Heston模型下最优投资-消费策略选择4

!*:F +*),? ? ￠ O $'P :F 3$P )!5*)>-6.Q N R 2,/S L 2$&T /@U N !3(.N D#@N V WN "

?(A P 5*)>

!!9:-* 5*);"*0"56

!!1;<. ìít O ??ew 2 ?w 4!);Q 85)F6"N

!!=>?@ ò!)HF6("&￡&ó? ?<&?￥&@ABC #s m #.D ' &V "4(@')Y (/S %,/S ;))X)>6N #$4N !!AB2C òN +,-.$/%&@ABCD "EFGHIJ +Q ,N ?!? "!# $"&5*)>&:F !)"))0(55N

N 4:;+.GH K

&!+

!í S ?×R # l !ìííT0)*)56"

PQ XT?.D??%&q & ?ABCD "EFGHIJTU N T?.D??ê? D D ??&X D '5=?x +,-.$/%&@&/k ABCD "EFGHA?t ?ìX÷ D EF',G?$T ×N X ùT × m a@&Fx /0S m ?x '+(4@'.$/"Q (#$\@"1,''4(/!+Q 1"56&kl?ABCD "EFGH ?S m'?x0N

RST +

,-.$/%&*+Q 156*ùT ×*CDGH *EFGH UVWXY Z

F6**=5))N >!!!!!CZ[\] C !!!!!C^7Y )>0)">F:)!5*)>"*)"**)0"*>P H R )*N 6H>H J [N @--/J )>0)">F:)N 5*);*0*6;

&$A_

?×T???#s ABCD "EFGHIJ Y m #.D q'1d áZ TU +)(:,

N +),Tà

?CD "EFTU &?aCD I 'D ? XEF CD g N b {>?&4?Y X d N N +*&a

,k +*&l " ú/AB'CD "EFGH S m'?x0N +5, ?e -?? 'CD "EFTU N +6,

?X÷K ·? a'CDEFTU N +:, ?f R á?μF×'CD "EFTU N +;, ?C D "

EFTU &?aX.D?? ?K#d D & v?RJ ?'mS0N +>,?×T??ü#s CD "EFTU N 3±ABCD "EFTU' &é?a.D D ?x 1'(#`"D#2$',-%&&"# I X ABCDTU dé±1'(#`"D#2$',-%&b {?B¨N +0, ?<

V O %&@!";?yF&t?.'ABCD N +F ,X ="T %&@& ?!";?yF&t?.'ABCD N +H ,X +,-.$/%&@!";?yF&t?.'ABCD N +)*,X +

,-.$/%&@ ?E \]'ABCD ?E & +)),X +$"^,,}~%&@ ?ABCD "EFTU N < " +,-.$/%&@&ABCD "EFG HIJTU N

? Y /0S m ?x '+Q 156&>t R K9?á?0'ax &/k S m ABCD "E FGH'?x0N

#$?￠GH

?a^K'T?kˉ x 6é??X -' ~MN !+&J &5"&?x J !D ?5U -R ?aS K?μF×&D Y? ?>'&X ~MN !+&J &5" ??üd ?,? ? 9)!!" 95!!

"&üd ?,? ? 'h?lm|%&×m 9)&95n !c %!R 1d.D??êüd.D D §?)1d Y? D !s "&dμ!'~?|`!

* 1d| D ! "&dμ!'~?a|:!!"R `!?x 56W `!c L `!W !&) m L n *|? D '}~&`*c )R ?a D 'vw 5=éY T?'&×:!!

"?x @y'+,-.$/%&W :!!"c :!!"+!L 7>,!!""W !7,!! "W 9)!!",&:!*"c N *n

*&!)"

q >Y 1dV m &,

!!"?x @y'

? ? ￠

*:F +

W ,!!"c !4I &,!!""W !7$,!! "W 95!!"&,!*"c ,*n

*&)(4

&&&$Y V' m &?x 54n $5

&w?2*C

*

,5!!"W []!^l &±^K'!

+*&K ,!!"n *R !)"xq !>c *&,!!"| m &(+,-.$/%&k|1'(#`"D#2$',-%&N qr +,-.$/%&? f HY 1'(#`"D#2$',-%&'B¨R

adμ!X D 'CD.p |-!!"&CD I '_X÷|$!!"&(X ? D CD'.p |$!!"I -!!"R O !!"ìàCD I X !dμ'EF~&(￥?CD /EF?&X÷x 6$!!"?x @y'T? >56

W $!!"c -!!"W :!!":!!"7+$!!"I -!!",W `!!"

`!!"

I O !!"W !c +L $!!"7>-!!",!!"I O !!",W !7-!!",!! "W 9)!!"R !5" ￡#$1GH (!%"c !O !%"&-!%"")|?{'&ùù(!%"?"v 1J !

2Y?ó'&?±"??d μ!

+*&x 6(!%"?x @y'K9))"±^K'C m l &

*

C

*+-!!",5

W !^l ytdd ? *5"±^K C m l &*C *

O !!"W !^l ytdd ? *6"x !5"±"(!!"K *1'#0&^K ?{GH a|.R

%$N /O 80;+.K */?+U 8K J 400O /. N *J

?aCD I '4'Y X N +*&C , A ?dμC A?t ,G?$EF T ×R aZ (

!!&,&Q "Ydμ!XC D U EFGH (!!"c !O !!"&-!!"" +,|Q &,!!"c ,d 'S m &×

Z (

!!&,&Q "P2&*

C

*

,T !!;)

!O !!""W !S !)T&",T !C

;5!$!C "",!!"P ,&$!!"[]

PQ &

!6"

q );)!Q

" ;5!Q "Y 5d]à? /è+íj '-'T × m *&'+*&),| m *7m !Y ?$~*;)!Q "Y :ˉEF'T × m *;5!Q "Y :ˉA ?X÷'T × m R ?aEF T × A ?X÷'T ×Y h ? '& 4?Y ú.l S m !

Z !!&,&Q "c -&%('.

Z (

!!&,&Q "&!:" AB'GH (!

U k

Z !!&,&Q "c Z (

!!!&,&Q "R !;"

+)5()6,34&?×T???#s &μk S m Z

!!&,&Q "?x @y'?#) ?#5R &#$?aê!:"x &Z ?"kˉ!Y "? ' m &fd ?"Q

&,']1 "? ' m &(Z ?x @y'+

Q 156-&%('.1Z !7+L Q 7>-!!",!!",Z Q 7)5-5!!",!!"Z Q Q 7+4I &,!!",Z ,7)5

$5,!!"Z ,,7%$-!!",!!"Z Q ,7

&,I !!

;)!O !!""I O !!"Z Q 2c *&!>"R K9

Z !C &,&Q "c !)I &",I !C

;5!Q "R

!0"

)(Z !/Z Q /Z Q Q >áaH )Z ?"!'11¤m &?"Q '11¤m &?"Q ']1¤m &Z ,/Z ,,

>áY ?",'11¤m /]1¤m R

&%$a9'?5

Y +Q 156!>"'0&Y è+/@'- m &??x R K9x !0"R (x !:"??'Z 0?u "9&b 15a(!

U @x ? &

Z !7+L Q 7>-!!!",!!",Z Q 7)5+-!

!!",5,!!"Z Q Q 7+4I &,!!",Z ,7)5

$5,!!"Z ,,7%$-!!!",!!"Z Q ,7

&,I !!;)!O !!!""I O !

!!"Z Q c *&F

)

!*),'!, +,-.$/23?ì?÷?"ùú?üyt

((!!%"Y ABGH &×9!!&Q "c Z !!&Q "c Z (

!!!&Q "R !>"xX -!!!"/O !

!!"dq k A?S &??x @x -!!!"cI >Z Q 7%$Z Q ,Z Q Q

&;Y )!O !

!!""c Z Q &,I !!&!F "!F "x C ?!>"x k

Z !7L Q Z Q 7+4I &,!!",Z ,7)5$5,!!"Z ,,I ,

!!"5Z Q Q

+>Z Q 7%$Z Q ,

,57&,I !!;)!O !!!""I O !

!!"Z Q c

*R !H "

@yi /0!H "x &kl S m'?x0&b 15kl ABCD U EFGH'?x0R

($ U ¤ ￥4

?aT × m ;)!Q "c ;5!Q "c Q /

/&*m /m )&?a6)c &7>$/%/I )&65c />55!)I /

"5&66c L /I !

)I /R |?/0!H "x &T? v 1y [#R

A&#$-!!"?x ù@'B @,,(.@56

-Y !!"7)5

$5+/!%5

I )"7),-5!!"I -!!"6)765c *&-!C "c *&!)*"

(56!)*"'0|)

)" 0n *d &

-!!"c *)I *5,

!C I !"

0)I *)*5

,!C I !

" 0R

5" 0c *d &

-!!"c !*6"5)5

$5+/!%5

I )"7),!C I !"*6

)5

$5+/!%5

I )"7),!C I !"7)R

6" 0m *d &

-!!"c *:I *;,

!C I !"

I 0

@

)I *:*;

,

!C I !"I 0@R

q )

0c !6)"5I 5$5+/!%5

I )"7),65*!*)c 6)7 0+/!%5I )"7),*!*5c 6)I 0+/!%5

I )"7),

**6c 6)+/!%5I )"7),*!*:c 6)7I 0@+/!%5I )"7),*!*;c 6)I I

0@+/!%5I )"7),

R $T?à?)5$5+/!%5I )"7),(*R !)5$5+/!%5I )"7),c *&(%5c )I )/R q |/m )&^ %5

m *R R qr

)5

$5+/!%5I )"7),(*R @y>67 /056!)*"R

)" 0n *d &56!)*"?ka|

W -!!"

)5

$5+/!%5I )"7),-5

!!"I -!!"6)765

cI W !&

!))"

q |0n *&^ )5

$5+/!%5

I )"7),-5!!"I -!!"6)765c *K üd e h u 'n >*)&*5R qr !))"x ?b 15ka|

)-!!"I *)I )-!!"I *[

]

5

W -!!"cI )5$5+/!%5

I )"7),!*)I *5"W !&!)5"

x !)5"üìX +!

&C , / >&!?-!C "c *&×?/v -!!"R 5" 0c *d &56!)*"?ka|

H

)

? ? ￠

*:F +

W -!!"

)5

$5+/!%5I )"7),-5

!!"I -!!"6)765

cI W !R

!)6"

q |0n

*&^ )5

$5+/!%5

I )"7),-5!!"I -!!"6)765c *K n >*6R qr !)6"x ?b 15ka|W -!!"+-!!"I *6

,5cI )5$5+/!%5

I )"7),W !&!):"

x !):"üìX +!

&C , / >&!?-!C "c *&×?/v -!!"R 6" 0m *d &rd )5$5+/!%5

I )"7),-5!!"I -!!"6)765c *K ü?"'a >*:&*;R *17 34&×?/v -!!

"R ) %$G

!!"?x >56G Y !!"7G !!"+667

4-!!",c *&G !C "c )&!);"

(56!);"'0|

G !!"P ,e %1

*

C

!

+66

S4-!N ",W N 2R [\$x !);"?ka|W G !!"

G !!"

cI +6674-!!",W !&x üìX +!&C , / >&!?G !C "c )&×?/v G !!"R

) ($a

3!!&,"P&)

)T /

*

C

!

3X

!;&,"W ;S !)T&")

)T /3X

!!&,"&

!)>"

!3!!

&,"o c 3?x 563!7!66765,"37!4I 6),"3,7)5$5,3,,

7)5/$5!%5I )",!3,"53

7&))I /c *&!)0"

R K93!C

&,"c !)I &")

)I /&(3d !!&,"o c 3d

?x 563d !

7!66765,"3d

7!4I 6),"3d ,

7)5$5,3d ,,7)5

/$5!%5

I )",!3d

,"5c *&3d !C &,"c )&!)F "

?

3d

!!&,"c G !!",-!!",

R !)H "

-!!"/G !!">á?x [#) [#5R

[\$)"êx !)>"ê3d

!C &,"c )R @y°|!)F "x ? &T?a m?\

3c !66765,"37!4I 6),"3,7)5$5,3,,7)5/$5!%5

I )",!3,"53

&(!)0"x ?#|3!7 37&)

)I /c *R ?×!)>"x &!)0"x ?#?

T&

)

)T /

P3!S 3PT&

))T /

3X

S !)T&"

))T /

3X

!S&

)

)T /

*

C

! 3X

!;&,

"W ;S !)T&")

)T / 3X

P T&)

)T /3X

S&)

)T /

*

C

!

3X !;&,

"W ;S !)T&")

)T /+3X

!S 3X

,&!5*"

$q |

T&)

)T /3X

T&)

)T /

*

C

!

3X ;!;&,"W ;PT&)

)T /3X

!!&,"T&)

)T /+3X

!C &,"T3X

!!&,",P

T&)

3X !!

&,"T&)

+)T3X

!!&,",PT&)

&!5)"

^ !5*"x u "!5)"&qr

T&

)

)T /

*

C

!

3X ;!;&,"W ;P&

)

)T /

*

C

!

3X !;&,

"W ;S !)T&")

)T /+3X !S 3X

,& E Y

T&)

)T /

*

C !

+3X ;S 3X

,W ;P !)T&")

)T /+3X !S 3X

,R

!55"

*

5

!*),

'!, +,-.$/23?ì?÷?"ùú?üyt

A U !5)"? &(}H K 3d

!7 3d

c

*&×!)F "x ? R 5"@y/0!)F "x R êR K93d

!C

&,"c )&?a3d

!!&,"c G !!",-!!",

&M ?C ?!)F "xK G Y !!"7G !!"+6674-!!",71-Y !!"7)5

$5+/!%5

I )"7),-5!!"I -!!"6)7652,!!"G !!"c *&!56"

A U !56"x ? &(

G Y !!"7G !!"+6674-!!",c *&-Y !!"7)5

$5+/!%5

I )"7),-5!!"I -!!"6)765c *R 3üx &X[#) [#5q <1/0&qr E ? /k 3d

!!&,"R b 15E ? ìàv 3!!

&,"R Fx 7y6d[#',-&@yE ? /056!H "?&êR K9x !0"&?a56!H "K ù@ax'0

Z !!&,&Q "c K !!&,"Q //

,I !!

&K !C &,"c )&

!5:"

^ &K

Z !c Q //,I !!+K !I !K ,&Z ,c K ,Q //,I !!&Z ,,c K ,Q //,I !!&Z Q c K Q /I

),I !!&Z Q Q c !/I )"K Q /I 5,I !!&Z Q ,c ,I !!Q /I )K ,

&O !!!"c &)

K I )

Q R C ?56!H "K

Q //

,I !{

!!)I /"1))I /K I /)I /7K !I !K 7L /K 7!4I &,!!""K ,7)5$5,!!"K ,,I )5,!!">5//I )K I )5,!!"$5%5

//I )!K ,"5K I ,!!">$%//I

)K }

,c *& xE 8±Q ' 9&kl

!)I /"&))I /K I /)I /7K !I !K 7L /K 7!4I &,!!""K ,7)5$5,!!"K ,,I )5,!!">5//I

)K I )5,!!"$5%5

//I )!K ,"5K I ,!!">$%//I

)K ,c *&!5;"

!5;"x "é:é/0&úAb {:t &a

K !!&,"c +3!!&,",)I /&

!5>"

^

K !c

!)I /"3I /

3!&!K ,c

!)I /"3I /

3,&!K ,,c

!)I /"3I /

+I /!3,"5

3

73,,

,R C ?!5;"x &kl

!)I /

"3I {/

&

))I /

73!I !3)I /7L /3)I /

7!4I &,!!""3,7)5$5,!!"+I /!3,"5373,,

,I )5,!!">5//I )3)I /I )5,!!"$5%5

/!3,"53I ,!!">$%

//I

)3}

,c *R x|;&×!)0"? &Fx [#6×?/k 3!!

&,"&b 15E ? ìàv Z !!&,&Q "R Fx 3

&($±"X÷x 6!5"AB'CDGH|

-!

!!"c %$3,37

>)I []

/

Q &!50"

AB'EFGH|

O !

!!"c &)

)I /3I )!!&,"Q &

!5F "

h?'S m|

Z !!&,&Q "c Q /

/

,I !!

+3!!&,",)I /&

!5H "

)

5

55

? ? ￠ *:F+ q)3!!&,"P&)*C!G!;",-!;",W;S!)T&")G!!",-!!",&

-!!"&G!!">á?x[#) [#5R

3yCZ

+),!?V a B=B

.@-.@#-&)H>H&;)!6")5:0(5;0N

+5,!<=iQ&+T C3]

+6,!]B=DD?C3D Q&R+=TR N=%.@4('@/U,-.4,/.-.A(.,S@,-_$A#$/.A$''@/S WA(fW$f/-+Q,N?(.2,4(.@#('_@/(/#,&)HH6&6!6") 5:)(50>N

+:,!?C]E^^?&<=3Da C3a E3E G V D]N9$A._$'@$-,',#.@$/f@.2.A(/-(#.@$/#$-.-+Q,N Q$&A/('$_,#$/$4@#.2,$A Y&)H0>&)6!5") 5:;(5>6N

+;,!C8E C3?&?V3C^G E Q&DT^V?C N=/.2,@/U,-.4,/.(/W#$/-&4%.@$/4$W,'f@.2.A(/-(#.@$/#$-.-+Q,N D@(4Q<$/.A$'=%"

.@4('&)HH>&6:!)")65H(6>:N

+>,!V C Da+C?Q&+C Da E3]D8N=%.@4('@4%&'-,#$/.A$'$_%$A._$'@$+Q,N?(.2,4(.@#-$_$%,A(.@$/-A,-,(A#2&)HFF&)6!:") ;FF(>*;

+0,!]C=Q N=%.@4('%$A._$'@$-_$A G<%,/-@$/%'(/-&/W,A

+F,!=>?&@AB&C K N=A/-.,@/"T2',/\,#`%&@G!:")

0);(05>N

+H,!′D& w O N+,-.$/T?5=%&@;?yF&t?.'ABCD+Q,N?×m &5*)*&56!5"):)6(:)FN

+)*,@E5&′D N+,-.$/T?5=%&@F"34 %&'ABCD ?E GH+Q,N1om &5**H&5>!:")65(:)N +)), G N+$"^,,}~%&@fK;H's%'CD"EF%&+Q,N q/?#2 &5*):&55!)*")5H(60N

+)5,IJK&LJM N CD vw@'?E ?~%&+Q,N ;￥N S "!g12 $"&5**F&5)!6")6;F(6>*N

+)6, ?O& P&QR&u N F"34 %&@'ABCD ?E GH+Q,N?!? "!# $"&5*);&:0!)");*(;:N

H L;8O/0R.94:;O4.;K?+.:,O L;8+.S+08?84:>404?;8+.1+-N4:;+.D+A40

h C3]9,/S

!@)<&L!G)-!'K.<<6%)B:!&!%N!%O N&-B:O%)-O)&$%1%-3#-%[)L N%!F&$%'&-&0)*)56&?A%-&"

M U:;-/?;)a2,$%.@4('@/U,-.4,/."#$/-&4%.@$/%$'@#@,--,',#.@$/%A$\',4-f,A,-.&W@,W f@.2-.$#2(-.@# _@/(/#@('4(A`,.N E/-.$#2(-.@#_@/(/#@('4(A`,.&(--,.-(A,#$4%$-,W$_A@-`"_A,,(/W A@-`Y(--,.&(/W.2, U$'(.@'@.Y$_.2,A@-`Y(--,.f(-W,-#A@\,W\Y(+,-.$/4$W,'N=%.@4('@/U,-.4,/."#$/-&4%.@$/%$'@#@,-f2@#24(e@4@L,.2,,e%,#.,W W@-#$&/.,W&.@'@.Y$_.,A4@/('f,('.2(/W(##&4&'(.@U,#$/-&4%.@$/f(-_$&/WN1Y-$'U@/S.2,#$A A,-%$/W@/S+(4@'.$/"Q(#$\@"1,''4(/!+Q1",K&(.@$/&#'$-,W"_$A4-$'&.@$/-_$A .2,U('&,_&/#.@$/(-f,''(-.2,@/U,-.4,/."#$/-&4%.@$/%$'@#@,-@/.2,%$f,A&.@'@.Y_&/#.@$/#(-,(A,$\"

.(@/,WN

X4

!`? )5 "