• No results found

On Backward Stochastic Partial Differential Equations.

N/A
N/A
Protected

Academic year: 2022

Share "On Backward Stochastic Partial Differential Equations."

Copied!
13
0
0

Laster.... (Se fulltekst nå)

Fulltekst

(1)

!#"%$&'

()+*-,/.0%1+2231#405516 ,87:9;4.

()),/<.<=1+0635>)?@+AB?CD?E0<<4

FHGJILKNM/OQPRKNSUTJVXWUYQM/ZXK\[]WU^8M`_aKNSUWU^8KNbdce^8fhgiSUgiGQWU^8KNb

jlkQm KNWU^8YNGX[

nQoprqtsvuhwUxoqzyz{}|+~8€ ƒ‚zx„zoqz…‡†„/{}qz…‰ˆ

Š

o‹ŒstoŽop’‘‰“}”•‘}–‰–/—

˜™/š+›œ8ž Ÿ›

¡£¢¤¦¥#§8¨r¢i©ª«¢¬­¯®#°:¢ª¦±¢©rª ²«³ ª¦­ ´³ ¢ª ¢®#®/¥#¢®#³¦µ °z¶§r¥/©ƒ·¢ª¦¢¥:©µ¸±µ¯©r®#®/§r¶

¹ ©r±;º»¼©r¥#²X®#°:§±;½ ©r®#°:­ ±\¤¦©¥:°#­¯©rµ¼² ­¯¾¿¢¥#¢ª°#­¯©rµ¼¢´³¦©r°#­¯§ª¦®#ÀÂÁ/½ ­¯®Ã­ ®Ã©Q°+Ĥ¿¢\§r¶

¢´³ ©r°#­¯§rª ®}»z½ ­¯±;½•©¤¦¤¿¢©r¥}©r®}©²-Å#§r­ ª°Œ¢´³ ©r°#­¯§rª ®}­¯ª•°#½¦¢¼Æƒ©¬­¯ÆÃ³ ÆÇ¤ ¥#­¯ª¦±­¯¤¦µ ¢

©r¤ ¤¦¥#§©r±;½È°:§§r¤ °:­¯ÆÈ©µ¦±§rª°#¥:§µ¦§¶¿®#Ä®#°:¢ÆÈ®U² ¢®#±¥#­¹ ¢² ¹ Äî#°:§±;½ ©r®#°:­¯±/¤ ©r¥#°:­¯©µ

² ­¯¾¿¢¥#¢ª°#­¯©rµÉ¢´³ ©r°:­ §rª ®#À

ÊÌËÎÍÏÍ]Ð]ÑUÒ-Ó Ô¦Õ×ÖרÙtÚrÚÜÛÝ]Ô¦Ù¸Õ8ÛÞÉß¼àÈá}â;ã:äæåâ&çéèëêëìÃíëî×ïðÜñ8òëótô]åâ;çéõëöë÷/øëêùúöëîëûèëêëü

ý þúÿ ×ÿ

ðaêŒð«åóôtã#äÃð8óã%òëó]å !Uâ;ò#"ótã:åó ä ò$&ãòóQòëó å%&';ð8â;ð8ô(tâ;ò$]å tã'%ã':ç

)(]åñ8ð*,+-.-/102;ü43Éã'57698 êXåó¸ô:ð<;=*)>2Œð åó?-A@BäÃð å$)Ctâå D%ðæâåótôtòŒä

E

åâ;ã:å D%ð ü

ð F

G&H

ê$6JIKMLONPKML NRQTSVU L N

ŒðÃåWã

E

ðÜó

E

ðÜñ;òëâ=%tð1%ô]üX]òëóã%ôtð8âY,Ztð[(¸â&ò$Dð ä\;òJ%tó¸ô." ò-/'åô]å](&;ð8ô(tâ;ò ñÜð^_ð`

a

*)b2c7L

N

.dD*)b2c7L

N^QTS

)Ctñ Ze,Z]å

f a

*)b2hg

F

*)

a

*)b2_.dD*)b2,2

f

ji7dD*)b2

f

k.cl*;êR6m2 *;íëü%í$2

a

*)6<2hg ;on=p'q#p *;íëü%ø$2

r

Ztã.zã. ås]åñ t^"Èåâ;ôuv;òñ]Z]å$);ã%ñe*òëâ&ô¸ã%ó]åâ;ç=2zôtã'wŒð8â;ðÜó^;ã:å ]ðx`C]å ;ã%òëóu*k!iïDy ÷2;ü{z|•ã

ñ å '%ðÜô}¼åñ t`"ƒåâ;ô4Œð8ñ å Cð ã'ã.j,Ztð<;ð8âä ã%ó]å

E

å 'C¸ð

a

*)6m2g~;,Z]å ã.jWëã

E

ð8ó]ù¿ótò$

,ZtðÈã%ótã|&ã#å]

E

å 'Ctð

a

*;ê$2;üiï#;ã''

a

*)b2¼ã.¼â;ðx^C¸ã%â;ð8ô};òŒð<-A|åô]å (D;ðÜô]üJzóeWëð8ó¸ð8âå =,Ztã.

ã.¼òëóD%çP(Œò€,ã'D%ðÈã'" ð’å .ò åâ;ðsâ;ðÜðs;òæñ]Ztòò€ð<dD*)b2*]ã%ódåó}-/|åô]å (D;ð8ô}"Èå;ç=2;ü

‚/ƒ…„k†D‡‰ˆ‰Š]‹{„ŒŠRŽj‡Š]‘,„‹‡Š]’“k”‰•j–Œ#’— „kˆ ”‰’Šv˜™RŽjš[”œ›R•[$žš[žŸ^_ 

~ ¡b¢ˆ

Ÿ£›¤’Œ#¥,„kˆ]Œ#•[¦D§

¡ˆ-~ ¨

š[”œ›$•^¦Dbˆœ©‡)˜žYª…‹‡b’›«`b¬^”­„Œ#¥#‡b›®j‹J‡‰Š]‘Tž¯#’$žŒ,

°¦& ˆ‰©€„k±R’‡ ŒA²#“.‘,1$› RŽ^ª…“³RŒ,$‹’¤“”=‡ ŒT¥Ÿ=¯#”‰’Œ,„”‰”£´¥#‹’Œ#’”­Šœˆ ‡‰Š ’$Œ#•$µ&„››„k— „’„Œ

ˆ ¡

•$¦D§

¢ ¡¶$¢

Ÿ^„kˆœ±b„Œ#•^¦D ˆ‰©‡)˜

·¸ƒ&„k†D‡ˆœŠ ‹{„ŒŠ¹RŽ‡‰Š]‘_„‹‡‰Š]’¤“”‰•–Œ#’—b„kˆ]”‰’Šv˜ºRŽ‡ Œ#“.‘,„”­Šœ„kˆ]•š€ ^Ž ˆ ¥™»BR‡ ¥T•{‡bŒ#“.‘,„”¤Š‰„ˆ



~-ˆ¼

€½`•^ª…Œ,±R›‡ Œ#¥T•R–ž¾jž^ª…‹J‡ ’›«`Šœ¿‘#‡bŒ,±R®j‹‡Š]‘#”‰ž‹‡bŒ#ž‡ “kž¯_¬

(2)

E H H'ÂÃ H

åótô

HÃÄ

I åótô},ZtðÈâ;ðð8â;ðÜótñ8ð`,Ztð8â;ð8ã%ó¼ü

r

Ztð8â;ð×åâ;ð×ä×åó ç å](D(&%ã%ñ å ;ã%òëóúò$=,Ztã.Y,Z¸ð8òëâ;ç ü÷=5‰åëä/(&%ð^úã%ótñ1'Ctôtð/,Ztð/ò$'%ò#"J

ã%óDW

G

*ã.2 r Z¸ð4(tâ;ò$D%ðä ò$s%tótô¸ã%óDW å£â;ð(Dã%ñ å ;ã%óDW?(Œòëâ,,ò$%ã%òò$QåeWëã

E

ð8óÏñ8òëó^;ã%óDWëðÜó^

ñ:åã:ä ã%óXå/ñÜòÉä¸(D%ð;ð%tó¼åó¸ñ8ã:å úäæåâktëð1¼ñ åóºŒð;âåóvòëâäÃð8ô«ã%ó`&òQå(tâ;ò$D%ðä

ò$JòÅ

E

ã%óDW å!ïDyƒ÷/ü

*ã%ã.2 r Z¸ðÌäæå]5ã:äC]äÆ(tâ;ã%ótñ8ã'(&%ðÌä ð,Ztòô4òëâºò$

E

ã%óDWéåv;òñ]Z¼åÅ);ã%ñ\ñ8òëó^;â;ò${(¸â&ò$D

%ð äÎã%ó

E

òÅ

E

ð`å!ïDyƒ÷7òëâ,Ztð’åô&Çòëã%ó`J(tâ;òñ8ð`,ð`

a

*)b2_dB*)2&ü

3òâ\ä òâ;ðƒãóDòâäæå];ã%òëóå Œò$CDJ,Ztð^ðQåótô£ò$,Ztð8â\å (D(D%ã%ñå];ã%òëó¼òŁJ!ïDyƒ÷" ðÈâ;ðð8â

H÷ áÁ¹I åó¸ô HÙÄ

I åótô£â;ð1ð8ótñ8ð`,Ztð8â;ðÜã%ó]ü

r

Ztð<(DCtâ,(Œò€ð ò$,Ztã.j(]å (Œð8âã.j;ò),Ctôtç]åñ]t`"Èåâ;ôÈv;òñ]Z¼åÅ);ã%ñ<É`ÊRË Ì‰ÍvÊ$ÎUôtã'wŒðÜâk

ð8ó`;ã:å ð1x^C]å ;ã%òëóO*,!ïtáy ÷,2;ü r ZtðÜçÏåâ;ð’ôtð1%tótð8ôdã%ó åMã:äÃã':åâO"ƒå;çÏå$O!ï&y ÷ ù

DC&"ã',Zl,Ztð™]å$ã%ñæðx`C]å ;ã%òólŒðÜã%óDW å});ò ñ Z]å$v;ã%ñ™(¼åâ,;ã:å /ôtã'wŒðÜâ;ð8ó^;ã:å /ð1x^C]å ;ã%òëó

âå ,Ztð8âOkZ¼åó åM)&òñ Z]å$v;ã%ñ’òâ;ô¸ã%ó]åâ;çhô¸ã'wŒð8â;ð8ó`;ã:å ðx`C]å ;ã%òëó]ü

Â

òëâ;ð(¸â&ðÜñ8ã.ð%çù{" ð

"ã''),CtôtçéåÃñ1:å$,¼ò$J!ï¸áy ÷"JZtãñ]ZÂã%ótñ1'Ctôtð^,Ztðò$'%ò#"ã%óDW

G

f€Ï

*)Ð2gÈÑ

Ï

*)Ð2

f

ji

F

*)1.Ðj

Ï

*)Ð2_1Ò/*).Ð2,2

f

iÓÒ/*)Ð2 f

*).Ð2c?*;ê$6m2K4LºN *;íëü%ö$2

Ï *)6/Ð2gÕԀ*)>AÐ2 *;íëüր2

ìzðÜâ;ð

f€Ï

*)Ð2tôtð8ótò$;ð^D,Ztðsz|×òÃôtã'wŒð8â;ðÜó^;ã:å ^"iã',Z\â;ð^v(ŒðÜñ&ò¸-ù1"Ztã'%ð¹Ñ ã.]åj(]åâ,;ã:å

ôtã'wŒðÜâ;ð8ó^;ã:å úò$(Œð8âå ;òâ"iã',ZÂâ;ð^)(Œð8ñ;ò™Ð]ü

r Ztð/vCtó¸ñ;ã%òëó

F

G&H

êR6I{KPL

N

KLØKPL U L ã.YWëã

E

ð8óvåó¸ô:òNã.{,Ztð/;ðÜâäÃã%ó]å

E

å 'Ctð³C¸ótñ;ã%òëóVԀ*)>AÐ2;ü™Ùéð å$,)C]äÃðkZ¼å]eԀ*)>AÐ2ã.-A@BäÃðåÅ)Ctâå D%ð™òëâÌå]|Ð

åótô},Z]å

Ú H Û¹Ü

Ԁ*)>AÐ2)Ý f Ð&IjÞÀßà *;íëü%î$2

"Ztð8â;ð Ú ôtð8ótò$;ð^ƒð5^(ŒðÜñå ;ã%òëóu"ã',Z‡â;ð^v(ŒðÜñ™;òá0ƒüÈÙéðdåâ;ð\ðÜðt ãóDW:,Ztð}." ò

(tâ;òñ8ð^_ð` Ï *)1.Ð2ƒåó¸ôàÒ/*).Ð2svCtñ Z7,Z]å *;íëü%ö$2 åótôà*&íëüր2Ztò$%ô]ü r Z¸ðº(tâ;ò ñÜð^_ð^

Ï *)1.Ð2\åótôÕÒ/*v1.Ð2\åâ;ðéåÅ_vC¼äÃð8ô\;òMŒð-A|åô]å (D;ðÜô]ùUã:üðü%ù Ï *).Ð2\åótôÕÒ/*)Ð2

åâ;ð<-A|ä ð å$vC¸ârå Dðsòëâ\å '{Ð7c7L

N

åótô}" ð’å].ò×â;ðx`Ctã%â;ð,Z]å

Ú

Û¹Ü

@

âäã

Ï *v1.Ð2 Ý iÓÒ/*)Ð2 ÝTåbf f Ð ÞÕßÓp *;íëü%è$2

÷£x`C]å ;ã%òëó¿òŁ`,Ztã`:ç`(ŒðÃåâ;ð/òŁã%ó`;ð8â;ð`vYŒð8ñå Cð,Ztð8çæå (D(Œð åâ å$úåôDÇòëã%ó`úðx`C]å];ãòó

ã%ó åvä×å 5 ã:äC]äæ(tâ;ã%ótñÜã'(D%ð å](&(tâ;òŒåñ Z~;òvò$(D;ã:äæå ñ8òëó`&â;ò$•òŁ™);ò ñ Z]å$v;ãñ}(]åâ,;ã:å

ôtã'wŒðÜâ;ð8ó^;ã:å úðx`C]å ;ã%òó;üïð8ð

H

!í I åó¸ô

H'ç

IYòâôtðåã'. ü

è¸é ÙDê?ë]ØÓMì…í)ìîX¼òóã%ôtðÜâJ,Ztðò$'%ò#"iãóDWM!ïtáy ÷ G

f…Ï *)Ð2gðï}ñ

Ý`ò

Ï *)Ð2 f jiÓÒ/*)Ð2 f kb*)1.Ð2cl*;êR6m2K4L N *;íëü'ó$2

Ï

*)6/Ð2gðԅ*)>/.Ð2 *;íëü'ô$2

(3)

ìzðÜâ;ð

ò Ï

*)Ð2g Nõö

ñ=÷

0³øjù

÷ û 0ý

ã.,Ztð

å (D#åñÜã:åóM"ã',Z â;ð`v(Œð8ñ1/;òЇå (D(D%ã%ðÜôM;ò

Ï ùúåótôÈԅ*)>/.Ð2J å &ãv%tð` Ú H Û¹Ü

Ԁ*v>/.Ð2

Ý f ÐDIjÞÕßÓp

zó:,Ztã.<ã:ä/(D%ðæñåÅð ù=" ð åâ;ð å D%ð™;ò}%tótôl,Ztðò$'CD;ã%òëóvð5^(&%ã%ñ8ã',%çüºÙéð%¸â#v

Cð,ZtðPz'#×ò×â;ð(¸â;ð^ðÜó^å ;ã%òëóe,Z¸ð8òëâ;ð äþ;ò"iâ;ã';ðù^òëâ\å :ä ò…)\å '{Ð]ù

Ԁ*)>AÐ2gÕÿ&*)Ð2Yi

@

â

*_q#Ðj.>A2f *;íëü%õ$2

"Ztð8â;ð

ÿ&*)Ð2g Ú H

Ԁ*|.Ð2vI *;íü%íëê$2

*_q#Ð'2]ã.j-.ä ð å$vCtâå D%ðòëâ\å 'q#.Ðdåótô

Ú H Û¹Ü

@

â

*_q#Ð'2Ý_f q f Ð&IjÞÀßàp *;íü%íëí$2

ð

J £*)Ð2g

ÛÜ

*;ø=b2 Ü0

£*£2kÐ

a

*_ï Ðï

Ý

øb

2f = 8 ê *;íü%íëø$2

ŒðP,ZtðP;âåóã|;ã%òódò$(Œð8âå ;òëâmòëâe!]â&òT"iótã:åó äÃò$;ã%òëódôtð%tó¸ð8ô\òâÂå 'ÈäÃð å$vC¸âå]&%ð

G L N U

LævC¸ñ]Z},Z]å J,ZtðÈã%ó`;ðWëâå ‰ñÜòó

E

ð8â,Wëð^ ü

r

Ztð8óÂã'iã." ð1'=tótò#"ió},Z]å

* £*)Ð2,2g ñ

ÝBò

*J €*)Ð2,2 *;íü%íëö$2

ò#" ôtð1%tótð

Ï *)Ð2[gJ@

.*

â

*_q#|.>A2f jiÓÿ…*2,2

g

â

*J@

*_q#|>2,2,*)Ð2f [i~*J@

kÿ&2,*)Ð2 *;íü%í ր2

r Ztð8ó

f…Ï

*)Ð2g

â

ï}ñ

Ý`ò

*@

*_q#|.>A2,2,*)Ð2

f

ï ñ

Ý`ò J@

ÿ…*2,*)Ð2

f *;íü%íëî$2

i~*J@

*)|.>A2,2k*)Ð2

f

¹ *;íü%íëè$2

gÕï4ñ

Ý ò Ï *)1.Ð2 f [iÓÒ/*).Ð2f *;íü%í$ó$2

"Ztð8â;ð

Ò/*)1.Ð2[gÕ*J@

*)'>2,2,*)Ð2_p *;íü%í$ô$2

ìzðÜótñ8ð\,Ztð\(¸â&òñ8ð`,ð^

Ï

*)1.Ð2_Ò/*)Ð2Wëã

E

ð8óþ çØ*;íü%í ր2 åó¸ô *;íëü%í$ô$2Mò$

E

ð\,Z¸ð

!ïtáy ÷Ó*;íëü'ó$2…*;íëü'ô$2;ü

(4)

ìzò#" ð E

ðÜârù ãóvïð8ñ;ã%òëóöA" ðs"ã''=(tâ;ò

E

ðQåóÌð15 ã.);ð8ótñÜðQåó¸ôeCtótã'x`Ctð8ótð`,]â;ð`vCD'Jòëâ å

Wëð8ótðÜâå ñ:å$_¸ò$)Ctñ ZNð1x^C]å ;ã%òëó;ü{Ùéð¹"ã''úåñ Ztã%ð

E

ð,Ztã.… ç â;ð1WÉåâ;ôtã%ó&W¸,Ztð!iïtáyƒ÷

ò$=#ç^(Œð*;íëü%ö$2€*&íëüր2 å$ åm)(Œð8ñÜã:å]ŒñåÅðÃò$ å]åñ t^"Èåâ;ô:v;ò ñ Z]å$v;ãñÃð

E

ò$'CD;ã%òëó’ðx^C¼å]

;ã%òëó:òâ’ìzã''ŒðÜâkP)(]åñ8ð

E

å 'Ctð8ô:(tâ;òñ8ð`,ð^;ü r Ztã.;ù}ãó7,C¸â;ó]ù}ã.v,C¸ôtã%ð8ô:çMå]tã%óDW

%tó¸ã';ð£ô¸ã:äÃð8óã%òëó]å (tâ;òDÇð8ñ;ã%òëóQåótô~,ZtðÜó~rå tã%óDW:,Z¸ðeã:äÃã'ü

r

Ztã. ã¸,Ztð}" ð1'

tótò#"ió Èå %ðÜâktã%óÇå (D(¸â;ò#5 ã:äæå];ã%òëó‡äÃð,ZtòôM"Ztã%ñ ZMZ]å$Œð8ð8óCðÜôMç?ð

E

ð8âå iå CD

,Ztòëâ#Uã%ó£òÅ,ZtðÜâñ8òëótótðÜñ;ã%òëó;üiïð8ð ð üWÉü

H

!í I:ù

H

!ø I/åó¸ô

H

á£I:üÙéðQå].ò«â;ðð8ââ;ð åôtðÜâ#

Há Ä

I¹òâO,ZtðWëðÜótð8âå ,ZtðÜòëâ&çéò$m);ò ñ Z]å$v;ã%ñ’ð

E

ò$'CD;ã%òëódðx^C¼å];ã%òëóòëó ìzã''Œð8â,

)(]åñ8ð^ ü

r ZtðQâ;ð^)\òŁJ,Ztð(]å (ŒðÜâ ã.òâ,WŒåótã"!8ðÜô å$JòÅ'ò#"m

G

z%ó ïð8ñ1&ãòó ø" ðWã

E

ð,Z¸ð

(tâ;ðÜñ8ã.ðsâåëäÃð1" òëâ,túü

r

Z¸ðQäæåã%ó£â;ð`vCD'NåótôÂã'#(tâ;ò ò$iåâ;ðWëã

E

ð8óÂã%óïð8ñ;ã%òëódöëü

# $4&%('*),+Ø.-

ð0/ ù21VŒð." òºð(]åâå D%ðƒìzã''Œð8â,)(]åñ8ð`{)Ctñ Z<,Z]å 3/vã.tñÜòó`;ã%ó^C¸òÅCv%çùÜôtð8óð%ç

ã:äŒð8ôtôtðÜôÂã%ó41XüJz%ô¸ð8ó`&ã|çã%óDW1Ø"ã',ZÂã'#¼ôDC]å =" ðZ]å

E ð

/6517g 198,5/:8# *;øëü%í$2

"Ztð8â;ð/

8

)åótô¹òëâs,Ztð;ò$(ŒòÅòÅWëã%ñå]zôDC]å zòŁ;/ ü

ðÑ Œð åmŒò$CtótôtðÜô:%ã%ótð åâ

ò$(Œð8âå ;òëââ;òŒä/V;ò</

8

å ;ã.vç ã%ó&W™,Ztðò$'%ò#"ã%óDWæñ8òð8â;ñ8ã

E

ã':çZ ç`(ŒòÅ,Z¸ð^ã.

G r ZtðÜâ;ð

ð5ã.)iñ8òëó)åó`#>=~8 ê×åótô@?P êvCtñ Z},Z]å

øBA)ÑCjDC>EYiF?

C Ý G =

H

C

H

Ý I

òëâNå]'JC7cK/O *;øëü%ø$2

"Ztð8â;ðLA)ÑCC>E[guÑC*C2]ôtðÜótò$;ð^,Ztð’åñ;ã%òëóÂòŁjÑC7cK/

8

òóC7cK/\ü

û ðä×åâ,tP,Z]å AÑÎã.jWëð8ótð8âå '%çÌótò$Œò$Ctótô¸ð8ôdå$•åóÂò$(Œð8âå ;òëââ;òÉä1 ã%ó^;ò<1Xü

ð@M ŒðÏåótò$,Ztð8â\ð1(]åâå]&%ðÏìzã''Œð8â,\v(¼åñÜð ü

ð

ã

k.Ó ê å

ŒðÏå ñÜç^%ã%óD

ôtâ;ã%ñå]e!Uâ;ò#"iótã:åóLä ò$&ãòóð"iã',ZañÜò

E

åâ;ã:åó¸ñ8ðá)(]åñ8ðNM òëóLå7(tâ;ò$]å tã'%ã':ç v(]åñÜð

*,+A.-02;ùÈã:üð ü%ùòëâ‡åóçPO cQM ùHA,kO2Eæã. åvâ;ð å

E

å 'Ctð8ô&]!Uâ;ò#"ótã:åó äÃò$;ã%òëó

"ã',Z

Ú H A_kO2E

Ý

I<gØ

O Ý R

üÈy ð8ótò$;ðPç?-AmgPS*_1qUTØb2,ZtðÌó]å ,Ctâå]#ù]ñ8òŒä/

(D%ð1;ð8ô~%D';ðÜâå];ã%òëó~Wëð8ó¸ð8âå ;ð8ô~ ç

ã

Hê å ü û¼ðÜñ å ',Z]å åP%ã%ó¸ð åâæòÅ(ŒðÜâå];òëâ

Vàâ;òŒäWM>ã%ó^;òX1 ã. ñ å '%ð8ôÎìzã''Œð8â,,ïñ]Z¼äÃã%ôD×ã'

Yõö

ñZ V&O

õ ÝG Þ ß òëâMòÉä ð

òëâ,,Ztòëótòëâä×å ]å$ã.

ã O õ

[/ í å

ò$M ü(\

Ý

*M142["iã'']ô¸ð8ótò$;ðm,Ztð£ì/ã''Œð8â,v(]åñÜð

ò$•ìzã''Œð8â,,ïñ Z]äÃã%ôDÃòÅ(ŒðÜâå];òëâ#[â;òŒä]M ã%ó`;ò<1 ð1x^Ctã'(&(ÉðÜô4"ã',Z4,Ztð\ã%ótótðÜâA(tâ;ò ôD

Ctñ1AV

ñ

V

Ý

E_^

0 ùR úG ü4g

Yõö ñ

AV

ñ O õ V

Ý O õ E G ü ð

F

*)=D`£>2Œð ådäÃð å$vC¸âå]&%ð

ä×å (D(¸ã%óDWºâ&òŒä

H

êR.6JIKU1 KU\

Ý

*M1}2K}+ã%ó^;òU1 vC¸ñ]Ze,Z]å]

F

*)=`€>2]ã.-/'

åô]å (D;ð8ô¼ùã:üð ü%ù

F

*)=`£'2¼ã.[-/'äÃðåÅ)Ctâå D%ð<òëâNå '=D`ŒüÃï#CD(D(Œò€ð" ð’åâ;ð<Wëã

E

ðÜó

åóe-A@PäÃð å$)Ctâå]&%ð ùa1º

E

å 'CtðÜôÌâåó¸ôtòŒä

E

åâ;ã#å]Dðԅ*)>A2;ü{Ùvðæåâ;ð¸%òò$t ã%ó&WOòëâ¹." ò

-/'åô]å (D;ð8ôm(tâ;òñ8ð^_ð` Ï 1ÒY^"ã',Z

E

å]'C¸ð^tã%ób1Låó¸ôb\

Ý

*M1}2;ùrâ;ð^)(Œð8ñ;ã

E

ð%çù`)Ctñ Z

,Z]å J,Z¸ðsò$'%ò#"ã%óDW¼åñ t^"Èåâ;ô\v;òñ]Z]å$);ã%ñÈð

E

òÅ|CD;ã%òëó£ð1x^C]å ;ã%òëó}Ztò$%ô

G

f…Ï g Ñ Ï f ji

F

*) Ï k1Ò=2f jiÓÒYf k.cl*;êR6m2 *;øëü%ö$2

Ï @ g Ԁ*v>A2våëü';ü *;øëüր2

(5)

G

r Ztð8â;ðÈð15 ã.)#åÃñ8òëó)åó`0cmÞÕß vCtñ Z},Z]å

F

*)

ñ

D`

ñ

2,*)>2{ï

F

*)

Ý

`

Ý

2,*)>2

G T cÅ*

ñ ï

Ý G i ` ñ ï<`

Ý ^ 0 ùR úG

ü,2 *;øëü%î$2

òëâ\å '¹*)=D`€2c

H

êR.6JIK1 K<\

Ý

*M142;ü

Ú H @

â F

*)1-êR8ê$2

Ý Gf IjÞÕß *;øëü%è$2

d ef)gjh1ig

ÙéðÈótò#" )å &ð’åótô}(tâ;ò

E

ð,Ztð’ä×åã%óÂâ;ð^vC&'iò$,Ztã(]å (ÉðÜâü

jlk Ó ÞnmrÓ^êpo…í)ì q:rrsutwv:Ì_xRÊRÌ Ú HÔ ÝG I4Þ ßFyfzJxBv|{VÌ}xBv1Ëv~v|1Ír]Ì}r}ÊB{€su{&́|suv

1 K<\

Ý

*M1}2‚DƒÊR΄s…v†É£Ë‡|ˆËv|rr]͉ƒ|v΋Š9twv]ÊBrsBËÊBŒ1΋vAɣˇ|v|rrs*

Ï

1ÒY.2rsu|xÌ_xRÊRÌ

Ž ͏ Ú H @

â Ï

ÝGf IjÞÀßà Ú H @

â ÒY

Ý^ 0 ùR úG ü f IjÞÀßFy

Ž Ív͏PÔPg

Ï

£i

@

Ñ Ï f qi

@

F

*_q#

Ï

11Òa2

f

qi

@

Ґ

f

‘ ê;Tu0Tu6ly

’ mrÞúÞZ“í¹Ùéð"ã''úñ8òŒä/(D%ð1;ðs,Z¸ðs(¸â;ò ò$çP,Ztâ;ð8ðPv;ð1(;ü

Í]ÕÜÓ^ëÕì…í•”,)C]äÃðO,Z¼å]

F

*)=`€>2Ag

F

*)>2/ã. ã%ótôtð1(ÉðÜótôtðÜó^ ò$åótô<`Œù åótô

Ú H @

â F

*)2

ÝGf

IjÞÕ߇ü

– Ír]Ì_v|{—v(‡˜r‡R΄sB̜͇B{¸ü

ïð1™*)Ѻ2Pg

㛚œš

cP/Ñ

š

cž1 å ü r ZtðÜóŸ™*)Ѻ2Ãã.æå’ôtðÜóð\)CDv(¼åñÜð£ò$91Xü

r

Z^C/" ð£ñ åó ñ Ztòò€ðåótôÈ%&5 åóÇòëâ,,Ztòëótòëâä×å ]å$ã.

ã ñ 1p'p'p"

N

1p'p'på ò$ 1 )Ctñ Z

,Z]å ¡

õ

cN™™*)Ѻ2;üÌïðL/

N g q a n£¢[*

ñ

Ý

1p'p'p'

N

2;üy/ðÜótòÅ;ðçu0

N

,Z¸ðº(tâ;òDÇðÜñ;ã%òëó

ò$(Œð8âå ;òë⺁â;òŒäW1 ã%ó^;òX/

N ü á=CD™Ñ

N g 0 N

Ñ«ü r Ztð8óÓÑ

N

ã.×åPŒòÅC¸ótôtð8ô~%ã%ótð åâ

ò$(Œð8âå ;òëâsâ;òÉä¤/

N

;ò4/

N

ü3òëâs,Ztð×ñ8ç`%ã%ótôtâ;ã%ñå]/!Uâ;ò#"ió¸ã:åóÏäÃò$;ã%òëóà#ù}ã'ƒã.¹" ð1'

tótò#"ió},Z]å J,Z¸ðsò$'%ò#"ã%óDW×ôtðÜñ8òŒä/(Œò€ã';ãòó}Ztò$%ô

G

¹g

Y

õö ñ ¥ õ O õ

*;öëü%í$2

"Ztð8â;ð

ã O ñ O

Ý

1p'p'p'O

õ

1p'p'p

å

ã.\åóòâ,,Ztòëótòëâä×å ]å$ã.òŁ(M ù åótô4¥

õ

[ºg íR8øR-öR1p'p'p

åâ;ðÂãótôtð(ŒðÜótôtð8ó`)råótô¼åâ;ôä!Uâ;ò#"ió¸ã:åó äÃò$;ã%òëó ü ïð7

N g *¥

ñ

p'p'p|D¥

N

2;üþy/ð

%tó¸ðP-

N

g¦S*_

N

qbT b2ñ8òŒä/(D%ð1;ð8ô?ç7,ZtðP(tâ;ò$]å tã'%ã'#ç ä ð å$vCtâ;ðl0ƒùåótô\(DCD

Ô N g Ú H0 N Ô - N

@ I åótô F N

*)b2[g

Ú H0 N F

*)2

- N

I:üX]òëóã%ôtðÜâ,Z¸ðAòÅ|%ò#"iã%óDWm]åñ t^"Èåâ;ô

);ò ñ Z]å$v;ã%ñÈôtã'wŒðÜâ;ð8ó^;ã:å úðx`C]å ;ã%òóÂòëó},Ztð%tótã';ðÈôtã:äÃðÜóãòó¼å]¹v(¼åñÜðL/

N G

f…ϺN

g Ñ N

ÏON

f ji F N

*)2 f [iÓÒ N f N

œ

Þ\6 *;öëü%ø$2

Ï N

@ g Ô N

*)>2Qåü|;ü *;öëü%ö$2

”¹Ñ

N

ã. å<ŒòÅC¸ótôtð8ôã%ótð åâ ò$(ÉðÜâå];òë⸁â;òŒä§/

N

;òK/

N

ùÉã'¸ò$'%ò#"Oç},Ztð«â&ð`vCD|

ò$£á åâ;ô¸òÅCD5 åótôRá¸ðÜóDW

H

á/á=Is,Z]å ?*;öëü%ø$2…*;öëü%ö$2£åô]ä ã'#£åMCtó¸ã'x^Ctðùñ8òëó`;ã%ó^Ctò$C ù

(6)

N

N

N

N

N N

N N©¨ N

\ Ý

*M

N

/

N

2;ù0M

N

g9v(¼åó*O

ñ

O

Ý

p'p'p|O

N

2;ü ð5^O" ðéåâ;ðWëòëã%óDWM;òuvZ¸ò#" ,Z¼å]m,Z¸ð

ðx^C¸ð8ótñÜðO*

Ï N

Ò

N

2/åô]ä ã'# åñ8òëó

E

ðÜâ,WðÜó^svCDðx`Ctð8ó¸ñ8ð ü«ª/ã%óDWz|#×òZ¬'=òëâäJC&:åëùÅ" ð

%tó¸ôe,Z]å

Ú H ÏON

Ý G Ig Ú HÔ N Ý G I£ï‡ø Ú H @

A ÏON

10

N Ñ ÏON Ef qkI

ï ø Ú H @

Þ ÏºN

F N

*_q28 f qkI£ï Ú H @

Ò N

Ý^ 0 ùR Ü úI Ü ü f qkI *;öëüր2

"Ztð8â;ð

Ò N

Ý^ 0 ùR Ü úI Ü ü g Nõ ú­ö ñ

*_Ò

N

*[bD®B2,2

Ý

)åó¸ô£òâ,ZtðNìzã''Œð8â,,ïñ]Z¼äÃã%ôD/ótòëâä×ü

z'òÅ'ò#"mâ;òŒä *;øëü%ø$2,Z¼å]

Ú H ÏON

Ý G I>T Ú HÔ Ý G I£ï<= Ú H @

"

ÏON

H

Ý I=f qkI€iF? Ú H @

ϺN

Ý Gf qkI

i Ú H @

ϺN

ÝG f qkI€i Ú H @

F N

*_q2

ÝG f qkI£ï Ú H @

Ò N

Ý^ 0 ùR Ü úI Ü üf qkI *;öëü%î$2

ìzðÜótñ8ðù

Ú H ÏON

Ý G I£i<=

Ú H @

"

ÏON

H Ý I=f

qkI€i

Ú H @

°¯

Ò

N

Ý^

0 ùR úG ü f qkI

T Ú HÔ

ÝGI€i~*?mi‡í$2

Ú H @

ÏON

ÝG

f

qkI€i

Ú H @

F

*_q2

ÝG

f

qkI

"Ztð8â;ð

¯

Ò

N gÕÒ

N

¯

0 N ù åótô

¯

0 N

ã.BkZ¸ð[(tâ;òDÇð8ñ1&ãòóâ;òŒä±M ã%ó^;ò(M

N

gðq

a

n£¢[*O

ñ

1p'p'p'O

N

2&ü

r Ztð8â;ðòëâ;ð ù

Ú H ÏON

ÝG[I>T

Ú HÔ

ÝG[I£i~*?mi‡íÅ2

Ú H @

ϺN

ÝG f qkI€i

Ú H @

F

*_q2

ÝG f qkI *;öëü%è$2

ïð1

¯

Ï N

g @

Ï N

ÝG ü r Z¸ð8ó\*;öü%è$2¼ã:ä/(D%ã%ð`,Z]å

ï f * ù²›³

ñ

ü|

¯

Ï N

2

f

T ù²›³

ñ

ü|

* Ú HÔ Ý G I€i Ú H @

F

*_q2

Ý Gf qkI2 *;öëü'ó$2

ìzðÜótñ8ðù

@

â Ú H ÏON

Ý G If q,T´* Ú HÔ Ý G I€i Ú H @

â F

*_q2

Ý Gf qkI.2_ *;öëü'ô$2

"Ztð8â;ðL´Ïã.åódå (D(tâ;ò$(tâ;ã:å ;ðÈñ8òëóvåó`ü r Z¸ã.;òÅWëð1,Ztð8â"iã',Z\*;öüîÅ2¼ç ã%ð1%ô,Z]å

)CD(

N ã @

â Ú H ÏON

ÝG[If q å ÞÕß *;öëü%õ$2

)CD(

N ã @

â Ú H H

Ï N

H

Ý I=If q å ÞÕß *;öü%íëê$2

)CD(

N ã @

â Ú H µ¯

Ò N

Ý^ 0 ùR úG ü If q å ÞÕß *;öü%íëí$2

(7)

Ý H

ò$j(tâ;ò$Wëâ;ð^,ã

E

ð%çäÃðå$vCtâå D%ðù{)x^C¼åâ;ð\ã%ó`&ð1Wëâå]&%ð ùi\€

E

å 'Ctð8ôM(tâ;ò ñÜð^_ð^zð1x^Ctã'(&(ÉðÜô

"ã',Z~,Z¸ðXã%ótótð8â(tâ;òôDCtñ:Þ n=

F

8«·g

Ú H @

â Þ n`k

F

8«^

f

I:üä!UçÈ,Z¸ð4" ðå]t

ñ8òŒä¸(]åñ;ótð^_Nò$£åÇìzã''ŒðÜâ,?v(¼åñÜð ù•ã'eò$'%ò#"Omâ;òŒä *;öüíê$2£åótô *;öëü%íëíÅ2O,Z]å å

)CDð1x^CtðÜótñ8ð

ã

¢¸$O™aí å ñ åóŒðºð%ð8ñ1;ð8ô7òm,Z]å Ï

Nº¹

»

O íñ8òëó

E

ðÜâ,Wð`=" ð å t^%ç

;òlòŒäÃðO%ã:ä ã' Ï ã%óF¶

Ý * H

êR6JI/m2;ù¼åótô

¯

Ò N ¹

»

O} íÈñ8òëó

E

ð8â,Wëð^" ð å t^ç};ò:òÉä ð

%ã:ä ã'OÒÇã%ó4¶

Ý * H êR6I\

Ý

*M142,2;ü ðJC(tâ;ò

E

ðs,Z]å m*iå

E

ð8â#ãòóÂòŁJ2,* Ï 1Ò/2]ã.iå

òÅ'C&;ã%òó;ò,ZtðO]åñ]t`"Èåâ;ô~);ò ñ Z]å$v;ã%ñNð

E

ò$'CD;ã%òëó ðx`C]å ;ã%òëóu*;øëü%ö$2 åó¸ôu*;øëüր2;üº3òëâ

¢?¼[aíëù`" ðZ]å

E

ð,Z]å

f AÏON

õ

Eg±A_0

N Ñ ÏON

õ Ef jiFA F N

*v2_

õ Ef ji½A

¯

Ò N

f k

õ E *;öü%íëø$2

g±A)Ñ ÏºN

õ E f jiFA F N

*)b2_

õ E f [iFA

¯

Ò N

f k

õ E *;öü%íëö$2

ðÿ&*)2[ŒðÂåó‡å ò$'CD;ð%çXñ8òëó^;ã%ó`Ctò$CvCtótñ1;ã%òëóâ;òŒä

H

êR6JI;òL "iã',ZÓÿº¾v*2/c

\ Ý * H

êR.6JI.2åó¸ô\ÿ…*&ê$2g êëü!UçP,ZtðPz'#×òòëâäJCD#åù

A ÏON

@

õ

E_ÿ&*)6m2 *;öü%í ր2

g @

â

ÿ…*)b2AvÑ ÏON

õ E f [i

@

â

ÿ&*)b2A F N

*)2_

õ E f *;öü%íëî$2

i @

â

ÿ&*)b2f A

â ¯

Ò N f 1

õ

EYi

@

â AϺN

õ

E,ÿ

¾

*)b2 f p *;öü%íëè$2

û ð(D#åñÜã%óDW9¢çL¢a¸Ãã%ó\*;öüíè$2åótô}%ð1,;ã%óDWO U ß ;òæòÅ&råã%ó

A_Ô£

õ

E_ÿ&*)6<2 *;öü%í$ó$2

g @

â

ÿ&*)b2A)Ñ Ï k õ E f ji

@

â

ÿ&*)2A

F

*)b2_

õ Ef *;öü%í$ô$2

i @

â

ÿ&*)b2f A

â

Òa f 1

õ

EYi

@

â A Ï

õ

E,ÿ

¾

*)b2 f p *;öü%íëõ$2

3â;òÉä *;öëü%íëè$2=;ò}*;öëü%íõ$2;ùR" ð/Z¼å

E

ð/CðÜô,Z¸ð/&åñ1,Z]å ¹,Ztð/%ã%ótðåâƒäæå](D(¸ã%óDWb¿~â;òŒä

¶ Ý * H êR6JI\

Ý

*M1}2,2]ãó^;ò\

Ý

*,+J2¼ôtð%¸ótð8ô} ç

¿*,Ò/2g

@

â

ÿ&*)b2f A

â

Ґ f 1

õ

E[g

Y

­ ö ñ @

â

ÿ&*)2A,ÒY*O­2_

õ E f ¥ ­

ã. ñ8òëó^;ã%ó`Ctò$C;ü×ïòŒù&,Ztð«ñ8òëó

E

ð8â,Wëð8ó¸ñ8ðNò$¸*&öëü%íëè$2;ò:*;öëü%íëõÅ2å tëð^(&:åñÜðO" ð å t^%çXã%ó

\ Ý

*,+J2;ü3¸ã'5cl*;êR6m2iåótôÂñ]Z¸ò ò€ð’ù`òâ:¢?aíëù

ÿ N

*_q2[g

íR q/~ji ñ

ÝN

íAï ñ

N

*)ji ñ

ÝN

ïàq2_ ï ñ

ÝkN

Tðq,Tuji ñ

ÝkN

êR q,Tu[ï ñ

ÝN

ÙÏã',ZÈÿ&*'2]â;ð1(D:åñ8ð8ô}çlÿ

N

*2¼ã%ó\*;öü%íëõ$2;ù ã|Jò$'%ò#"m,Z]å

A_Ԁ

õ

Eg

@

â ÿ N

*_q2AvÑ Ï 1 õ Ef qi

@

â ÿ N

*_q2A

F

*_q2_

õ E f q *;öü%øëê$2

i @

â ÿ N

*_q2f A

â

ҐÀ f À`

õ

EYi

í

¢ ³ 4

4

0Ü AÏ

õ Ef q#p *;öü%øëí$2

(8)

A_Ô£

õ

E[g

@

A)Ñ Ï 1"

õ Ef q¹i

@

A F

*_q2_

õ Ef q *;öü%øëø$2

i @

f A

â

ÒaÀ f À`

õ

EYiFA Ï k

õ

E_p *;öü%øëö$2

òëâ\å :äÃò€)\å '{Jc

H

ê$6JI*"iã',ZÂâ;ð^)(Œð8ñ;ò

ðŒð`vW$Ctð’äÃðåÅ)Ctâ;ð^2;ü

”™>[ ã.åâ,tã';âåâ;ç ù`,Ztã.¼ã:ä¸(D%ã%ð^,Z]å

ÔPg

@

Ñ Ï f qi

@

F

*_q2 f q¹i

@

Ò f i Ï p *;öü%ø ր2

òëâ\å :äÃò€)\å '{Jc

H

ê$6JI*"iã',ZÂâ;ð^)(Œð8ñ;ò

ðŒð`vW$Ctð’äÃðåÅ)Ctâ;ð^2;ü

3òëâc

H

êR6I#ù ôtð1%tótð

×

Ï jgÕÔºï

@

Ñ Ï f q¹ï

@

F

*_q2 f q¹ï

@

Ґ f

r Ztð8óm" ðð8ð¹,Z]å J*

×

Ï k1Ò=2]å .ò å ;ã.v%tð`*ã%ã.2tã%óO,Ztð r Ztð8òëâ;ð äÎöëü%í["iã',Z Ï â;ð(D:åñ8ðÜô

ç

×

Ï òëâƒå]'€c

H

êR6I:üiìzð8ótñÜð ù…*

×

Ï kÒY2}ã. åmò$'CD;ã%òëó;ò<,Ztð•ðx`C]å ;ã%òëó*;øüöÅ2 åótô

*;øëüր2;ü

Á {D́|suv|{—v|rr

G

ðO* Ï k1ÒY.2åó¸ô?*

¯

Ï

¯

ÒY2Œð." òMòÅ'C&;ã%òó¼òŁ,ZtðÈðx`C]å &ãòó\*;øëü%ö$2;ü r Z¸ð8ó

@

Ѻ* Ï [ï

¯

Ï 2f qi

@

*_Ґï

¯

Ґ2f [i~* Ï =ï

¯

Ï 2[gaê *;öü%øëî$2

”/(D(&%ç ã%ó&WMz'#×òn¬'òâäCD:åëù`" ðWëð

ê<g

Ï £ï

¯

Ï Ý G i‡ø

@

A Ï ï

¯

Ï 1 f ¶bE *;öü%øëè$2

i‡ø

@

AvÑ* Ï ï

¯

Ï 2_ Ï ï

¯

Ï Af qi

@

Òaï

¯

Ґ

Ý^ 0 ùR úG ü f q *;öü%ø$ó$2

"Ztð8â;ðL¶}jg

â

*_Ґï

¯

Ґ2f 8üJ!Uç:*;øëü%ø$2;ù" ðWëð,Z]å

Ú HÏ ï ¯ Ï

Ý G[IgÕï ø @

Ú H

A)Ѻ*Ï ï ¯

Ï 2_ Ï ï ¯

Ï E³I f q¹ï Ú H @

Ò ï ¯

Ò Ý^ 0 ùR úG üf qkI

Tðï3=

@

Ú H

H

Ï ï

¯

Ï

H

Ý I If qiF?

@

Ú HÏ ï

¯

Ï Ý G If q

T?

@

Ú HÏ ï

¯

Ï Ý G If q#p

!UçÌå,â;òëó^"Èå '`:ç`(Œðã%ótðx`C]å %ã':çùã'ò$'%ò#"m…,Z]å

Ú HÏ 1ï

¯

Ï ÝG Igaê ù#"Ztã%ñ ZO(tâ;ò

E

ð^

,ZtðCtó¸ã'x^CtðÜótð^_;ü

(9)

Í]ÕÜÓ^ë½Â…íԙ,)C]äÃð,Z]å *)=D`€2,*)>2g *)`€2,*)>2]ã.¼ã%ó¸ôtð(Œð8ó¸ôtð8ó`iò$i}ü

ïðMÒ

â

g êëüày ð8ó¸òÅ;ðeçV* Ï N 1Ò

N

2/,ZtðeCtótã'x`Ctð?ò$'C&&ãòóÇò$A,Ztðe]åñ t^"Èåâ;ô

);ò ñ Z]å$v;ã%ñÈð

E

ò$'CD;ã%òëóÂðx`C]å &ãòó

G

f…ÏON

g Ñ ÏºN

f [i

F

*)1Ò N£ ñ

2f jiÓÒ N

f *;öü%ø$ô$2

ÏON

@ g Ԁ*)>2_p *;öü%øëõ$2

r Ztð•ð5ã.v;ðÜótñ8ð•ò$vC¸ñ]Zhåmò$'C&&ãòó7* Ï N Ò

N

2=Z]å$=ŒðÜð8ó(tâ;ò

E

ð8ôQãó7v;ð(híëüá=CD,;ã%óDW

¶ N

g

â Ò N

f 8ùúåótô} ç:z'#×òn¬'òëâäJCD:å¸" ðWëð,Z]å

êg

ϺN

³ ñ

@ ï ϺN

@ Ý G *;öü%öëê$2

g ÏON

³ ñ

ï ϺN

Ý G i‡ø

@

AvѺ* ϺN

³ ñ

ï ÏON

2_ ϺN

³ ñ

ï ÏON

E f q *;öü%öëí$2

i‡ø

@

A F

*)Ò N

2{ï

F

*v1Ò N£ ñ

2_ ÏON

³ ñ

ï ϺN

Ef q *;öü%öëø$2

i‡ø

@

AÏON

³ ñ

ï ÏON

f *¶ N ³ ñ

ï½¶ N

2EYi

@

Ò N ³ ñ

ïÓÒ N

Ý^ 0 ùR úG ü f q *;öü%öëö$2

z%ó

E

ã%â,kC¸ðÈòŁ*;øü%ø$2;ù`òâ:Ä8 êù

Ú H ϺN

³ ñ

ï ϺN

Ý G I€i Ú H @

Ò N ³ ñ

ïàÒ N

Ý^ 0 ùR úG üf qkI *;öü%ö ր2

gðïÈø Ú H @

A)Ѻ* ÏON

³ ñ

ï ÏON

2_ ÏON

³ ñ

ï ÏON

Ef qkI *;öü%öëî$2

ï‡ø Ú H @

AF

*)1Ò N

2{ï

F

*)1Ò N£ ñ

2_ ÏON

³ ñ

ï ÏON

E f qkI *;öü%öëè$2

T?

Ú H @

ϺN

³ ñ

ï ÏON

ÝG f qkI=ï<=

Ú H @

H ÏON

³ ñ

ï ÏON

H

ÝI f qkI *;öü%ö$ó$2

i<Ä

Ú H @

F

*)1ÒN

2{ï

F

*)1ÒNB ñ

2 Ý G f qkI£i

í

Ä Ú H @

Ï N ³ ñ

ï Ï N

Ý G f qkI *;öü%ö$ô$2

XZtò ò€ð¡Ä?Þ

ñ

ÝÅ

ù{"JZ¸ð8â;ðLc×ã.J,Z¸ð ã'(ñ Ztã'!QñÜòóvåó^\ãóV*;øëü%îÅ2;ü?z'mò$'%ò#"mâ;òŒä

*;öëü%ö$ôÅ2,Z]å

Ú H ÏON

³ ñ

ï ÏON

Ý G I£i Ú H @

Ò N ³ ñ

ïÓÒ N

Ý^ 0 ùR úG ü f qkI€i= Ú H @

"

ÏON

³ ñ

ï ÏON

H Ý IYf qkI

TÕ*?mi

í

Ä 2 Ú H @

ÏON

³ ñ

ï ÏON

Ý Gf qkI€i ñ

Ý Ú H @ Ò N ³ ñ

ïàÒ N

Ý^ 0 ùR úG ü f qkI *;öü%öëõ$2

(10)

ï f

f

* ù²³ 4Æ

ü³

Ú H @

ÏON

³ ñ

ï ÏON

Ý Gf qI *;öü֌ê$2

iF ù²³

4

Æ

ü|

Ú H @

Ò N ³ ñ

ïàÒ N

Ý^ 0 ùR úG üf qkI *;öü֌í$2

i<=0 ù²³ 4Æ

ü|

Ú H @

H Ï N ³ ñ

ï Ï N

H

Ý I f qkI *;öü֌ø$2

T4ñ

Ý ù²³ 4Æ ü|

Ú H @

Ò N ïàÒ N£

ñ

Ý^ 0 ùR úG ü f qkI *;öü֌ö$2

3â;òÉäØZtð8â;ðù£ò$'%ò#"ã%óDWdå}ã#äÃã':åâ(tâ;òòŁÈå$ã%ó

Há á£I" ð™"iã''¸vZtòT"Æ,Z]å P*Ï N 1Ò

N 2

ñ8òëó

E

ð8â,Wëð^;òÀòŒä ðM%ã:äÃã':*

Ï

1Ò/2Èã%óà,ZtðM(¸â&òôDCtñ1:v(]åñ8ð òŁL¶

Ý * H

êR6JI/m2’åótô

¶ Ý * H êR6JI\

Ý

*M1}2,2;ü

ð10¥\g?mi ñ

Ç

üz%ó^;ð1Wâå ;ã%óDW™ŒòÅ,Z\ã%ôtð^¼ã%ó\*;öëü֌öÅ2" ðsWëð1J,Z]å

Ú H @

â

ÏON

³ ñ

ï ϺN

Ý Gf qkI€i

@

â Ú H @

Ò N ³ ñ

ïÓÒ N

Ý^ 0 ùR úG üf qkIÈ f

i<=

@

â Ú H @

"

ÏON

³ ñ

ï ÏON

H Ý IYf qkIÈ

f

T ñ

Ý @

â Ú H @

Ò N

ïÓÒ N£ ñ

Ý^ 0 ùR úG ü f qkIÈ

f *;öüÖ$ր2

z%ó}(]åâ,;ã%ñC&:åâù

@

â Ú H @

Ò N ³ ñ

ïÓÒ N

Ý^ 0 ùR úG ü f qkIÈ f

T ñ

Ý @

â Ú H @

Ò N

ïÓÒ N£ ñ

Ý^ 0 ùR úG ü f qIÈ

f *;öü֌î$2

r

Ztã.¼ã:ä/(Dã%ð^,Z]å

@

â Ú H @

Ò N ³ ñ

ïàÒ N

Ý^ 0 ùR úG ü f qkIÈ

f 0TÕ*$ñ

Ý 2 N ´

òëâmòÉä ðÈñ8òëóvåó`(´Èü r Z^C;ùã'Jò$'%ò#"mâ;òŒä *;öëüÖÅր2,Z]å

Ú H @

â

ÏON

³ ñ

ï ÏON

Ý Gf qkI>TÕ*

íø 2 N ´ *;öü֌è$2

ìzðÜótñ8ðù^" ðÈñ8òëótñ|Ctôtðâ;òÉä *;öëü%öëõÅ2,Z]å

Ú H @

â Ò N ³ ñ

ïÓÒ N

Ý^ 0 ùR úG üf qkI>TÕ*$ñ

Ý 2 N ´«¥i ñ Ý Ú H @

â Ò N ïàÒ NB

ñ

Ý^ 0 ùR úG üf qkI

*;öüրó$2

(11)

E E

Ú H @

â Ò N ³ ñ

ïÓÒ N

Ý^ 0 ùR úG ü f qkI>TÕ*$ñ

Ý 2 N ¢&´«¥i Ú H @

â Ò ñ

Ý^ 0 ùR úG ü f qkI *;öüրô$2

X¼òÉätã%ótãóDWM*;öëü%öëõÅ2åótô?*;öëüրó$2" ðZ]å

E

ð,Z]å

Ú H @

â

"

ÏON

³ ñ

ï

ÏON

H ÝI f

qkI>TÕ*

íø 2 N ã

*¢}i‡í$2´«¥Mi

Ú H @

Ò ñ

Ý^ 0 ùR úG üf qkI

å

*;öü֌õ$2

z'ȁò$'%ò#"OXótò#" â;òÉä *;öüրô$2 åótô *;öëü֌õ$2},Z¼å]È,ZtðV,Ztð ðx`Ctð8ótñÜð *Ï N

N 2;ù

¢?aí ñ8òëó

E

ðÜâ,Wð`¿ã%ób¶

Ý * H

êR.6JI/<2RK¶

Ý * H êR.6JI\

Ý

*MP1}2,2B;òºòÉä ð%ã:ä ã'* Ï k1Ò=2;ü

ð,;ã%ó&W9¢ U ß ã%ó\*;öëü%ø$ôÅ2;ù`" ðPð8ð,Z]å O*Ï kÒY2J å ;ã.v%tð`

Ï £i

@

Ñ Ï f q¹i

@

F

*_q#1Ґb2 f qi

@

Ґ f JgðÔ *;öü%îëê$2

ã:üðü%ù=*Ï kÒY2]ã.åò$'CD;ã%òëó}&òæðx`C]å ;ã%òó\*;øëü%ö$2;ü

Á

{D́|suv|{—v|rr

ð* Ï k1Ò=2×ù/*

¯

Ï k

¯

ÒY2/Œð}." òàò$'CD;ã%òëó¸;òà*;öëü%îëê$2;üÀ!UçÀz'#×òZ¬|/òëâäJC&:åéùÃå$ ã%ó

*;öëü%ø$óÅ2" ðZ]å

E ð

Ú HÏ ï ¯ Ï

Ý G[I€i

Ú H @

Ò ï ¯

Ò Ý^ 0 ùR úG ü f

qkI *;öü%îëí$2

gÕï ø Ú H @

ÞÈѺ* Ï ï

¯

Ï b2_ Ï [ï

¯

Ï 8 f qkI *;öü%îëø$2

ï ø Ú H @

AF

*)1Ò=2{ï

F

*)

¯

ÒY.2_ Ï ï

¯

Ï Ef qkI *;öü%îëö$2

T?

Ú H @

Ï ï

¯

Ï

Ý Gf qkI£ï=

Ú H @

H

Ï ï

¯

Ï

H

Ý IYf qkI *;öü%î ր2

i ñ

Ý Ú H @

Ò==ï

¯

ÒY

Ý^ 0 ùR úG ü f qkI€i<c

Ú H @

Ï

ï

¯

Ï Ý Gf

qkI *;öü%îëî$2

X¼òóðx`Ctð8ó`,%ç ù

Ú HÏ =ï

¯

Ï Ý G I>TÕ*?mi<c$2

Ú H @

Ï [ï

¯

Ï

Ý Gf qkI *;öü%îëè$2

!Uçâ;òó`"Èå]|D¬'¼ã%ótðx`C]å %ã':çéù

Ï g

¯

Ï

"Ztã%ñ ZevCtâ,,ZtðÜâiã:ä/(&%ã%ð^AÒYg

¯

ÒYY ç:*;öëü%îëî$2;ü

(12)

Í]ÕÜÓ^ë½o…íɐð8ó¸ð8âå ‰ñåÅð *)У2,*)>A2&ü

ð Ï â

g>êü y ð%tótðÇùòëâ¡¢ >íëù* ÏON

³ ñ

1Ò N ³ ñ

2s;ò?Œð4,Ztðuò$'CD;ãòóÏòŁs,Z¸ð

ðx`C]å ;ã%òëó

G

f…Ï N ³ ñ

g Ñ Ï N ³ ñ

f ji

F

*) Ï N

ÒN

³ ñ

2 f jiàÒN

³ ñ

f *;öü%î$ó$2

ÏON

³ ñ

@ g Ô *;öü%î$ô$2

r ZtðÈð5 ãv;ð8ó¸ñ8ðÈò$J*ÏON

³ ñ

1Ò N ³ ñ

2]ã¼ñ8òëó^åã%ótð8ôÂã%ó\)&ð1(døü

ïã:äÃã':åâ,%çdå$zã%ó~v;ð1(Çø" ðNñ åóu)Ztò#"þ,Z]å *ÏON

³ ñ

1Ò N ³ ñ

2zñÜòëó

E

ð8â,Wëð^;ò:òÉä ð

%ã:ä ã'm*Ï 1Ò 2&ùúåótôdäÃòëâ;ð8ò

E

ðÜâm* Ï 1Ò 2¼ã.,ZtðCtó¸ã'x^CtðPò$'CD;ã%òëó};ò×ðx`C]å ;ã%òëó?*;øëü%ö$2;ü

ÙéðÈòŒäÃã',ZtðÈôtðåã'.ü

è¸é ÙDê?ë]ØÓUo…í ðw1 g\

Ý

*LwÊ#2;ùúåótô\ð

/ðg1 ñ

Ý

*L

Ê

2[g

ã

C7cK\

Ý

*Ë Ê 2

œ|Ì C7cK\

Ý

*L

Ê U L Ê 2 å

y ð8ótò$;ðçn=*vÐ2gð*)n

õ

­*)Ð2,2åæä×å ;â;ã'5^

E

å 'Ctð8ô4³C¸ótñ;ã%òëóÂòëólLwÊO å ;ã.v%tó&W,Z¸ðCtótã'

òëâä ð1'%ã'(D;ã%ñÜã':ç’ñ8òëótô¸ã';ã%òó

G

í

cZÍ

Ê

Tun£*)Ð2T¼c

Í Ê

òâOòŒäÃðÈñÜòóvåó^Fcmcl*;êR1ß~2_p

ðw £*)Ð2Œð’å

E

ðÜñ;òëâJ%tð1%ô£òëólL Ê "iã',Z@ ecK\Î*L Ê 2òâOòŒäÃð

a 8 f üy/ð%¸ótð

ÑCgðï f [ š

*)n£*)Ð2

Ì

C*)Ð2,2YiF £*)Ð2Ï

Ì

C*)Ð2

r Ztð8ó7*;øëü%ø$2‰ã.YvCD'%D'%ðÜôòëâs*1/.Ñ2&ü r Z`C;ù$òëâƒåó ç×ñ]Z¸òãñ8ð•ò$‰ñ8ç`%ã%ótô¸â&ãñ å j!Uâ;ò#"J

ótã:åóéäÃò$;ã%òól\ùŒåó ç×ôtâ;ã'v ñ8òðÐ×ñÜã%ð8ó`

F

*)=`£.>A2;å ;ã.)çã%óDWe*;øëü%î$2 åótô7*;øëü%è$2 åótô

;ð8âä ã%ó]å ‰âåótôtòŒä

E

åâ;ã:å D%ðԌù^,Ztð’ä×åã%óÂâ;ð^)CD'ã%óïð8ñ;ã%òëódö×å (D(D%ã%ð`;ü

рÒZÓ0Ô,Õ>ּרÏÙÛÚØJÜPØÏÔ0Ý

r Ztã." òëâ,tP"Èå$¼ã%ó¸ã';ã:å &ðÜô}"Ztð8ó}Œò$,ZÂò$Cå ,;ð8ó¸ôtð8ôdå¸" òëâ,t=)Ztò$(Âòëódïtáyƒ÷ ã%ó

ÙÏåâ,"iã%ñ tQã%ó

Â

å;ç ù‰øëêëêëíü

r ü Ä

Z¼åó&WXå .òWŒå

E

ðQåAå 'tQòëóe,Ztã." òëâ,tQã%óe,Ztð’ïtáyƒ÷

ñ8òëóDðÜâ;ð8ótñ8ð\ã%óMÙÏåâ,"iã%ñ t£ã%ó¼ÞbCD%çøëêëêëíüÙéðm" ò$CDô4%ã'tëðm;ò,Z]åóDte,Ztð\òëâ,WŒåótã"!8ðÜâ

Þ-òUß ð8ótótðÜôtç ù‰û ò$Wëð8â r â;ã'ŒðQåótôKÞ-ðÜâ!8ç

Ä

å Œñ!8ç`t‰ù‰å$" ð|/å$y å E

ã%ô÷£'" òëâ,,Z ç ù^òëâ

,Ztð«ã%ó

E

ã'å ;ã%òó åótôZtò€)(tã'å %ã':ç ü¸ÙéðÂå .ò,Z]åóDt},Z¸ð

Â

å ,Ztðä×å ;ã%ñ^sz%ó);ã',CD;ðÂå

ªzótã

E

ðÜâ#ã':ç’ò$ÙÏåâ,"ã%ñ tPòëâ(tâ;ò

E

ã%ô¸ã%óDW,ZtðP);ã:äJCD:å ;ã%óDWæâ;ð^ð åâ;ñ]ZÂð8ó

E

ã%â;òëó]ä ð8ó^ü

ef)àº)¸&) ÿM :),g

H

!í Iá”×ü'!UðÜóò$C_;åó

G Â

å 5 ã:äC]ä (tâ;ã%ótñ8ã|(D%ðlåó¸ôRôtçó]åëäÃã%ñá(tâ;ò$WëâåäæäÃã%óDW å (D

(tâ;òŒåñ Ztð`Œò$€,Ztðiò$(D;ã:äæå]tñÜòëó^;â;ò$tò$€(]åâk;ã#å]'%çNò$ðÜâ

E

ð8ô×ôtã'w€Cã%òëó ütï#;ò ñ Z]å$v

;ã%ñ^«â7*;íõ$ôëö$2;ù%íèëõ øëøëøëü

H

!ø Iá”×ü'!UðÜóò$C_;åó

G

ï#;ò ñ Z]å$v;ã%ñÂä×å 5 ã#äJC]ä (tâ;ã%ótñ8ã|(D%ðmòëâçYv;ðä["ã',ZM(]åâ,;ã:å

ã%óDòëâräæå &ãòóhåótô å (D(Dã%ñ å ;ã%òëó™;ò<,Ztðºð1(]åâå];ã%òëó(tâ;ã%ótñÜã'(D%ð ü…z%ó

Â

ü'y å E ã. åótô

ûü÷£'%ã%ò$,:*ðÜôtã';òëâ#,2

G

”/(D(D%ã%ðÜô ïT&òñ Z]å$v;ã%ñ”Ãó]å %çYã. üòëâ;ôtòëó åótô !Uâ;ð åñ Z

íõëõëíëù^(&(íëî$ó í$óëøü

(13)

H Â G

ð8ó^;ã#å]úðx`C]å ;ã%òëó¼ã%ó}%tó]åótñ8ðü

Â

å],Ztðä×å ;ã%ñ å ¹3¸ã%ó¼åó¸ñ8ð9ã*;íëõëõ$óÅ2훅óëíëü

H'™à I¡Þ¦ü

Â

åÂåótô½Þ¦ü

Ã

òëóDW

G

3òâ,"Èåâ;ôD !åñ t`"ƒåâ;ô ïT;ò ñ Z]å$v;ã%ñMy/ã'wŒðÜâ;ð8ó^;ã:å Ã÷£x^C¼å]

;ã%òó]åótô r Z¸ð8ã%âϔ/(D(D%ãñ å ;ã%òëó;ü ï#(¸â&ãóDWëð8â J

Â

í$óëêøëù¿ï#(tâ;ã%óDWëðÜâk}äzð8â,:å W×íëõõëõëü

H'ç Iþ!ü

ç

tYðÜótô]å

(D;ã:ä×å ¸ñ8òëó`&â;ò$¸ò$[);ò ñ Z]å$v;ã%ñ(]åâ,;ã:å ¸ôtã'wŒð8â;ðÜó^;ã:å ¸ðx`C]å];ãòó

åó¸ô å (D(D%ãñ å ;ã%òóÈ;ò(]åâ,;ã:å ò$ð8â

E

å ;ã%òëódñÜòó`;â;ò$:ü á‰â;ð(tâ;ã%ó`ù3ªzó¸ã

E

ðÜâ#ã'#çéò$

å

v%ò í$óºæëøêëêëíëü

H

á=I ÷/üÈá¼åâ;ôtò$CD5

G

ïT;ò ñ Z]å$v;ã%ñ:(]åâ,;ã:å Èôtã'wŒð8â;ð8ó`;ã:å ƒð1x^C]å ;ã%òëóÂåótôÀ%D';ðÜâ;ã%óDWÇò$

ôtã'w€C_ã%òëó}(tâ;ò ñÜð^,ð^ ü‰ï#;òñ Z]å$v;ã%ñ^«o7*;íëõÅóëõ$2iíëø$ó›Œíè$óëü

Há á£I’÷/üÃá¼åâ;ôtò$CD5 åótô ïtü•áÉðÜóDW

G

”Ãô]å (D;ð8ôÆò$'CD;ãòóƒò$s¼åñ t`"ƒåâ;ôÆv;ò ñ Z]å$);ã%ñ

ôtã'wŒð8â;ð8ó`;ã:å ‰ð1x^C]å ;ã%òëó ü‰ïçYv;ðäåótô\X]òëó^;â;ò$

ð,;ð8â#ìZç7*;íõëõëê$2iîëèíëü

Há Ä

I¡ ü'y üá}âå ;ò×åótôbÞ¦ü

Ä

å Œñ!Üç^t

G

ï#;òñ]Z]å$);ã%ñ ðx^C¼å];ã%òëó¸ã%óNã%óD%¸ótã';ð ôtã:äÃðÜóã%òëó ü

XiåëäJ¸â;ã%ôDWëð¡ªzótã

E

ð8â#ã|:çéá}â;ð^_;ùúíëõëõøëü

HÃÄ I¡Þ¦ü

Ã

òëóDW åótôbè×ü

à üÄ Ztò$C

G

ï#;ò ñ Z]å$v;ã%ñPX¼òó`;â;ò$.;üúï#(tâ;ã%óDWëðÜâk}äzð8â,:å W íëõëõëõëü

Referanser

RELATERTE DOKUMENTER

We prove an existence and uniqueness result for non-linear time-advanced backward stochastic partial differential equations with jumps (ABSPDEJs).. We then apply our results to study

[8] Gozzi F., Marinelli C., Stochastic optimal control of delay equations arising in advertis- ing models, Da Prato (ed.) et al., Stochastic partial differential equations and

[r]

Stochastic control of the stochastic partial differential equations (SPDEs) arizing from partial observation control has been studied by Mortensen [M], using a dynamic

model uncertainty; stochastic differential game; stochastic maximum principle; operator- valued backward stochastic differential equation; optimal consumption of a mean-field cash

Backward stochastic differential equations with respect to general filtrations and applications to insider finance.. Bernt Øksendal 1,2 and Tusheng Zhang 3,1 14

We prove an existence and uniqueness result for a general class of backward stochastic partial differential equations with jumps.. See [FØS], [NS], [S] and the

The aim of this paper is to study the approximations of stochastic evolu- tion equations of the above type by solutions of stochastic evolution equations driven by pure jump