均值-方差准则下再保险双方联合收益的最优投资再保险策略 Optimal Investment Reinsurance Strategy for the Joint Benefits of the Insurer and the Reinsurer under the Mean-Variance

doi:10.12677/PM.2021.1111208

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PureMathematicsnØêÆ,2021,11(11),1850-1870

PublishedOnlineNovember2021inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2021.1111208

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Ï"¤n§2x§þŠ-•OK§éÜÂÃ§HJB•§

OptimalInvestmentReinsuranceStrategy

fortheJointBeneﬁtsoftheInsurerand

theReinsurerundertheMean-Variance

TingtingSun,HuihuiWang,HuishengShu

∗

CollegeofScienceandTechnology,DonghuaUniversity,Shanghai

Received:Oct.15

,2021;accepted:Nov.15

,2021;published:Nov.22

,2021

∗ÏÕŠö"

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2021,11(11):1850-1870.DOI:10.12677/pm.2021.1111208

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Abstract

Underthebackgroundoftheriskmodel relatedtotwotypes ofinsurancebusiness, the

optimalinvestmentreinsuranceofthejointbeneﬁtsoftheinsurerandthereinsurer

isstudied.Assumingthattheinsurercanbuyproportionalreinsurancefromthe

reinsurerandinvestinaﬁnancialmarketconsistingofrisk-freeandriskassets,the

reinsurer can use the expected premium principle to charge premiums and reduce risk

byinvesting inrisk-freeassets.Underthemean-variancecriterion,theexpressionsof

the optimal investment strategy and the optimal reinsurance strategy and the optimal

valuefunctionofthecombinedreturnsareobtained by solvingtheextended Hamilton-

Jacobi-Bellmanequationsystem,andthevalidity oftheresultisveriﬁedbyexample.

Keywords

ExpectedPremiumPrinciple,Reinsurance,Mean-VarianceCriterion,JointBeniﬁts,

HJBEquation

2021byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

http://creativecommons.org/licenses/by/4.0/

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DOI:10.12677/pm.2021.11112081852nØêÆ

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g(t,x)

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g(t,x)}= 0,(3.4)

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∗

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∗

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g(t,x)

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DOI:10.12677/pm.2021.11112081855nØêÆ

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t,x

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+(1+η

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−(λ

+λ

+λ)[U(t,x)+

g(t,x)

]

+λ

E[U(t,x−θ

X)−

g(t,x−θ

X)(g(t,x−θ

X)−2g(t,x))]

+λ

E[U(t,x−θ

Y)−

g(t,x−θ

Y)(g(t,x−θ

Y)−2g(t,x))]

+λE[U(t,x−θ

X−θ

Y)−

g(t,x−θ

X−θ

Y)(g(t,x−θ

X−θ

Y)−2g(t,x))] = 0,(4.1)

+{rx+cα+α(β−r)π+(1−2α)[(1+η

)(1−q

+(1+η

)(1−q

]}g

+λ

E[g(t,x−θ

X)−g(t,x)]

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E[g(t,x−θ

Y)−g(t,x)]

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X−θ

Y)−g(t,x)] = 0,(4.2)

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b(t),g

(T,x) = a(t),g

(T,x) = 0,(4.4)

DOI:10.12677/pm.2021.11112081856nØêÆ

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A(t) = dA(t)/dt§

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A(t)[rx+cα+α(β−r)π+(1−2α)((1+η

)(1−q

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)(1−q

)]+

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A(t)x−A(t)θ

−A(t)θ

−

a(t)

(θ

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+2θ

λµ

+α

) = 0.(4.5)

Ù¥b

= (λ

+λ)E(X

)§b

= (λ

+λ)E(Y

)"-

f(q

) = −

a(t)

−

a(t)

−A(t)θ

−γa(t)

λµ

+A(t)(1−2α)(1+η

)(1−q

+A(t)(1−2α)(1+η

)(1−q

.(4.6)











∂f(q

)

∂q

= (2α−1)(−γa(t)

+A(t)η

−γa(t)

λµ

∂f(q

)

∂q

= (2α−1)(−γa(t)

+A(t)η

−γa(t)

λµ

∂

f(q

)

∂q

= −γa(t)

(2α−1)

∂

f(q

)

∂q

= −γa(t)

(2α−1)

∂

f(q

)

∂q

= −γa(t)

(2α−1)

λµ

(4.7)

f(q

)HessianÝŒ±L«•µ







∂

f(q

)

∂q

∂

f(q

)

∂q

∂

f(q

)

∂q

∂

f(q

)

∂q













−γa(t)

(2α−1)

−γa(t)

(2α−1)

λµ

−γa(t)

(2α−1)

λµ

−γa(t)

(2α−1)







Šâ…Ü-–]Øª§·‚•b

= (λ

+λ)E(X

)(λ

+λ)E(Y

) >λ

§γ

a(t)

(2α−

−λ

) >0§¤±f(q

)Hessian Ý½§=f(q

)´'uq

§q

à¼ê"u











−γa(t)

+A(t)θ

−γa(t)

λµ

= 0,

−γa(t)

+A(t)θ

−γa(t)

λµ

= 0.

(4.8)











A(t)(−a

λµ

)

γa(t)

−λ

)

A(t)(−a

λµ

)

γa(t)

−λ

)

(4.9)

dθ

= 1−α−(1−2α)q

,i= 1,2§·‚kq

−(1−α)

2α−1

,i= 1,2"Šâ(4.5)§

eπ=

A(t)(β−r)

αγa(t)

,(4.10)

DOI:10.12677/pm.2021.11112081857nØêÆ

šxx

òq

,eπ“\(4.5)Ú(4.2)§k

(

A(t)+rA(t))x+

B(t)+A(t)(αc−(1+η

)αa

−(1+η

)αa

A(t)

2γa(t)

ξ= 0,(4.11)

(˙a(t)+ra(t))x+

b(t)+a(t)(αc−(1+η

)αa

−(1+η

)αa

A(t)

γa(t)

ξ= 0,(4.12)

Ù¥

ξ= a

−2a

λµ

(β−r)

,(4.13)

‡Ž(4.11)Ú(4.12)¤á§K

A(t)+rA(t) = 0,A(T) = 1,

B(t)+A(t)(αc−(1+η

)αa

−(1+η

)αa

A(t)

2γa(t)

ξ(t) = 0,

˙a(t)+ra(t) = 0,a(T) = 1,

b(t)+a(t)(αc−(1+η

)αa

−(1+η

)αa

A(t)

γa(t)

ξ(t) = 0.

)þã‡©•§§k

A(t) = e

r(T−t)

,(4.14)

B(t) = (αc−(1+η

)αa

−(1+η

)αa

)

r(T−t)

−1)+

2γ

ξ(T−t),(4.15)

a(t) = e

r(T−t)

,(4.16)

b(t) = (αc−(1+η

)αa

−(1+η

)αa

)

r(T−t)

−1)+

ξ(T−t).(4.17)

ò(4.14)Ú(4.16)“\(4.9)§Œ

2α−1

[

−a

λµ

γe

r(T−t)

−λ

)

−(1−α)],(4.18)

2α−1

[

−a

λµ

γe

r(T−t)

−λ

)

−(1−α)].(4.19)

=−a

λµ

=−a

λµ

§q{üyŒ•

λµ

µ21

λµ

"-t

(

)´¦q

=0(bq

=0,eq

=0)¤ážm:§t

(

)´¦q

=1(bq

1,eq

= 1)¤ážm:§t

´¦q

(T−t)=0¤ážm:§i=1,2§t

´¦eq

(T−t)=0

DOI:10.12677/pm.2021.11112081858nØêÆ

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¤ážm:i= 1,2§u´·‚Œ±Xeü‡½nµ

½n4.1e

<α<1§K•`2xüÑ(q

∗

)•

£1¤η

λµ

µ21

λµ

ž§=£m

>0,m

<0¤

∗

) =











(0,0),t≤

(bq

,0),

<t<

(1,0),t≥

(4.20)

Ù¥bq

2α−1

[

−γe

r(T−t)

(1−α)λµ

+η

γe

r(T−t)

−(1−α)]

£2¤

λµ

µ21

>η

λµ

ž§=£m

>0,m

>0¤

≥m

§k

∗

) =











(0,0),t≤

(0,bq

<t≤t

),t

<t<t

(1,eq

),t

≤t<

(1,1),t≥

(4.21)

Ù¥bq

2α−1

[

−γe

r(T−t)

(1−α)λµ

+η

γe

r(T−t)

−(1−α)]§eq

2α−1

[

−γe

r(T−t)

αλµ

+η

γe

r(T−t)

−(1−α)]¶

§k

∗

) =











(0,0),t≤

(0,bq

<t≤t

),t

<t<t

(eq

,1),t

≤t<

(1,1),t≥

(4.22)

Ù¥bq

2α−1

[

−γe

r(T−t)

(1−α)λµ

+η

γe

r(T−t)

−(1−α)]§eq

2α−1

[

−γe

r(T−t)

αλµ

+η

γe

r(T−t)

−(1−α)]¶

£3¤

λµ

µ21

λµ

>η

ž§=(m

<0,m

>0)

∗

) =











(0,0),t≤

(0,bq

<t<

(0,1),t≥

(4.23)

DOI:10.12677/pm.2021.11112081859nØêÆ

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½n4.2e0 <α<

§K•`2xüÑ(q

∗

)•

£1¤η

λµ

µ21

λµ

ž§=£m

>0,m

<0¤

∗

) =











(1,1),t≤

(eq

,1),

<t<

(0,1),t≥

(4.24)

Ù¥eq

2α−1

[

−γe

r(T−t)

αλµ

+η

γe

r(T−t)

−(1−α)]

£ii¤

λµ

µ21

>η

λµ

ž§=£m

>0,m

>0¤

≥m

§k

∗

) =











(1,1),t≤

(1,eq

<t≤t

),t

<t<t

(0,bq

),t

≤t<

(0,0),t≥

(4.25)

Ù¥eq

2α−1

[

−γe

r(T−t)

αλµ

+η

γe

r(T−t)

−(1−α)]§bq

2α−1

[

−γe

r(T−t)

(1−α)λµ

+η

γe

r(T−t)

−(1−α)]

§k

∗

) =











(1,1),t≤

(1,eq

<t≤t

),t

<t<t

(bq

,0),t

≤t<

(0,0),t≥

(4.26)

Ù¥eq

2α−1

[

−γe

r(T−t)

αλµ

+η

γe

r(T−t)

−(1−α)]§bq

2α−1

[

−γe

r(T−t)

(1−α)λµ

+η

γa(t)

−(1−α)]

£iii¤

λµ

µ21

λµ

>η

ž§=£m

<0,m

>0¤

∗

) =











(1,1),t≤

(1,eq

<t<

(1,1),t≥

(4.27)

DOI:10.12677/pm.2021.11112081860nØêÆ

šxx

54.1XJEÜPossionL§L

(t)ÚL

(t)©ÙƒÓ§=λ

= λ

§µ

= µ

§µ

= µ

§u´a

§b

= b

§K(4.9)¥2xúiéu11ax«SKÖη

Úéu12ax«SKÖη

ž§kθ

=θ

§?˜Úq

∗

¶2xúiéu11ax«SKÖη

Úéu12ax«

SKÖη

Øƒž§kη

= η

½n4.3é¯K(3.1)§·‚d(4.10)ªŒ••`Ý]üÑ•π

∗

(β−r)

αγe

r(T−t)

§•`2xüÑ

d½n4.1Ú½n4.2‰Ñ§…•`Š¼êV(t,x)•

£1¤•`2xüÑ(q

∗

) = (q

)ž§

V(t,x) = e

r(T−t)

x+(αc−(1+η

)αa

−(1+η

)αa

)

r(T−t)

−1)+

2γ

ξ(T−t)(4.28)

Ù¥ξ= a

−2a

λµ

(β−r)

£2¤•`2xüÑ(q

∗

) = (0,0)ž§

V(t,x) = ((1−2α)(1+η

+(1−2α)(1+η

−(1−α)a

)

r(T−t)

−1)

αc

r(T−t)

−1)+e

r(T−t)

x−

2r(T−t)

−1)(1−α)

+2λµ

)

2γ

(β−r)

(T−t)(4.29)

£3¤•`2xüÑ(q

∗

) = (0,bq

)ž§

V(t,x) = (αc−(1+η

)αa

+(1−2α)(1+η

−

(1−α)λµ

)

r(T−t)

−1)

−(1−α)a

r(T−t)

−1)+e

r(T−t)

x−

2r(T−t)

−1))(1−α)

−

]

2γ

[

(β−r)

](T−t)(4.30)

£4¤•`2xüÑ(q

∗

) = (bq

,0)ž§

V(t,x) = (αc−(1+η

)αa

+(1−2α)(1+η

−

(1−α)λµ

)

r(T−t)

−1)

−(1−α)a

r(T−t)

−1)+e

r(T−t)

x−

2r(T−t)

−1))(1−α)

−

]

2γ

[

(β−r)

](T−t)(4.31)

DOI:10.12677/pm.2021.11112081861nØêÆ

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£5¤•`2xüÑ(q

∗

) = (eq

,1)ž§

V(t,x) = e

r(T−t)

x+(αc−(1+η

)αa

−αa

−

αλµ

)

r(T−t)

−1)

−

2r(T−t)

−1))α

−

2γ

[

(β−r)

](T−t)(4.32)

£6¤•`2xüÑ(q

∗

) = (1,eq

)ž§

V(t,x) = e

r(T−t)

x+(αc−(1+η

)αa

−αa

−

αλµ

)

r(T−t)

−1)

−

2r(T−t)

−1))α

−

2γ

[

(β−r)

](T−t)(4.33)

£7¤•`2xüÑ(q

∗

) = (1,1)ž§

V(t,x) = e

r(T−t)

x+(αc−αa

−αa

)

r(T−t)

−1)+

2γ

(β−r)

(T−t)

−

2r(T−t)

−1)α

+2λµ

)(4.34)

5.êŠ©Û

ù˜Ü©¥§·‚òÏLêŠ~f`²ëêé•`üÑK•"bλ=1,λ

=2,λ

3,T=10,r=0.03,α=0.6,µ

=0.045,µ

=0.053,µ

=0.045,µ

=0.055,η

=0.35,η

0.35,γ= 0.5,σ= 0.25"ëêXkCz§,Š`²"

ã1`²•`2xüÑ(q

∗

)‘XžmtOŒOŒ§3vkÄœ¹e§xúi¬

éüax«2xÝ\O\"ã2¥n^-‚©O•σ=0.25§0.35§0.45ž•`Ý]üÑ-‚§

•`Ý]üÑπ

∗

‘XÅÄÇσOŒ~§σŒž§ºx]ÂÃÅÄÒ¬Œ§¤±§

ºxƒÓž§xúi3ºx]þÝ]7¬~§…•`2xüÑ†ÅÄÇσÃ'§

•`2xüÑ†ºx]ëêÃ'"

e¡·‚•©Ûëêé•`2xüÑq

∗

K•§éq

∗

©ÛŒ±aqÑ"ã3n^-‚

L«ºxXêγ=0.5§0.51§0.52ž•` 2xüÑq

∗

-‚§γŒž§• `2xü

Ñ§g3'~§=xúiºxž§•Žï2x§42xúi«ú •õº

x"ã4n^-‚L«ºxXêγ=0.5§0.51§0.52ž•`Ý]üÑ-‚§γŒ§=

ºx§Ý•pž§•`Ý]üÑπ

∗

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ã5L«•`2xüÑq

∗

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2xüÑq

∗

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õž§2xúiÒ¬áýÉxúiï•õ2x§…•`Ý]üÑ†ëêλÃ'•`Ý

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∗

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DOI:10.12677/pm.2021.11112081862nØêÆ

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Figure1.Theeﬀectoftonoptimalreinsurancestrategy

ã1.té•`2xüÑK•

Figure2.Theeﬀectofγonoptimalreinsurancestrategy

ã2.γé•`2xüÑK•

Figure3.Theeﬀectofγonoptimalinvestmentstrategy

ã3.γé•`Ý]üÑK•.

Figure4.Theeﬀectofσonoptimalinvestmentstrategy

ã4.σé•`Ý]üÑK•

Figure5.Theeﬀectofλonoptimalreinsurancestrategy

ã5.λé•`2xüÑK•

Figure6.Theeﬀectofαonoptimalinvestmentstrategy

ã6.αé•`Ý]üÑK•

DOI:10.12677/pm.2021.11112081863nØêÆ

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6.o(

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ë•©z

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Economics,27,215-228.https://doi.org/10.1016/S0167-6687(00)00049-4

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[19]Bai,Y.,Zhou,Z.,Gao,R.,etal.(2020)NashEquilibriumInvestment-ReinsuranceStrate-

giesforanInsurerandaReinsurerwithIntertemporalRestrictionsandCommonInterests.

Mathematics,8,139.https://doi.org/10.3390/math8010139

[20]Liang,Z.andYuen,K.C.(2016)OptimalDynamicReinsurancewithDependentRisks:Vari-

ancePremiumPrinciple.ScandinavianActuarialJournal,2016,18-36.

https://doi.org/10.1080/03461238.2014.892899

[21]Liang, Z., Bi, J. and Yuen, K.C., etal.(2016) OptimalMean-VarianceReinsurance andInvest-

mentinaJump-DiﬀusionFinancialMarketwithCommonShockDependence.Mathematical

MethodsofOperationsResearch,84,155-181.https://doi.org/10.1007/s00186-016-0538-0

DOI:10.12677/pm.2021.11112081865nØêÆ

šxx

[22]Bjork, T.andMurgoci, A.(2010)AGeneralTheoryofMarkovianTimeInconsistentStochastic

ControlProblems.SSRNElectronicJournal.

DOI:10.12677/pm.2021.11112081866nØêÆ

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N¹

y²½n£4.3¤µ

£1¤òq

∗

= q

∗

= q

,π

∗

“\(4.5)Ú(4.2)§k

(

A(t)+rA(t))x+

B(t)+A(t)(αc−(1+η

)αa

−(1+η

)αa

A(t)

2γa(t)

ξ= 0,(6.1)

(˙a(t)+ra(t))x+

b(t)+a(t)(αc−(1+η

)αa

−(1+η

)αa

A(t)

γa(t)

ξ= 0,(6.2)

Ù¥

ξ= a

−2a

λµ

(β−r)

,(6.3)

‡Ž(4.11)Ú(4.12)¤á§K

A(t)+rA(t) = 0,A(T) = 1,

B(t)+A(t)(αc−(1+η

)αa

−(1+η

)αa

A(t)

2γa(t)

ξ(t) = 0,

˙a(t)+ra(t) = 0,a(T) = 1,

b(t)+a(t)(αc−(1+η

)αa

−(1+η

)αa

A(t)

γa(t)

ξ(t) = 0.

)þã‡©•§§k

A(t) = e

r(T−t)

(6.4)

B(t) = (αc−(1+η

)αa

−(1+η

)αa

)

r(T−t)

−1)+

2γ

ξ(T−t)(6.5)

a(t) = e

r(T−t)

(6.6)

b(t) = (αc−(1+η

)αa

−(1+η

)αa

)

r(T−t)

−1)+

ξ(T−t)(6.7)

u´§•`Š¼ê•

V(t,x) = U(t,x) = e

r(T−t)

x+(αc−(1+η

)αa

−(1+η

)αa

)

r(T−t)

−1)+

2γ

ξ(T−t)(6.8)

d§

t,x

∗

] = g(t,x) = e

r(T−t)

x+(αc−(1+η

)αa

−(1+η

)αa

)

r(T−t)

−1)+

ξ(T−t)(6.9)

DOI:10.12677/pm.2021.11112081867nØêÆ

šxx

Var

t,x

∗

] =

t,x

∗

]−V(t,x)}=

ξ(T−t)(6.10)

Ïd

Var

t,x

∗

]/ξ(T−t)§“\£6.9¤µ

t,x

∗

] = e

r(T−t)

x+(αc−(1+η

)αa

−(1+η

)αa

)

r(T−t)

−1)+

Var

t,x

∗

]ξ(T−t)(6.11)

£2¤òq

∗

= 0,q

∗

= 0,π

∗

“\(4.5)Ú(4.2)§k

B(t)+A(t)(αc+(1−2α)(1+η

+(1−2α)(1+η

−(1−α)a

)

A(t)+rA(t))x−

a(t)

(1−α)

+2λµ

A(t)

2γa(t)

(β−r)

= 0,(6.12)

b(t)+a(t)(αc+(1−2α)(1+η

+(1−2α)(1+η

−(1−α)a

)

+(˙a(t)+ra(t))x+

A(t)

γa(t)

(β−r)

= 0,(6.13)

aq£1¤¥){§·‚Œ±•`Š¼ê

V(t,x) = (αc+(1−2α)(1+η

+(1−2α)(1+η

−(1−α)a

)

r(T−t)

−1)

−(1−α)a

r(T−t)

−1)+e

r(T−t)

x−

2r(T−t)

−1)(1−α)

+2λµ

)

2γ

(β−r)

(T−t)(6.14)

£3¤òq

∗

= 0,q

∗

=bq

,π

∗

“\(4.5)Ú(4.2)§k

(

A(t)+rA(t))x+

B(t)+A(t)(αc−(1+η

)αa

+(1−2α)(1+η

−

(1−α)λµ

)

−A(t)(1−α)a

−

a(t)

(1−α)

−

A(t)

2γa(t)

[

(β−r)

] = 0,

(˙a(t)+ra(t))x+

b(t)+a(t)(αc−(1+η

)αa

+(1−2α)(1+η

−

(1−α)λµ

)

−a(t)(1−α)a

A(t)

γa(t)

[

(β−r)

] = 0,(6.15)

DOI:10.12677/pm.2021.11112081868nØêÆ

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aq£1¤¥){§·‚Œ±•`Š¼ê

V(t,x) = (αc−(1+η

)αa

+(1−2α)(1+η

−

(1−α)λµ

)

r(T−t)

−1)

−(1−α)a

r(T−t)

−1)+e

r(T−t)

x−

2r(T−t)

−1))(1−α)

−

]

2γ

[

(β−r)

](T−t)(6.16)

£4¤òq

∗

=bq

∗

= 0,π

∗

“\(4.5)Ú(4.2)§k

(

B(t)+A(t)(αc−(1+η

)αa

+(1−2α)(1+η

−(1−α)a

−

(1−α)λµ

)

A(t)+rA(t))x−

a(t)

(1−α)

−

A(t)

2γa(t)

[

(β−r)

] = 0,(6.17)

b(t)+a(t)(αc−(1+η

)αa

+(1−2α)(1+η

−(1−α)a

−

(1−α)λµ

)

+(˙a(t)+ra(t))x+

A(t)

γa(t)

[

(β−r)

] = 0,(6.18)

aq£1¤¥){§·‚Œ±•`Š¼ê

V(t,x) = (αc−(1+η

)αa

+(1−2α)(1+η

−

(1−α)λµ

)

r(T−t)

−1)

−(1−α)a

r(T−t)

−1)+e

r(T−t)

x−

2r(T−t)

−1))(1−α)

−

]

2γ

[

(β−r)

](T−t)(6.19)

£5¤òq

∗

=eq

∗

= 1,π

∗

“\(4.5)Ú(4.2)§k

(

A(t)+rA(t))x+

B(t)+A(t)(αc−(1+η

)αa

−αa

−

αλµ

)

−

a(t)

−

A(t)

2γa(t)

[

(β−r)

] = 0,(6.20)

(˙a(t)+ra(t))x+

b(t)+a(t)(αc−(1+η

)αa

−αa

−

αλµ

)

A(t)

γa(t)

[

(β−r)

] = 0,(6.21)

DOI:10.12677/pm.2021.11112081869nØêÆ

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aq£1¤¥){§·‚Œ±•`Š¼ê

V(t,x) = e

r(T−t)

x+(αc−(1+η

)αa

−αa

−

αλµ

)

r(T−t)

−1)

−

2r(T−t)

−1))α

−

2γ

[

(β−r)

](T−t)(6.22)

£6¤òq

∗

= 1,q

∗

=eq

,π

∗

“\(4.5)Ú(4.2)§k

(

A(t)+rA(t))x+

B(t)+A(t)(αc−(1+η

)αa

−αa

−

αλµ

)

−

a(t)

−

A(t)

2γa(t)

[

(β−r)

] = 0,(6.23)

(˙a(t)+ra(t))x+

b(t)+a(t)(αc−(1+η

)αa

−αa

−

αλµ

)

A(t)

γa(t)

[

(β−r)

] = 0,(6.24)

aq£1¤¥){§·‚Œ±•`Š¼ê

V(t,x) = e

r(T−t)

x+(αc−(1+η

)αa

−αa

−

αλµ

)

r(T−t)

−1)

−

2r(T−t)

−1))α

−

2γ

[

(β−r)

](T−t)(6.25)

£7¤òq

∗

= 1,q

∗

= 1,π

∗

“\(4.5)Ú(4.2)§k

(

A(t)+rA(t))x+

B(t)+A(t)(αc−(1+η

)αa

−αa

−

αλµ

)

−

a(t)

+2λµ

A(t)

2γa(t)

(β−r)

= 0,(6.26)

(˙a(t)+ra(t))x+

b(t)+a(t)(αc−(1+η

)αa

−αa

−

αλµ

)

A(t)

γa(t)

(β−r)

= 0,(6.27)

aq£1¤¥){§·‚Œ±•`Š¼ê

V(t,x) = e

r(T−t)

x+(αc−αa

−αa

)

r(T−t)

−1)+

2γ

(β−r)

(T−t)

−

2r(T−t)

−1)α

+2λµ

)(6.28)

T½ny²."

DOI:10.12677/pm.2021.11112081870nØêÆ