$6HTXHQWLDO)RUPXODWLRQIRU&RPSRVLWLRQDO5HVHUYRLU6LPXODWLRQ
8VLQJ3HQJ5RELQVRQ(TXDWLRQRI6WDWH
$6KDKUDEDGL%'DELU
&KHPLFDO(QJLQHHULQJ'HSDUWPHQW$PLU.DELU8QLYHUVLW\RI7HFKQRORJ\7HKUDQ,UDQ
$EVWUDFW
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
.H\ZRUGV &RPSRVLWLRQDO 5HVHUYRLU VLPXODWRU 6OLP WXEH (TXDWLRQ RI VWDWH 3HQJ
5RELQVRQ
(PDLO DBVKDKUDEDGL#\DKRRFRP ,QWURGXFWLRQ
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
DQG5VVROXWLRQJDVRLOUDWLRZKLFKDUHRQO\
IXQFWLRQV RI SUHVVXUH :KHQHYHU LW LV LQDS
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
SUHGLFWLRQ RI WKH YDSRU OLTXLG HTXLOLEULXP EHWZHHQ RLO DQG JDV *HQHUDOO\ WKLV LV GRQH E\XVLQJDQHTXDWLRQRIVWDWH5HFHQWO\FXELF HTXDWLRQVRIVWDWHVXFKDV3HQJ5RELQVRQDQG 6RDYH 5HGOLFK .ZRQJ DSSHDU WR EH PRUH SRSXODUIRUWKHFRUUHODWLRQRIIOXLGSURSHUWLHV )RU FRPSRVLWLRQDO VLPXODWLRQ WZR EDVLF VR OXWLRQ PHWKRGV KDYH EHHQ SURSRVHG 7KHVH PHWKRGV DUH WKH ³1HZWRQ5DSKVRQ´ DQG ³1RQ 1HZWRQ5DSKVRQ´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
XQNQRZQV 3 SUHVVXUH6R RLO VDWXUDWLRQ
DQG6Z ZDWHUVDWXUDWLRQ+DYLQJWKHVHSDUD
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
&RPSRVLWLRQDOPRGHOHTXDWLRQV
7KH VWDUWLQJ SRLQW IRU WKH FRPSRVLWLRQDO IRUPXODWLRQ LV WKHPRODUFRQWLQXLW\ HTXDWLRQ IRUDQ\FRPSRQHQWLDQGZDWHU
1 L L QF
L W T
w
w
1 QF
Q W T F
w
w
L
RI UHVHUYRLU YROXPH DQG 1QF LV WKH PROHV
RIZDWHUSHUXQLWRIUHVHUYRLUYROXPHUHODWHG WRWKHSKDVHPRODUGHQVLWLHVVDWXUDWLRQVDQG FRPSRVLWLRQDVIROORZV
1L I [R R6 [L [J J6 \L
F
Q Z Z
1 I[ 6
L QF
T T LV WKH UDWH RI WKH DFFXPXODWLRQ RI FRPSRQHQW L ZDWHUHTXDOWRWKHGLYHUJHQFH RI WKH IOX[ RI FRPSRQHQWL SOXV WKH UDWH RI LQMHFWLRQRUSURGXFWLRQ7KXV
UJ UR
L R L R J L J L
R J
NN NN
[ \ T
T [ [
P P ª º « ) ) » « » ¬ ¼ F UZ
Q Z Z Z
Z NN T T [ P ª º « ) » ¬ ¼ ,QWKHDERYHHTXDWLRQV
R 3R JR K
)
J 3R 3FRJ JJ K
)
Z 3R 3FRZ JZ K
)
M M
F
J
& M R J Z J
J U
7KH PRODU FRQWLQXLW\ HTXDWLRQ IRU WKH K\ GURFDUERQ V\VWHP LV REWDLQHG E\ VXPPLQJ
HTXDWLRQRYHUQFK\GURFDUERQFRPSRQHQWV
1 W T w w ,QWKLVHTXDWLRQ
1 I [R R6 [J J6
UJ UR
R R J J W
R J
NN
NN T
T [ [
P P ª º « ) ) » « » ¬ ¼ F Q W L L
T
¦
T0XOWLSO\LQJ (TXDWLRQV DQG E\ JULG EORFN YROXPH 9E DQG ZULWLQJ LQ ILQLWH GLIIHUHQFHIRUP
Q Q Q Q Q Q
E
L L R L R R
Q Q Q Q Q
J L R FRJ R L E
9 1 1 7 [ 3 K
W
7 \ 3 3 K T 9
J J ' ' ' ' ' ' ' '
Q Q Q Q Q QE
Z Z Z R FRZ Z Z E
9
1 1 7 3 3 K T 9
W J ' ' ' ' '
Q Q Q Q Q
E
R R R
Q Q Q Q
J R FRJ R W E
9 1 1 7 3 K
W
7 3 3 K T 9
J J ' ' ' ' ' ' ' ' ,QWKHDERYHHTXDWLRQV UM M M W NN $
& DQG & DUH FRQYHUVLRQ FRQVWDQWV WKDW
GHSHQGRQWKHXQLWVLQYROYHG
,W LV DVVXPHG WKDW JDV DQG RLO DUH LQ SKDVH HTXLOLEULXPDWWKHHQGRIHDFKWLPHVWHS7KLV PHDQV WKDW WKH LQWHUSKDVH WKHUPRG\QDPLF H[FKDQJH LQ WKH UHVHUYRLU LV UDSLG FRPSDUHG ZLWK IOXLG IORZ 7KH FRQGLWLRQ IRU WKHU PRG\QDPLFHTXLOLEULXPLVHVWDEOLVKHGE\WKH HTXDOLW\ RI WKH RLO DQG JDV SKDVH FKHPLFDO SRWHQWLDORUIXJDFLW\IRUHDFKFRPSRQHQW
R J
L L L L
I 7 3 [ I 7 3 \
)URP D PDWHULDO EDODQFH RQ RLO DQG JDV SKDVHVWKHIROORZLQJHTXDWLRQLVREWDLQHG
L L L F
)] /[ 9\ L Q
7KHIROORZLQJHTXDWLRQVDUHXVHGIRUVROYLQJ HTXDWLRQ F Q L L [
¦
F Q L L \¦
F Q L L L L ] . 9 .¦
7KH GHILQLWLRQ RI VDWXUDWLRQ JLYHV WKH FRQ VWUDLQWHTXDWLRQ
R J Z
6 6 6
(DFKJULGEORFNKDVLWVRZQVHWRIWKHDERYH HTXDWLRQV7KHEDVLFLGHDXVHGLQWKLVZRUNLV WR UHGXFH WKHQXPEHURI XQNQRZQV LQWR RQH XQNQRZQIRUHDFKJULGEORFNLQWHUPVRIJULG EORFNSUHVVXUH
&RQVWUXFWLRQRISUHVVXUHHTXDWLRQ
$GGLQJ HTXDWLRQV DQG UHVXOWV LQ WKH SUHVVXUHHTXDWLRQ
Q Q Q Q Q
E
Z Z R R R
Q Q Q Q Q Q Q
J R FRJ J Z R FRJ
9 1 1 1 1 7 3 K
W
7 3 3 K 7 3 3
J J ª º ' ' ' ¬ ¼ ' ' ' ' ' ' ' '
QZ K 9 T TE W Z
J
'
7KH IROORZLQJ DSSUR[LPDWLRQV DSSO\ WR WKH IORZWHUPV
Q O
M M
7 7 M R J Z
Q O L L [ [ Q O L L \ \
O GHQRWHV WKH ODVW LWHUDWLRQ 7KH SUHVVXUH HTXDWLRQ LV VROYHG E\ XVLQJ WKH LWHUDWLYH EL FRQMXJDWH JUDGLHQW PHWKRG IRU VSDUVH PDWUL FHV>@
&RPSRVLWLRQDQGVDWXUDWLRQHTXDWLRQV
7KH PROH IUDFWLRQ RI HDFK FRPSRQHQW LV FDOFXODWHG DW HDFK LWHUDWLRQ E\ XVLQJ WKH SUHVVXUH RI WKH VDPH LWHUDWLRQ )LUVW WKH QXPEHURIPROHVRIFRPSRQHQWLLVFDOFXODWHG IURPHTXDWLRQDQGWKHQWRWDOPROHVZLOOEH REWDLQHG IURP HTXDWLRQ DQG ]L REWDLQHG
E\WKHUDWLRRI1LWR1
O Q Q Q Q Q Q
L R L R R J L
E
W
1 7 [ 3 K 7 \
9 J
' ª' ' ' '
¬
Q Q Q QR FRJ J W E L
3 3 J K T 9 º 1
' ' ' ¼
D
O Q Q Q Q Q
R R R J R
E
W
1 7 3 K 7 3
9 J
' ª' ' ' ' '
¬
Q Q Q
FRJ J W
3 J K T 9 º 1
' ' ¼ E O O L L O 1 ] 1 F
ZDWHU VDWXUDWLRQ LV REWDLQHG IURP HTXDWLRQ
Q Z E Z Q Z Q FRZ Q R Q Z E O Z 1 9 T K 3 3 7 9 W 1 » ¼ º « ¬ ª ' ' ' ' ' J O O Z Z O R 1 6 I[ $IWHUFDOFXODWLQJFRPSRVLWLRQVIODVKFDOFXOD WLRQLVSHUIRUPHGRQFDOFXODWHGFRPSRVLWLRQV LQ HDFK JULG EORFN DW EORFN SUHVVXUH DQG WHPSHUDWXUHWRREWDLQ/DQG97KHQJDVDQG RLOVDWXUDWLRQVDUHFDOFXODWHGDV O O R Z OJ O O
J R J
9 6 6 9 [ [ [ [
O O O
R J R
6 6 6
:KHUH
J J
R R J J
6 9 6 6 [ [ [ 6ROXWLRQ3URFHGXUH
7KH IROORZLQJ SURFHGXUH WR VROYH WKH DERYH
HTXDWLRQVRYHUDWLPHVWHSPXVWEHXVHG 6ROYLQJWKHSUHVVXUHHTXDWLRQHTXDWLRQ
E\ XVLQJ WKH LWHUDWLYH ELFRQMXJDWH JUDGLHQWPHWKRGIRUVSDUVHPDWUL[>@ 2EWDLQLQJ ZDWHU VDWXUDWLRQ HTXDWLRQ
DQG FRPSRVLWLRQ RI HDFK FRP SRQHQW HTXDWLRQ LQ HDFK JULG EORFN XVLQJ WKH EORFN SUHVVXUHV FRP SXWHGLQ6WHS
3HUIRUPLQJIODVKFDOFXODWLRQRQFDOFXO DWHG FRPSRVLWLRQ LQ 6WHS DW EORFN WHPSHUDWXUH DQG SUHVVXUH LQ HDFK JULG EORFNWRREWDLQ/DQG9
&DOFXODWLQJ RLO DQGJDVVDWXUDWLRQVHT XDWLRQV
8SGDWLQJ7M[L\L1DQG1Z
7KH DERYH VWHSV DUH UHSHDWHG XQWLO FRQYHU JHQFHLVDFKLHYHG
5HVXOWV
D 6OLPWXEHVLPXODWLRQRQHGLPHQVLRQDO FDVH
7KH UHFRYHU\ FXUYHV REWDLQHG LQ WKLV VWXG\ ZHUH GHWHUPLQHG E\ XVLQJ WKH VOLP WXEH DSSDUDWXV7KHVSHFLILFDWLRQVRIWKHDSSDUDWXV DUHJLYHQLQ7DEOH 7DEOH6SHFLILFDWLRQVRIVOLPWXEH ,QLWLDO'LDPHWHUPP /HQJWKP 3RUH9ROXPHFP 3RURVLW\ 3HUPHDELOLW\'DUF\ 3RURXV0HGLD *ODVVEHDGV
7KH WXEH ZDV LQLWLDOO\ VDWXUDWHG ZLWK UHVHU YRLU RLO 2LO ZDV GLVSODFHG E\ JDV DW D FRQVWDQWLQMHFWLRQUDWHRIFPVHF7KHRLO
VDPSOHXVHGLQWKLVZRUNZDVREWDLQHGIURP DQ ,UDQLDQ RLO UHVHUYRLU $OVR D JDV VDPSOH IURPDJDVUHVHUYRLURI,UDQ ZDVXVHGDVWKH GLVSODFLQJDJHQW7KHFRPSRVLWLRQDODQDO\VLV RIRLODQGJDVDVZHOODVWKHFKDUDFWHULVWLFVRI WKHSOXVIUDFWLRQDUHJLYHQLQ7DEOHVDQG %XEEOH SRLQW SUHVVXUH DQG VZHOOLQJ GDWD DUH
DOVR LQFOXGHG $ERXW D SRUH YROXPH RI JDV ZDV LQMHFWHG LQ HDFK WHVW $W WKH HQG RI HDFKVWHSSURGXFHGWKHYROXPHRIRLODQGJDV ZHUH PHDVXUHG 7KH UHFRYHU\ IDFWRU ZDV FDOFXODWHGIURPWKHPHDVXUHGYROXPHVRIWKH SURGXFHGRLODQGJDV7KLVVWXG\LVSHUIRUPHG WR GHWHUPLQH WKH HIIHFW RI LQWHUIDFLDO WHQVLRQ ,)7 RQ VLPXODWLQJ UHVXOWV LQ VOLP WXEH H[SHULPHQWV )LYH GLVSODFHPHQW WHVWV DW GLI IHUHQW SUHVVXUHV ZHUH SHUIRUPHG DQG WKH RLO UHFRYHU\ DV D IXQFWLRQ RI WKH LQMHFWHG SRUH YROXPH ZDV UHFRUGHG 7KH LQIOXHQFH RI LQ
WHUIDFLDO WHQVLRQ RQ WKH UHVXOWV RI WKH GHYH ORSHGPRGHOZDVH[DPLQHGLQWZRVLPXODWLRQ UXQVZLWKGLIIHUHQWUHODWLYHSHUPHDELOLW\FXU YHV7KHILUVWVLPXODWLRQZDVSHUIRUPHGZLWK IL[HG UHODWLYH SHUPHDELOLW\ FXUYHV LP PLVFLEOH FRQGLWLRQ ZKLOH WKH YDULDEOH UHOD WLYHSHUPHDELOLW\FXUYHVDVDIXQFWLRQRIERWK VDWXUDWLRQ DQG LQWHUIDFLDO WHQVLRQ ZHUH XVHG PLVFLEOH FRQGLWLRQ LQ WKH VHFRQG UXQ 7R LQFOXGH LQWHUIDFLDO WHQVLRQ V LQ UHODWLYH SHUPHDELOLWLHV WKH IROORZLQJ FXUYHV ZHUH XVHG>@
7DEOH&KDUDFWHULVWLFVRISOXVIUDFWLRQ
0ROHFXODU:HLJKW
6SHFLILF*UDYLW\
D UR D UJUR UR R
N H N H 6
D UJ D UJUJ UJ J
N H N H 6
,QWKHDERYHHTXDWLRQVU V
V DQGD DJ R DUH
SRVLWLYH QXPEHUV DQG XVHG DV DGMXVWDEOH SDUDPHWHUV,QWHUIDFLDOWHQVLRQEHWZHHQYDSRU DQG OLTXLG SKDVHV DW N3DSVLD DQG R& R) DW HTXLOLEULXP ZDV FRQ
VLGHUHGDVEDVH,)7V*HQHUDOO\LQDWZR SKDVH V\VWHP DV WKH ,)7 EHWZHHQ SKDVHV GHFUHDVHV WKH FXUYDWXUH DV ZHOO DV WKH UHVLGXDOVDWXUDWLRQ GHFUHDVHV ,Q WKH H[WUHPH FDVH RI ]HUR ,)7 WKH UHODWLYH SHUPHDELOLW\ FXUYHV EHFRPH WZR VWUDLJKW OLQHV DQG WKH YDOXH RI WKH UHODWLYH SHUPHDELOLW\ RI HDFK SKDVH ZLOO EH HTXDO WR WKH YDOXH RI LWV VDWXUDWLRQ 7KH IROORZLQJ HTXDWLRQV ZHUH XVHGIRU N NUJ UR
UJ J
N 6
R UR
6 N ¨§ ·¸
© ¹
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
)LJXUH5HFRYHU\FXUYHYV39LQMHFWHGDWSUHVVXUHN3DSVLD
2
LO
5
HF
RY
HU
\
)LJXUH5HFRYHU\FXUYHYV39LQMHFWHGDWSUHVVXUHN3DSVLD
)LJXUH5HFRYHU\FXUYHYV39LQMHFWHGDWSUHVVXUHN3DSVLD
)LJXUH5HFRYHU\FXUYHYV39LQMHFWHGDWSUHVVXUHN3DSVLD
2
LO
5
HFR
YHU
\
39,QMHFWHG
2
LO
5
HFR
YHU
\
39,QMHFWHG
2
LO
5
HFR
YHU
\
)LJXUH5HFRYHU\FXUYHYV39LQMHFWHGDWSUHVVXUHN3DSVLD
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
)LJXUH2LOUHFRYHU\FXUYHVDWGLIIHUHQWSUHVVXUHV
2
LO
5
HF
RY
HU
\
39,QMHFWHG
2
LO
5
HF
RY
HU
\
E$UHDOUXQWZRGLPHQVLRQDOFDVH
7KH SDWWHUQ VKRZQ LQ )LJXUH ZDV FRQ VLGHUHG IRU WKH JDV LQMHFWLRQ SURFHVV 7KLV SDWWHUQ ZDV VLPXODWHGXVLQJDQDUHDO u
JULGV\VWHP7ZRLQMHFWRUVDQGWZRSURGXFHUV ZHUH FRQVLGHUHG 7KH ERWWRP KROH SUHVVXUH IRULQMHFWRUVDQGSURGXFHUVZHUHN3D SVLD DQG N3D SVLD UHVSHFWLYHO\
)LJXUH7KHSDWWHUQXVHGLQDUHDOUXQ
$GGLWLRQDOUHTXLUHGGDWDDUHJLYHQLQ7DEOH
7DEOH5HTXLUHGGDWDIRUDUHDOUXQ
/HQJWKRIUHVHUYRLUPIW
:LGWKRIUHVHUYRLUPIW
7KLFNQHVVPIW
3HUPHDELOLW\PGDUF\
3RURVLW\
,QLWLDO UHVHUYRLU SUHVVXUH N3D
SVLD
5HVHUYRLUWHPSHUDWXUHR&R)
,QLWLDOZDWHUVDWXUDWLRQ
)LJXUHVKRZVWKHJDVVDWXUDWLRQFRQWRXUVDW GD\VDIWHU JDV LQMHFWLRQ ,Q WKLV ILJXUH QR PLVFLEOH UHJLRQ ZDV IRXQG DQG WKHUHIRUH WKHSURFHVVZDVLPPLVFLEOH
)LJXUH *DV VDWXUDWLRQ FRQWRXUV DW GD\VDIWHU
JDVLQMHFWLRQ
7KH SUHVVXUH DQG PHWKDQH PROH IUDFWLRQ GLVWULEXWLRQV RI WKH UHVHUYRLU DUH VKRZQ LQ )LJXUHVDQG7KHVHFXUYHVFRQILUPHDFK RWKHU DQG VKRZ JRRG WUHQGV FRPSDUHG ZLWK DQWLFLSDWHG UHDO UHVHUYRLU EHKDYLRU +HQFH WKH DSSOLFDWLRQ RI WKH GHYHORSHG PRGHO IRU WKHSUHGLFWLRQRIUHVHUYRLUEHKDYLRUVHHPVWR EHVXFFHVVIXO
)LJXUH 3UHVVXUH FRQWRXUV DW GD\V DIWHU JDV
LQMHFWLRQ
)LJXUH 0HWKDQH PROH IUDFWLRQ FRQWRXUV DW GD\VDIWHUJDVLQMHFWLRQ
&RQFOXVLRQ
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
1RPHQFODWXUH
$ DUHDSHUSHQGLFXODUWRIORZIW
%R RLOIRUPDWLRQYROXPHIDFWRUEEO67%
& FRQYHUVLRQIDFWRULQHTXDWLRQ
& FRQYHUVLRQIDFWRULQHTXDWLRQ
) IHHG
M L
I IXJDFLW\ RI FRPSRQHQW L LQ SKDVH M
DWP
J JUDYLWDWLRQDODFFHOHUDWLRQIWV
K GHSWKIW
N DEVROXWHSHUPHDELOLW\PGDUF\
UM
N UHODWLYH SHUPHDELOLW\ RI SKDVH M LP PLVFLEOHFRQGLWLRQ
UM
N UHODWLYH SHUPHDELOLW\ RI SKDVH MPLV FLEOHFRQGLWLRQ
L
. .YDOXHRIFRPSRQHQWL
/ UDWLR RI PROHV LQ OLTXLG SKDVH WR WKH WRWDOQXPEHURIPROHVLQWKHPL[WXUH
/W OHQJWKIW
1 PROHVRIWRWDOK\GURFDUERQSHUXQLWRI UHVHUYRLUYROXPHOE PROH
IW
L
1 PROHV RI FRPSRQHQW L SHU XQLW RI
UHVHUYRLUYROXPHOE PROH
IW
F Q
1 PROHV RI ZDWHU SHU XQLW RI UHVHUYRLU
YROXPHOE PROH
IW
3 SUHVVXUH RI PL[WXUH DW HTXLOLEULXP SVLD
M
3 SUHVVXUHRISKDVHMSVLD
FRJ
3 RLOJDVFDSLOODU\SUHVVXUHSVLD
FRZ
3 ZDWHURLOFDSLOODU\SUHVVXUHSVLD
L
T PRODU LQMHFWLRQ SURGXFWLRQ UDWH RI FRPSRQHQW L SHU XQLW YROXPH
OE PROH IW GD\
W
T PRODU LQMHFWLRQ SURGXFWLRQ UDWH RI K\GURFDUERQ SHU XQLW YROXPH
OE PROH IW GD\
Z
T PRODU LQMHFWLRQ SURGXFWLRQ UDWH RI
ZDWHUSHUXQLWYROXPH
OE PROH IW GD\
U UDWLRRI,)7WREDVH ,)7
5V JDVRLOUDWLR6&)67%
M
6 VDWXUDWLRQRISKDVHM
W WLPHGD\
7 WHPSHUDWXUHR5
M
7 WUDQVPLVVLELOLW\RISKDVHM
OE PROH SVLD GD\
9 UDWLR RI PROHV LQ YDSRU SKDVH WR WKH WRWDOQXPEHURI PROHVLQWKHPL[WXUH
E
9 EORFNYROXPH IW
L
[ PROH IUDFWLRQ RI FRPSRQHQW L LQ RLO SKDVH
L
\ PROH IUDFWLRQ RI FRPSRQHQW L LQ JDV SKDVH
L
] PROH IUDFWLRQ RI FRPSRQHQW L LQ K\GURFDUERQ
*UHHN6\PEROV L
T WKHUDWHRIDFFXPXODWLRQRIFRPSRQHQW
L
OE PROH IW GD\
F Q
T WKH UDWH RI DFFXPXODWLRQ RI ZDWHU
OE PROH IW GD\
T WKH UDWH RI DFFXPXODWLRQ RI WRWDO K\GURFDUERQ
OE PROH IW GD\
I SRURVLW\
M
J VSHFLILFZHLJKWRISKDVHM OE
IW
M
P YLVFRVLW\RISKDVHMF3
M
U PDVVGHQVLW\RISKDVHM OE
IW
M
[ PRODUGHQVLW\RISKDVHMOE PROH
IW
' GLIIHUHQFHRSHUDWRU
JUDGLHQWRSHUDWRU
GLYHUJHQFHRSHUDWRU
M
) SRWHQWLDORISKDVHMSVLD
V LQWHUIDFLDOWHQVLRQG\QH FP
V EDVHLQWHUIDFLDOWHQVLRQG\QH FP
5HIHUHQFHV
&DR + ³'HYHORSPHQW RI 7HFKQLTXHV IRU *HQHUDO 3XUSRVH 6LPXODWRUV´ 3K' 'LV VHUWDWLRQ6WDQIRUG8QLYHUVLW\-XQH .D]HPL + 9HVWDO &5 DQG 6KDQN *'
³$Q (IILFLHQW 0XOWLFRPSRQHQW 1XPHULFDO 6LPXODWRU´ 6RF 3HW (QJ - 2FW
)XVVHOO / 7 DQG )XVVHOO ' ' ³$Q ,WHUDWLYH7HFKQLTXHIRU&RPSRVLWLRPO5HVHU YRLU 0RGHOV´ 6RF 3HW (QJ - $XJ
<RXQJ / & DQG 6WHSKHQVRQ 5 ( ³$ *HQHUDOL]HG &RPSRVLWLRQDO $SSURDFK IRU 5HVHUYRLU6LPXODWLRQ´6RF3HW(QJ-2FW
1HORQ - 6 ³1XPHULFDO 6LPXODWLRQ RI &RPSRVLWLRQDO 3KHQRPHQD LQ 3HWUROHXP 5HVHUYRLU´SDSHU63(SUHVHQWHGDWUG
63( 6\PSRVLXP RQ 5HVHUYRLU 6LPXODWLRQ +RXVWRQ7H[DV
$FV * 'ROHVFKDOO 6 DQG )UDNDV ( ³*HQHUDO 3XUSRVH &RPSRVLWLRQDO 0RGHO´ 3DSHU 63( SUHVHQWHG DW WK 63(
6\PSRVLXP RQ 5HVHUYRLU 6LPXODWLRQ 1HZ 2UOHDQV
:DWWV -0 ³$ &RPSRVLWLRQDO )RUPXODWLRQ RI WKH 3UHVVXUH DQG 6DWXUDWLRQ (TXDWLRQV´ 3DSHU 63( SUHVHQWHG DW WK 63(
6\PSRVLXP RQ 5HVHUYRLU 6LPXODWLRQ 6DQ )UDQFLVFR
4XDQGDOOH 3 DQG 6DYDU\ ' ³$Q ,PSOLFLW LQ 3UHVVXUH DQG 6DWXUDWLRQV $SSURDFK WR )XOO\&RPSRVLWLRQDO6LPXODWLRQ´SDSHU63( SUHVHQWHGDW6\PSRVLXPRQ5HVHUYRLU 6LPXODWLRQ+RXVWRQ7H[DV
&HOVR %UDQFR 0 DQG 5RGULJXH] ) ³$ 6HPL,PSOLFLW)RUPXODWLRQIRU&RPSRVLWLRQDO 5HVHUYRLU6LPXODWLRQ´63($GYDQFHG7HFK QRORJ\6HULHV
&RPSRVLWLRQDO 6LPXODWRU IRU /DUJH 6FDOH 5HVHUYRLU 6LPXODWLRQ´ SDSHU 63( SUHVHQWHG DW WKH WK 6\PSRVLXP RQ 5HVHU
YRLU6LPXODWLRQ+RXVWRQ7H[DV -HVVHQ . DQG 2UU -U )0 ³*UDYLW\
6HJUHJDWLRQ DQG &RPSRVLWLRQDO 5HVHUYRLU 6LPXODWLRQ´ SDSHU 63( SUHVHQWHG DW WKH WK V\PSRVLXP RQ LPSURYHG RLO UHFR
YHU\7XOVD2NODKRPD
'DWWD*XSWD $ DQG .LQJ 0- ³$ 6HPL $QDO\WLFDO $SSURDFK WR 7UDFHU )ORZ 0R GHOLQJLQ+HWHURJHQHRXV3HUPHDEOH 0HGLD´ $GYDQFHVLQ:DWHU5HVRXUFHV 7KLHOH05%DW\FN\53%OXQW0-DQG
2UU -U )0 ³6LPXODWLQJ )ORZ LQ +HWHUR JHQHRXV 0HGLD 8VLQJ 6WUHDPWXEHV DQG 6WUHDPOLQHV´ 63( 5HVHUYRLU (QJLQHHULQJ
&UDQH 0 %UDWYHGW ) %UDWYHGW . DQG 2OXIVHQ 5 ³$ )XOO\ &RPSRVLWLRQDO 6WUHD POLQH 6LPXODWRU´ SDSHU 63( SUH VHQWHG DW WKH 63( $QQXDO 7HFKQLFDO &RQ IHUHQFH DQG ([KLELWLRQ 'DOODV 7H[DV
,FKLUR 2 'DWWD*XSWD $ DQG .LQJ0-³7LPH 6WHS 6HOHFWLRQ 'XULQJ 6WUHDPOLQH 6LPXODWLRQ9LD7UDQVYHUVH)OX[&RUUHFWLRQ´ SDSHU 63( SUHVHQWHG DW WKH 63( 5HVHUYRLU 6LPXODWLRQ 6\PSRVLXP +RXVWRQ 7H[DV
6HWR &- -HVVHQ . DQG 2UU -U )0 ³&RPSRVLWLRQDO 6WUHDPOLQH 6LPXODWLRQ RI )LHOG6FDOH&RQGHQVDWH9DSRUL]DWLRQE\*DV ,QMHFWLRQ´SDSHU63(SUHVHQWHGDWWKH 63( 5HVHUYRLU 6LPXODWLRQ 6\PSRVLXP +RXVWRQ7H[DV
1JKLHP /; )RQJ '. DQG $]L] . ³&RPSRVLWLRQDO 0RGHOLQJ ZLWK DQ (TXDWLRQ RI6WDWH´6RF3HW(QJ-'HF 3UHVV :+ 7HXNROVN\ 6$ 9HWWHUOLQJ