ΛT≤J}+(AT−LT)1{ΛT>J}]}=EQ{e−rTLT[ω(ΛT−1ω)++(1−ω)ΛT1{ΛT>J}]},ఃᇏΛ0=A0L>1,rູڄག০ੱ.03SRଆ֥ᇶေࢲݔᄝՎ,קၬྍ֥ڄགᇏྟۀੱҩ؇Q˜.ഡλ˜1,···,λ˜M൞ၘᆩᆞӈඔ.Z0(t)=exp{−θW(t)−1θ2t},θ=−σL,2ZmmNt)(t)=e(λm−λ˜m)tλ˜()m(,m=1,···,M,λmZ(t)=Z0(t)ΠMm=1Zm(t),∫Q˜=Z(T)dQ,∀A∈۴ऌ໓ང[7]֥ק,ંބંॖᆩ,ᄝۀੱҩ؇Q˜༯Ⴕ(i)ݖӱW˜Λt=WΛt+θt൞҃ᄎ;(ii)ૄ۱Nm൞఼؇ູλ˜֥ѻݖӱ;(iii)W˜ΛtބN1,···,Nmཌྷ৫.קၬMλ˜=∑λ˜m,p˜(ym)=λ˜m,Ⴟ൞ᄝQ˜༯,N(t)=∑MNλ˜m(t)൞఼؇ູλ˜֥ѻݖӱ,m=1m=1Q˜{Yi=ym}=p˜(ym),β˜=EQ˜Yi.ၹՎ,Λtᄝྍ֥ڄགᇏྟۀੱҩ؇Q˜༯֥ݖӱູdΛt=µΛΛtdt+σΛΛtdWΛt+ΛtdQt=µΛΛtdt+σΛΛt(dW˜Λt−θdt)+ΛtdQt=µ˜ΛΛtdt+σ˜ΛΛtdW˜Λt+ΛtdQt,ఃᇏµ˜Λ=µΛ−σΛθ,σ˜Λ=σΛ.ഡᄝڄགᇏྟۀੱҩ؇Q༯,ႵڵᅏLTડቀLT=L0eµLTZ(T).طڄགᇏྟۀੱҩ؇Q˜ડቀdQ˜dQFT=Z(T),EQ˜[Z(T)|Ft]=Z(t),∀t∈[0,T].
556ඔ࿐ᄖᆽؓΛT֥ၩݦඔ০Ⴈႄ[8],ႵEQ˜[exp[−r(T−t)]f(ΛT)|Ft]=EQ[exp[−r(T−Z(T)t)]f(ΛZT)|Ft].(t)ၹՎEQ[e−r(T−t)LTf(ΛT)|Ft]=EQ˜[e−r(T−t)f(ΛT)|Ft]Z(t)L0eµLT=LteµL(T−t)EQ˜[e−r(T−t)f(ΛT)|Ft].ႮႿW˜ΛTაQT֥ٳ҃ݦඔၘᆩ,ॖၛԛΛT֥ٳ҃ݦඔ,ᆃ҂ٞഡPJ=Q˜(ΛT>J)ູၘᆩ.ູਔᆣૼᇶေࢲં,༵ႄೆ၂۱ႄēႄ[7]ഡ(Wt,0≤t≤T)ᄝջੀۀੱॢࡗ(Ω,F,(Ft),P˜)༯൞ѓሙ҃ᄎ,ఃᇏP˜൞ڄགᇏྟۀੱҩ؇.ڄགሧӁࡎ۬ݖӱStڛՖෛࠏັٳٚӱdSt=µStdt+σStdWt+StdQt,ᄵ๋ঔݖӱ༯֥ൔुᅨ௹ಃקࡎ܄ൔູ∑∞−t)j∏jc(t,x)=e−λ˜(T−t)λ˜j(TE0jP˜κ(T−t,xe(µ−r)(T−t)(Yi+1)),()!j=i=1ൔ()ડቀٚӱ−rc(t,x)+ct(t,x)+µxcx(t,x)+1σ2x2cxx(t,x)∑2M+λ˜[p˜(ym)c(t,(ym+1)x)−c(t,x))]=0,0≤t≤T,x≥0,m=1ᇔ؊่ࡱູc(T,x)=(x−K)+,x≥0.ൔ()ᇏc(t,x)ູtൈख़ुᅨ௹ಃ֥ࡎᆴ,Kູ௹ಃᆳྛࡎ۬,STູTൈख़ܢௐ֥ࡎ۬,λູѻݖӱN(t)఼֥؇,rູڄག০ੱ,κ(τ,x)=xN(d+(τ,x))−Ke−rτN(d−(τ,x)),xd±(τ,x)=√1[logστK+(r±1σ2)τ],2∫yN(y)=√1e−12z2dz2pi∞൞ᆞٳ҃ݦඔ,τ=T−t.༯૫۳ԛЧ໓֥ᇶေࢲݔ.קᄝഈඍ่ࡱ༯,Ќགഅᇔ௹൬ၭ֥གྷᆴ∑∞jVSR0=L0erTe−λ˜Tλ˜TjEΛ−∏r)TQ˜{ωκ1(T,Λ0e(µ˜(Yi+1))j!j=0i=1j+(1−ω)κ−∏r)T2(T,Λ0e(µ˜Λ(Yi+1))}+L0(1−ω)JPJ,()i=1
༱֨ڂ֩:๋ঔݖӱ༯֥Ќགഅӊᅏੱଆ࣮557ఃᇏκ1(τ,x)=xN(d+(τ,x))−1e−rτN(d−(τ,x)),ωκ2(τ,x)=xN(l+(τ,x))−Je−rτN(l−(τ,x)),d±(τ,x)=1√x[log+(µ˜σ˜Λ±1σ˜22Λ)τ],Λτ1/ωxl±(τ,x)=1√[log+(µ˜σ˜ΛτJΛ±1σ˜22Λ)τ].ᆣႮႿЌགഅᇔ௹൬ၭ֥གྷᆴູVSR0=EQ{e−rTLT[ω+(ΛT−1)+(1−ω)ΛωT1ΛT>J]},ഈൔࣜݖGirsanovҩ؇эߐ֤VSR0=L0eµLT{ωEQ˜[e−rT(Λ+T−1)]+(1−ω)EωQ˜[e−rTΛT1ΛT>J]}.``C1=E+Q˜[e−rT(ΛT−1)],ωC2=EQ˜[e−rTΛT1ΛT>J]=EQ˜[e−rT[(ΛT−J+)+J1ΛT>J]]=EQ˜[e−rT(ΛT−J+)]+e−rTJPJ.۴ऌႄ,ॖ֤∑∞C1=e−∏jλ˜Tλ˜TjETµ˜Λ−r)T,Λ0e(jQ˜{κ1((Yi+1))},!j=0i=1∑∞jCλ˜Tj2=e−λ˜TEΛ−∏r)T,Λ0e(µ˜jQ˜{κ−rT2(T(Yi+1))}+eJPJ.!j=0i=1ࣉ၂҄,োරႿKrvavych[5]֥ษંႵµL=r,Ⴟ൞VSR0=L0eµLTωEQ˜[{e−rT[(ΛT−1+)]+(1−ω)EQ˜[e−rTΛωT1ΛT>J]}∑∞=Lλ˜Tλ˜Tj∏jT0erTe−EjQ˜{ωκ1(T,Λ0e(µ˜Λ−r)(Yi+1))!j=0i=1+(1−ω)κ2(T,Λ0e(µ˜Λ−∏jr)T(Yi+1))}+L0(1−ω)=1```Ֆطק֤ᆣēࢲંЧ໓ᄝ໓ང[5]෮ҐႨ֥Ⴎ҃ᄎ౺֥ӊᅏੱଆࠎԤഈ,ࡹ৫Ⴎ҃ᄎބѻݖӱ܋౺֥ӊᅏੱଆ.ҐႨGirsanovקࣉྛҩ؇эߐ,০Ⴈ๋ঔݖӱ༯֥ुᅨ௹ಃקࡎ܄ൔ,࠹ෘԛਔЌགഅᇔ௹൬ၭ֥གྷᆴ,ھଆܼਔགྷႵࢲં.
558ඔ࿐ᄖᆽҕॉ໓ང[1][J].QuaterlyJournalofEconomics,1991,106:709–737.[2]ઔ༭,༱֨ڂ,ٮູၿ.э০ੱ༯֥Ќགഅӊᅏੱଆ࣮[J].ࣜ࠶ඔ࿐,2007,24(4):358–362.[3]OplerT,[J].JournalofFinance,1994,49:1015–1040.[4]BriysE,:anextension[J].JournalofFinancialandQuantitativeAnalysis,1997,32:239–248.[5][D].Australia:UniversityofNewSouthWales,2005.[6]KrvavychY,[J].Insur-anceğMathematicsandEconomics,2006,38:495–517.[7][M].NewYork:Springer,2004.[8]ޅല,ລࡁۏ,ࡆν.϶᷉აෛࠏٳ༅[M].Кࣘ:॓࿐ԛϱഠ,’SSOLVENCYRATIOMODELUNDERJUMPDIFFUSIONPROCESSXIADeng-feng,FEIWei-yin,LIUHong-jian(SchoolofMathematicsandPhysics,AnhuiPolytechnicUniversity,Wuhu241000,China)Abstract:Thisarticleconstructstheinsurer’ssolvencyratiomodelunderjumpdiffusionprocessinthepresenceoffinancialdistresscostēByGirsanov’stheoremandtheoptionpricingformulaunderjumpdiffusionprocess,themaximizationofshareholders’’:jumpdiffusionprocessĠsolvencyratioĠGirsanov’stheoremĠfinancialdistresscost2010MRSubjectClassification:60H10;91B30
