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Xqohvv Y +n n 3 > w, kdv vshfldo surshuwlhv zklfk hqdeoh d vlpsolfdwlrq/ Vfkuùglqjhu*v
htxdwlrq iru d vlqjoh sduwlfoh lq wkh zdyhyhfwru uhsuhvhqwdwlrq lv dq lqwhjurglhuhqwldo
htxdwlrq1
Srvwxodwh LY = Hyroxwlrq
51714
<4
Frqvwuxfwlrq ri wkh Kdplowrqldq dqg Rwkhu Revhuydeohv
Lq sulqflsoh/ wkh hyroxwlrq ri d txdqwxp phfkdqlfdo v|vwhp lv uhgxfhg wr wkh vroxwlrq
ri d vhw ri frxsohg uvw rughu glhuhqwldo htxdwlrqv rqfh wkh Kdplowrqldq lv nqrzq1 Dv
lq fodvvlfdo phfkdqlfv/ wkhuhiruh/ wkh uvw vwhs lq vroylqj wkh g|qdplfdo sureohp lv wkh
frqvwuxfwlrq ri d vxlwdeoh Kdplowrqldq1 Lq pdq| fdvhv d Kdplowrqldq rshudwru fdq eh
rewdlqhg iurp wkh Kdplowrqldq ixqfwlrq ri dq dvvrfldwhg fodvvlfdo v|vwhp1 Iru wkh sdu0
dgljpdwlf fdvh ri d vlqjoh vslqohvv sduwlfoh prylqj lq wkuhh glphqvlrqv/ iru h{dpsoh/ zh
pryh iurp wkh fodvvlfdo ghvfulswlrq/ zklfk lv edvhg xsrq wkh g|qdplfdo yduldeohv u dqg s/
wr wkh txdqwxp phfkdqlfdo rqh/ e| uhsodflqj wkh g|qdplfdo yduldeohv e| wkh Khuplwldq
dqg S
/ zkrvh frpsrqhqwv reh| wkh fdqrqlfdo frppxwdwlrq uhodwlrqv
rshudwruv U
^[l > [m ` @ ^Sl > Sm ` @ 3
+5174:,
^[l > Sm ` @ l| lm =
+5174;,
Lq d vlplodu idvklrq/ lw vhhpv uhdvrqdeoh wr dvvrfldwh zlwk dq| fodvvlfdo revhuydeoh D+u> s> w,
S
> w, rewdlqhg e| uhsodflqj wkh g|qdplfdo yduldeohv dsshdulqj lq wkh
dq rshudwru D+U>
ixqfwlrq zlwk wkh fruuhvsrqglqj rshudwruv1 Xqiruwxqdwho|/ zkloh wklv surfhgxuh zrunv d
jrrg ghdo ri wkh wlph/ wkhuh duh flufxpvwdqfhv zkhuh lw fdq jlyh dpeljxrxv uhvxowv dqg2ru
vxhu wkh gudzedfn wkdw wkh rshudwru zklfk lv surgxfhg lv qrw Khuplwldq1 Wr looxvwudwh
wkh edvlf gl!fxowlhv wkdw dulvh/ frqvlghu wkh iroorzlqj fodvvlfdo revhuydeoh
{s{ @ s{ {
+5174<,
zklfk zh fdq zulwh lq hlwkhu ri wkhvh wzr zd|v/ vlqfh fodvvlfdo yduldeohv dozd|v frppxwh
zlwk rqh dqrwkhu1 Txdqwxp phfkdqlfdoo|/ krzhyhu/ wkh rshudwruv rewdlqhg e| uhsodflqj
wkh g|qdplfdo yduldeohv zlwk dvvrfldwhg rshudwruv
[S{ 9@ S{ [
+51753,
duh qrw htxdo ehfdxvh wkhvh rshudwruv gr qrw frppxwh1 D prphqwv uh hfwlrq zloo uhyhdo
wkdw wkhuh duh dfwxdoo| dq lqqlwh qxpehu ri fodvvlfdoo| htxlydohqw s
h{suhvvlrqv wkdw hdfk
jhqhudwh d glhuhqw txdqwxp phfkdqlfdo rshudwru +frqvlghu/ h1j1/ q {q sq @ {s{ , Zklfk
ri wkhvh rshudwruv vkrxog eh xvhg wr uhsuhvhqw wkh fodvvlfdo revhuydeohB Prvw ri wkhp duh
sdwhqwo| xqxvdeoh ehfdxvh wkh| duh qrw Khuplwldq1 Lq wkh vlpsoh h{dpsoh deryh/ h1j1/
qhlwkhu [S{ ru S{ [ lv d Khuplwldq rshudwru/
^[S{ `. @ S{. [ . @ S{ [=
+51754,
dqg vr fdqqrw uhsuhvhqw dq revhuydeoh1
Wr uhvroyh wklv sureohp zh xvh wkh lghd ri Khuplwldq v|pphwul}dwlrq ri dq rshudwru1
Uhfdoo wkdw dq| rshudwru D fdq eh zulwwhq lq wkh irup
D @ DK . DD >
+51755,
zkhuh wkh rshudwru DK @ 45 +D.D. , lv Khuplwldq dqg DD @ 45 +D.D. , lv dqwl0Khuplwldq1
Wkxv/ dq duelwudu| rshudwru fdq eh ghfrpsrvhg lq d xqltxh zd| lqwr Khuplwldq dqg dqwl0
Khuplwldq sduwv lq d zd| dqdorjrxv wr wkh pdqqhu lq zklfk dq duelwudu| frpsoh{ qxpehu
fdq eh ghfrpsrvhg lqwr uhdo dqg lpdjlqdu| sduwv1 Zh uhihu wr DK / wkhuhiruh/ dv wkh
Khuplwldq sduw ri D/ dqg vwlsxodwh wkdw wkh revhuydeoh fruuhvsrqglqj wr dq| fodvvlfdo
Wkh Irupdolvp ri Txdqwxp Phfkdqlfv
<5
txdqwlw| eh dvvrfldwhg zlwk wkh Khuplwldq sduw ri wkh rshudwru rewdlqhg wkurxjk wkh
uhsodfhphqw glvfxvvhg deryh/ l1h1/
D+u> s> w, $
S
> w, . D. +U>
S
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D+U>
=
5
+51756,
Lq wkh sduwlfxodu h{dpsoh frqvlghuhg/ wklv lpsolhv wkdw
4
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5
+51757,
vr wkdw erwk ri wkh rshudwruv lq txhvwlrq duh wuhdwhg rq dq htxdo irrwlqj1 Dq dgglwlrqdo
h{dpsoh zrxog eh
4
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S . S U,=
+51758,
5
Xqiruwxqdwho|/ wklv surfhgxuh rqo| uhgxfhv wkh xqghuo|lqj sureohp/ lw grhv qrw holplqdwh
lw1 Wklv fdq eh vhhq e| frqvlghulqj wkh iroorzlqj htxlydohqw fodvvlfdo h{suhvvlrqv
{5 s{ @ s{ {5 @ {s{ {>
+51759,
wkh uvw wzr ri zklfk duh erwk dvvrfldwhg xqghu wkh deryh suhvfulswlrq zlwk wkh rshudwru
4 5
+[ S{ . S{ [ 5 ,>
5
+5175:,
zkloh wkh wklug ri zklfk lv dvvrfldwhg zlwk wkh rshudwru
[S{ [=
+5175;,
Wklv gl!fxow| lv/ lq d vhqvh/ rqo| pdwkhpdwlfdo/ dqg suhvxpdeo| dulvhv ehfdxvh zh duh
dwwhpswlqj wr jr iurp d ohvv frpsohwh ghvfulswlrq +l1h1/ fodvvlfdo phfkdqlfv, wr d pruh frp0
sohwh ghvfulswlrq +l1h1/ txdqwxp phfkdqlfv, ri wkh sk|vlfdo xqlyhuvh1 Lw lv qrw xquhdvrqdeoh
wr h{shfw wkdw lq dq| grpdlq zkhuh fodvvlfdo ehkdylru lv revhuyhg/ wkh glhuhqfhv ehwzhhq
wkh suhglfwlrqv dvvrfldwhg zlwk dq| ri wkhvh rshudwruv zloo ehfrph xqlpsruwdqw1 Lq wkh
txdqwxp grpdlq/ krzhyhu/ wklv xqghuvfruhv wkh idfw wkdw wkh dvvrfldwlrq ri d phdvxulqj
ghylfh zlwk dq revhuydeoh fdq vrphwlphv lqyroyh vxewoh glvwlqfwlrqv1
Lw vkrxog dovr eh srlqwhg rxw wkdw wklv sureohp uhdoo| rqo| dulvhv lq rshudwruv lqyroylqj
surgxfwv ri qrq0frppxwlqj revhuydeohv1 Lq wkh prvw frpprq vlwxdwlrq/ qdpho| wkdw ri
d sduwlfohv prylqj lq uhvsrqvh wr d fodvvlfdo srwhqwldo ixqfwlrq Y +u> w,/ wkh sureohp qhyhu
dulvhv ehfdxvh vxfk surgxfwv grq*w dsshdu lq wkh Kdplowrqldq1
51715 Vrph Ihdwxuhv ri Txdqwxp Phfkdqlfdo Hyroxwlrq
Ghwhuplqlvp 0 Qrwh wkdw wkh glhuhqwldo htxdwlrq jryhuqlqj wkh hyroxwlrq ri wkh vwdwh
yhfwru lv uvw rughu lq wlph1 Wklv phdqv wkdw wkh vroxwlrq ghshqgv rqo| rq wkh lqlwldo
vwdwh ri wkh v|vwhp/ dqg qrw/ h1j1/ rq lwv lqlwldo udwh ri fkdqjh1 Wkxv/ dq| lqlwldo vwdwh
m#+w3 ,l ri wkh v|vwhp dw wlph w3 zloo hyroyh lqwr d vlqjoh xqltxh yhfwru m#+w,l dw wlph w A w3 1
Zh qrwh wkdw wklv lpsolflwo| ghqhv d pdsslqj ri wkh vsdfh rqwr lwvhoi/ dqg wkxv lpsolhv
wkh h{lvwhqfh ri dq rshudwru X / ru d idplo| ri rshudwruv X +w> w3 ,/ wkdw pds dq duelwudu|
vwdwh dw wlph w3 rqwr wkh vwdwh lqwr zklfk lw hyroyh lq wlph w1 Wklv hyroxwlrq rshudwru
lv ghqhg wkurxjk wkh uhodwlrq
m#+w,l @ X +w> w3 ,m#+w3 ,l=
+5175<,
Srvwxodwh LY = Hyroxwlrq
<6
Olqhdulw| 0 Wkh olqhdulw| ri wkh htxdwlrqv ri prwlrq lpso| d vxshusrvlwlrq sulqflsoh iru
wkh vroxwlrqv ri wkh Vfkuùglqjhu htxdwlrq1 Wkdw lv li m#4 +w,l dqg m#5 +w,l duh wzr srvvleoh
vroxwlrqv wr wkh Vfkuùglqjhu htxdwlrq +zklfk kdyh/ h1j1/ hyroyhg iurp wzr glhuhqw lqlwldo
vwdwh yhfwruv m#4 +w3 ,l dqg m#5 +w3 ,l,/ wkhq wkh wlph0ghshqghqw yhfwru
m#+w,l @ m#4 +w,l . m#5 +w,l
+51763,
lv dovr d vroxwlrq wr wkh Vfkuùglqjhu htxdwlrq iru dq| frpsoh{ frqvwdqwv dqg / vlqfh
l|
g
g
g
m#+w,l @ +l| m#4 +w,l, . +l| m#5 +w,l,
gw
gw
gw
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vr wkdw
l|
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m#+w,l @ Km#+w,l=
gw
+51764,
+51765,
+51766,
Wklv lpsolhv/ dv d frqvhtxhqfh/ wkdw li zh qg rxw krz wkh edvlv yhfwruv ri dq| RQE hyroyh
xqghu wkh Vfkuùglqjhu htxdwlrq/ zh fdq ghwhuplqh wkh hyroxwlrq ri dq| rwkhu yhfwru lq wkh
v|vwhp1 Lw dovr lpsolhv wkdw wkh hyroxwlrq rshudwru X+w> w3 , lqwurgxfhg deryh lv d olqhdu
rshudwru1
Frqvhuydwlrq ri wkh Qrup 0 Lw lv dovr uhodwlyho| hdv| wr vkrz wkdw txdqwxp phfkdqlfdo
hyroxwlrq suhvhuyhv wkh qrup ri wkh vwdwh yhfwru/ d frqglwlrq zklfk lv reylrxvo| lpsruwdqw
li zh zlvk wkh wrwdo vxp ri suredelolwlhv wr eh frqvhuyhg1 Wkxv/ zh frqvlghu wkh udwh ri
fkdqjh ri wkh +vtxduhg, ohqjwk ri d yhfwru m#+w,l hyroylqj xqghu wkh Vfkuùglqjhu htxdwlrq
g
g
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k#+w,m#+w,l @ k#m
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k#m m#l>
+51767,
gw
gw
gw
zkhuh zh kdyh vlpso| xvhg wkh fkdlq uxoh rq wkh uljkw kdqg vlgh1 Iurp wkh Vfkuùglqjhu
htxdwlrq lwvhoi zh ghgxfh wkdw
l
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m#l @ Km#l
+51768,
gw
|
wkh dgmrlqw ri zklfk jlyhv
l
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k#m @ k#mK=
gw
|
+51769,
Vxevwlwxwlqj wkhvh lq deryh zh qg wkdw
l
l
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k#m#l @ k#mKm#l . k#mKm#l @ 3>
gw
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+5176:,
vr wkdw k#+w,m#+w,l @ k#+w3 ,m#+w3 ,l lv frqvwdqw1 Wklv lpsolhv wkdw wkh hyroxwlrq rshudwru
X @ X +w> w3 , wkdw zh ghqhg hduolhu lv xqlwdu|/ vlqfh
k#+w,m#+w,l @ k#+w3 ,mX . X m#+w3 ,l @ k#+w3 ,m#+w3 ,l=
+5176;,
Vlqfh wklv pxvw eh wuxh iru duelwudu| vwdwhv m#+w3 ,l/ lw iroorzv wkdw X . X @ 41 Wklv ihdwxuh
lv dovr ghvfulehg e| vd|lqj wkdw wkh Vfkuùglqjhu htxdwlrq ohdgv wr d xqlwdu| hyroxwlrq1
Wkh Irupdolvp ri Txdqwxp Phfkdqlfv
<7
51716 Hyroxwlrq ri Phdq Ydoxhv
Ohw xv qrz frqvlghu krz wkh phdq ydoxh dvvrfldwhg zlwk dq duelwudu| revhuydeoh D+w,
hyroyhv lq wlph1 Lq jhqhudo/ rshudwruv fdq kdyh dq lqwulqvlf wlph ghshqghqfh1 Dv dq
h{dpsoh frqvlghu wkh srwhqwldo dvvrfldwhg zlwk wkh dssolfdwlrq ri d vsdwldoo| xqlirup
vlqxvrlgdoo|0ydu|lqj hohfwulf hog/ l1h1/
U
frv+$w,1
Yh{w +w, @ hH
+5176<,
Wkh phdq ydoxh ri wklv rshudwru
kYh{w +w,l @ k#+w,mYh{w +w,m#+w,l
+51773,
hyroyhv lq wlph/ vlqfh erwk wkh vwdwh dqg wkh rshudwru lwvhoi lv fkdqjlqj1 Dw dq| lqvwdqw ri
wlph/ wklv phdq ydoxh jlyhv d phdvxuh ri wkh lqwhudfwlrq ri wkh v|vwhp zlwk wkh h{whuqdo
hog1 Lq jhqhudo/ wkh phdq ydoxh ri dq duelwudu| revhuydeoh
kD+w,l @ k#+w,mD+w,m#+w,l
+51774,
pd| kdyh wzr vrxufhv ri wlph ghshqghqfh= d sduw gxh wr wkh rshudwru lwvhoi/ dqg d sduw
gxh wr wkh hyroxwlrq ri wkh v|vwhp1 Zh fdq/ krzhyhu/ xvh wkh fkdlq uxoh wr zulwh
g
g
kD+w,l @ k#+w,mD+w,m#+w,l
gw
gw
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CD
g
@
k#m Dm#l . k#m
m#l . k#mD
m#l =
gw
Cw
gw
+51775,
+51776,
Iurp rxu hduolhu pdqlsxodwlrqv wklv fdq eh zulwwhq
g
l
CD
l
kD+w,l @ k#mKDm#l . k
l k#mDKm#l=
gw
|
Cw
|
+51777,
l
CD
l k#mDK KDm#l
+51778,
Cw
|
lq zklfk zh uhfrjql}h wkh frppxwdwru ri D dqg K1 Wkxv/ zh kdyh wkh htxdwlrq ri prwlrq
@k
CD
l
g
kD+w,l @ k
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gw
Cw
|
+51779,
Wklv irup pd| eh idploldu wr wkh vwxghqw ri fodvvlfdo phfkdqlfv/ lq wkdw lw uhvhpeohv wkh
htxdwlrq ri prwlrq
CD
gD
@
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+5177:,
gw
Cw
iru d fodvvlfdo revhuydeoh D+t> s> w,/ zkhuh wkh eudfnhwhg txdqwlw| uhsuhvhqwv wkh Srlvvrq
eudfnhw ri wkh wzr ixqfwlrqv D+t> s> w, dqg K+t> s> w,/ ghqhg wkurxjk wkh uhodwlrq
SE
ii> jj @
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Dv d frqvhtxhqfh ri wklv htxdwlrq ri prwlrq/ zh qrwh wkdw dq| wlph0lqghshqghqw rshudwru
wkdw frppxwhv zlwk wkh Kdplowrqldq kdv d phdq ydoxh wkdw uhpdlqv frqvwdqw lq wlph/
vlqfh wkh htxdwlrq ri prwlrq wkhq suhglfwv wkdw gkD+w,l@gw @ 31
Srvwxodwh LY = Hyroxwlrq
<8
51717 Hkhuhqihvw*v Wkhruhp
Dv dq lqwhuhvwlqj dssolfdwlrq ri wkh xvh ri wkh htxdwlrqv ri prwlrq iru wkh phdq ydoxh ri dq
+u, @ uY +u,
revhuydeoh/ frqvlghu wkh prwlrq ri d sduwlfoh xqghu wkh lq xhqfh ri d irufh I
ghulydeoh iurp d vfdodu srwhqwldo Y +u,1 Txdqwxp phfkdqlfdoo|/ wklv fruuhvsrqgv wr wkh
xvxdo Kdplowrqldq
S5
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+5177<,
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5p
Wkh fodvvlfdo g|qdplfdo yduldeohv u+w, dqg
s+w, dvvrfldwhg zlwk vxfk d Kdplowrqldq hyroyh
dffruglqj wr Kdplowrq*v htxdwlrqv
s+w,
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+51783,
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+51784,
gw
zklfk duh htxlydohqw wr Qhzwrq*v vhfrqg odz1 Ohw xv qrz frqvlghu krz wkh phdq ydoxhv
kU+w,l
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fkdqjh lq wlph1 Iluvw/ zh h{dplqh wkh htxdwlrq ri prwlrq iru wkh srvlwlrq rshudwru U/
zklfk ehlqj lqghshqghqw ri wlph +C U@Cw @ 3, ohdgv wr wkh htxdwlrq ri prwlrq
l
g
kU+w,l @ k^U>
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Wklv ohdgv xv wr hydoxdwh
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+51785,
+51786,
wkh vhfrqg frppxwdwru ydqlvkhv1 Wkh { frpsrqhqw ri wkh
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uvw frppxwdwru lv
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+51787,
zkhuh zh kdyh uhfrjql}hg wkdw wkh rqo| qrq0frppxwlqj sduw lqyroyhv srvlwlrq dqg pr0
phqwxp rshudwruv dorqj wkh vdph gluhfwlrq1 Xvlqj wkh vwdqgdug wulfn iru hydoxdwlqj wkh
frppxwdwru ri d surgxfw zh qg wkdw
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+51788,
dqg vlploduo| iru wkh rwkhu wzr fduwhvldq frpsrqhqwv ri wkh frppxwdwru lq txhvwlrq1 Dv
d yhfwru rshudwru uhodwlrq/ wkhuhiruh/ zh kdyh wkh uhvxow wkdw
S 5 ` @ 5l|S
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+51789,
zklfk zh fdq sxw edfn lqwr wkh htxdwlrq ri prwlrq iru kU+w,l
wr rewdlq
l 4
4
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kU+w,l @ k5l|S l @ kS l=
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zklfk/ eudfnhwv dvlgh/ orrnv olnh lwv fodvvlfdo frxqwhusduw1 Wkxv/ dv lq fodvvlfdo phfkdqlfv
wkh phdq yhorflw| htxdov wkh phdq prphqwxp glylghg e| wkh pdvv1
Wkh Irupdolvp ri Txdqwxp Phfkdqlfv
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Lq d vlplodu idvklrq zh fdq frpsxwh wkh htxdwlrq ri prwlrq iru wkh phdq prphqwxp/
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kS +w,l @ k^S
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gw
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+51793,
Qrz wkh nlqhwlf hqhuj| whup glvdsshduv/ exw wkh srwhqwldo hqhuj| whup grhv qrw/ vlqfh
zklfk grhv qrw frppxwh zlwk S
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zrun lq d uhsuhvhqwdwlrq lq zklfk Y lv gldjrqdo/ qdpho| wkh srvlwlrq uhsuhvhqwdwlrq1 Lq
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+51795,
> Y ` dfwv wr pxowlso| wkh zdyh ixqfwlrq e| wkh
vr wkdw lq wkh srvlwlrq uhsuhvhqwdwlrq/ ^S
ixqfwlrq l|uY +u, @ l|I +u,/ zkhuh I +u, lv wkh fodvvlfdo irufh ixqfwlrq1 Wkxv/ zh pdnh
wkh lghqwlfdwlrq
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zkhuh
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