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PureMathematics
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PublishedOnlineNovember2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1111208
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OptimalInvestmentReinsuranceStrategy
fortheJointBenefitsoftheInsurerand
theReinsurerundertheMean-Variance
TingtingSun,HuihuiWang,HuishengShu
∗
CollegeofScienceandTechnology,DonghuaUniversity,Shanghai
Received:Oct.15
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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
Copyright
c
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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+
λ
)
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(
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)
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η
i
•
2
x
ú
i
é
u
1
i
a
x
«
S
K
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(
i
= 1
,
2)
"
2.3.
7
K
½
|
x
ú
i
Ø
Ï
L
ï
2
x
=
£
º
x
§
„
Œ
±
3
7
K
½
|
?
1
Ý
]
§
5
J
p
€
U
å
"
b
7
K
½
|
d
˜
‡
Ã
º
x
]
(
Å
)
Ú
˜
‡
º
x
]
(
¦
)
|
¤
"
3
t
ž
•
§
Ã
º
x
]
d
‚
S
0
(
t
)
•
(
dS
0
(
t
) =
rS
0
(
t
)
dt,
S
0
(
t
) =
S
0
.
(2.3)
Ù
¥
r>
0
´
Ã
º
x
]
(
Å
)
|
Ç
¶
º
x
]
d
‚
S
1
(
t
)
•
(
dS
1
(
t
) =
S
1
(
t
)[
βdt
+
σdW
(
t
)]
,
S
1
(
t
) =
S
1
.
(2.4)
DOI:10.12677/pm.2021.11112081853
n
Ø
ê
Æ
š
xx
Ù
¥
β
•
º
x
]
(
¦
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x
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(
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1
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2
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(
t
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á
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2.4.
ã
L
L
§
©
¥
§
x
ú
i
Œ
±
Ý
]
u
Ã
º
x
]
Ú
º
x
]
§
2
x
ú
i
Œ
±
Ý
]
u
Ã
º
x
]
5
5
;
º
x
"
3
t
ž
•
§
π
(
t
)
•
x
ú
i
Ý
]
u
º
x
]
7
§
x
ú
i
ã
L
L
§
R
1
t
•
dR
1
t
= [
c
+
r
(
R
1
t
−
π
)
−
δ
(
q
1
t
,q
2
t
)]
dt
+
π
[
βdt
+
σdW
(
t
)]
−
q
1
t
dL
1
(
t
)
−
q
2
t
dL
2
(
t
)
.
(2.5)
2
x
ú
i
ã
L
L
§
R
2
t
•
dR
2
t
= [
rR
2
t
+
δ
(
q
1
t
,q
2
t
)]
dt
−
(1
−
q
1
t
)
dL
1
(
t
)
−
(1
−
q
2
t
)
dL
2
(
t
)
.
(2.6)
©•
Ó
ž
o
x
ú
i
Ú
2
x
ú
i
|
Ã
§
ò
ü
ö
ã
L
L
§
±
α
∈
[0
,
1]
\
§
P
•
R
t
=
αR
1
t
+ (1
−
α
)
R
2
t
"
\
X
ê
α
=0
ž
§
R
t
•
x
2
x
ú
i
ã
L
L
§
§
\
X
ê
α
=1
ž
§
R
t
•
x
x
ú
i
ã
L
L
§
§
…
α
Œ
•
x
•
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ü
Ñ
Ï
é
L
§
¥
x
ú
i
|
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'
-
§
Œ
K
•
ý
-
x
ú
i
|
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§
‡
ƒ
K
•
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-
2
x
ú
i
|
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±
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)
¤
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x
ú
i
Ú
˜
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2
x
ú
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á
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x
8
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α
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−
α
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¤
±
k
°
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t
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x
8
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ã
L
L
§
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Ï
d
§
d
ž
B
ú
ª
§
·
‚
k
dR
t
= [
rR
t
+
αc
+
α
(
β
−
r
)
π
+(1
−
2
α
)
δ
(
q
1
t
,q
2
t
)]+
ασπdW
(
t
)
−
2
X
i
=1
[1
−
α
−
(1
−
2
α
)
q
it
]
dL
i
(
t
)
=
{
rR
t
+
αc
+
α
(
β
−
r
)
π
+(1
−
2
α
)[(1+
η
1
)(1
−
q
1
)
a
1
+(1+
η
2
)(1
−
q
2
)
a
2
]
}
dt
+
ασπdW
(
t
)
−
2
X
i
=1
[1
−
α
−
(1
−
2
α
)
q
it
]
dL
i
(
t
)
.
(2.7)
d
þ
ª
Œ
•
§
α
=
1
2
ž
§
ã
L
L
§
†
q
1
t
§
q
2
t
Ã
'
§
Ù
•
`
2
x
ü
Ñ
•
?
¿
Š
§
©
3
α
6
=
1
2
…
α
∈
(0
,
1)
Ä
:
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Ï
¦
•
`
ü
Ñ
"
½
Â
2.1
X
J
ü
Ñ
Q
= (
q
1
t
,q
2
t
,π
(
t
))
´
'
u
6
{F
t
}
t
≥
0
-
·
A
‘
Å
L
§
§
…
÷
v
±
e
^
‡
µ
1.
é
u
6
{F
t
}
t
≥
0
§
(
q
1
s
,q
2
t
,π
(
t
))
´
ÌS
Œ
ÿ
¶
2.
é
∀
s
∈
[
t,T
]
§
0
≤
q
is
≤
1
,i
= 1
,
2
§
E
[
R
T
t
(
q
2
1
s
+
q
2
2
s
+
π
(
s
)
2
)]
<
+
∞
¶
3.
(
Q,R
Q
t
)
´
‘
Å
‡
©•
§
(2.7)
•
˜
)
"
K
ü
Ñ
Q
= (
q
1
t
,q
2
t
,π
(
t
))
•
Œ
N
N
ü
Ñ
§
¤
k
Œ
N
N
ü
Ñ
8
Ü
P
•
Q
"
3.
½
Â
Ú
Ú
n
Š
â
©
z
[22]
§
·
‚
3
Æ
‰
Ø
µ
e
e
•
x
8
ì
ï
˜
‡
Ý
]
2
x
¯
K
§
x
8
ì
8
I
´
•
Œ
z
ª
à
ž
•
T
ã
L
Ï
"
§
b
Ù
^
¼
ê
æ
^
þ
Š
-
•
O
K
§
=
é
∀
(
t,x
)
∈
([0
,T
]
×
R
)
§
DOI:10.12677/pm.2021.11112081854
n
Ø
ê
Æ
š
xx
8
I
¼
ê
•
sup
Q
∈
Q
J
(
t,x,Q
) =sup
Q
∈
Q
{
E
t,x
[
R
Q
T
]
−
γ
2
Var
t,x
[
R
Q
T
]
}
.
(3.1)
Ù
¥
E
t,x
[
·
] =
E
[
·|
R
Q
t
=
x
]
§
Var
t,x
[
·
] =
Var
[
·|
R
Q
t
=
x
]
§
x
´
Ð
©
J{
§
γ>
0
•
º
x
X
ê
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P
C
1
,
2
´
˜
‡
¼
ê
˜
m
§
÷
v
é
∀
ψ
(
t,x
)
∈C
1
,
2
§
ψ
§
ψ
t
§
ψ
x
Ú
ψ
xx
´
[0
,T
]
×
R
þ
ë
Y
¼
ê
§
Ù
¥
ψ
t
•
ψ
é
t
˜
§
ψ
x
Ú
ψ
xx
•
ψ
é
x
˜
Ú
§
…
A
Q
ψ
(
t,x
) =
ψ
t
+
{
rx
+
cα
+
α
(
β
−
r
)
π
+(1
−
2
α
)[(1+
η
1
)(1
−
q
1
)
a
1
+(1+
η
2
)(1
−
q
2
)
a
2
]
}
ψ
x
+
1
2
α
2
σ
2
π
2
V
xx
+
λ
1
E
[
ψ
(
t,x
−
(1
−
α
−
(1
−
2
α
)
q
1
)
X
)
−
ψ
(
t,x
)]
+
λ
2
E
[
ψ
(
t,x
−
(1
−
α
−
(1
−
2
α
)
q
2
)
Y
)
−
ψ
(
t,x
)]
+
λE
[
ψ
(
t,x
−
(1
−
α
−
(1
−
2
α
)
q
1
)
X
−
(1
−
α
−
(1
−
2
α
)
q
2
)
Y
)
−
ψ
(
t,x
)]
,
(3.2)
½
Â
3.1
(
þ
ï
ü
Ñ
)
X
J
Q
∈
R
+
×
R
+
×
R
§
h>
0
Ú
(
t,x
)
∈
[0
,T
]
×
R
§
liminf
h
→
0
J
(
t,x,Q
∗
)
−
J
(
t,x,Q
h
)
h
≥
0
.
Ù
¥
Q
h
(
s,
e
x
) =
Q,s
∈
[
t,t
+
h
]
,
e
x
∈
R,
Q
∗
(
s,
e
x
)
,s
∈
[
t
+
h,T
]
,
e
x
∈
R,
K
Q
∗
(
t,x
)
•
þ
ï
ü
Ñ
§
ƒ
A
þ
ï
Š
¼
ê
•
V
(
t,x
) =
J
(
t,x,Q
∗
) =
E
t,x
[
R
Q
∗
T
]
−
γ
2
Var
t,x
[
R
Q
∗
T
]
.
(3.3)
Ú
n
3.1
(
y
½
n
)
é
¯
K
(3.1)
§
X
J
•
3
ü
‡
¢
¼
ê
U
(
t,x
)
,g
(
t,x
)
∈
C
1
,
2
([0
,T
]
×
R
)
÷
v
e
HJB
•
§
|
µ
sup
Q
∈
Q
{A
Q
U
(
t,x
)
−A
Q
γ
2
g
(
t,x
)
2
+
γg
(
t,x
)
A
Q
g
(
t,x
)
}
= 0
,
(3.4)
U
(
T,x
) =
x,
(3.5)
A
Q
∗
g
(
t,x
) = 0
,
(3.6)
g
(
T,x
) =
x.
(3.7)
Ù
¥
Q
∗
=
arg
sup
Q
∈
Q
{A
Q
U
(
t,x
)
−A
Q
γ
2
g
(
t,x
)
2
+
γg
(
t,x
)
A
Q
g
(
t,x
)
}
,
(3.8)
DOI:10.12677/pm.2021.11112081855
n
Ø
ê
Æ
š
xx
K
V
(
t,x
) =
U
(
t,x
)
§
E
t,x
[
R
Q
∗
T
] =
g
(
t,x
)
§
•
`
ü
Ñ
´
Q
∗
"
þ
ã
½
n
y
²
§
„
©
z
[22]
¥
½
n
4.1
"
4.
.
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)
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§
·
‚
3
þ
Š
-
•
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K
e
¦
)
x
8
ì
•
`
Ý
]
Ú
2
x
ü
Ñ
"
b
U
(
t,x
)
Ú
g
(
t,x
)
´
ü
‡
÷
v
½
n
3.1
¥
^
‡
¼
ê
§
u
´
(3.4)
Ú
(3.6)
Œ
±
¤
X
e
/
ª
µ
sup
Q
∈
Q
{
U
t
+
{
rx
+
cα
+
α
(
β
−
r
)
π
+(1
−
2
α
)[(1+
η
1
)(1
−
q
1
)
a
1
+(1+
η
2
)(1
−
q
2
)
a
2
]
}
U
x
+
1
2
(
U
xx
−
γg
x
(
t,x
)
2
)
α
2
σ
2
π
2
−
(
λ
1
+
λ
2
+
λ
)[
U
(
t,x
)+
γ
2
g
(
t,x
)
2
]
+
λ
1
E
[
U
(
t,x
−
θ
1
X
)
−
γ
2
g
(
t,x
−
θ
1
X
)(
g
(
t,x
−
θ
1
X
)
−
2
g
(
t,x
))]
+
λ
2
E
[
U
(
t,x
−
θ
2
Y
)
−
γ
2
g
(
t,x
−
θ
2
Y
)(
g
(
t,x
−
θ
2
Y
)
−
2
g
(
t,x
))]
+
λE
[
U
(
t,x
−
θ
1
q
1
X
−
θ
2
Y
)
−
γ
2
g
(
t,x
−
θ
1
X
−
θ
2
Y
)(
g
(
t,x
−
θ
1
X
−
θ
2
Y
)
−
2
g
(
t,x
))] = 0
,
(4.1)
g
t
+
{
rx
+
cα
+
α
(
β
−
r
)
π
+(1
−
2
α
)[(1+
η
1
)(1
−
q
1
)
a
1
+(1+
η
2
)(1
−
q
2
)
a
2
]
}
g
x
+
1
2
α
2
σ
2
π
2
g
xx
+
λ
1
E
[
g
(
t,x
−
θ
1
X
)
−
g
(
t,x
)]
+
λ
2
E
[
g
(
t,x
−
θ
2
Y
)
−
g
(
t,x
)]
+
λE
[
g
(
t,x
−
θ
1
X
−
θ
2
Y
)
−
g
(
t,x
)] = 0
,
(4.2)
Ù
¥
θ
i
= 1
−
α
−
(1
−
2
α
)
q
i
,i
= 1
,
2
"
Š
â
Bjork
Ú
Murgoci(2010)
±
9
½
n
3.1
¥
^
‡
§
·
‚
ßÿ
)
X
e
µ
U
(
T,x
) =
A
(
t
)
x
+
B
(
t
)
,A
(
T
) = 1
,B
(
T
) = 0
,
g
(
T,x
) =
a
(
t
)
x
+
b
(
t
)
,a
(
T
) = 1
,b
(
T
) = 0
,
é
Ù
¦
‡
©
Œ
µ
U
t
(
T,x
) =
˙
A
(
t
)
x
+
˙
B
(
t
)
,U
x
(
T,x
) =
A
(
t
)
,U
xx
(
T,x
) = 0
,
(4.3)
g
t
(
T,x
) =˙
a
(
t
)
x
+
˙
b
(
t
)
,g
x
(
T,x
) =
a
(
t
)
,g
xx
(
T,x
) = 0
,
(4.4)
DOI:10.12677/pm.2021.11112081856
n
Ø
ê
Æ
š
xx
Ù
¥
§
˙
A
(
t
) =
dA
(
t
)
/dt
§
˙
B
(
t
) =
dB
(
t
)
/dt
§
˙
a
(
t
) =
da
(
t
)
/dt
§
˙
b
(
t
) =
db
(
t
)
/dt
§
“
\
(4.1)
A
(
t
)[
rx
+
cα
+
α
(
β
−
r
)
π
+(1
−
2
α
)((1+
η
1
)(1
−
q
1
)
a
1
+(1+
η
2
)(1
−
q
2
)
a
2
)]+
˙
B
(
t
)
˙
A
(
t
)
x
−
A
(
t
)
θ
1
a
1
−
A
(
t
)
θ
2
a
2
−
γ
2
a
(
t
)
2
(
θ
2
1
b
2
1
+
θ
2
2
b
2
2
+2
θ
1
θ
2
λµ
11
µ
21
+
α
2
σ
2
π
2
) = 0
.
(4.5)
Ù
¥
b
2
1
= (
λ
1
+
λ
)
E
(
X
2
)
§
b
2
2
= (
λ
2
+
λ
)
E
(
Y
2
)
"
-
f
(
q
1
,q
2
) =
−
γ
2
a
(
t
)
2
θ
2
1
b
2
1
−
γ
2
a
(
t
)
2
θ
2
2
b
2
2
−
A
(
t
)
θ
1
a
1
−
A
(
t
)
θ
2
a
2
−
γa
(
t
)
2
θ
1
θ
2
λµ
11
µ
21
+
A
(
t
)(1
−
2
α
)(1+
η
1
)(1
−
q
1
)
a
1
+
A
(
t
)(1
−
2
α
)(1+
η
2
)(1
−
q
2
)
a
2
.
(4.6)
K
∂f
(
q
1
,q
2
)
∂q
1
= (2
α
−
1)(
−
γa
(
t
)
2
θ
1
b
2
1
+
A
(
t
)
η
1
a
1
−
γa
(
t
)
2
θ
2
λµ
11
µ
21
)
,
∂f
(
q
1
,q
2
)
∂q
2
= (2
α
−
1)(
−
γa
(
t
)
2
θ
2
b
2
2
+
A
(
t
)
η
2
a
2
−
γa
(
t
)
2
θ
1
λµ
11
µ
21
)
,
∂
2
f
(
q
1
,q
2
)
∂q
2
1
=
−
γa
(
t
)
2
(2
α
−
1)
2
b
2
1
,
∂
2
f
(
q
1
,q
2
)
∂q
2
2
=
−
γa
(
t
)
2
(2
α
−
1)
2
b
2
2
,
∂
2
f
(
q
1
,q
2
)
∂q
1
q
2
=
−
γa
(
t
)
2
(2
α
−
1)
2
λµ
11
µ
21
.
(4.7)
f
(
q
1
,q
2
)
Hessian
Ý
Œ
±
L
«
•
µ
∂
2
f
(
q
1
,q
2
)
∂q
2
1
∂
2
f
(
q
1
,q
2
)
∂q
1
∂q
2
∂
2
f
(
q
1
,q
2
)
∂q
1
∂q
2
∂
2
f
(
q
1
,q
2
)
∂q
2
1
=
−
γa
(
t
)
2
(2
α
−
1)
2
b
2
1
−
γa
(
t
)
2
(2
α
−
1)
2
λµ
11
µ
21
−
γa
(
t
)
2
(2
α
−
1)
2
λµ
11
µ
21
−
γa
(
t
)
2
(2
α
−
1)
2
b
2
2
Š
â
…
Ü
-
–
]
Ø
ª
§
·
‚
•
b
2
1
b
2
2
= (
λ
1
+
λ
)
E
(
X
2
)(
λ
2
+
λ
)
E
(
Y
2
)
>λ
2
µ
2
11
µ
2
21
§
γ
2
a
(
t
)
4
(2
α
−
1)
2
(
b
2
1
b
2
2
−
λ
2
µ
2
11
µ
2
21
)
>
0
§
¤
±
f
(
q
1
,q
2
)
Hessian
Ý
½
§
=
f
(
q
1
,q
2
)
´
'
u
q
1
§
q
2
à
¼
ê
"
u
´
−
γa
(
t
)
2
θ
1
b
2
1
+
A
(
t
)
θ
1
a
1
−
γa
(
t
)
2
θ
2
λµ
11
µ
21
= 0
,
−
γa
(
t
)
2
θ
2
b
2
2
+
A
(
t
)
θ
2
a
2
−
γa
(
t
)
2
θ
1
λµ
11
µ
21
= 0
.
(4.8)
K
θ
1
=
A
(
t
)(
−
a
2
η
2
λµ
11
µ
21
+
a
1
b
2
2
η
1
)
γa
(
t
)
2
(
b
2
1
b
2
2
−
λ
2
µ
2
11
µ
2
21
)
,
θ
2
=
A
(
t
)(
−
a
1
η
1
λµ
11
µ
21
+
a
2
b
2
1
η
2
)
γa
(
t
)
2
(
b
2
1
b
2
2
−
λ
2
µ
2
11
µ
2
21
)
.
(4.9)
d
θ
i
= 1
−
α
−
(1
−
2
α
)
q
i
,i
= 1
,
2
§
·
‚
k
q
i
=
θ
i
−
(1
−
α
)
2
α
−
1
,i
= 1
,
2
"
Š
â
(4.5)
§
e
π
=
A
(
t
)(
β
−
r
)
αγa
(
t
)
2
σ
2
,
(4.10)
DOI:10.12677/pm.2021.11112081857
n
Ø
ê
Æ
š
xx
ò
q
1
,q
2
,
e
π
“
\
(4.5)
Ú
(4.2)
§
k
(
˙
A
(
t
)+
rA
(
t
))
x
+
˙
B
(
t
)+
A
(
t
)(
αc
−
(1+
η
1
)
αa
1
−
(1+
η
2
)
αa
2
)+
A
(
t
)
2
2
γa
(
t
)
2
ξ
= 0
,
(4.11)
(˙
a
(
t
)+
ra
(
t
))
x
+
˙
b
(
t
)+
a
(
t
)(
αc
−
(1+
η
1
)
αa
1
−
(1+
η
2
)
αa
2
)+
A
(
t
)
2
γa
(
t
)
2
ξ
= 0
,
(4.12)
Ù
¥
ξ
=
a
2
1
η
2
1
b
2
2
+
a
2
2
η
2
2
b
1
2
−
2
a
1
a
2
η
1
η
2
λµ
11
µ
21
+
(
β
−
r
)
2
σ
2
,
(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
1
)
αa
1
−
(1+
η
2
)
αa
2
)+
A
(
t
)
2
2
γa
(
t
)
2
ξ
(
t
) = 0
,
˙
a
(
t
)+
ra
(
t
) = 0
,a
(
T
) = 1
,
˙
b
(
t
)+
a
(
t
)(
αc
−
(1+
η
1
)
αa
1
−
(1+
η
2
)
αa
2
)+
A
(
t
)
2
γa
(
t
)
2
ξ
(
t
) = 0
.
)
þ
ã
‡
©•
§
§
k
A
(
t
) =
e
r
(
T
−
t
)
,
(4.14)
B
(
t
) = (
αc
−
(1+
η
1
)
αa
1
−
(1+
η
2
)
αa
2
)
1
r
(
e
r
(
T
−
t
)
−
1)+
1
2
γ
ξ
(
T
−
t
)
,
(4.15)
a
(
t
) =
e
r
(
T
−
t
)
,
(4.16)
b
(
t
) = (
αc
−
(1+
η
1
)
αa
1
−
(1+
η
2
)
αa
2
)
1
r
(
e
r
(
T
−
t
)
−
1)+
1
γ
ξ
(
T
−
t
)
.
(4.17)
ò
(4.14)
Ú
(4.16)
“
\
(4.9)
§
Œ
q
1
=
1
2
α
−
1
[
−
a
2
η
2
λµ
11
µ
21
+
a
1
b
2
2
η
1
γe
r
(
T
−
t
)
(
b
2
1
b
2
2
−
λ
2
µ
2
11
µ
2
21
)
−
(1
−
α
)]
,
(4.18)
q
2
=
1
2
α
−
1
[
−
a
1
η
1
λµ
11
µ
21
+
a
2
b
2
1
η
2
γe
r
(
T
−
t
)
(
b
2
1
b
2
2
−
λ
2
µ
2
11
µ
2
21
)
−
(1
−
α
)]
.
(4.19)
P
m
1
=
−
a
2
η
2
λµ
11
µ
21
+
a
1
b
2
2
η
1
,m
2
=
−
a
1
η
1
λµ
11
µ
21
+
a
2
b
2
1
η
2
§
q
{
ü
y
Œ
•
a
2
λµ
11
µ
21
a
1
b
2
2
<
a
2
b
2
1
a
1
λµ
11
µ
21
"
-
t
i
0
(
b
t
i
0
,
e
t
i
0
)
´¦
q
i
=0(
b
q
i
=0
,
e
q
i
=0)
¤
á
ž
m
:
§
t
i
1
(
b
t
i
1
,
e
t
i
1
)
´¦
q
i
=1(
b
q
i
=
1
,
e
q
i
= 1)
¤
á
ž
m
:
§
t
0
i
´¦
q
i
(
T
−
t
)=0
¤
á
ž
m
:
§
i
=1
,
2
§
t
e
0
i
´¦
e
q
i
(
T
−
t
)=0
DOI:10.12677/pm.2021.11112081858
n
Ø
ê
Æ
š
xx
¤
á
ž
m
:
i
= 1
,
2
§
u
´
·
‚
Œ
±
X
e
ü
‡
½
n
µ
½
n
4.1
e
1
2
<α<
1
§
K
•
`
2
x
ü
Ñ
(
q
∗
1
,q
∗
2
)
•
£
1
¤
η
1
>
a
2
b
2
1
a
1
λµ
11
µ
21
η
2
>
a
2
λµ
11
µ
21
a
1
b
2
2
η
2
ž
§
=
£
m
1
>
0
,m
2
<
0
¤
(
q
∗
1
,q
∗
2
) =
(0
,
0)
,t
≤
b
t
10
(
b
q
1
,
0)
,
b
t
10
<t<
b
t
11
(1
,
0)
,t
≥
b
t
11
(4.20)
Ù
¥
b
q
1
=
1
2
α
−
1
[
−
γe
r
(
T
−
t
)
(1
−
α
)
λµ
11
µ
21
+
η
1
a
1
γe
r
(
T
−
t
)
b
2
1
−
(1
−
α
)]
£
2
¤
a
2
b
2
1
a
1
λµ
11
µ
21
η
2
>η
1
>
a
2
λµ
11
µ
21
a
1
b
2
2
η
2
ž
§
=
£
m
1
>
0
,m
2
>
0
¤
e
m
1
≥
m
2
§
k
(
q
∗
1
,q
∗
2
) =
(0
,
0)
,t
≤
b
t
10
(0
,
b
q
2
)
,
b
t
10
<t
≤
t
20
(
q
1
,q
2
)
,t
20
<t<t
11
(1
,
e
q
2
)
,t
11
≤
t<
e
t
21
(1
,
1)
,t
≥
e
t
21
(4.21)
Ù
¥
b
q
2
=
1
2
α
−
1
[
−
γe
r
(
T
−
t
)
(1
−
α
)
λµ
11
µ
21
+
η
1
a
1
γe
r
(
T
−
t
)
b
2
2
−
(1
−
α
)]
§
e
q
2
=
1
2
α
−
1
[
−
γe
r
(
T
−
t
)
αλµ
11
µ
21
+
η
2
a
2
γe
r
(
T
−
t
)
b
2
2
−
(1
−
α
)]
¶
e
m
1
<m
2
§
k
(
q
∗
1
,q
∗
2
) =
(0
,
0)
,t
≤
b
t
20
(0
,
b
q
2
)
,
b
t
20
<t
≤
t
10
(
q
1
,q
2
)
,t
10
<t<t
21
(
e
q
1
,
1)
,t
21
≤
t<
e
t
11
(1
,
1)
,t
≥
e
t
11
(4.22)
Ù
¥
b
q
2
=
1
2
α
−
1
[
−
γe
r
(
T
−
t
)
(1
−
α
)
λµ
11
µ
21
+
η
2
a
2
γe
r
(
T
−
t
)
b
2
2
−
(1
−
α
)]
§
e
q
1
=
1
2
α
−
1
[
−
γe
r
(
T
−
t
)
αλµ
11
µ
21
+
η
1
a
1
γe
r
(
T
−
t
)
b
2
1
−
(1
−
α
)]
¶
£
3
¤
a
2
b
2
1
a
1
λµ
11
µ
21
η
2
>
a
2
λµ
11
µ
21
a
1
b
2
2
η
2
>η
1
ž
§
=
(
m
1
<
0
,m
2
>
0
)
(
q
∗
1
,q
∗
2
) =
(0
,
0)
,t
≤
b
t
20
(0
,
b
q
2
)
,
b
t
20
<t<
b
t
21
(0
,
1)
,t
≥
b
t
21
(4.23)
DOI:10.12677/pm.2021.11112081859
n
Ø
ê
Æ
š
xx
½
n
4.2
e
0
<α<
1
2
§
K
•
`
2
x
ü
Ñ
(
q
∗
1
,q
∗
2
)
•
£
1
¤
η
1
>
a
2
b
2
1
a
1
λµ
11
µ
21
η
2
>
a
2
λµ
11
µ
21
a
1
b
2
2
η
2
ž
§
=
£
m
1
>
0
,m
2
<
0
¤
(
q
∗
1
,q
∗
2
) =
(1
,
1)
,t
≤
e
t
11
(
e
q
1
,
1)
,
e
t
11
<t<
e
t
10
(0
,
1)
,t
≥
e
t
10
(4.24)
Ù
¥
e
q
1
=
1
2
α
−
1
[
−
γe
r
(
T
−
t
)
αλµ
11
µ
21
+
η
1
a
1
γe
r
(
T
−
t
)
b
2
1
−
(1
−
α
)]
£
ii
¤
a
2
b
2
1
a
1
λµ
11
µ
21
η
2
>η
1
>
a
2
λµ
11
µ
21
a
1
b
2
2
η
2
ž
§
=
£
m
1
>
0
,m
2
>
0
¤
e
m
1
≥
m
2
§
k
(
q
∗
1
,q
∗
2
) =
(1
,
1)
,t
≤
e
t
11
(1
,
e
q
2
)
,
e
t
11
<t
≤
t
21
(
q
1
,q
2
)
,t
21
<t<t
10
(0
,
b
q
2
)
,t
10
≤
t<
b
t
20
(0
,
0)
,t
≥
b
t
20
(4.25)
Ù
¥
e
q
2
=
1
2
α
−
1
[
−
γe
r
(
T
−
t
)
αλµ
11
µ
21
+
η
1
a
1
γe
r
(
T
−
t
)
b
2
2
−
(1
−
α
)]
§
b
q
2
=
1
2
α
−
1
[
−
γe
r
(
T
−
t
)
(1
−
α
)
λµ
11
µ
21
+
η
2
a
2
γe
r
(
T
−
t
)
b
2
2
−
(1
−
α
)]
e
m
1
<m
2
§
k
(
q
∗
1
,q
∗
2
) =
(1
,
1)
,t
≤
e
t
21
(1
,
e
q
2
)
,
e
t
21
<t
≤
t
11
(
q
1
,q
2
)
,t
11
<t<t
20
(
b
q
1
,
0)
,t
20
≤
t<
b
t
10
(0
,
0)
,t
≥
b
t
10
(4.26)
Ù
¥
e
q
2
=
1
2
α
−
1
[
−
γe
r
(
T
−
t
)
αλµ
11
µ
21
+
η
2
a
2
γe
r
(
T
−
t
)
b
2
2
−
(1
−
α
)]
§
b
q
1
=
1
2
α
−
1
[
−
γe
r
(
T
−
t
)
(1
−
α
)
λµ
11
µ
21
+
η
1
a
1
γa
(
t
)
2
b
2
1
−
(1
−
α
)]
£
iii
¤
a
2
b
2
1
a
1
λµ
11
µ
21
η
2
>
a
2
λµ
11
µ
21
a
1
b
2
2
η
2
>η
1
ž
§
=
£
m
1
<
0
,m
2
>
0
¤
(
q
∗
1
,q
∗
2
) =
(1
,
1)
,t
≤
e
t
21
(1
,
e
q
2
)
,
e
t
21
<t<
e
t
20
(1
,
1)
,t
≥
e
t
20
(4.27)
DOI:10.12677/pm.2021.11112081860
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5
4.1
X
J
E
Ü
Possion
L
§
L
1
(
t
)
Ú
L
2
(
t
)
©
Ù
ƒ
Ó
§
=
λ
1
=
λ
2
§
µ
11
=
µ
21
§
µ
12
=
µ
22
§
u
´
a
1
=
a
2
§
b
2
1
=
b
2
2
§
K
(4.9)
¥
2
x
ú
i
é
u
1
1
a
x
«
S
K
Ö
η
1
Ú
é
u
1
2
a
x
«
S
K
Ö
η
2
ƒ
ž
§
k
θ
1
=
θ
2
§
?
˜
Ú
q
∗
1
=
q
∗
2
¶
2
x
ú
i
é
u
1
1
a
x
«
S
K
Ö
η
1
Ú
é
u
1
2
a
x
«
S
K
Ö
η
2
Ø
ƒ
ž
§
k
η
1
θ
1
=
η
2
θ
2
½
n
4.3
é
¯
K
(3.1)
§
·
‚
d
(4.10)
ª
Œ
•
•
`
Ý
]
ü
Ñ
•
π
∗
=
(
β
−
r
)
αγe
r
(
T
−
t
)
σ
2
§
•
`
2
x
ü
Ñ
d
½
n
4.1
Ú
½
n
4.2
‰
Ñ
§
…
•
`
Š
¼
ê
V(t,x)
•
£
1
¤
•
`
2
x
ü
Ñ
(
q
∗
1
,q
∗
2
) = (
q
1
,q
2
)
ž
§
V
(
t,x
) =
e
r
(
T
−
t
)
x
+(
αc
−
(1+
η
1
)
αa
1
−
(1+
η
2
)
αa
2
)
1
r
(
e
r
(
T
−
t
)
−
1)+
1
2
γ
ξ
(
T
−
t
)(4.28)
Ù
¥
ξ
=
a
2
1
η
2
1
b
2
2
+
a
2
2
η
2
2
b
1
2
−
2
a
1
a
2
η
1
η
2
λµ
11
µ
21
+
(
β
−
r
)
2
σ
2
£
2
¤
•
`
2
x
ü
Ñ
(
q
∗
1
,q
∗
2
) = (0
,
0)
ž
§
V
(
t,x
) = ((1
−
2
α
)(1+
η
1
)
a
1
+(1
−
2
α
)(1+
η
2
)
a
2
−
(1
−
α
)
a
1
−
(1
−
α
)
a
2
)
1
r
(
e
r
(
T
−
t
)
−
1)
αc
1
r
(
e
r
(
T
−
t
)
−
1)+
e
r
(
T
−
t
)
x
−
γ
4
r
(
e
2
r
(
T
−
t
)
−
1)(1
−
α
)
2
(
b
2
1
+
b
2
2
+2
λµ
11
µ
21
)
+
1
2
γ
(
β
−
r
)
2
σ
2
(
T
−
t
)(4.29)
£
3
¤
•
`
2
x
ü
Ñ
(
q
∗
1
,q
∗
2
) = (0
,
b
q
2
)
ž
§
V
(
t,x
) = (
αc
−
(1+
η
2
)
αa
2
+(1
−
2
α
)(1+
η
1
)
a
1
−
(1
−
α
)
λµ
11
µ
21
η
2
a
2
b
2
2
)
1
r
(
e
r
(
T
−
t
)
−
1)
−
(1
−
α
)
a
1
1
r
(
e
r
(
T
−
t
)
−
1)+
e
r
(
T
−
t
)
x
−
γ
4
r
(
e
2
r
(
T
−
t
)
−
1))(1
−
α
)
2
[
b
2
1
−
λ
2
µ
2
11
µ
2
21
b
2
2
]
+
1
2
γ
[
(
β
−
r
)
2
σ
2
+
η
2
2
a
2
2
b
2
2
](
T
−
t
)(4.30)
£
4
¤
•
`
2
x
ü
Ñ
(
q
∗
1
,q
∗
2
) = (
b
q
1
,
0)
ž
§
V
(
t,x
) = (
αc
−
(1+
η
1
)
αa
1
+(1
−
2
α
)(1+
η
2
)
a
2
−
(1
−
α
)
λµ
11
µ
21
η
1
a
1
b
2
1
)
1
r
(
e
r
(
T
−
t
)
−
1)
−
(1
−
α
)
a
2
1
r
(
e
r
(
T
−
t
)
−
1)+
e
r
(
T
−
t
)
x
−
γ
4
r
(
e
2
r
(
T
−
t
)
−
1))(1
−
α
)
2
[
b
2
2
−
λ
2
µ
2
11
µ
2
21
b
2
1
]
+
1
2
γ
[
(
β
−
r
)
2
σ
2
+
η
2
1
a
2
1
b
2
1
](
T
−
t
)(4.31)
DOI:10.12677/pm.2021.11112081861
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£
5
¤
•
`
2
x
ü
Ñ
(
q
∗
1
,q
∗
2
) = (
e
q
1
,
1)
ž
§
V
(
t,x
) =
e
r
(
T
−
t
)
x
+(
αc
−
(1+
η
1
)
αa
1
−
αa
2
−
αλµ
11
µ
21
η
1
a
1
b
2
1
)
1
r
(
e
r
(
T
−
t
)
−
1)
−
γ
4
r
(
e
2
r
(
T
−
t
)
−
1))
α
2
[
b
2
2
−
λ
2
µ
2
11
µ
2
21
b
2
1
]+
1
2
γ
[
(
β
−
r
)
2
σ
2
+
η
2
1
a
2
1
b
2
1
](
T
−
t
)(4.32)
£
6
¤
•
`
2
x
ü
Ñ
(
q
∗
1
,q
∗
2
) = (1
,
e
q
2
)
ž
§
V
(
t,x
) =
e
r
(
T
−
t
)
x
+(
αc
−
(1+
η
2
)
αa
2
−
αa
1
−
αλµ
11
µ
21
η
2
a
2
b
2
2
)
1
r
(
e
r
(
T
−
t
)
−
1)
−
γ
4
r
(
e
2
r
(
T
−
t
)
−
1))
α
2
[
b
2
1
−
λ
2
µ
2
11
µ
2
21
b
2
2
]+
1
2
γ
[
(
β
−
r
)
2
σ
2
+
η
2
2
a
2
2
b
2
2
](
T
−
t
)(4.33)
£
7
¤
•
`
2
x
ü
Ñ
(
q
∗
1
,q
∗
2
) = (1
,
1)
ž
§
V
(
t,x
) =
e
r
(
T
−
t
)
x
+(
αc
−
αa
1
−
αa
2
)
1
r
(
e
r
(
T
−
t
)
−
1)+
1
2
γ
(
β
−
r
)
2
σ
2
(
T
−
t
)
−
γ
4
r
(
e
2
r
(
T
−
t
)
−
1)
α
2
(
b
2
1
+
b
2
2
+2
λµ
11
µ
21
)(4.34)
5.
ê
Š
©
Û
ù
˜
Ü
©
¥
§
·
‚
ò
Ï
L
ê
Š
~
f
`
²
ë
ê
é
•
`
ü
Ñ
K
•
"
b
λ
=1
,λ
1
=2
,λ
2
=
3
,T
=10
,r
=0
.
03
,α
=0
.
6
,µ
11
=0
.
045
,µ
11
=0
.
053
,µ
21
=0
.
045
,µ
22
=0
.
055
,η
1
=0
.
35
,η
2
=
0
.
35
,γ
= 0
.
5
,σ
= 0
.
25
"
ë
ê
X
k
C
z
§
,
Š
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²
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ã
1
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2
x
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q
∗
1
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∗
2
)
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m
t
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§
3
v
k
Ä
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x
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ã
2
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n
^
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§
0
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35
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Ý
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ü
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º
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Ã
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i
3
º
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x
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Å
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x
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º
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ë
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x
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q
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2
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±
a
q
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3
n
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‚
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x
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5
§
0
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51
§
0
.
52
ž
•
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2
x
ü
Ñ
q
∗
1
-
‚
§
γ
Œ
ž
§
•
`
2
x
ü
Ñ
§
g
3
'
~
§
=
x
ú
i
º
x
ž
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ï
2
x
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4
2
x
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i
«
ú
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õ
º
x
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ã
4
n
^
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‚
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«
º
x
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ê
γ
=0
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5
§
0
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51
§
0
.
52
ž
•
`
Ý
]
ü
Ñ
-
‚
§
γ
Œ
§
=
º
x
§
Ý
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ž
§
•
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Ý
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ü
Ñ
π
∗
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x
ú
i
¬
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3
º
x
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þ
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5
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x
ü
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q
∗
1
†
ë
ê
λ
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n
^
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•
λ
= 1
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1
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5
§
2
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Œ
ž
§
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2
x
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q
∗
1
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x
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x
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x
ú
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ï
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2
x
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•
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Ý
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ë
ê
λ
Ã
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ë
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ã
6
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n
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7
§
0
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ž
•
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2
x
ü
Ñ
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1
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Ý
\
'
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~
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DOI:10.12677/pm.2021.11112081862
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Figure1.
Theeffectof
t
onoptimalreinsurancestrategy
ã
1.
t
é
•
`
2
x
ü
Ñ
K
•
Figure2.
Theeffectof
γ
onoptimalreinsurancestrategy
ã
2.
γ
é
•
`
2
x
ü
Ñ
K
•
Figure3.
Theeffectof
γ
onoptimalinvestmentstrategy
ã
3.
γ
é
•
`
Ý
]
ü
Ñ
K
•
.
Figure4.
Theeffectof
σ
onoptimalinvestmentstrategy
ã
4.
σ
é
•
`
Ý
]
ü
Ñ
K
•
Figure5.
Theeffectof
λ
onoptimalreinsurancestrategy
ã
5.
λ
é
•
`
2
x
ü
Ñ
K
•
Figure6.
Theeffectof
α
onoptimalinvestmentstrategy
ã
6.
α
é
•
`
Ý
]
ü
Ñ
K
•
DOI:10.12677/pm.2021.11112081863
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6.
o
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·
‚
3
þ
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•
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e
ï
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x
ú
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InsuranceMathematicsand
Economics
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[2]Schmidli,H.(2002)OnMinimizingtheRuinProbabilitybyInvestmentandReinsurance.
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DOI:10.12677/pm.2021.11112081866
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DOI:10.12677/pm.2021.11112081868
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DOI:10.12677/pm.2021.11112081869
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e
r
(
T
−
t
)
−
1)
−
γ
4
r
(
e
2
r
(
T
−
t
)
−
1))
α
2
[
b
2
2
−
λ
2
µ
2
11
µ
2
21
b
2
1
]+
1
2
γ
[
(
β
−
r
)
2
σ
2
+
η
2
1
a
2
1
b
2
1
](
T
−
t
)(6.22)
£
6
¤
ò
q
∗
1
= 1
,q
∗
2
=
e
q
2
,π
∗
“
\
(4.5)
Ú
(4.2)
§
k
(
˙
A
(
t
)+
rA
(
t
))
x
+
˙
B
(
t
)+
A
(
t
)(
αc
−
(1+
η
2
)
αa
2
−
αa
1
−
αλµ
11
µ
21
η
2
a
2
b
2
2
)
−
γ
2
a
(
t
)
2
α
2
[
b
2
1
−
λ
2
µ
2
11
µ
2
21
b
2
2
]+
A
(
t
)
2
2
γa
(
t
)
2
[
(
β
−
r
)
2
σ
2
+
η
2
2
a
2
2
b
2
2
] = 0
,
(6.23)
(˙
a
(
t
)+
ra
(
t
))
x
+
˙
b
(
t
)+
a
(
t
)(
αc
−
(1+
η
2
)
αa
2
−
αa
1
−
αλµ
11
µ
21
η
2
a
2
b
2
2
)
+
A
(
t
)
γa
(
t
)
[
(
β
−
r
)
2
σ
2
+
η
2
2
a
2
2
b
2
2
] = 0
,
(6.24)
a
q
£
1
¤
¥
)
{
§
·
‚
Œ
±
•
`
Š
¼
ê
V
(
t,x
) =
e
r
(
T
−
t
)
x
+(
αc
−
(1+
η
2
)
αa
2
−
αa
1
−
αλµ
11
µ
21
η
2
a
2
b
2
2
)
1
r
(
e
r
(
T
−
t
)
−
1)
−
γ
4
r
(
e
2
r
(
T
−
t
)
−
1))
α
2
[
b
2
1
−
λ
2
µ
2
11
µ
2
21
b
2
2
]+
1
2
γ
[
(
β
−
r
)
2
σ
2
+
η
2
2
a
2
2
b
2
2
](
T
−
t
)(6.25)
£
7
¤
ò
q
∗
1
= 1
,q
∗
2
= 1
,π
∗
“
\
(4.5)
Ú
(4.2)
§
k
(
˙
A
(
t
)+
rA
(
t
))
x
+
˙
B
(
t
)+
A
(
t
)(
αc
−
(1+
η
2
)
αa
2
−
αa
1
−
αλµ
11
µ
21
η
2
a
2
b
2
2
)
−
γ
2
a
(
t
)
2
α
2
(
b
2
1
+
b
2
2
+2
λµ
11
µ
21
)+
A
(
t
)
2
2
γa
(
t
)
2
(
β
−
r
)
2
σ
2
= 0
,
(6.26)
(˙
a
(
t
)+
ra
(
t
))
x
+
˙
b
(
t
)+
a
(
t
)(
αc
−
(1+
η
2
)
αa
2
−
αa
1
−
αλµ
11
µ
21
η
2
a
2
b
2
2
)
+
A
(
t
)
γa
(
t
)
(
β
−
r
)
2
σ
2
= 0
,
(6.27)
a
q
£
1
¤
¥
)
{
§
·
‚
Œ
±
•
`
Š
¼
ê
V
(
t,x
) =
e
r
(
T
−
t
)
x
+(
αc
−
αa
1
−
αa
2
)
1
r
(
e
r
(
T
−
t
)
−
1)+
1
2
γ
(
β
−
r
)
2
σ
2
(
T
−
t
)
−
γ
4
r
(
e
2
r
(
T
−
t
)
−
1)
α
2
(
b
2
1
+
b
2
2
+2
λµ
11
µ
21
)(6.28)
T
½
n
y
²
.
"
DOI:10.12677/pm.2021.11112081870
n
Ø
ê
Æ