6�?�|i�,*i4i?| 4i|�L_t ��|� *LU@*

|h�@?}�*@|�L? hi�?i4i?| uLh UL?|�?�L�t

+i�?uLhUi4i?| wi@h?�?} ThLM*i4t

U�hpl Pxqrv

#�55�NwA��VW�AW��c #BA�#A��,w�,|���@?Uitc.H ^�@� �@hUi* #@tt@�*|c b22�e 5@�?|��*L�_c 6+���,

@?_�,��B+,6c wW5�c �@hU _i AL�h�L�ic�� �2�c b2�HD �?|L?) �i_i c 6+���,

Ai* G Ef�� ef bS S� .b� 6@ G Ef�� ef bS Sf Hf,�4@�*G +i4����?Lt9Ui4@}hiu�uh

�Mt|h@U|� A��t T@Tih Thiti?|t @ hi�?uLhUi4i?| *i@h?�?} @*}Lh�|�4 _i�t�}?i_ uLh tL*��?} LT|�4@* UL?|hL* ThLM*i4t uLh ���U� |�i t|@|i tT@Ui @?_|�i |�4i @hi UL?|�?�L�t �@h�@M*it� w�!i #)?@4�U �hL}h@44�?} 4i|�L_tchi�?uLhUi4i?| *i@h?�?} |iU�?�^�it }i?ih@|i @? LT|�4@* uii_�M@U! TL*�U)M) |�i 4i@? Lu |�i �@*�i u�?U|�L? ���U� it|�4@|it |�i Mit| i TiU|@|�L? LuU�4�*@|��i hi�@h_ @t @ u�?U|�L? Lu �?�|�@* t|@|i� A�i @*}Lh�|�4 ThLTLti_�ihi �tit �?�|i�i*i4i?|t 4i|�L_t uLh @TThL �4@|�?} |��t u�?U|�L?� W| �tUL4TLti_ Lu |�L _)?@4�Ut G |�i *i@h?�?} _)?@4�Utc U@**i_6�?�|i�,*i4i?|

+i�?uLhUi4i?| wi@h?�?}c ���U� it|�4@|it |�i �@*�it @| |�i �ih|�Uit Lu @|h�@?}�*@|�L? _i�?i_ �TL? |�i t|@|i tT@Uic @?_ |�i t|h�U|�h@* _)?@4�Utc���U� hi�?it |�i |h�@?}�*@|�L? �?t�_i hi}�L?t ��ihi |�i �@*�i u�?U|�L? �t�hhi}�*@h� A��t 4it� hi�?i4i?| @*}Lh�|�4 �?|i?_t |L tL*�i |�i ThLM*i4Lu |�i UL4M�?@|Lh�@* i T*Lt�L? Lu |�i ?�4Mih Lu �@*�it |L Mi it|�4@|i_� �uLh4@*�t4 uLh hi�?uLhUi4i?| *i@h?�?} �? |�i UL?|�?�L�t U@ti �t ThLTLti_c|�i O@4�*|L?�a@ULM���i**4@? i^�@|�L? �t t|@|i_c |�i? |�i @*}Lh�|�4 �tThiti?|i_ @?_ @TT*�i_ |L @ t�4T*i |�L�_�4i?t�L?@* |@h}i| ThLM*i4�

� W?|hL_�U|�L?

Lq wklv sdshu/ zh duh frqfhuqhg zlwk dgdswlyh qrq0olqhdu frqwuro sureohpv/ olnhwdujhw ru revwdfoh sureohpv/ yldelolw| sureohpv ru rswlpl}dwlrq sureohpv/ iruzklfk wkh vwdwh vsdfh dqg wkh wlph duh frqwlqxrxv yduldeohv1 Lq rughu wr gh�qhdq rswlpdo frqwuro/ uhlqirufhphqw ohduqlqj +UO, exlogv wkh ydoxh ixqfwlrq zklfkhvwlpdwhv wkh ehvw h{shfwdwlrq ri ixwxuh uhlqirufhphqw iru doo srvvleoh frqwurovdv d ixqfwlrq ri lqlwldo vwdwh1

Lq wkh frqwlqxrxv fdvh/ wkh ydoxh ixqfwlrq kdv wr eh uhsuhvhqwhg e| d jhq0hudo dssur{lpdwlrq v|vwhp xvlqj d �qlwh vhw ri sdudphwhuv1 Vhyhudo whfkqltxhvkdyh ehhq sursrvhg/ iru h{dpsoh e| xvlqj qhxudo qhwzrunv +vhh ^Edu<3`/ ^Jxo<5`/^Olq<6` dqg pdq| rwkhuv,/ ix}}| frqwuroohuv +vhh ^Qrz<8`, ru rwkhu dssur{lpd0wlrq v|vwhpv +vhh wkh vsduvh0frduvh0frghg FPDFv ri ^Vxw<9`,1 Krzhyhu/ dv lw kdv

ehhq srlqwhg rxw lq ^Edl<8` dqg ^EP<8`/ lq jhqhudo/ wkh frpelqdwlrq ri uhlqirufh0phqw ohduqlqj dojrulwkpv zlwk vxfk ixqfwlrq dssur{lpdwruv grhv qrw frqyhujh1Wkh Uhvlgxdo0judglhqw dgydqwdjh xsgdwlqj sursrvhg lq ^KEN<9` lv d frqyhujhqwdojrulwkp lq wkh vhqvh ri wkh frqyhujhqfh ri judglhqw ghvfhqw phwkrgv1 Exw wkhsureohp zlwk wkhvh phwkrgv lv krz wr �qg d vxlwdeoh dufklwhfwxuh iru wkh qhw0zrun/ l1h1 zklfk shuplwv wr dssur{lpdwh wkh ydoxh ixqfwlrq1 Ehvlghv/ judglhqwghvfhqw phwkrgv rqo| lqvxuh orfdo rswlpxp1

Khuh/ zh suhvhqw d gluhfw uhlqirufhphqw ohduqlqj dojrulwkp wkdw xvhv �qlwh0hohphqw phwkrgv zlwk d orfdo phvk uh�qhphqw surfhvv iru dssur{lpdwlqj wkhydoxh ixqfwlrq1 Wkh dojrulwkp frqvlvwv ri wkh frpelqdwlrq ri wkhvh wzr g|qdplfv =

~ Wkh ohduqlqj g|qdplfv = iru d jlyhq wuldqjxodwlrq ri wkh vwdwh vsdfh/ wkhIlqlwh0Hohphqw Uhlqirufhphqw Ohduqlqj +IHUO, prgl�hv wkh ydoxhv ri wkhyhuwlfhv dffruglqj wr wkh uhlqirufhphqw rewdlqhg gxulqj wkh uxqqlqj ri wudmhf0wrulhv/ vr wkh ydoxh ixqfwlrq lv dssur{lpdwhg zlwk d slhfhzlvh olqhdu ixqfwlrq1Wkh IHUO dojrulwkp xvhg khuh lv d frqyhujhqw UO dojrulwkp +vhh ^Pxq<9`,1

~ Wkh vwuxfwxudo g|qdplfv wkdw orfdoo| uh�qhv wkh phvk ri wkh wuldqjxod0wlrq lq rughu wr exlog dq uhohydqw wuldqjxodwlrq zkrvh suhflvlrq ghshqgv rqwkh uhjxodulwlhv ri wkh ydoxh ixqfwlrq1 Lwv vwduwv zlwk d urxjk wuldqjxodwlrqfrpsrvhg ri d vpdoo qxpehu ri odujh vlpsoh{hv +nqrzohgjh ri d qrylfh, dqgexlogv dq dffxudwh wuldqjxodwlrq +nqrzohgjh ri dq h{shuw, dffruglqj wr wkhohduqlqj g|qdplfv1 Wklv vwuxfwxudo g|qdplfv fdq eh vhhq dv d fdwhjrul}dwlrqsurfhvv yld uhlqirufhphqw iurp wkh hqylurqphqw1

Vhfwlrq 5 lqwurgxfhv d irupdolvp iru wkh vwxg| ri uhlqirufhphqw ohduqlqjlq wkh frqwlqxrxv fdvh1 Vhfwlrq 6 ghvfulehv wkh IHUO dojrulwkp iru d jlyhqwuldqjxodwlrq ri wkh vwdwh vsdfh +l1h1 wkh ohduqlqj g|qdplfv,1 Vhfwlrq 7 suhvhqwvwkh vwuxfwxudo g|qdplfv1 Vhfwlrq 8 looxvwudwhv wklv dojrulwkp zlwk dq h{dpsoh riwdujhw sureohp lq d wzr glphqvlrqdo vsdfh1

2 +i�?uLhUi4i?| wi@h?�?}c |�i �L?|�?�L�t �@ti

Lq rughu wr hvwlpdwh wkh shuirupdqfhv ri d uhlqirufhphqw ohduqlqj dojrulwkp/ zhqhhg wr frpsduh wkh ixqfwlrq frpsxwhg e| wkh dojrulwkp wr wkh ydoxh ixqfwlrqri wkh frqwlqxrxv surfhvv1

Lq wkh iroorzlqj/ zh frqvlghu ghwhuplqlvwlf v|vwhpv zlwk lq�qlwh wlph krul}rq

dqg glvfrxqwhg uhlqirufhphqw1 Ohw {+w, 5 �R eh wkh vwdwh ri wkh v|vwhp zlwk wkhvwdwh vsdfh R erxqghg/ rshq vxevhw ri LUg1 Wkh hyroxwlrq ri wkh v|vwhp ghshqgvrq wkh fxuuhqw vwdwh {+w, dqg frqwuro x+w, > lw lv gh�qhg e| d gl�huhqwldo htxdwlrq =

g

gw{+w, @ i+{+w,> x+w,,

zkhuh wkh frqwuro x+w, lv d erxqghg/ Ohehvjxh phdvxudeoh ixqfwlrq zlwk ydoxhvlq d frpsdfw X 1 Iurp dq| lqlwldo vwdwh {/ wkh fkrlfh ri wkh frqwuro x+w, ohdgvwr d xqltxh wudmhfwru| {+w,1 Ohw � eh wkh h{lw wlph ri {+w, iurp �R +zlwk wkh

frqyhqwlrq wkdw li {+w, dozd|v vwd|v lq �R/ wkhq � @ 4,1 Wkhq/ zh gh�qh wkhglvfrxqwhg uhlqirufhphqw ixqfwlrqdo ri vwdwh {/ frqwuro x+=, =

M+{>x+=,, @

] �

3

�wu+{+w,> x+w,,gw. ��U+{+� ,,

zkhuh u+{> x, lv wkh uxqqlqj uhlqirufhphqw dqg U+{, wkh whuplqdo uhlqirufhphqw1� lv wkh glvfrxqw idfwru +3 � � ? 4,1

Wkh remhfwlyh ri wkh frqwuro sureohp lv wr �qg wkh rswlpdo ihhg0edfnfrqwuro x�+{, wkdw rswlpl}hv wkh uhlqirufhphqw ixqfwlrqdo iru lqlwldo vwdwh {1

UO whfkqltxhv ehorqjv wr wkh fodvv ri GSphwkrgv zklfk frpsxwh wkh rswlpdofrqwuro e| wkh phdqv ri wkh ydoxh ixqfwlrq =

Y +{, @ vxsx+=,

M+{>x+=,, +4,

zklfk lv wkh pd{lpxp ydoxh ri wkh ixqfwlrqdo dv d ixqfwlrq ri lqlwldo vwdwh {1

Lq wkh UO dssurdfk/ wkh v|vwhp wulhv wr dssur{lpdwh wklv ixqfwlrq zlwkrxwnqrzlqj wkh vwdwh g|qdplfv i qru wkh uhlqirufhphqw ixqfwlrqv u>U= Wkxv/ UO ds0shduv dv d frqvwuxfwlyh dqg lwhudwlyh surfhvv/ edvhg rq h{shulhqfh/ wkdw hvwlpdwhvwkh ydoxh ixqfwlrq e| vxffhvvlyh dssur{lpdwlrqv1

Iroorzlqj wkh g|qdplf surjudpplqj sulqflsoh/ wkh ydoxh ixqfwlrq vdwlv�hvd �uvw0rughu qrqolqhdu sduwldo gl�huhqwldo htxdwlrq fdoohg wkh Kdplowrq0Mdfrel0

Ehoopdq htxdwlrq +vhh ^IV<6` iru d vxuyh|,1

Wkhruhp4 +Kdplowrq0Mdfrel0Ehoopdq,1 Li Y lv gl�huhqwldeoh dw { 5 R/ ohw

GY +{, eh wkh judglhqw ri Y dw {/ wkhq wkh iroorzlqj KME htxdwlrq krogv dw {1

Y +{, oq � . vxsx5X

^GY +{,=i+{> x, . u+{>x,` @ 3

Ehvlghv/ Y vdwlv�hv wkh iroorzlqj erxqgdu| frqglwlrq =

Y +{, � U+{, iru { 5 CR

Wkh fkdoohqjh ri ohduqlqj wkh ydoxh ixqfwlrq lv prwlydwhg e| wkh idfw wkdwiurp Y / zh fdq ghgxfh wkh iroorzlqj rswlpdo ihhg0edfn frqwuro srolf| =

x�+{, @ duj vxsx5X

^GY +{,=i+{>x, . u+{>x,`

Lq wkh iroorzlqj/ zh lqwhqg wr dssur{lpdwh wkh ydoxh ixqfwlrq zlwk slhfhzlvholqhdu ixqfwlrqv gh�qhg e| wkhlu ydoxhv dw wkh yhuwlfhv ri d wuldqjxodwlrq ri wkhvwdwh vsdfh1

6�}� �� Ah�@?}�*@|�L? N Lu |�i t|@|i tT@Ui �� � |h@�iU|Lh) %E|� UhLttit |�i t�4T*i E1�c 12c 1���

� A�i wi@h?�?} #)?@4�Ut

Ohw xv frqvlghu d wuldqjxodwlrq vxfk wkdw wkh vhw ri vlpsoh{hv fryhuv R +vhh �j0xuh 4,1 E| xvlqj d �qlwh0hohphqw frqyhujhqw dssur{lpdwlrq vfkhph +ghulyhg iurp^Nxv<3`,/ wkh frqwlqxrxv frqwuro sureohp pd| eh dssur{lpdwhg e| d PdunryldqGhflvlrq Surfhvv zkrvh vwdwh vsdfh lv wkh vhw ri yhuwlfhv i�j1

Wkh ydoxh ixqfwlrq Y lv dssur{lpdwhg e| d slhfhzlvh olqhdu ixqfwlrq Y

gh�qhg e| lwv ydoxhv dw wkh yhuwlfhv i�j1 Wkh ydoxh ri Y dw dq| srlqw { lqvlghvrph vlpsoh{ +�3> ===> �g, lv d olqhdu frpelqdwlrq ri Y dw wkh yhuwlfhv �3> ===> �g =

Y +{, @g[

l@3

��l +{,Y+�l, iru doo { 5 vlpsoh{ +�3> ===> �g,

zlwk ��l+{, wkh edu|fhqwulf frruglqdwhv ri { lqvlgh wkh vlpsoh{ +�3> ===> �g, 6 {1+Zh uhfdoo wkdw wkh gh�qlwlrq ri wkh edu|fhqwulf frruglqdwhv ��l+{, lv vxfk wkdw =Sg

l@3 ��l +{,=+�l � {, @ 3 dqg =Sg

l@3 ��l+{, @ 4,1Zh dssur{lpdwh wkh Kdplowrq0Mdfrel0Ehoopdq htxdwlrq zlwk wkh Ilqlwh0Hohphqw

vfkhph =Y +�, @ vxs

x5X

k�� +�>x,=Y +�+�> x,, . �+�> x,u+�> x,

l+5,

zkhuh �+�> x, lv wkh surmhfwlrq ri � lq d gluhfwlrq sdudooho wr i+�> x, rqwr wkhrssrvlwh vlgh ri wkh vlpsoh{ +vhh �jxuh 5, dqg � +�> x, lv vxfk wkdw =

�+�> x, @ � . � +�> x,i+�> x,

Iurp wkh olqhdulw| ri Y xsrq wkh vlpsoh{hv/ +5, lv htxlydohqw wr =

Y +�, @ vxsx5X

57�� +�>x,=

g[m@4

��m +�+�> x,,=Y+�m, . � +�> x,u+�> x,

68

6�}� 2� � |h@�iU|Lh) }L�?} |�hL�}� @ t�4T*i � #E1c �� �t |�i ThL�iU|�L? Lu 1 �? @ _�hiU|�L?T@h@**i* |L sE1c �� L?|L |�i LTTLt�|i t�_i Lu |�i t�4T*i � +3%

b�E%��t @ }LL_ @TThL �4@|�L?

Lu #E1c ��3 1�

zklfk lv d G|qdplf Srjudpplqj htxdwlrq iru d �qlwh Pdunry Ghflvlrq Surfhvvzkrvh vwdwh vsdfh lv wkh vhw ri yhuwlfhv i�j> dqg wkh suredelolwlhv ri wudqvlwlrqiurp vwdwh � wr wkh dgmdfhqw vwdwhv i�mjm@4==g zlwk frqwuro x duh wkh edu|fhqwulffrruglqdwhv ��m +�+�> x,,=

E| lqwurgxflqj wkh T0ydoxhv T+�> x, vxfk wkdw Y +�, @ vxsx5X T+�> x,dqg wkdqnv wr d frqwudfwlrq surshuw| gxh wr wkh glvfrxqw idfwru �/ G|qdplfSurjudpplqj wkhru| +vhh ^Ehu;:`, lqvxuhv wkdw wkhuh lv d xqltxh vroxwlrq/ fdoohgY > wkdw vdwlv�hv htxdwlrqv =

T+�> x, @ ��+�>x,=Y +�+�> x,, . � +�> x,u+�> x, iru � 5 R +6,

T+�> x, @ U+�, iru � @5 R

Khuh/ lw lv qrw srvvleoh wr xvh gluhfwo| d Uhdo Wlph G|qdplf Surjudpplqjdojrulwkp +vhh ^EEV<4`, iru vroylqj lwhudwlyho| htxdwlrq +6, ehfdxvh wkh g|qdplfvi +wkxv �+�> x, dqg � +�> x,, lv xqnqrzq1 Wkh prgho0edvhg dssurdfk vkrxog ehwr exlog lq d �uvw wlph d prgho ri wkh g|qdplfv i dqg wkhq wr xvh GS phwkrgvzlwk wklv prgho1 Lq wklv sdshu/ zh duh lqwhuhvwhg lq wkh prgho0iuhh dssurdfkzklfk frqvlvwv lq dq rq0olqh dqg gluhfw ohduqlqj/ wkdw lv zklfk grhv qrw exlog dq|prgho ri wkh g|qdplfv1 Wkh Ilqlwh0Hohphqw Uhlqirufhphqw Ohduqlqj +lqwurgxfhglq ^Pxq<9`, lv d prgho0iuhh UO dojrulwkp wkdw xvhv dssur{lpdwlrq ri �+�> x, dqg�+�> x, wkdqnv wr wkh dydlodeoh rq0olqh nqrzohgjh1

614 Suhvhqwdwlrq

Vxssrvh wkdw d wudmhfwru| {+w, hqwhuv lqvlgh vlpsoh{ W dw srlqw { @ {+w4,1 Dw wlphw4 vxssrvh wkdw d frqwuro x lv fkrvhq dqg nhsw xqwlo wkh wudmhfwru| ohdyhv W dw| @ {+w5, +vhh �jxuh 5,1 Ohw �{ @ w5�w41 Ohw Wlq 6 { eh wkh +g�4,0lqsxw0vlpsoh{dqg Wrxw 6 | wkh +g� 4,0rxwsxw0vlpsoh{1

Wkh dojrulwkp suhvhqwhg lq wkh qh{w vhfwlrq lv wkh lwhudwhg yhuvlrq ri htxdwlrq+6, zklfk xvhv =

Y +|, � Y +{,

��+{,. Y +�, dv dq dssur{lpdwlrq ri Y +�+�> x,,

�{��+{,

dv dq dssur{lpdwlrq ri � +�> x,=

Wkhvh dssur{lpdwlrqv frph iurp wkh olqhdulw| ri Y lqvlgh W dqg wkdw iurpWkdohv* wkhruhp/ |�{

��+{,lv dq dssur{lpdwlrq ri �+�> x,� �1

615 Wkh Ilqlwh0Hohphqw Uhlqirufhphqw Ohduqlqj

Lq wkh iroorzlqj/ zh dvvxph wkdw wkh dfwlrq vsdfh X lv �qlwh1 Ohw Tq +�> x, dqg

Y q +�, eh wkh lwhudwhg ydoxhv ri T+�> x, dqg Y +�,1 Ohw xv fkrrvh d frqvwdqw

� 5 +3> 4` forvh wr }hur1 Lqlwldo ydoxhv T3 +�> x, duh lqlwldol}hg wr dq| ydoxh1

Frqvlghu d wudmhfwru| {+=, jrlqj wkurxjk d vlpsoh{ W zlwk d frqwuro x1 Zkhqwkh wudmhfwru| ohdyhv W dw |/ li wkh iroorzlqj frqglwlrqv =

- ��+{, � � +wklv uhodwlrq holplqdwhv fdvhv iru zklfk ��+{, lv wrr vpdoo,

- ;�l 5 Wlq_Wrxw> ��l+|, � ��l+{,.� +wkhvh uhodwlrqv lpso| wkdw |�{ vwulfwo|ehorqjv wr wkh frqh ri yhuwh{ � dqg edvh Wrxw dqg lqvxuh wkdw iru vpdoovlpsoh{hv/ �+�> x, 5 Wrxw,1

duh vdwlv�hg/ wkhq xsgdwh wkh T0ydoxh ri yhuwh{ � rssrvlwh wr wkh h{lw vlghWrxw iru frqwuro x zlwk =

Tq.4+�> x, @ �

�{��+{,

�Y q +|,�Y

q +{,��+{,

. Y q +�,

�. �{

��+{,u+{> x, +7,

Zkhq wkh wudmhfwru| uhdfkhv wkh erughu ri wkh vwdwh vsdfh/ dw wlph � / wkhforvhvw yhuwh{ �m @5 R iurp wkh h{lw srlqw {+� , lv xsgdwhg zlwk =

Y q.4+�m, @ U+{+� ,,

Uhpdun1 Zlwk vrph dgglwlrqdo k|srwkhvhv rq wkh g|qdplfv i dqg wkh uhjx0odulwlhv ri u/ U dqg CR/ wklv dojrulwkp frqyhujhv wr wkh ydoxh ixqfwlrq ri wkhfrqwlqxrxv sureohp dv q whqgv wr lq�qlw| dqg wkh vl}h ri wkh vlpsoh{hv whqgv wr}hur +vhh ^Pxq<9`,1

Phdqzkloh/ iru d jlyhq wuldqjxodwlrq/ wkh Y q 0ydoxhv gr qrw frqyhujh dv q

whqgv wr lq�qlw|1 Lq rughu wr lqvxuh wkh frqyhujhqfh/ zh qhhg wr frpelqh wkhohduqlqj g|qdplfv zlwk d wuldqjxodwlrq uh�qhphqw surfhvv/ fdoohg vwuxfwxudo g|0qdplfv1

e A�i 5|h�U|�h@* #)?@4�Ut

Wkh vwuxfwxudo g|qdplfv lqwhqgv wr exlog d wuldqjxodwlrq vxfk wkdw wkh vlpsoh{hvhqforvh vwdwhv zkrvh ydoxh ixqfwlrq lv doprvw olqhdu1 Khuh/ zh fkrrvh wkh %jhq0hudo wrzdugv vshfldol}hg% dssurdfk = wkh lqlwldo wuldqjxodwlrq lv frpsrvhg ri dvpdoo qxpehu ri odujh vlpsoh{hv > wkhq wkh vwuxfwxudo g|qdplfv uh�qhv wkh wul0dqjxodwlrq dw sodfhv zkhuh wkh ydoxh ixqfwlrq lv luuhjxodu +wkh vlpsoh{hv ri orzuhlqirufhphqw glvfulplqdqw vnloo,1

714 Wkh Ghodxqd| Wuldqjxodwlrq

Lq wklv sdshu/ zh fkrvh wkh Ghodxqd| wuldqjxodwlrq/ zklfk lv d yhu| xvhg wuldq0jxodwlrq whfkqltxh iru �qlwh0hohphqwv phwkrgv zkhq wkh vwdwh vsdfh lv ri orzglphqvlrqdolw|1 Ghodxqd| wuldqjxodwlrq lv exlow iurp wkh edvlv ri d vhw ri yhuwlfhvdqg lv forvh uhodwhg wr wkh Yrurqrl gldjudp +wkhuh lv d gxdo uhodwlrqvkls,1 Wkhvwxg| ri Ghodxqd| wuldqjxodwlrqv dqg wkhlu surshuwlhv duh eh|rqg wkh vfrsh riwklv sdshu +vhh iru h{dpsoh vrph uhfhqw zrun = ^Plg<6`/ ^Uxs<7`,1 Zh zloo rqo|jlyh d gh�qlwlrq lq wkh 50glphqvlrqdo fdvh =

Jlyhq d �qlwh vhw ri srlqwv/ 6 srlqwv frqwulexwh d wuldqjoh wr wkh Ghodxqd|wuldqjxodwlrq li wkh flufxpvfulelqj flufoh wkurxjk wkrvh srlqwv frqwdlqv qr rwkhusrlqwv lq lwv lqwhulru1

715 Wkh Ghodxqd| Uh�qhphqw Surfhvv

Wkh uh�qhphqw surfhvv frqvlvw ri dgglqj qhz yhuwlfhv dqg frpsxwh wkh Ghodx0qd| wuldqjxodwlrq dvvrfldwhg wr wkh qhz vhw ri yhuwlfhv +zh xvh wkh lqfuhphqwdoZdwvrq* dojrulwkp ^Zdw;4`/ vhh �jxuh 6,1

Wkh fkrlfh ri dgglqj d qhz yhuwh{ lqvlgh d jlyhq vlpsoh{ ghshqgv rq d fulwh0ulrq edvhg rq wkh h{shfwhg uhlqirufhphqw dyhudjh yduldwlrq = wkhT0ghyldwlrq1 Iruhdfk yhuwh{ � dqg frqwuro x/ wkh T0ghyldwlrq H

q +�> x, lv lqfuhphqwdoo| xsgdwhgdw wkh vdph wlph dv wkh T0ydoxh lv e| =

Hq.4+�> x, @

4

q.4

�q=H

q +�> x, .�Tq.4+�> x, �T

q +�> x,�5�

dqg wkh T0ghyldwlrq Hq +W , ri d vlpsoh{ W lv wkh vxp ri wkh T0ghyldwlrqv ri lwv

yhuwlfhv � iru frqwurov x vxfk wkdw wkh g|qdplfv i+�> x, %jrhv lqvlgh% W 1Wkh uh�qhphqw uxoh lv wkh iroorzlqj =Li wkh T0ghyldwlrq ri d vlpsoh{ W lv vxshulru wr vrph ydoxh/ dgg wr wkh vhw ri

yhuwlfhv wkh edu|fhqwhu ri W +iru h{dpsoh/ lq �jxuh 6/ wkh edu|fhqwhu ri wkh jud|vlpsoh{ lv dgghg wr wkh olvw ri yhuwlfhv,1

Uhpdun1 Wkh uh�qhphqw ri erxqgdu| vlpsoh{hv +wkrvh zkrvh doo yhuwlfhv h{fhswrqh duh rxwvlgh R, frqvlvwv lq dgglqj wkh edu|fhqwhu ri wkh yhuwlfhv wkdw duhrxwvlgh R +vhh dq looxvwudwlrq lq vhfwlrq 8,1

Uhpdun1 Gxulqj wkh wuldqjxodwlrq uh�qhphqw surfhvv/ vrph lqlwldo dyhudjhg T0ydoxhv +iru wkh qh{w ohduqlqj g|qdplfv, duh dwwulexwhg wr wkh srlqwv dgghg wr wkholvw ri yhuwlfhv1 Wklv lqvxuhv d frqwlqxrxv ohduqlqj wkurxjk vxffhvvlyh vwuxfwxudog|qdplfv surfhvvhv1

6�}� �� �__�|�L? Lu @ ?i� TL�?| E|�i t^�@hi _L|� |L |�i ti| Lu �ih|�Uit @?_ |�i hit�*|�?}#i*@�?@) |h�@?}�*@|�L?

Iru d jlyhq wuldqjxodwlrq/ zh kdyh wr glvwlqjxlvk wzr skdvhv gxulqj wkh ohduq0lqj g|qdplfv =

~ Wkh wudqvlwlrqdo skdvh gxulqj zklfk uhlqirufhphqwv duh sursdjdwhg xsrq wkhzkroh vwdwh vsdfh dffruglqj wr wkh g|qdplfv1

~ Wkh doprvw vwdwlrqdu| skdvh gxulqj zklfk wkh T0ydoxhv rvfloodwh durxqg dqdyhudjh ydoxh1

Hlwkhu d T0ydoxh frqyhujhv/ vr lwv T0ghyldwlrq whqgv wr 3/ ru lw grhv qrw dqgwkh T0ghyldwlrq grhv qrw whqg wr 31 Vr wkh T0ghyldwlrqv duh d phdvxuh ri wkhluuhjxodulw| ri wkh h{shfwhg uhlqirufhphqw +wkxv lqgluhfwo| ri wkh ydoxh ixqfwlrq,1

Wkh vwuxfwxudo g|qdplfv frqvlvwv lq uh�qlqj wkh wuldqjxodwlrq dw sodfhv zkhuhwkh T0ghyldwlrq ri wkh vlpsoh{hv duh kljk1 Wkhq wkh qhz wuldqjxodwlrq ehfrphvpruh suhflvh dw wkh luuhjxodulwlhv ri wkh ydoxh ixqfwlrq1

D vxffhvvlrq ri ohduqlqj dqg vwuxfwxudo g|qdplfv lv h{hfxwhg xqwlo wkh T0ghyldwlrqv ri doo vlpsoh{hv duh vpdoo hqrxjk1

Uhpdun1 Wkh uh�qhphqw fulwhulrq xvhg khuh/ edvhg rq wkh T0ghyldwlrqv/ lv yhu|vlpsoh dqg pd| eh lpsuryhg e| wdnlqj lqwr dffrxqw rwkhu idfwruv olnh wkh h{0lvwhqfh ri d fkdqjh lq wkh rswlpdo frqwuro lqvlgh d vlpsoh{/ ru wkh frkhuhqfh riwkh ydoxhv dw wkh yhuwlfhv ri d vlpsoh{ ghshqglqj rq wkh orfdo rswlpdo frqwuro1Wkh T0ghyldwlrq fulwhulrq lv rqo| dq looxvwudwlrq ri d srvvleoh orfdo phdvxuh iruwuldqjxodwlrq uh�qhphqw surfhvv/ dqg lw lv xvhg lq wkh iroorzlqj vlpxodwlrq1

D W**�t|h@|�L? ��|� @ 5�4T*i A@h}i| �hLM*i4

814 Ghvfulswlrq

Ohw xv frqvlghu d pdvv prylqj rq d edu +vhjphqw ^�4>4`, zklfk kdv wr vwrs dwrqh ri lwv h{wuhplwlhv1 Wkh frqwuro frqvlvwv lq sxvklqj wkh pdvv zlwk d frqvwdqwvwuhqjwk hlwkhu rq wkh ohiw ru rq wkh uljkw +wkh frqwuro lv x @ 4,1 Wkh vwdwh riwkh v|vwhp lv = { @ +|> y, zlwk | +5 ^�4> 4`, wkh srvlwlrq dqg y +5 ^�5>5`, wkhyhorflw| ri wkh pdvv1 Wkxv/ wkh g|qdplfv ri wkh v|vwhp lv =

�|3+w, @ y+w,y3+w, @ x+w,

Ohw wkh uxqqlqj uhlqirufhphqw u @ 3 dqg wkh whuplqdo uhlqirufhphqw U ghshqgvrq wkh vlgh iurp zklfk wkh pdvv ohdyhv wkh edu +vhh �jxuh 7, = li wkh edu uhdfkhvwkh ohiw h{wuhplw| zlwk d qxoo yhorflw| +ohiw wdujhw, wkhq lw uhfhlyhv U @ .4/ li lwuhdfkhv wkh uljkw h{wuhplw| zlwk d qxoo yhorflw| +uljkw wdujhw,/ lw uhfhlyhv U @ .51Li lw uhdfkhv dq h{wuhplw| zlwk d srvlwlyh yhorflw|/ wkh whuplqdo uhlqirufhphqw zlooghfuhdvh zlwk wkh yhorflw| +xqwlo U @ �4 iru wkh pd{lpdo yhorflw|,1

+1

-1

+2

y

R(-1,v)

R(1,v)

-1

-1-1v

6�}� e� A�i |ih4�?@* hi�?uLhUi4i?| -E+c ��� - ' n� EhitT� - ' n2� @| |�i *iu| EhitT�h�}�|� i |hi4�|) Lu |�i M@h uLh @ ?�** �i*LU�|)� - _iUhi@tit �?|�* 3� uLh |�i 4@ �4@*�i*LU�|)�

Wkxv/ wkh remhfwlyh ri wkh ohduqlqj lv wzlfh = �uvw/ wkh v|vwhp kdv wr ohduq wruhdfk/ dv idvw dv srvvleoh/ hdfk h{wuhplw| ri wkh edu zlwk d orz yhorflw|/ dv pxfkdv srvvleoh1 Wkhq/ lw kdv wr ohduq wr fkrrvh zklfk h{wuhplw| +wkh uljkw rqh zlwkd srvvleoh uhlqirufhphqw ri .5 ru wkh ohiw rqh zlwk .4, lv wkh ehvw ghshqglqj rqlwv srvlwlrq dqg yhorflw| +wkxv rq wkh wlph uhtxluhg iru uhdfklqj wkh wdujhw,1

-1 -0.5 0 0.5 1 -2

0

2

-1

0

1

2

6�}� D� A�i Ei @U|� �@*�i u�?U|�L? T E%��

815 Wkh Rswlpdo Vroxwlrq

Lq wklv h{dpsoh/ lw lv hdv| wr frpsxwh h{dfwo| wkh ydoxh ixqfwlrq +vhh �jxuh8, dqg wkh rswlpdo frqwuro1 Wkxv zh fdq hvwlpdwh wkh gl�huhqfh ehwzhhq wkhfrpsxwhg Yq0ydoxhv dqg wkh rswlpdo ydoxh ixqfwlrq1 D phdvxuh ri wklv dssur{l0pdwlrq huuru lv wkh huuru vxs =

Huuru vxs @vxs{5R

��Y q +{, � Y +{,��

816 Qxphulfdo Uhvxowv

0 200 400 600 8000

0.2

0.4

0.6

0.8

1

1.2

6�}� S� A�i ihhLh t�T @t @ u�?U|�L? Lu |�i ?�4Mih Lu h�??�?} |h@�iU|Lh�it�

Wkh uhvxowv ri wkh vlpxodwlrq duh ghslfwhg lq �jxuh 91 Wklv dssur{lpdwlrqhuuru lv jlyhq dv d ixqfwlrq ri wkh qxpehu q ri uxqqlqj wudmhfwrulhv iru vxffhvvlyhwuldqjxodwlrqv 4/ 5/ 6/ 7 +vhh �jxuh :,1

Wuldqjxodwlrq 4 5 6 7

Iljxuh := Vxffhvvlyh wuldqjxodwlrqv gxulqj wkh vlpxodwlrq1

Lqlwldo wuldqjxodwlrq +4, fruuhvsrqgv wr 4 � q � 4331 Iru q @ 433/ wkhvlpsoh{hv zkrvh T0ghyldwlrq lv kljkhu wkdq 3=334 duh uh�qhg/ zklfk ohdgv wrwuldqjxodwlrq 51 Lq d vdph zd|/ wuldqjxodwlrq 6 rffxuv dw q @ 533/ dqg wuldq0jxodwlrq 7 rffxuv dw q @ 733 dqg lv nhsw xqwlo wkh hqg ri wkh vlpxodwlrq1

817 Dqdo|}h ri wkh Uhvxowv

Joredoo|/ wkh huuru vxs whqgv wr 3 dv vxffhvvlyh ohduqlqj dqg vwuxfwxudo g|qdplfvduh h{hfxwhg1 Dw wkh hqg ri wkh vlpxodwlrq/ wkh huuru vxs lv orzhu wkdq 3=4 dqgfrqwlqxhv wr rvfloodwh durxqg 3=381

E| frpsdulvrq wr wkh +h{dfw, ydoxh ixqfwlrq ri �jxuh 8/ zh revhuyh wkdw wkhuh�qhphqw surfhvv rffxuv dw sodfhv zkhuh Y lv wkh prvw luuhjxodu1

Frpsdulvrq zlwk d frqvwdqw dqg xqlirup wuldqjxodwlrq = Zh kdyh uxq d vlpx0odwlrq zlwk d frqvwdqw wuldqjxodwlrq frpsrvhg ri 493 +qxpehu oljkwo| vxshulruwr wkh 486 vlpsoh{hv ri wuldqjxodwlrq 7, xqlirupo| glvwulexwhg vlpsoh{hv dqgrewdlqhg zlwk ;33 wudmhfwrulhv dq huuru vxs ri 3=551 Wklv uhvxow lqglfdwhv wkhehqh�w ri xvlqj wkh orfdo uh�qhphqw vwuxfwxudo g|qdplfv1

S �L?U*�t�L?

Wkh frpelqdwlrq ri wkh ohduqlqj dqg vwuxfwxudo g|qdplfv surylghv dq lqwhuhvwlqjuhlqirufhphqw ohduqlqj dojrulwkp iru wkh frqwlqxrxv dqg ghwhuplqlvwlf fdvh1 D�uvw lpsuryhphqw vkrxog eh wr vwxg| wkh vwrfkdvwlf fdvh +zkhq wkh hyroxwlrq riwkh v|vwhp lv jryhuqhg e| d vwrfkdvwlf gl�huhqwldo htxdwlrq, iru zklfk zh frxogxvh wkh T0ohduqlqj +vhh ^Zdw;<`, yhuvlrq ri IHUO uxoh =

�Tq+�> x, @ �q

��

�{��+{,

�Y q +|,� Y

q +{,

��+{,. Y

q +�,

��T

q +�> x, .�{

��+{,u+{>x,

�

zlwk vrph ghfuhdvlqj ohduqlqj udwh �1Wkh orfdo uh�qhphqw fulwhulrq edvhg rq wkh T0ghyldwlrqv jhqhudwhv d wuldq0

jxodwlrq wkdw dgdswv wr wkh uhjxodulwlhv ri wkh ydoxh ixqfwlrq1 Phdqzkloh/ wkhuh�qhphqw surfhvv xvhg khuh lv yhu| vlpsoh dqg vrphwlphv jhqhudwhv pruh vlp0soh{hv wkdq qhfhvvdu|1 D srvvleoh lpsuryhphqw vkrxog eh wr frqvlghu d vwuxfwxudog|qdplfv lqfoxglqj erwk %erwwrp0xs% dqg %wrs0grzq% surfhvvhv/ iru h{dpsoh e|vxssuhvvlqj vrph srlqwv dw sodfhv zkhuh orfdoo|/ wkh frpsxwhg T0ydoxhv duh uhj0xodu/ ru e| prylqj vrph yhuwlfhv dffruglqj wr wkh g|qdplfv i 1 Dqrwkhu dssurdfkvkrxog frqvlvw rq d wuldqjxodwlrq lqlwldol}hg durxqg whuplqdo uhlqirufhphqwv dqgsurjuhvvlyho| lqfuhdvlqj lqvlgh wkh vwdwh vsdfh gxulqj wkh uxqqlqj ri wudmhfwrulhv1

+iuihi?Uit

d�@�bDo wii4L? �� �@�h_� +it�_�@* @*}Lh�|�4t G +i�?uLhUi4i?| *i@h?�?}��|� u�?U|�L?

@TThL �4@|�L?� �@U��?i wi@h?�?} G ThLUii_�?}t Lu |�i A�i*u|� W?|ih?@|�L?@*

�L?uihi?Uic �bbD�

d�@hbfo �?_hi� B� �@h|L� �i�h@* ?i|�Lh!t uLh UL?|hL*� `�A� ��**ihc +�5�5�||L?c ��a�

ihMLt i_�|Lht� �WA Thittc �@4Mh�_}ic �@tt@U��tti||tc �bbf�

d��5b�o �?_hi� B� �@h|Lc 5|i�i? a� �h@_|!ic @?_ 5@|�?_ih �� 5�?}�� +i@*�|�4i *i@h?�

�?} @?_ UL?|hL* �t�?} @t)?U�hL?L�t _)?@4�U ThL}h@44�?}� AiU�?�U@* +iTLh|b��D.c �L4T�|ih 5U�i?Ui #iT@h|4i?|c N?��iht�|) Lu �@tt@U��ti||tc �bb��

d�ihH.o #�4�|h� �� �ih|ti!@t� #)?@4�U �hL}h@44�?} G #i|ih4�?�t|�U @?_ 5|LU�@t|�U

�L_i*t� �hi?|�Ui O@**c �bH.�

d��bDo a��� �L)@? @?_ ��`� �LLhi� Bi?ih@*�3@|�L? �? hi�?uLhUi4i?| *i@h?�?} G5@ui*) @TThL �4@|�?} |�i �@*�i u�?U|�L?� �_�@?Uit �? �i�h@* W?uLh4@|�L?

�hLUitt�?} 5)t|i4tc .c �bbD�

d65b�o i?_i** O� 6*i4�?} @?_ O� �i|i 5L?ih� �L?|hL**i_ �@h!L� �hLUittit @?_

V�tULt�|) 5L*�|�L?t� �TT*�U@|�L?t Lu �@|�i4@|�Ut� 5Th�?}ih�Vih*@}c �bb��

dB�*b2o V��@) B�**@T@**�� +i�?uLhUi4i?| wi@h?�?} @?_ �|t @TT*�U@|�L? |L UL?|hL*� ��#

|�it�tc N?��iht�|) Lu �@tt@U��tti||tc �4�iht|�c �bb2�

dO�kbSo �@?Ui ,� O@h4L?c wii4L? �� �@�h_c @?_ �� O@hh) k*LTu� +i�?uLhUi4i?|

*i@h?�?} @TT*�i_ |L @ _�gihi?|�@* }@4i� �_@T|��i �i�@��Lhc eG��2Hc �bbS�

dk�tbfo O@hL*_ a� k�t�?ih� ��4ih�U@* 4i|�L_t uLh t|LU�@t|�U UL?|hL* ThLM*i4t �?

UL?|�?�L�t |�4i� 5W�� a� �L?|hL* @?_ �T|�4�3@|�L?c 2HGbbb��feHc �bbf�

dw�?b�o wL?}�a� w�?� +i�?uLhUi4i?| wi@h?�?} uLh +LML|t �t�?} �i�h@* �i|�Lh!t� ��#

|�it�tc �@h?i}�i �i**L? N?��iht�|)c ��||tM�h}c �i??t)*�@?�@c �bb��

d��_b�o Aih�i ��_|ML� 5T@|�@* �L_i**�?} M) #i*@�?@) �i|�Lh!t Lu A�L @?_ A�hii

#�4i?t�L?t� ��# |�it�tc �Lh�i}�@? W?t|�|�|i Lu AiU�?L*L})c �bb��

d��?bSo +�i4� ��?Lt� � UL?�ih}i?| hi�?uLhUi4i?| *i@h?�?} @*}Lh�|�4 �? |�i UL?|�?��

L�t U@ti G |�i �?�|i�i*i4i?| hi�?uLhUi4i?| *i@h?�?}� W?|ih?@|�L?@* �L?uihi?Ui

L? �@U��?i wi@h?�?}c �bbS�

d�L�bDo �?? �L��i� 6�33) hi�?uLhUi4i?| *i@h?�?}� @? L�ih��i�� �_�@?Uit �? 6�33)

A�iLh) @?_ AiU�?L*L})c �bbD�

d+�Tbeo a�4 +�TTih|� � _i*@�?@) hi�?i4i?| @*}Lh�|�4 uLh ^�@*�|) 2�_�4i?t�L?@*4it� }i?ih@|�L?� aL�h?@* Lu �*}Lh�|�4tc �bbe�

d5�|bSo +�U�@h_ 5� 5�||L?� Bi?ih@*�3@|�L? �? hi�?uLhUi4i?| *i@h?�?} G 5�UUittu�* i �

@4T*it �t�?} tT@hti UL@hti UL_�?}� �_�@?Uit �? �i�h@* W?uLh4@|�L? �hLUitt�

�?} 5)t|i4tc Hc �bbS�

d @|H�o #�6� @|tL?� �L4T�|�?} |�i ?�_�4i?t�L?@* _i*@�?@) |itti**@|�L? ��|� @TT*��

U@|�L? |L �LhL?L� TL*)|LTit� A�i �L4T�|ih aL�h?@*c 2eG�S.��.2c �bH��d @|Hbo ��h�t|LT�ih a���O� @|!�?t� wi@h?�?} uhL4 _i*@)i_ hi�@h_� ��# |�it�tc

�@4Mh�_}i N?��iht�|)c �bHb�

A��t @h|�U*i �@t ThLUitti_ �t�?} |�i w�A,j 4@UhL T@U!@}i ��|� ww��5 t|)*i