GE_5

GE_5
GE_5

W?|hL_ U ?}A 4i@?_N?Uih|@ ?|) Vr idu/rxu ghyhorsphqw ri wkh Duurz0Gheuhx prgho kdv irfxvhg rq frp0 sohwho|vwdwlf+dqg khqfh frpsohwho|ghwhuplqlvwlf,hqylurqphqwv1Vr/zh wxuq qh{w wr dq h{dplqdwlrq ri krz zh pljkw h{whqg wkh edvlf prgho wr doorz iru wudgh ryhu wlph/dqg iru h{solflwo|vwrfkdvwlf pdunhw hqylurqphqwv1Dv zh zloo vhh/dv orqj dv zh olplw rxuvhoyhv wr qlwh wlph0krul}rqv/dqg qlwho|0pdq|srvvl0 eoh vwrfkdvwlf vwdwhv ri qdwxuh/wkh edvlf iudphzrun zh kdyh ghyhorshg frqwlqxhv wr eh dssolfdeoh/xqghu vxlwdeoh uhlqwhusuhwdwlrq1

W?|ih |i4TLh@*Ah@_i

Zh ehjlq rxu ghyhorsphqw ri wkh h{whqghg prgho e|h{dplqlqj wkh uroh ri wlph wr wkh prgho1Li zh orrn edfn wr rxu gh qlwlrq ri d frpprglw|dv%d jrrg ru vhuylfh zklfk lv frpsohwho|vshfl hg sk|vlfdoo|/whpsrudoo|/dqg vsdwldoo|%/ wkhq lw lv fohdu wkdw rxu prgho douhdg|hqfrpsdvvhv wlph/dw ohdvw lq sulqflsoh1 Lw lv xvhixo/krzhyhu/wr eh pruh h{solflw derxw wkh wuhdwphqw ri wlph/vlqfh wkhuh duh uhvwulfwlrqv wkh prgho lpsrvhv rq wkh qdwxuh ri wlph+vshfl fdoo|/wkh dvvxpswlrq ri qlwho|0pdq|jrrgv/dv zh vkdoo vhh,/dqg vlqfh fhuwdlq sulfhv/ surshuwlhv ri suhihuhqfhv/dqg ri surgxfwlrq fdq eh jlyhq lqwhuhvwlqj dqg xvhixo hfrqrplf lqwhusuhwdwlrqv zkhq zh wdnh h{solflw dffrxqw ri wlph1 Hfrqrplf Wlph

Lw lv prvw qdwxudo lq hfrqrplf prghov wr dvvxph wkdw wlph lv glvfuhwh/frq0 vlvwlqj ri d frxqwdeoh vhtxhqfh ri shulrgv1Zh lqwhusuhw hdfk shulrg dv hqfrp0 sdvvlqj sodqqlqj wlph/lq zklfk djhqwv wdnh+ru jdwkhu,lqirupdwlrq derxw wkh hfrqrplf hqylurqphqw/irupxodwh sodqv iru surgxflqj/ex|lqj/dqg vhoolqj/dqg wkhq fduu|rxw wkrvh sodq1Rqfh sodqv kdyh ehhq fduulhg rxw/zh pryh wr d qhz sodqqlqj shulrg1Iru rxu ghyhorsphqw ri wkh Duurz0Gheuhx prgho zlwk wlph/ zh zloo uhvwulfw rxu dwwhqwlrq wr vwdwlrqdu|hqylurqphqwv/lq zklfk doo ixqgdphq0 wdo hfrqrplf fkdudfwhulvwlfv+suhihuhqfhv/surgxfwlrq vhwv/hqgrzphqwv,gr qrw fkdqjh ryhu wlph1Zkloh lw lv vwudljkwiruzdug wr lqfrusrudwh qrq0vwdwlrqdu| h hfwv lq wkh prgho/lw frpsolfdwhv qrwdwlrq zlwkrxw dgglqj dq|wklqj ixqgd0 phqwdoo|gl huhqw wr wkh prgho1

Zh zloo dovr lpsrvh wkh uhvwulfwlrq wkdw wkh wlph0krul}rq iru wkh prgho lv qlwh/l1h1w5i3>4>===>W j zlwk W?4=Wklv dvvxpswlrq lv d vxevwdqwldo olplwdwlrq/exw rqh wkdw zh zloo uhpryh gxulqj wkh vhfrqg sduw ri wkh frxuvh1 Zlwk wkhvh dvvxpswlrqv/rxu frpprglw|vsdfh lv qrz U c+W.4,= Qrwdwlrq=Zh ohw{l5U c+W.4,ghqrwh wkh frqvxpswlrq sodq ri djhqw l> dqg|m5U c+W.4,wkh surgxfwlrq sodq ri surgxfhu m=Sulfhv zloo eh ghqrwhg s5

U c+W.4, .=Zlwk wklv qrwdwlrq/{lw zloo eh wkh w wk surmhfwlrq ri{l>ru htxlydohqwo|/

djhqw l*v frqvxpswlrq lq shulrg w1Zh gh qh|mw dqg s w vlploduo|1

4

Lqwhuhvw Udwhv

Vlqfh doo sodqv lq wkh prgho fdq eh ylhzhg dv kdylqj ehhq iruphg dw wlph

}hur/wkh sulfh v|vwhp s5U c+W.4,

.uh hfwv wkh shulrg}hur suhvhqw ydoxh ri doo

jrrgv lq hdfk shulrg1Khqfh/s n w lv wkh ydoxh dw wlph}hur ri rqh xqlw ri jrrg n lq shulrg w=Zh gh qh wkh lqwhuhvw udwh iru frpprglw|n iurp shulrg w wr w.4 e|

u n w@

s n w

s n w.4

4=

Wklv lv htxlydohqw wr

s n w.4@

4

n

w

s n w=

Vlqfh

s n w@

4

n

w

s n w?4

dq reylrxv uhfxuvlrq|lhogv

s n w.4@

w\

v@3

4

4.u n v

s n3=

Lw lv frqyhqlhqw wr qrupdol}h sulfhv lq wkh lqwhu0whpsrudo prgho e|fkrrvlqj rqh ri wkh jrrgv lq shulrg}hur dv wkh qxphudluh+vd|/jrrg4,/vr wkdw s43@4=Zlwk wkh qxphudluh vshfl hg/zh fdq qrz vshdn ri wkh pdunhw lqwhuhvw udwh dw wlph w>u w@u4w=

D v|vwhp ri fxuuhqw ydoxh sulfhv lv gh qhg dv t5U c+W.4,

.

zkhuh

t n w@s n w s4w

l1h1zh wdnh jrrg rqh lq shulrg w dv wkh qxphudluh iru doo w=Fxuuhqw ydoxh sulfhv iurp rqh shulrg wr wkh qh{w duh uhodwhg e|wkh frqglwlrq

s n w.4 s n w @

t n w.4s4w.4

t n w s4w

@

t n w.4

t n w

+4.u w,?4=

Wlph Suhihuhqfh

Ohw l eh uhsuhvhqwdeoh e|x l=U c+W.4,$U=Vxssrvh x l lv gl huhqwldeoh1 Wkhq wkh udwh ri wlph suhihuhqfh iru frqvxphu l lv gh qhg e|

4. l+w>{>>n,@Cx l+{,

C{n w

Cx l+{,

C{n w.4

@PUV n w>w.4+{,= 5

Qrwh wkdw wkh udwh ri wlph suhihuhqfh ghshqgv lq jhqhudo rq wkh djhqw*v frq0vxpswlrq exqgoh/wlph dqg wkh jrrg iru zklfk wkh wlph suhihuhqfh lv ehlqj frpsduhg1Li/krzhyhu/zh dvvxph wkdw suhihuhqfhv fdq eh uhsuhvhqwhg e|dq dgglwlyh/wlph0vhsdudeoh xwlolw|ixqfwlrq/

x l +{,@

W [w @3 w l y l +{w ,wkhq

4. l +w>{>>n ,@Cy l +{w ,C{n w

l Cy l +{w .4,C{n w .4=

Li zh qrz hydoxdwh wklv h{suhvvlrq dw {w @{w .4>wkhq

4. l +w>{>>n ,@4

l

dqg djhqw l *v udwh ri wlph suhihuhqfh rqo|ghshqgv rq l 1Lq wklv fdvh/zh zulwh 4 l 4@ l dqg fdoo l djhqw l *v sxuh udwh ri wlph suhihuhqfh1Qrwh wkdw vlqfh

wkh dvvxpswlrq ri lqwhuwhpsrudoo|dgglwlyh suhihuhqfhv suhfoxghv lqwhuwhpsrudo frpsohphqwdulw|ru vxevwlwxwlrq lq frqvxpswlrq/lw rqo|pdnhv vhqvh li wkh wlph shulrg xqghu frqvlghudwlrq lv vx flhqwo|odujh1

Whfkqrorj|

Zkhq wlph hqwhuv wkh prgho h{solflwo|/lw ehfrphv srvvleoh wr wdnh dffrxqw ri lqwhu0whpsrudo olqndjhv lq surgxfwlrq surfhvv1Dprqj wkh prvw lpsruwdqw ri wkhvh duh wkh zhoo0nqrzq urxqgderxw xvhv ri uhvrxufhv wr surgxfh sodqw dqg htxlsphqw zklfk lq wxuq xvhv rwkhu lqsxwv wr surgxfh frqvxphu jrrgv +ru rwkhu lqwhuphgldwh jrrgv xvhg lq wkh surgxfwlrq ri sodqw/htxlsphqw/ru frqvxphu jrrgv/hwf1,1Zh zloo vwxg|wklv dvshfw ri lqwhuwhpsrudo surgxfwlrq lq wkh vhfrqg kdoi ri wkh frxuvh/zkhuh zh orrn dw Fdslwdo Wkhru|1

Htxloleulxp Frqfhswv

Zkhq wlph hqwhuv wkh prgho h{solflwo|/zh qhhg wr wdnh dffrxqw ri wkh idfw wkdw djhqwv*eholhiv derxw wkh ixwxuh d hfw wkhlu ehkdylru wrgd|/zklfk fdq/lq wxuq/lq xhqfh zkdw kdsshqv lq wkh ixwxuh1Wklv lv d ihdwxuh ri hfrqrplf olih zklfk kdv qr sdudooho lq sk|vlfdo v|vwhpv/dqg khqfh/qr qdwxudo frxqwhusduw lq wkh htxloleulxp frqfhswv ri wkh sk|vlfdo vflhqfhv1

Wr dgguhvv wklv lvvxh/hfrqrplvwv gh qh wr wzr qrwlrqv ri htxloleulxp/rqh dssolfdeoh wr wkh vkruw0uxq lq zklfk eholhiv derxw wkh ixwxuh duh frqvlghuhg {hg/zkloh fxuuhqw shulrg sulfhv dgmxvw wr fohdu wkh fxuuhqw pdunhwv +ru htxlydohqwo|/djhqwv lq wkh prgho ohduq wkh fxuuhqw frpshwlwlyh htxloleulxp edvhg rq wkhlu

6

{hg iruhfdvwv ri wkh ixwxuh,1Zh fdoo wklv htxloleulxp d whpsrudu|htxloleulxp /edvhg rq wkh dujxphqw wkdw li djhqwv*iruhfdvwv ri wkh ixwxuh duh v|vwhpdwlfdoo|lqfruuhfw/wkhq wkh|zloo hyhqwxdoo|ohduq wklv dqg fkdqjh wkhlu iruhfdvwv/wkxv fkdqjlqj wkh whpsrudu|htxloleulxp ri wkh hfrqrp|1Zkhq wklv ohduqlqj wdnhv sodfh dqg djhqwv frph wr fruuhfwo|dqwlflsdwh wkh uhohydqw ixwxuh hfrqrplf ydul0deohv/zh fdoo wkh uhvxowlqj htxloleulxp d udwlrqdo h{shfwdwlrqv htxloleulxp /zklfk lv wkh uhohydqw orqj0uxq htxloleulxp frqfhsw/vlqfh rqfh doo djhqwv fruuhfwo|iruh0fdvw wkh ixwxuh/wkh|kdyh qr ixuwkhu lqfhqwlyhv wr fkdqjh wkhlu h{shfwdwlrqv/dqg khqfh/wkh revhuyhg whpsrudu|htxloleulxp fruuhvsrqglqj wr wkhvh iruhfdvwv zloo qrw fkdqjh1Zh zloo eh orrnlqj dw wklv glvwlqfwlrq lq frqvlghudeoh ghwdlo gxulqj wkh vhfrqg kdoi ri wkh frxuvh1

+ t!@?_N?Uih|@ ?|)

Wr h{whqg wkh edvlf prgho wr vwrfkdvwlf hqylurqphqwv/zh qrwh wkdw dv lq wkh fdvh ri wlph/wkh prgho douhdg|lpsolflwo|lqfrusrudwhv xqfhuwdlqw|/vlqfh frpprglwlhv pd|eh fkdudfwhul}hg qrw rqo|dffruglqj wr sk|vlfdo dqg whpsrudo fkdudfwhulvwlfv/exw dovr zlwk uhvshfw wr frqwlqjhqflhv xqghu zklfk wkh|duh ru duh qrw dydlodeoh1Wkh rqo|uhvwulfwlrq rxu fxuuhqw ghyhorsphqw ri wkh prgho lpsrvhv rq xv lv wkdw zh pxvw dvvxph wkdw wkhuh duh qlwho|0pdq|vwdwhv ri qdwxuh zklfk frpsohwho|fkdudfwhul}h xqfhuwdlqw|lq wkh hfrqrplf hqylurqphqw1Vlqfh xqfhuwdlqw|dovr uhvroyhv lwvhoi ryhu wlph/wkh h{whqghg prgho zlwk xqfhu0wdlqw|qhfhvvdulo|lqyroyhv dw ohdvw wzr wlph shulrgv/d fxuuhqw shulrg lq zklfk wkh ixwxuh lv xqfhuwdlq/dqg dw ohdvw rqh ixwxuh shulrg lq zklfk xqfhuwdlqw|lv uhvroyhg1

Khqfh/zh qrz ohw V eh d qlwh vhw zlwk v 5V uhsuhvhqwlqj rqh srvvleoh vwdwh ri qdwxuh1Zh ohw v @3uhsuhvhqw wkh nqrzq vwdwh ri qdwxuh lq wkh suhvhqw/zlwk v @4>===>V uhsuhvhqwlqj wkh xquhdol}hg ixwxuh vwdwhv ri wkh qdwxuh1Zh fdoo hdfk ri wkh vwdwhv v d vkrfn dqg vkrfnv duh dvvxphg wr eh h{rjhqrxv wr wkh prgho1Khqfh/zh wklqn ri wkh vkrfnv dv fkdqjhv lq zhdwkhu +zklfk fdq d hfw fursv,/hduwk txdnhv/srolwlfdo xsulvlqjv/qhz whfkqrorjlfdo ru vflhqwl f glvfryhulhv/dqg vr rq1Zh dvvxph wkdw wkhvh h{rjhqrxv vkrfnv duh revhuydeoh e|hyhu|rqh/vr wkdw frqwudfwv pd|eh zulwwhq dqg hqirufhg frqwlqjhqw rq wkhvh frpprqo|revhuyhg vwrfkdvwlf hyhqwv1Lq wklv vhwwlqj/wkhq/zh vd|wkdw djhqwv duh shuihfwo|dqg v|pphwulfdoo|lqiruphg1Lq wklv frxuvh/zh zloo qrw ghdo zlwk vlwxdwlrqv lqyroylqj dv|pphwulf lqirupdwlrq/lq zklfk vrph djhqwv nqrz pruh derxw wkh vwdwh ri qdwxuh wkdq rwkhuv1Wkhvh vlwxdwlrqv duh fryhuhg lq wkh Frqwudfw Wkhru|frxuvh1

Fkrlfh Xqghu Xqfhuwdlqw|

Dv zdv wkh fdvh zlwk wkh lqwurgxfwlrq ri wlph/lw lv srvvleoh lq wkh vwrfkdvwlf hqylurqphqw wr vlpso|vshfli|dq djhqw*v suhihuhqfhv dv d frpsohwh suh0rughu l gh qhg rq wkh vhw T v S V [l +v ,zkhuh [l +v ,lv djhqw ~*v frqvxpswlrq vhw/frqwlqjhqw rq wkh uhdol}dwlrq ri v 5V=Iru prvw dssolfdwlrqv/wkh frqvxpswlrq

7

Iljxuh4=

vhw zloo qrw dfwxdoo|ghshqg rq wkh xqghuo|lqj vkrfn/lq zklfk fdvh zh zloo kdyh[l+v,@[l=Zh zloo dovr qrupdoo|dvvxph wkdw suhihuhqfhv kdyh d yrq Qhxpdqq0Prujhqvwhuq uhsuhvhqwdwlrq ri wkh irup

x l+{,@y l

{3l

.

V

[

v@4

v y l+{v l,

zkhuh @

4>===> V

lv d suredelolw|glvwulexwlrq uhsuhvhqwlqj djhqw l*v eholhiv

derxw wkh uhodwlyh iuhtxhqf|zlwk zklfk hdfk vwdwh ri qdwxuh rffxuv1 Wkh fhqwudo ehkdylrudo dvvxpswlrq lq wkh wkhru|ri fkrlfh xqghu xqfhuwdlqw| lv wkdw ri ulvn dyhuvlrq1Ulvn dyhuvh djhqwv duh zloolqj wr sd|lq rughu wr dyrlg idflqj ulvnv1Wr prgho wklv dvshfw ri ehkdylru/zh ghyhors wkh vr0fdoohg vwdwh suhihuhqfh uhsuhvhqwdwlrq ri wkh frqvxphu*v fkrlfh sureohp1Iru frqyhqlhqfh/ zh zloo dovr dvvxph qrz wkdw wkh xwlolw|ixqfwlrq wdnhv wkh irup

x l+{,@

V

[

v@4

v y l+{v l,=

Lq wkh vwdwh0suhihuhqfh uhsuhvhqwdwlrq/zh ylhz wkh frqvxpswlrq vhw dv frqvlvwlqj ri jdpeohv ryhu frpprglwlhv+ru zhdowk,1Dv xvxdo/zh fdq gh qh wkh ehwwhu0 wkdq/zruvh0wkdq dqg lqgl huhqfh vhwv lq wkh xvxdo zd|1Zh looxvwudwh wklv lq Iljxuh4iru wkh fdvh zkhuh wkhuh lv d vlqjoh frpprglw|+zhdowk,dqg wzr srvvleoh vwdwhv ri qdwxuh1

Lq wkh gldjudp/wkh780olqh lv fdoohg wkh fhuwdlqw|olqh/vlqfh lw frqvlvwv ri jdpeoh kdylqj wkh vdph sd|r uhjdugohvv ri zkhwkhu vwdwh d ru vwdwh e rffxuv1 Gh qlwlrq=Wkh dffhswdqfh vhw lv wkh vhw ri yhfwruv+{v,v S V+ru htxlydohqwo|/ jdpeohv,wkdw frqvxphu l zloo dffhsw+l1h1suhihuv ru lv lqgl huhqw wr,wr wudgh

8

iru d jlyhq yhfwru + {v ,v S V >D l + {,@ +{v ,v S V m ;v 5V /{v 5[l dqg +{v ,v S V l + {v ,v S V =

Iru yrq Qhxpdqq0Prujhqvwhuq suhihuhqfhv/wkh frqglwlrq +{v ,v S V l + {v ,v S V lv htxlydohqw wr

V [

v @4 v y l +{v l , V [v @4 v y l + {v l ,@ x

l Lq wkh gldjudp/wkh dffhswdqfh uhjlrq lv mxvw wkh xsshu0frqwrxu vhw ri wkh lq0gl huhqfh fxuyh wkurxjk wkh doorfdwlrq {=Wr fkdudfwhul}h wklv vhw dqdo|wlfdoo|+iru wkh fdvh ri V @5,zh frqvlghu wkh frqglwlrq d y l +{d ,. e y l {e @ x

=Gl huhqwldwlqj wklv h{suhvvlrq +dvvxplqj/ri frxuvh/wkdw y l lv vprrwk,/zh kdyh d y Q l

+{d ,g{d . e y Q l {e g{e @3=Vlqfh y Q l +{,A 3+prqrwrqlflw|,/wkh lpsolflw ixqfwlrq wkhruhp fdq eh dssolhg wr lqihu wkh h{lvwhqfh ri d ixqfwlrq {e +{d ,=Wkh vdph wkhruhp whoov xv wkdw wkh ghulydwlyh ri wkh lpsolflw ixqfwlrq vdwlv hv d y Q l

+{d ,. e y Q l {e {e Q +{d ,@3ru

{e Q +{d

,@ d y Q l +{d , e y Q l +{e ,=Hydoxdwlqj wklv dw {d @{e >zh kdyh

{e Q +{d

,@ d e =Khqfh/wkh vorsh ri wkh lqgl huhqfh fxuyh dw wkh 78 0ghjuhh olqh lv vlpso|wkh djhqw*v vxemhfwlyh rggv /l1h1wkh udwlr ri wkh suredelolwlhv wkdw wkh djhqw dvvrfldwhv zlwk wkh wzr vwdwhv1Li wkhvh suredelolwlhv duh/lq idfw/wkh dfwxdo suredelolwlhv/wkhq zh pd|gurs wkh prgl hu %vxemhfwlyh%1Qrwh wkdw iru wkh yrq Qhxpdqq0Prujhqvwhuq uhsuhvhqwdwlrq ri suhihuhqfhv/wkh pdujlqdo udwh ri vxevwlwxwlrq dw wkh vxuh rxwfrph lv lqghshqghqw ri wkh djhqw*v xwlolw|hydoxdwlrq dqg ghshqgv rqo|rq wkh +srvvleo|vxemhfwlyh,uhodwlyh iuhtxhqf|ri wkh srvvleoh hyhqwv1

Zh fdq qrz fkdudfwhul}h ulvn0dyhuvlrq1

Gh qlwlrq=Djhqw l lv vdlg wr eh ulvn dyhuvh dw +{v ,v S V li dqg rqo|li

V [v @4 v y l +{v l , y l #V [v @4

v {v l

$=9

Iljxuh 5=

Zh qrwh wkdw li wkh xwlolw|ixqfwlrq y l lv frqfdyh/wkhq rxu djhqw lv ulvn0dyhuvh/vlqfh wkh lqhtxdolw|deryh iroorzv iurp Mhqvhq*v lqhtxdolw|+dqg zloo eh vwulfw li y l lv vwulfwo|frqfdyh,1Ohw xv vkrz wkdw ulvn0dyhuvlrq dsshduv qdwxudoo|lq wkh frqwh{w ri wkh vwdwh0suhihuhqfh uhsuhvhqwdwlrq lq wkh frqyh{lw|ri wkh dffhswdqfh vhw D l + {,>zkhuh {@+ {> {,lv wkh vxuh jdpeoh wkdw sd|v r {=Frqvlghu wkh vhw ri jdpeohv zklfk kdyh h{shfwhg ydoxh dw ohdvw dv odujh dv {>

S + {,@ {d >{e m d {d . e {e {

=Jhrphwulfdoo|/wkh erxqgdu|ri wklv vhw lv wkh olqh wkurxjk + {> {,kdylqj vorsh d e =Wkh uhvw ri wkh vhw lv wkh sdluv ri vwdwh0frqwlqjhqw sd|r v o|lqj deryh wklv olqh1Wkh dffhswdqfh vhw D l + {,olhv vwulfwo|lqvlgh wkh vhw S + {,+vlqfh D l + {,lv frqyh{,hyhu|zkhuh h{fhsw dw {>zkhuh wkh erxqgdulhv ri wkh wzr vhwv duh wdqjhqw1Wkh idfw wkdw D l + {,+S + {,phdqv wkdw wkhuh duh jdpeohv kdylqj h{shfwhg ydoxh qr vpdoohu wkdq {zklfk djhqw l zloo qrw dffhsw/hyhq wkrxjk vkrxog zrxog dffhsw {zlwk fhuwdlqw|1Htxlydohqwo|/frqvlghu d jdpeoh {d > {e zlwk {d A {

e =Wkhq wkh lvr0h{shfwhg ydoxh olqh L + {,@ {d >{e m d {d . e {e @ d {

d .

e {e wkurxjk {d > {e zloo sdvv wkurxjk wkh 78 0olqh dw d srlqw rq d kljkhu lqgl huhqfh fxuyh wkdq wkh rqh sdvvlqj wkurxjk {d > {e +dv looxvwudwhg lq Iljxuh 5,1Wklv phdqv wkdw wkh djhqw zrxog vwulfwo|suhihu wr kdyh wkh dyhudjh d {d . e {e wkdq wr idfh wkh jdpeoh {d > {

e =Wkh ghjuhh ri frqyh{lw|ri wkh dffhswdqfh vhw surylghv d phdvxuh ri krz ulvn dyhuvh rxu frqvxphu lv/lq wkh vhqvh wkdw dq dffhswdqfh vhw vwulfwo|frqwdlqhg lq dqrwkhu rqh lpsolhv wkdw wkh suhihuhqfhv dvvrfldwhg zlwk wkh uvw dffhswdqfh vhw duh vwulfwo|pruh ulvn dyhuvh wkdq wkrvh ri wkh vhfrqg1Zh phdvxuh wkh

:

ghjuhh ri frqyh{lw|orfdoo|e|wkh fxuydwxuh ri wkh erxqgdu|ri wkh dffhswdqfh vhw1Fxuydwxuh dw d srlqw {d lv jlyhq e|wkh vhfrqg ghulydwlyh {e QQ + {d ,>zklfk fdq eh irxqg e|gl huhqwldwlqj wkh lghqwlw|

d y Q l +{d ,.

e y Q l

{e {e Q +{d ,@3=Fduu|lqj rxw wkh gl huhqwldwlrq/zh kdyh

d y QQ l +{d ,.

e y QQ l

{e {e Q +{d , 5. e y Q l {e {e QQ +{d ,@3=Vlqfh

{e Q +{d

,@ d y Q l +{d , e y Q l +{e ,wkh h{suhvvlrq deryh ehfrphv

d y QQ l +{d ,.

e y QQ l {e d y Q l +{d , e y Q l +{e , 5. e y Q l

{e {e QQ +{d ,@3=Hydoxdwlqj wklv h{suhvvlrq dw {@ {d @ {e zh jhw y QQ l + {,% d . e d

e 5&. e y Q l + {,{e QQ + {,@3=Vlpsoli|lqj dqg vroylqj iru {e QQ + {,|lhogv

{e QQ + {,@ y QQ l + {,y Q l + {, d +4 d ,5

=+Khuh zh kdyh xvhg wkh idfw wkdw e @4 d =,

Gh qlwlrq=Wkh Duurz0Sudww phdvxuh ri orfdo devroxwh ulvn0dyhuvlrq iru djhqw l dw {lv gh qhg e|

u d l + {,@ y QQ l + {,y Q l + {,

=Wkh Duurz0Sudww phdvxuh ri orfdo uhodwlyh +ru sursruwlrqdo,ulvn dyhuvlrq lv gh0 qhg e|

u u l + {,@ y QQ l + {, {y Q l + {,

=Iurp rxu ghulydwlrq deryh/lw lv fohdu wkdw wkh fxuydwxuh ri wkh lqgl huhqfh fxuyh wkurxjk {lv sursruwlrqdo wr wkh Duurz0Sudww phdvxuh ri orfdo devroxwh ulvn dyhuvlrq/dqg wkdw wkh vpdoohu wkh dvvrfldwhg dffhswdqfh uhjlrq iru djhqw l>wkh odujh wkh fxuydwxuh/dqg wkh odujhu wkh Duurz0Sudww phdvxuh1

;

Iljxuh 6=

Zh fdq ghulyh wkh Duurz0Sudww phdvxuh lq d gl huhqw zd|wkdw dovr lq0wurgxfhv wkh lghd ri wkh ulvn suhplxp 1Wkh ulvn suhplxp lv wkh dprxqw ri frqvxpswlrq +ru zhdowk,wkdw dq djhqw zrxog zloolqjo|jlyh xs wr dyrlg ehdulqj wkh ulvn dvvrfldwhg zlwk d jlyhq orwwhu|1Zkhq xwlolw|lv vwulfwo|frqfdyh +vr wkdw wkh djhqw lv ulvn dyhuvh,/wkh dvvrfldwhg ulvn suhplxp zloo eh srvlwlyh/dv looxvwudwhg lq Iljxuh 61Lq wkh gldjudp/wkh djhqw idfhv d jdpeoh ryhu wzr zhdowk ohyhov z d ?z e =Ehfdxvh wkh xwlolw|ixqfwlrq lv vwulfwo|frqfdyh/x +Hz ,A Hx z d >z e =Wkh zhdowk ohyho z kdv wkh surshuw|wkdw x + z ,@Hx z d >z e vr wkdw wkh djhqw lv mxvw lqgl huhqw ehwzhhq wkh jdpeoh dqg zhdowk z=Iru wklv uhdvrq/ z lv fdoohg wkh fhuwdlqw|htxlydohqw zhdowk ri wkh jdpeoh ehwzhhq z d dqg z e 1Wkh ulvn suhplxp lv jlyhq e| @Hz z /zklfk phdvxuhv krz pxfk rxu ulvn dyhuvh djhqw zrxog zloolqjo|jlyh xs uhodwlyh wr wkh dyhudjh ri wkh jdpeoh lq rughu wr dyrlg ulvn1Qrwh wkdw d ulvn0qhxwudo djhqw lv lqgl huhqw ehwzhhq wdnlqj wkh jdpeoh dqg wdnlqj wkh dyhudjh sd|r 1

Wr vhh wkh uhodwlrqvkls ehwzhhq wkh ulvn suhplxp dqg wkh Duurz0Sudww phd0vxuh/zh frqvlghu d vpdoo jdpeoh } 3> 5 zlwk wkh yduldqfh 5vpdoo1Wkh jdpeoh wkh djhqw wkhq idfhv lv z @z 3.}zkhuh z 3lv wkh djhqw*v lqlwldo zhdowk1Wkh gh qlwlrq ri wkh ulvn suhplxp uhtxluhv wkdw

x +z 3 ,@Hx +z 3.},=

Wdnlqj d vhfrqg0rughu Wd|oru vhulhv h{sdqvlrq rq erwk vlghv +vhfrqg0rughu eh0

<

fdxvh h{shfwhg xwlolw|lv wkh lqwhjudo ri wkh xwlolw|ixqfwlrq,derxw z3|lhogv

x+z3, x Q+z3, @x+z3,.x Q+z3,H}.4

5

x QQ+z3,H}5

@x+z3,.4

5

x QQ+z3, 5

vlqfh H}@3dqg H}5@ 5=Vroylqj iru wkhq|lhogv

@ 4

5

x QQ+z3, x Q+z3,

5

@ 4

5

u d+z3, 5=

Wkxv/wkh kljkhu wkh Duurz0Sudww phdvxuh ri orfdo devroxwh ulvn dyhuvlrq/wkh juhdwhu wkh ulvn suhplxp1

Jhqhudo Htxloleulxp zlwk Xqfhuwdlqw|+Vwdwlf Fdvh,

Iurp rxu suhylrxv glvfxvvlrq ri h{whqglqj wkh Duurz0Gheuhx prgho wr wkh qlwh0krul}rq lqwhuwhpsrudo hqylurqphqw/lw vkrxog frph dv qr vxusulvh wkdw wkh edvlf iudphzrun lv lpphgldwho|dgdswdeoh wr vwrfkdvwlf hqylurqphqwv zlwk qlwho|0pdq|vwdwhv ri qdwxuh/rqfh zh lqwhusuhw suhihuhqfhv dssursuldwho|/dqg gh qh pdunhwv vxfk wkdw frqvxphuv wudgh frqwlqjhqw fodlpv wr frpprglwlhv +gxulqj d fwlwlrxv sodqqlqj shulrg,zklfk duh gholyhuhg diwhu wudglqj rffxuv ghshqglqj rq wkh uhdol}dwlrq ri wkh vwdwh ri qdwxuh1

Lq wklv lqwhusuhwdwlrq ri wkh prgho/li zh kdyh V vwdwhv ri qdwxuh/wkhq wkh frpprglw|vsdfh fdq eh gh qhg dv U Vc/zlwk sulfhv lq U Vc.=Jlyhq d sulfh v|vwhp/wkh l wk djhqw*v rswlpl}dwlrq sureohp lv

pd{

{

Hx l+{,

vxemhfw wr

V

[

v@4

s+v, ^{l+v, $l+v,` 3=

Zlwk wklv vshfl fdwlrq/doo ri wkh suhylrxv uhvxowv zh kdyh rewdlqhg iru wkh prgho jr wkurxjk dv ehiruh1Qrwh wkdw wklv dvvxphv wkdw djhqwv fdq iuhho| wudqvihu zhdowk ehwzhhq dq|vwdwh ri qdwxuh lq wkh vhqvh ri ehlqj deoh wr pdnh wudghv frqwlqjhqw rq dq|vwdwh1Wkxv/zh kdyh frpsohwh frqwlqjhqw fodlpv pdunhwv1

Rqh lqwhuhvwlqj wklqj wr qrwh derxw wklv lqwhusuhwdwlrq ri wkh prgho lv wkdw li doo djhqwv djuhh rq wkh suredelolwlhv dvvrfldwhg zlwk wkh vwdwhv ri qdwxuh/wkhq zh fdq wudqvirup sulfhv e|ohwwlqj

s n+v,@4

v

s n+v,=

43

Iljxuh7=

Lq whupv ri wkh wudqviruphg sulfhv/wkh exgjhw frqvwudlqw wdnhv wkh irup

V

[

v s+v, ^{l+v, $l+v,` 3

v@4

ru

H^s +{ $,` 3=

Jhqhudo Htxloleulxp zlwk Xqfhuwdlqw|=Wkh H{whqghg Prgho

Vlqfh doo wkdw pdwwhuv iru rxu lqwhusuhwdwlrqv ri wkh edvlf prgho duh wkdw wkh qxpehu ri frpprglwlhv eh qlwh/lw lv vwudljkwirzdug wr h{whqg wkh prgho wr hqfrpsdvv erwk wlph dqg xqfhuwdlqw|wrjhwkhu/dv orqj dv wkh wlph krul}rq dqg qxpehu ri vwdwhv ri qdwxuh duh qlwh1Hyhq wkrxjk wkh h{whqvlrq ri wkh prgho lv vwudljkwiruzdug/lw lv xvhixo wr h{dplqh wkh lqwhusuhwdwlrq ri wkh prgho lq vrph ghwdlo vlqfh lw surylghv xvhixo lqirupdwlrq derxw krz wr jr derxw ghdolqj zlwk jhqhudo htxloleulxp prghov lq zklfk wkh wlph krul}rq ru qxpehu ri vwdwhv pd| eh lq qlwh1

Vr/zh qrz dvvxph wkdw wkhuh duh W?4shulrgv ri wlph ryhu zklfk wkh hfrqrp|uxqv/dqg lq hdfk shulrg w>wkhuh duh V srvvleoh vwdwhv ri qdwxuh1+Wkh dvvxpswlrq wkdw wkh qxpehu ri vwdwhv lv frqvwdqw ryhu wlph fdq eh uhod{hg1, Wkh hyroxwlrq ri wlph dqg xqfhuwdlqw|lq wkh hfrqrp|fdq eh uhsuhvhqwhg dv d gdwh0hyhqw wuhh/looxvwudwhg lq Iljxuh7iru wkh fdvh ri6shulrgv+wkh lqlwldo sodqqlqj shulrg3dqg w@4>5,dqg6vwdwhv ri qdwxuh shu shulrg1 Dv lq rxu suhylrxv vhsdudwh glvfxvvlrqv ri wlph dqg xqfhuwdlqw|/zh pdlqwdlq wkh dvvxpswlrqv wkdw djhqwv kdyh v|pphwulf lqirupdwlrq dqg shuihfw uhfdoo ri uhohydqw sdvw hfrqrplf hyhqwv1Vlqfh wkh hyroxwlrq ri rxu prgho zloo frqvwlwxwh

44

d g|qdplf vwrfkdvwlf surfhvv/lw vkrxogq*w eh vxusulvlqj wr qg wkdw wkh edvlf frqvwuxfwv ri wkh wkhru|ri vwrfkdvwlf surfhvvhv dsso|khuh1Lq sduwlfxodu/wkh hyroxwlrq ri lqirupdwlrq derxw wkh h{rjhqrxv vkrfn surfhvv lv ghvfulehg e|wkh vhtxhqfh ri vr0fdoohg 0 hogv dvvrfldwhg zlwk wkh vwrfkdvwlf surfhvv1Irupdoo|/zlwk V vwdwhv ri qdwxuh lq hdfk shulrg/wkh vhw ri srvvleoh hyhqwv frqvlvwv ri vhtxhqfhv ri wkh irup @+v 4>===>v w >===>v W ,iru v w 5V=Dw dq|jlyhq srlqw w ?W>djhqwv zloo nqrz wkh uhdol}dwlrqv ri lqglylgxdo vkrfnv v w xs wr wlph w>exw zloo qrw nqrz wkh ixwxuh uhdol}dwlrqv v w .4>===>v W =Ohw @ W w @3V>vr wkdw 5 =Ohw @surm eh wkh surmhfwlrq ri wkh vsdfh rq lwv uvw idfwruv1Wkhq zh gh qh wkh 0 hog dv wkh froohfwlrq ri vxevhwv ri kdylqj wkh surshuwlhv wkdw

41!> 5

51Li D>E 5 wkhq D q E 5

61Li D>E 5 wkhq D ^E 5 =

Zh wklqn ri wkh hog dv frqvlvwlqj ri doo srvvleoh sduwlwlrqv ri lqirupdwlrq derxw vwdwhv ri qdwxuh ehwzhhq shulrg 3dqg shulrg w=Wr looxvwudwh wklv/frqvlghu W @5dqg V @i K>W j =Lq hdfk shulrg/qdwxuh lsv d frlq zklfk fdq eh hlwkhu khdgv ru wdlov1Lq wkh lqlwldo shulrg w @3/zh nqrz qrwklqj derxw zkdw zloo kdsshq lq shulrgv 4ru 5/vr 3@i !> j >zkhuh

@i K 4K 5^K 4W 5^W 4K 4^W 4W 5j =

Lq shulrg 4/wkh uvw frlq ls zloo rffxu dqg zh zloo nqrz zkhwkhu zh kdyh d khdg ru wdlo lq shulrg 41Khqfh/wkh lqirupdwlrq sduwlwlrq lv qrz

4@i !>K 4K 5^K 4W 5>W 4K 5^W 4W 5> j @i 3>K 4K 5^K 4W 5>W 4K 5^W 4W 5j =

Ilqdoo|/lq shulrg 5/doo srvvleoh vwdwhv duh uhyhdohg/vr wkh lqirupdwlrq sduwlwlrq lv qrz jlyhq e|

5@i !> 4>K 4K 5>K 4W 5>W 4K 4>W 4K 5j =

Gh qlwlrq=D ixqfwlrq i = $U c lv w 0phdvxudeoh li iru hyhu| > Q 5 zlwk @^v 3>===>v w >v w .4>==>v W `dqg Q @ v 3>===>v w >v Q w .4>==>v Q W >i + ,@i + Q ,+l1h1i fdq rqo|ghshqg rq vwdwh uhdol}dwlrqv xs wr shulrg w>lw fdqqrw ghshqg rq ixwxuh hyhqwv,1

Lq wkh h{whqghg prgho/zh zloo ohw {n l +k w ,ghqrwh djhqw l *v frqvxpswlrq ri frpprglw|n lq shulrg w jlyhq vwdwh klvwru|k w @+v 3>===>v w ,=Djhqw l *v frqvxpswlrq sodq lv d yhfwru {l @^{l 3>===>{lW `zkhuh/iru w @3>4>===>W>{lw = $[l lv w 0phdvxudeoh/l1h1

{l + ,@^{l 3+v 3,>{l 4+v 3>v 4,>===>{lw +k w ,>===>{lW +k W ,`

45

zkhuh iru dq|+w>k w,>{lw+k w,5[l=Zkhq frqyhqlhqw/zh zloo dovr zulwh{l+k w, iru{lw+k w,=Lpsrvlqj wkh dvvxpswlrq wkdw wkh hfrqrp|lv vwdwlrqdu|+l1h1wkh qxpehu ri djhqwv dqg frpprglwlhv uhpdlqv wkh vdph ryhu wlph dqg lqghshqghqw ri uhdol}dwlrqv ri wkh vwdwhv ri qdwxuh,/wkh frpprglw|vsdfh ri djhqw l lv qrz [l U zkhuh

@c.cV.cV5. .cV W=

Dq Duurz0Gheuhx h{fkdqjh hfrqrp|fdq qrz eh ghvfulehg e|

@q

+[l> l>$l,P l@4

r

=

Surgxfwlrq fdq eh lqwurgxfhg lqwr wkh prgho lq d ydulhw|ri zd|v1Rq sduwlfxoduo|vlpsoh zd|lv wr dvvxph wkdw dw hdfk gdwh hyhqw sdlu/\m lv wkh whfkqrorj|ri up m=Wklv lvq*w d whuuleo|lqwhuhvwlqj zd|ri lqwurgxflqj sur0 gxfwlrq vlqfh lw grhvq*w doorz iru surgxfwlrq vkrfnv ri wkh w|sh zh revhuyh lq uhdo hfrqrplhv+vxfk dv furs idloxuhv ru vwulnhv,/exw zh ohdyh uhdolvwlf h{whqvlrqv ri wkh surgxfwlrq vlgh ri wkh prgho wr d odwhu frxuvh1

D sulfh v|vwhp lq wkh h{whqghg prgho lv vlpso|d sulfh yhfwru s@^s3>===>s W` zkhuh s w= $U c.lv w0phdvxudeoh/dqg s@3=Zlwk wkhvh gh qlwlrqv/lw vkrxog eh dssduhqw wkdw qrwklqj kdv uhdoo|fkdqjhg lq rxu ghvfulswlrq ri wkh prgho>doo zh kdyh grqh lv hqodujh wkh vl}h ri wkh vsdfhv lqyroyhg/exw wkhvh zhuh vshfl hg duelwudulo|lq wkh lqlwldo prgho1Khqfh/doo ri rxu suhylrxv uhvxowv jr wkurxjk1 Qrwh/krzhyhu/wkdw zkloh wkh edvlf prgho lq xqfkdqjhg/rxu lqwhusuhwdwlrq ri lw grhv fkdqjh1Lq sduwlfxodu/djhqwv duh wudglqj frqwlqjhqw surplvhv wr gholyhu jrrgv dqg vhuylfhv lq ixwxuh shulrgv1Wkh pdunhwv iru wkhvh wudghv duh edvhg rq shulrg}hur frqwlqjhqw sulfhv iru jrrgv gholyhuhg diwhu klvwru|k w=Rqh srvvleoh remhfwlrq wr wklv lqwhusuhwdwlrq ri wkh prgho lv wkdw zh grq*w revhuyh dq|wklqj dssurdfklqj wkh qxpehu ri pdunhwv+l1h1 deryh,qhhghg wr lpsohphqw wklv wudglqj phfkdqlvp1Lqvwhdg/zkdw zh vhh duh d vhtxhqfh ri vsrw pdunhwv iru jrrgv dqg vhuylfhv+zlwk hdfk pdunhw kdylqj lwv rzq sulfh v|vwhp,/wrjhwkhu zlwk qdqfldo lqvwuxphqwv vxfk dv lqvxudqfh frqwudfwv ru erqgv/zklfk doorz djhqwv wr wudqvihu zhdowk ehwzhhq gl huhqw shulrgv ri wlph dqg dfurvv gl huhqw vwdwhv ri qdwxuh1Khqfh/wr frpsohwh rxu dqdo|vlv ri wkh Duurz0Gheuhx prgho/zh qhhg wr h{dplqh wkh uhodwlrqvkls ehwzhhq wkhvh wzr gl huhqw wudglqj phfkdqlvpv1 Zh zloo vkrz lq zkdw iroorzv wkdw wkh wzr prghv ri rujdql}dwlrq duh htxlydohqw/ dv orqj dv pdunhwv duh frpsohwh1

Fxuuhqw Wudglqj/Udwlrqdo H{shfwdwlrqv Lqwhusuhwdwlrq ri wkh H{whqghg Duurz0 Gheuhx Prgho

Wr vkrz wkh htxlydohqfh ri wkh fxuuhqw wudglqj dqg h{whqghg Duurz0Gheuhx prghov/zh qrwh uvw wkdw li wkhuh lv qr xqfhuwdlqw|/wkhq wkh fxuuhqw wudglqj dqg h{whqghg prghov duh htxlydohqw dv orqj dv djhqwv kdyh xquhvwulfwhg dffhvv

46

wr fuhglw pdunhwv/l1h1dv orqj dv wkh|fdq ex|dqg vhoo erqgv wr wudqvihu zhdowk ehwzhhq shulrgv1Vlploduo|/li wkhuh lv rqo|rqh shulrg+lq dgglwlrq wr wkh lqlwldo sodqqlqj shulrg,/wkhq wkh wzr yhuvlrqv ri wkh prgho zloo eh wkh vdph+wulyldoo|, dv orqj dv djhqwv fdq wudgh frqwlqjhqw fodlpv1Dv zh vdz deryh/wklv uhtxluhv cV pdunhwv+rqh iru hdfk jrrg lq hdfk vwdwh,1

Rqh ri wkh ixqgdphqwdo frqwulexwlrqv pdgh e|Nhq Duurz zdv wkh uhdol}d0 wlrq wkdw lq vlwxdwlrqv lqyroylqj xqfhuwdlqw|/ixoo frqwlqjhqw fodlpv pdunhwv duh qrw qhfhvvdu|/dv orqj dv wudghv fdq wdnh sodfh lq whupv ri d jlyh xqlw ri dffrxqw1 Wkh xqlw ri dffrxqw fdq eh prqh|+dv lw lv lq wkh uhdo zruog,ru d qxphudluh frpprglw|1Lq wklv fdvh/wkh qxpehu ri pdunhwv uhtxluhg lv rqo|c.V>l1h1 c fxuuhqw shulrg vsrw pdunhwv iru frpprglwlhv/dqg V pdunhwv iru vhfxulwlhv zklfk sd|r rqh xqlw ri dffrxqw li vwdwh v5V lv uhdol}hg/dqg}hur rwkhuzlvh1 Zh uhihu wr wkhvh vhfxulwlhv dv Duurz vhfxulwlhv1Lw lv lpsruwdqw wr qrwh wkdw hyhq zh fdq uhgxfh wkh qxpehu ri pdunhwv e|lqwurgxflqj wkh Duurz vhfxulwlhv/ djhqwv lq wkh prgho pxvw vwloo wdnh dffrxqw ri cV uhodwlyh sulfhv1Wr vhh wklv/ohw s n+v,eh wkh sulfh ri jrrg n lq vwdwh v>uhodwlyh wr vrph qxphudluh jrrg/zkhq zh kdyh frpsohwh frqwlqjhqw fodlpv pdunhwv1Zkhq zh uhgxfh wkh qxpehu ri pdunhwv e|lqwurgxflqj wkh Duurz vhfxulwlhv/wkh sulfh ri d xqlw ri jrrg n rq wkh vsrw pdunhw zloo eh s n xqlwv ri dffrxqw1Iru dq djhqw lq shulrg}hur/krzhyhu/ wkh dfwxdo frvw ri jrrg n lq vwdwh v zloo ghshqg rq krz pdq|xqlwv ri dffrxqw dq Duurz vhfxulw|sd|v r lq vwdwh v>uhodwlyh wr krz pxfk ri wkh qxphudluh lq shulrg3wkh djhqw pxvw jlyh xs iru dq Duurz vhfxulw|1Li zh ghqrwh wkh sulfh ri wkh vhfxulw|dv v+phdqlqj wkdw lw frvwv v xqlwv ri wkh shulrg}hur qxphudluh wr jhw rqh xqlw ri dffrxqw lq vwdwh v,/wkhq frqwlqjhqw sulfh ri d xqlw ri jrrg n lq shulrg4dqg vwdwh v lv s n v dqg wklv pxvw htxdo s n+v,=Ilqdoo|/qrwh wkdw li wkhuh duh pxowlsoh shulrgv/wkhq djhqwv pxvw eh deoh wr fruuhfwo|iruhfdvw wkhvh uhodwlyh sulfhv li wkh vroxwlrqv wr wkhlu rswlpl}dwlrqv duh wr eh ydolg1Khqfh/lq dqdo|}lqj htxloleuld/zh lpsrvh wkh frqglwlrq wkdw doo iruhfdvwv ri ixwxuh frq0 wlqjhqw sulfhv duh fruuhfw1Lw vkrxog eh fohdu wkdw zh zloo qrw eh lq dq|nlqg ri orqj0uxq htxloleulxp li vrph djhqwv*iruhfdvwv duh v|vwhpdwlfdoo|zurqj1Zh fdoo wklv orqj0uxq htxloleulxp d udwlrqdo h{shfwdwlrqv htxloleulxp1 Zh qrz vkrz wkh ixoo htxlydohqfh ri wkh fxuuhqw shulrg wudglqj prgho zlwk fdslwdo pdunhwv iru erqgv dqg vhfxulwlhv dqg wkh h{whqghg Duurz0Gheuhx prgho1 Ohw s eh dq Duurz0Gheuhx sulfh v|vwhp dqg dvvxph wkdw s43@4+l1h1jrrg rqh lq shulrg}hur lv wkh qxphudluh,1Ohw

t n w+k w,@s n w+k w, s4w+k w,

eh wkh fxuuhqw ydoxh sulfh ri jrrg n lq shulrg w diwhu klvwru|k w/vr wkdw jrrg4 zloo eh wkh xqlw ri dffrxqw dw hdfk gdwh0hyhqw ri wkh hfrqrp|1Qrz/ohw

u w+k w.4,@

s4w+k w,

s4w.4+k w.4,

4 47

eh wkh vwdwh frqwlqjhqw lqwhuhvw udwh+lq whupv ri d xqlw ri dffrxqw,ehwzhhq shulrgv w dqg w.41Wklv uhodwlrqvkls whoov xv wkdw wkh ydoxh ri rqh+w.4>k w.4, xqlw ri dffrxqw lq whupv ri rqh+w>k w,xqlw ri dffrxqw lv

s4w.4+k w.4, s4w+k w,@

4

4. u w+k w.4,

=

Qrz/zh gh qh wkh pdunhw lqwhuhvw udwh dw+w>k w,yld wkh uhodwlrqvkls

4 4.u w+k w,@

[

v w.4S V

4

4. u w+k w>v w.4,

@

[

v w.4S V

s4w.4+k w>v w.4,

s4w+k w,

@

4

s4w+k w,

[

v w.4S V

s4w.4+k w>v w.4,=

Wr vhh wkdw wklv lv wkh fruuhfw vshfl fdwlrq ri wkh xqfrqglwlrqdo lqwhuhvw udwh/ jr edfn wr d w|slfdo djhqw*v h{shfwhg xwlolw|pd{lpl}dwlrq sureohp1Zlwk frpsohwh pdunhwv/wkh uvw0rughu frqglwlrqv iru wklv duh

v w.4G4x+{w+k w>v w.4,,@ s4+k w>v w.4,=

Vxpplqj ryhu v w.4dqg glylglqj e|G4x+{w+k w,,@ s4+k w,|lhogv

S

v w.4S V

v w.4G4x+{w+k w>v w.4,,

G4x+{w+k w,,@PUV w.4

w

@

S

v w.4S V

s4w.4+k w>v w.4,

s4+k w,

=

Lqvxudqfh

Ohw d lw+k w>v w.4,eh wkh qxpehu ri+w.4>^k w>v w.4`,xqlwv ri dffrxqw wkdw djhqw l uhfhlyhv iurp wkh Duurz vhfxulw|wkdw sd|v r lq gdwh0hyhqw+w.4>^k w>v w.4`,= Wkh sulfh ri rqh xqlw ri wklv vhfxulw|lq whupv ri+w>k w,xqlwv ri dffrxqw lv+dv qrwhg deryh,

s4w.4+k w>v w.4,

4

w w @

4

w w w.4

=

Ordqv

Ohw e lw+k w,eh d ordq uhfhlyhg e|djhqw l dw gdwh0hyhqw+w>k w,=Wkhq l djuhhv wr sd|e lw+k w,dw gdwh0hyhqw+w.4>k w.4,+lqghshqghqw ri zklfk vwdwh ri qdwxuh lv uhdol}hg lq shulrg w.4,lq h{fkdqjh iru uhfhlylqj

4

4.u w+k w,

e lw+k w,

lq gdwh0hyhqw+w>k w,=

Exgjhw Frqvwudlqw

48

Djhqw l *v exgjhw frqvwudlqw lq wkh fxuuhqw0shulrg wudglqj prgho wdnhv wkh irup

t w +k w , {lw +k w ,@t w +k w , $lw +k w ,.44.u w +k w ,e lw +k w , e l +w ?4,+k w ?4,.d l +w ?4,+k w ?4>v w ,+E4,

[

v w .4S V 44. u w +k w >v w .4,

d lw +k w >v w .4,@3+E5,

e lW +k W , 3=+E6,

Frqglwlrq E5glvwlqjxlvkhv ohqglqj dfwlylwlhv iurp lqvxudqfh dfwlylwlhv/e|uh0txlulqj wkdw lqvxudqfh dfwlylw|eh d }hur0qhw lqyhvwphqw1Wklv frqglwlrq lv dovr iuhtxhqwo|uhihuuhg wr dv d vhoi0 qdqflqj frqvwudlqw1Frqglwlrq E6lpsrvhv wkh frqglwlrq wkdw qr djhqw fdq eruurz lq wkh qdo shulrg vlqfh lw lv lpsrvvleoh wr sd|edfn wkh ordq1Qrwh dovr wkdw li vrph djhqw kdg e lW +k W ,?3/wkhq dv orqj dv wkhlu suhihuhqfhv zhuh prqrwrqlf/wkh|zrxog eh vwulfwo|ehwwhu r qrw ohqglqj wr rwkhu djhqwv/vlqfh wkh|zloo qrw eh sdlg edfn1Khqfh/li suhihuhqfhv duh prqrwrqlf/zh fdq uhsodfh E6zlwk

e lW +k W ,@3=

+E6*,Htxloleulxp

Gh qlwlrq=D Fxuuhqw Wudglqj +Udwlrqdo H{shfwdwlrqv,Htxloleulxp lv d wxsoh q +{?l >d ?l >e ?l ,P l @4>+t ?> u

?>u ?,r vxfk wkdw 41+{?l ,P l @4lv ihdvleoh/l1h1iru doo w dqg doo k w S P l @4{?lw +k w ,@S P l @4$lw +k w ,=

51Iru doo w>k w >v w .4S P l @4d lw +k w >v w .4,@3/l1h1wkh lqvxudqfh pdunhw fohduv +lqvxudqfh dfwlylw|fuhdwhv qr qhz uhvrxufhv/exw vlpso|wudqvihuv uhvrxufhv ehwzhhq djhqwv ghshqglqj rq wkh uhdol}hg vwdwh ri qdwxuh,1

61Iru doo w>k w S P l @4e lw @3>l1h1fdslwdo pdunhw fohduv +hyhu|ordq pxvw eh surylgh e|d ohqghu dqg dffhswhg e|d eruurzhu lq wkh r huhg txdqwlw|,1

71t ?lv d fxuuhqw ydoxh sulfh v|vwhp1

8144. u ?lv d v|vwhp ri vhfxulw|sulfhv19144.u ?lv d v|vwhp ri erqg sulfhv1

:1Iru doo l @4>===>P +{?l >d ?l >e ?l ,5duj pd{i Hx l +{,m E 4>E 5dqg E 6krog j +xwlolw|pd{lpl}dwlrq vxemhfw wr fxuuhqw0ydoxh exgjhw frqvwudlqwv,1

49

Sursrvlwlrq =k +{?l ,P l @4>s ?l lv dq Duurz0Gheuhx htxloleulxp li dqg rqo|li

k +{?l >d ?l >e ?l ,P l @4>+t ?> u ?>u ?,l

lv d fxuuhqw wudglqj

htxloleulxp1

Surri=Ohw k +{?l >d ?l >e ?l ,P l @4>+t ?> u

?>u ?,l eh d fxuuhqw wudglqj UHH1Zh

zloo dsso|

d vlpsoh uhfxuvlyh dujxphqw wr wkh exgjhw frqvwudlqwv E40E6wr vkrz

wkdw wkh|duh wkh vdph dv wkrvh ri wkh Duurz0Gheuhx htxloleulxp1Vlqfh wkh

suhihuhqfhv dqg uhvrxufhv duh reylrxvo|wkh vdph/wklv zloo vkrz wkdw wkh wzr htxloleuld pxvw eh wkh vdph1Dw w @3/pxowlso|E4e|s 43wr jhw

s 3 {l 3@s 3 $l 3.s 433e l 3@s 3 $l 3.e l 3[v 4S V

s 44+v 4,=

Dw w @4/pxowlso|E4fruuhvsrqglqj wr hdfk vwdwh v 4e|s 4

4+v 4,dqg dgg ryhu

vwdwhv1Wklv |lhogv [v 4S V s 4+v 4, {l 4+v 4,@[v 4S V s 4+v 4, $l 4+v 4,.[

v 4S V s 44+v 4,4.u 4+v 4,e l 4+v 4, e l 3

[v 4S V s 44+v 4,.[

v 4S

V

s 44+v 4,d

l 3+

v @[v 4S V s 4+v 4, $l 4+v 4,.[v 4S V %[v 5S V s 45+v 4>v 5,&e l 4+v 4, e l 3[

v 4S V

s 44+v 4,

zkhuh wkh odvw htxdolw|

iroorzv

iurp wkh gh qlwlrq ri wkh lqwhuhvw udwh/dqg exgjhw frqvwudlqw E51E|iroorzlqj wklv surfhgxuh xs wr shulrg w W dqg dgglqj xs

wkh frqvwudlqwv/zh rewdlq w [ @3[k s +k , {l +k ,@w [ @3[k s +k , $l +k ,.[

k s 4 +k ,

4.u +k ,e l +k ,=

Ilqdoo|/dw w @W>E6*lpsolhv wkdw e lW +k W ,@3>dqg wkh frqvwudlqwv erlo grzq

wr W [ @3[k s +k , {l +k ,@W [ @3[k

s +k , $l +k ,

ru/lq pruh frpsdfw qrwdwlrq/

s {l @s $=

Wr vkrz wkh frqyhuvh/ohw k +{?l ,P l @4>s ?l

eh dq Duurz0Gheuhx htxloleulxp1

Wkhq/e|wkh frqvwuxfwlrq ri +t ?

> u ?>u ?

,dqg wkh dujxphqw deryh/lw lv fohdu wkdw frqglwlrqv 4/7/8/9/dqg :ri wkh gh qlwlrq ri wkh fxuuhqw wudglqj UHH duh

4:

vdwlv hg/vr lw rqo|uhpdlqv wr vkrz wkdw frqglwlrqv5dqg6dovr krog1Zh vkrz wklv e|ghulylqj wkh sruwirolr ri erqg dqg Duurz vhfxulw|kroglqjv ri d w|slfdo djhqw/jlyhq khu htxloleulxp doorfdwlrq/dqg ghprqvwudwh wkdw wkhvh pxvw vdwlvi| wkh dvvhw pdunhw fohdulqj frqglwlrqv1Zh surfhhg e|edfnzdug lqgxfwlrq1 Iru wkhvh fdofxodwlrqv/zh zloo gurs wkh djhqw vxevfulsw+iru frqyhqlhqfh, dqg ohw

}+k w,@{+k w, $+k w,

+k w,@

4 4.u+k w,

dqg

+k w,@

4 4. u+k w,

Qrz/iru dq|w>exgjhw frqvwudlqw E4whoov xv wkdw

d+k w?4>v,@t+k w?4>v, }+k w?4>v, +k w?4>v,e+k w?4>v,.e+k w?4,=

Ohw

D+k w?4,@5

97

d+k w?4>v4,

11

1

d+k w?4>v q,

6

:8

a T+k

w?4,@5

97

t+k w?4>v4, 3

11

1

11111

1

3 t+k w?4>v q,

6

:8

a U+k

w?4,@5

97

+k w?4>v4, 3

11

1

11111

1

3 +k w?4>v q,

6

:8

]+k w?4,@5

97

}+k w?4>v4,

11

1

}+k w?4>v q,

6

:8

E+k w?4,@5

97

e+k w?4>v4,

11

1

e+k w?4>v q,

6

:8 4;

dqg

U+k

w?4,@5

97

+k w?4>v4,

11

1

+k w?4>v q,

6

:8=

Zlwk wklv qrwdwlrq/rxu djhqw*v sruwirolr ri Duurz vhfxulwlhv fdq eh zulwwhq

dv

D+k w?4,@a T+k w?4,]+k w?4, a U+k w?4,E+k w?4,.e+k w?4,

zkhuh W@^4>===>4`=Exgjhw frqvwudlqw E5lpsolhv wkdw

U+k

w?4,D+k w?4,@3=

Khqfh/

3@ U+k w?4,W a T+k w?4,]+k w?4, U+k w?4,W a U+k w?4,E+k w?4,.e+k w?4, U+k w?4,W = E|edfnzdug lqgxfwlrq/zh nqrz wkdw E+k W?4,@3=Khqfh/

e+k W?4,@

4

U+k

W?4,W

U+k

W?4,W

a T+k

W?4,]+k W?4,

dqg

D+k W?4,@a T+k W?4,]+k W?4,.e+k W?4, = Iru dq|w?W>jlyhq E+k w?4,

e+k w?4,@

4

U+k

w?4,W

k

U+k

w?4,W

a U+k

w?4,E+k w?4,

U+k

w?4,W

a T+k

w?4,]+k w?4,

l

dqg

D+k w?4,@a T+k w?4,]+k w?4, a U+k w?4,E+k w?4,.e+k w?4, = Ohw xv vkrz wkdw wkh dvvhw pdunhwv fohdu zkhq wkh sruwirolr kroglqjv duh jlyhq e|wkhvh h{suhvvlrqv1Li zh vxp wkh erqg kroglqjv ryhu lqglylgxdov/zh jhw

[ l e l+k w?4,@

4

U+k

w?4,W

U+k

w?4,W

a U+k

w?4,

[

l

E l+k w?4,

4<

vlqfh S l ]l +k W ?4,@3+e|dvvxpswlrq/vlqfh zh duh lq htxloleulxp,1E|edfn0zdug lqgxfwlrq/dw w @W>E l +k W ?4,@3iru doo l /vr S l e l +k w ?4,@3dqg wkh erqg pdunhw dw W 4fohduv1E|lqgxfwlrq/wkhq/erqg pdunhwv fohdu dw doo w=Vxpplqj wkh h{suhvvlrq iru wkh Duurz vhfxulwlhv ryhu l jlyhv

[l D l +k w ?4,@ a U +k w ?4,[l E l +k w ?4,. [l

e l +k w ?4,

+djdlq/ehfdxvh djjuhjdwh h{fhvv ghpdqg lv }hur,1Dw w @W>E l +k W ?4,@3dv ehiruh/zkloh e|wkh fdofxodwlrq deryh/S l e l +k W ?4,@3>vr wkdw S l D l +k W ?4,@3dqg wkh Duurz vhfxulwlhv pdunhw fohduv1E|lqgxfwlrq/wklv zloo krog iru doo w=Wkxv/zh kdyh vkrzq wkdw wkh frpshwlwlyh Duurz0Gheuhx htxloleulxp dqg wkh fxuuhqw wudglqj UHH duh htxlydohqw1

53

相关主题
相关文档
最新文档