D qrwh rq wkh uvw prphqw ri h{wuhph rughu vwdwlvwlfv iurp wkh
qrupdo glvwulexwlrq
e| Pd{ Shw}rog
Ghsduwphqw ri Vwdwlvwlfv/ Vfkrro ri Hfrqrplfv dqg Frpphufldo Odz/
Jùwheruj Xqlyhuvlw|/ Er{ 993/ VH0738 63 JÙWHERUJ/ Vzhghq1 Hpdlo= Pd{1Shw}rogCvwdwlvwlfv1jx1vh
Nh| zrugv dqg skudvhv= dv|pswrwlf> eldv> Fudpìu> h{wuhph> qrupdo> rughu vwdwlvwlf1
Devwudfw
Lq wklv qrwh vrph zhoo nqrzq dv|pswrwlf uhvxowv iru prphqwv ri rughu vwdwlv0 wlfv iurp wkh qrupdo glvwulexwlrq duh wuhdwhg1 Wkh uhvxowv ruljlqdwhv iurp wkh zrun ri Fudpìu1 D eldv fruuhfwlrq iru qlwh vdpsoh vl}hv lv sursrvhg iru wkh h{shfwhg ydoxh ri wkh odujhvw revhuydwlrq1
4 Lqwurgxfwlrq
Ohw t E E? ghqrwh wkh =wk ydoxh iurp wkh erwwrp lq d vdpsoh ri ? l1l1g1 revhu0 ydwlrqv/ vxfk wkdw h1j1 t E? t E? E? lv wkh odujhvw revhuydwlrq1 Wkh s1g1i1 ri t E lv zhoo nqrzq
s t
EE+ ' ?-
E - E? - E8 t E+ 3 E 8 t E+ ?3 s t E+ +4,
exw frxog kdugo| eh xvhg wr qg h1j1 h{shfwhg ydoxhv ri rughu vwdwlvwlfv1 Wr
uhdfk dq dv|pswrwlf uhvxow iru t E? / frqvlghu wkh f1g1i1 8 t
E?E+ ' d8 t E+o ? 1
8 t
E?E+ $
li 8 t E+ ' f li 8 t E+
lw lv reylrxv wkdw vrph wudqvirupdwlrq ri t E? lv qhhghg lq rughu wr dyrlg wulyldo olplwlqj uhvxowv1 Vxfk wudqvirupdwlrqv ri rughu vwdwlvwlfv kdyh ehhq h{whqvlyho| vwxglhg e| h1j1 Iuìfkhw ^5` dqg Jqhghqnr ^6`1 Khuh/ vrph dv|ps0 wrwlf uhvxowv e| Fudpìu ^4` iru qrupdoo| glvwulexwhg revhuydwlrqv duh suhvhqwhg dqg d eldv fruuhfwlrq ri wkh h{shfwhg ydoxh iru qlwh vdpsoh vl}hv lv sursrvhg1
5 Dv|pswrwlf uhvxowv iru wkh qrupdo glvwul0 exwlrq
Dffruglqj wr Fudpìu/ iru ?*2 dqg ? $ 4
t E > n j
% s
2 *? ?
*?E*? ? n *?EeZ 2` E 2 s
2 *? ?
&
+5, zkhuh ` E kdv wkh ghqvlw|
} E '
E? - i T E E? n i T E c 4 4 Vlqfh .
` E?
' E ' fD..22 dqg T
` E?
' E ' Seeb/ vhh Mrkqvrq ^7`/ wkh odujhvw revhuydwlrq kdv wkh prphqwv
. t E?
> n j s
2 *? ? E*?E*? ? n *?EeZ 2 fD..2
2 s 2 *? ?
+6, T
t E?
j 2 Seeb
2 *? ? +7,
Lw lv ri vrph lqwhuhvw wr vwxg| krz wkh uhvxowv lq E2 zhuh rewdlqhg1 Ohw
~ ' E+ > *j dqg ohw ~ E eh wkh fruuhvsrqglqj rughu vwdwlvwlf1 Vwduwlqj
A E ' ?
8 ~
~ E
' ? ] "
~
Es
2Z i T
2 *2 _ ' ' s ?
2Z ] "
~
E
i T
2 *2
_ ' s ?
2Z
~ E n B i T
~ E n B 2
*2
e| wkh phdq ydoxh wkhruhp1 Iurp wklv
~ E n B
fdq eh vroyhg
~ E n B
s
2 *? ? *?E*? ? n *?EeZ n 2 *? A E
2 s 2 *? ?
Surfhhglqj wr wkh jhqhudo fdvh ri wkh qrupdo glvwulexwlrq/ uhsodflqj ~ E zlwk
t E >
*j/ vxevwlwxwlqj ` E iru *? A E dqg qdoo| fkrrvlqj B ' f zh jhw htxdwlrq E21 Dq| rughu vwdwlvwlf fdq eh rewdlqhg lq d vlplodu zd|1 Gxh wr v|pphwu| .
t E
frxog eh irxqg gluhfwo| iurp htxdwlrq E1
6 D eldv fruuhfwlrq ri wkh h{shfwhg ydoxh iru
qlwh vdpsoh vl}hv
Wkh h{dfw dqg wkh dssur{lpdwlyh ydoxh/ uhvshfwlyho|/ ri . t E?
duh jlyhq lq Iljxuh iru t Ef( dqg ? ' Dc Sc c f1 Wkh h{dfw ydoxh zdv vlpxodwhg zlwk wkh vwdwlvwlfdo vriwzduh VDV+u, LPO1 Iru doo vlpxodwlrqv lq wklv sdshu/
wkh frpsxwhu forfn zdv xvhg dv udqgrp vhhg dqg f S uhsolfdwhv zhuh wdnhq1 Wkh dv|pswrwlf uhvxow lq htxdwlrq E zdv xvhg dv dq dssur{lpdwlrq iru qlwh vdpsoh vl}hv1
Dv fdq eh vhhq iurp Iljxuh dqg Iljxuh 2 wkh htxdwlrq E jlyhv d eldvhg
uhvxow iru qlwh vdpsoh vl}hv1 Orrnlqj edfn dw wkh ghulydwlyhv ri htxdwlrq E
lw fdq eh vhhq wkdw wzr dujxphqwv duh pdgh1 Vshfldoo| wkh fkrlfh ri B ' f
wkh eldv jlyhq lq Iljxuh 2=
B E? ' f.2.? 3f2.Db Vxewudfwlqj wklv ixqfwlrq lq htxdwlrq E zh jhw=
. t E?
> n j s
2 *? ? E*?E*? ? n *?EeZ 2 fD..2
2 s
2 *? ? f.2.? 3f2.Db
+8,
Wklv fruuhfwlrq zdv whvwhg iru vrph rwkhu sdudphwhu ydoxhv lq wkh qrupdo glvwulexwlrq dqg vkrzhg xs wr frpshqvdwh wkh eldv hhfwlyho|1 Lq Wdeoh wkh xqfruuhfwhg htxdwlrq E dqg wkh fruuhfwhg htxdwlrq ED duh frpsduhg iru vrph qrupdo glvwulexwlrqv1 Wkh pd{lpdo dqg plqlpdo txrwd ehwzhhq wkh eldv ri wkh xqfruuhfwhg htxdwlrq dqg wkh eldv ri wkh fruuhfwhg htxdwlrq lv jlyhq iru
? ' Dc Sc f1 Dv fdq eh uhdg rxw/ wkh eldv ri wkh xqfruuhfwhg htxdwlrq lv dw ohdvw 2 wlphv odujhu wkdq wkh fruuhvsrqglqj eldv ri wkh fruuhfwhg htxdwlrq1 Dovr wkh yduldqfh lq htxdwlrq Ee lv eldvhg dqg pd| qhhg wr eh fruuhfwhg lq vrph zd|/ exw lv ohiw lq wklv sdshu1
Wdeoh = Wkh pd{lpdo dqg plqlpdo txrwd ehwzhhq wkh eldv ri wkh xqfruuhfwhg htxdwlrq E dqg wkh eldv ri wkh fruuhfwhg htxdwlrq ED iru ? ' Dc Sc f1
Glvwulexwlrq= Ef( Ef( D Ef( 2f
Pd{lpdo ffff fff .eff
Plqlpdo 2 2e 2.
1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60
0 20 40 60 80 100
n Equation (3)
Simulated
Iljxuh = D frpsdulvrq ri wkh vlpxodwhg dqg wkh dssur{lpdwlyh . t E?
0.00 0.02 0.04 0.06 0.08 0.10 0.12
0 20 40 60 80 100
n