HAL Id: inria-00072163
https://hal.inria.fr/inria-00072163
Submitted on 23 May 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,
Backward Stochastic Differential Equations Associated to a Symmetric Markov Process
Vlad Bally, Etienne Pardoux, L. Stoica
To cite this version:
Vlad Bally, Etienne Pardoux, L. Stoica. Backward Stochastic Differential Equations Associated to a Symmetric Markov Process. [Research Report] RR-4425, INRIA. 2002. �inria-00072163�
ISRN INRIA/RR--4425--FR+ENG
a p p o r t
d e r e c h e r c h e
THÈME 4
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Backward Stochastic Differential Equations Associated to a Symmetric Markov Process
V. Bally — E. Pardoux — L. Stoica
N° 4425
Avril 2002
Unité de recherche INRIA Rocquencourt
2354762689/ !:;1 =<?>@ 9#/
ACBEDFHGIGIJLKEMBENOFHPRQ!SUTVWKEX1BEYZ[SU\I]^F
_ `badce+f5gih^jkclbmonqprjtsvuwep#syxzprjtcjo{|n}p|jksyu
~ e{|^{p|aRceR{#svcxbmtedzeR{
Ers} edpnqpr`z
=n}xbxHsy|p ~ e1|e`bed`beuH!fyfvgi/^rjtm/yy+gibOxn}ye{
Rdqb yy¡£¢ e dsyu¤{|j~ edn={|eRsvu ~ sv ~ ed{|edcjt¥¦eRmtmkjkxzp|jo ~ j¨§Uedredup|jon}msyxHednqprsy©ª#jtp|`ceny{|lbrny«bmke
ds^e¬7jkedupr{Rzjku ~ jtvedryedue¯®°syrcbnyu ~ p|`be{|edcjkmtjkubeRny#x¤n}n}«Hsymkjo=²±¯³
´Iµv¶·
©0¸º¹
´°»¼|½
¸
·¿¾!´°»¼|½/¼
¹
¼À
¹UÁ¸Ã Ä
¼ºÅ
ÄÆ
»
ÆÈÇ
¹ ´Ç
¼|½
¸Ã É
´Ê½
¸
nyu ~ eRc5x¤mtsqËp|`¤e*{|ccepr|joÌn}rÍysqÌx¤|sze{|{s}®jkuz¤u¤j¨preR{|jtcn}m svxeRrn}p|sy©Wjku&sy~ eRp|syjkyen
x¤|sv«¤n}«bjkmkjk{p|jo5jtup|eR|x¤|edprnqprjtsvuË®°sy1pr`be{|symklzp|jksyuC¹/j¦ÎevÎ*ª9e7{svmtvep|`be*syrreR{|xsvu ~ jkubÏ#hz±¯³Î ¢ e
sv«zprnyjtuÐnymk{|sÌnÑredx¤|e{eRuvpnqprjtsvupr`bedsv|eRcÒ®°svcn}|p|jkubn}mkeR{1ª#`bjo`ÓredxbreR{|edupr{nÑyeRubedn}mkjtÔnqprjtsvuËsy®
pr`beredxbreR{|eduprn}p|jksyuÌp|`beRsyredcÒvjtveduË«^Õbl¤Íl{`bjkcn®°syn ~b~ jtp|jkye®°lbu¤p|jksyu¤nymEcn}|p|jkubvnymte{dÎ7_ `¤jk{
xHedrcj¨p{l¤{p|s{svmtve7yeRubedn}m=ÖÊubsyub¥×ny|ÍvsqbدÏ#hz±¯³Ù{ª#jtp|`¿{eRc5jt¥×mtjkubeRny1p|eR|c{RÎ_ `beu¤syubmkjtu¤eRn}
predrc
¾
{rnqprjk{¤e{n7csyu¤s}p|svubjojtp*svu ~ jtp|jksyuª#jtp|`Ì|e{xHeRpp|s*¹n}u ~ nÚjkx¤{r`bjtp|Ôdsyu ~ jtp|jksyuѪ#jtp|`
reR{|xHeRp*p|s
À
¹ÎÛÕjtun}mkmtܪ9eÌxbrsqyen&svcx¤n}rjk{|syuÝpr`bedsv|eRcÞn}u ~ l¤{eËjtpjkußsv ~ eR7prsÈ{|symkyeÌn
{p|sz`¤nv{prjkOsvup|rsymàxbrsy«bmkedcwÎ
áÑâãUäråæ qçy¡ h^^cepr|joÌn}rÍysq7xbrsze{|{|eR{RHh^eRcj¨¥×mtjkuben}=0±=³Ù{RUÏ nvͪ n} ~ {prs^`ny{p|jo ~ jt§HeR¥
eRup|jon}màeRèl¤n}p|jksyu¤{RbÕblÍl{`bjkcn{prs^`ny{p|joORn}molbmkl¤{
,.//9
#2)"Ã>@ !0#/) :;1 < 476
/ d'5)
¡ uÓsyu{j~ adrelbuÓsvxRrn}p|edl¤ ~ jt§HeR|eRuvprjteRm ~ e*{esyu ~ sv~ |e7{|edcjt¥¦eRmtmkjtxbp|joèvl¤e©n[ye
~ eR{#s^ed¬djteRupr{ ceR{|lbn}«bmkeR{Rzedu*®°syrce ~ jtvedryedupreyzep#m¦Ù³0±=(xn}r«svmtmkjkèlbe
´Iµv¶·
©0¸º¹
´°»¼|½
¸
·¿¾!´°»¼|½/¼
¹
¼À
¹UÁ¸Ã Ä
¼ºÅ
ÄÆ
»
ÆÈÇ
¹ ´Ç ¼|½ ¸ÃÂ É ´Ê½ ¸
edp1syuÌeRcxbmtsvjtemkexb|szdeR{r{l¤{ ~ eny|Ívsqw{|^cp|rjoèvl¤e ~ Ùsyxdnqpredlb+jkuz¤ubjtp|e{jkcn}mE©Lxsvlb ~ svububed
l¤ubejkup|edrxbpnqp|jksyuwxbrsy«n}«bjkmtjo{p|exHsylbOmon7{svmtlbp|jksyuѹCeèlbj|eR^jteRuvp7R{|syl ~ re1m¦Ù³²±+hz syrreR{¥
xHsyu ~ n}up|evÎ uËsy«zprjteRuvp1n}l{|{|j!lbuÌpr`dsv|aRce ~ eredxbR{|edupnqp|jksyu ~ eR{+cn}|p|jkubvnymte{¯èlbj!redxbR{|edupre
l¤ubeduRrnymtjo{|n}p|jksyu ~ lÌp|`Rsyradce ~ e|eRxb{eRuprnqprjtsvu ~ syu¤ux¤ny¯Õ¤l¤Í^l¤{`¤jtcn7xHsyl¤ ~ e{=cnyprjtuz¥
n}mkeR{+®°svu¤prjtsvububeRmtmkeR{n ~b~ jtp|jkye{dÎ0eRdj²xHedrcep ~ e{svl ~ remIÙ³²±+hzÛRudn}m¯Ö~ nyu¤{1mke*svup|e^pre
u¤syuÑn}rÍysq^jkeduØÎ0Ú/e1p|eR|ceubsvuwmtjkuRnyjtre
¾
{|n}p|jo{®Ênyj¨pl¤ubesvu ~ jtp|jksyu ~ ec5svubs}prsyubjke1x¤nyrnyxbxHsy|p
Â@¹Ëep ~ e5Újkx¤{|`¤jtÔRjknyubj¨px¤ny¯n}x¤xsvp Â
À
¹UÁ!Î!=syl¤{ ~ dcsvuvpr|svu¤{=nyl¤{r{jlbupr`dsv|aRce
~ e+dsycx¤nyrnyjk{|syuèvl/Ùsvu*eRc5x¤mtsvjàxsvlbR{|syl ~ reOlbu*x¤|sv«bmtaRce ~ esyupr"vmte+{prsz`¤ny{p|joèlbeyÎ
#æ ä $ Ë¡ ErszeR{r{|l¤{ ~ e7ny|Ívsq{^cpr|joèlbeR{R/³0±=W{|edcjt¥¦mkjtu%Rn}jkreR{R³²èl¤nqprjtsvu¤{ ~ jt§&dreduz¥
prjteRmtmkeR{#{p|sz`¤nv{prjkèlbe{ dp|rsyvrn ~ eR{R'9nymkdlbm/{p|sz`¤ny{p|joèlbe ~ eÕbl¤Í^l¤{|`bjkcn
? @
"/ !)*²"
Ï nvͪ n} ~ hp|sz`¤nv{prjk0±=jt§HeR|eRup|jon}m³²èl¤nqprjtsvu¤{0ÖÊjtu{`bsvpÏ hb±=³Ù{Ø`¤n[veE«Hededujtupr|s ~ l¤de ~ ÖÊjtup|`beRjt
nvprl¤n}m®°syrc7د«Ì³Î n} ~ sylbÌn}u ~ hUÎ!edu¤BAkRC²n}u ~ ®°sylbu ~ nyxbxbmkjkRnqp|jksyu{+jtuC{prsz`¤ny{p|jodsyup|rsym
nyu ~ cn}p|`beRcn}p|jodnym/¤un}u¤deyÎ ube5cs}p|jkqnqprjtsvuÌjo{Onymk{|s7prs*vjtven*xbrsy«¤ny«bjtmkjo{prjkjtup|eR|x¤|edprnqprjtsvuÑp|s
pr`be{|symklzp|jksyu¤{9sy®!n{|^{p|eRc s}®!x¤nyrny«svmtjo¯{|edcjkmtjkuben}#²±¯³Ù{ÖÊ{|ede2AkyC°Ø9s}®p|`¤e+®°symkmtsqª#jkub®°syrc
´Êµ¶·
©9¸º¹
·¿¾!´°»¼|½/¼
¹
¼rÀ
¹UÁ¸EÂÜÄ5syuEDÄ ¼
Ç)FHGJIK
¼
¹'LÌÂ ÉM
N×® ¾ n}u ~ p|`be&ds^e¬7jkedupr{Ìs}®p|`¤e&{esvu ~ sy~ ed ~ j¨§Uedredup|jon}msyxHednqprsy©ny|eÐ{|lz¬7jkedup|mk
{|cs^s}p|`wpr`bedup|`be²±=³L`¤ny{¯n7mony{r{|jkRn}m/{|symklzp|jksyunyu ~ syu¤ecn[*dsyu¤{p|rl¤p=p|`bex¤n}jk=sy®Exb|szdeR{r{e{
O
¶0PQ
R
ÂTS}¹
´Uv¼WV
¶0PQ
R ¸
¼X
¶0PQ
R
ÂTS
À
¹HÁ
´-Uv¼WV
¶0PQ
R
¸9ª#`bedre
V
¶0PQ
R
¼»
Æ U ÆÈÇ
¼
jo{9p|`be ~ j¨§Ul¤{|jtsvuwxb|szdeR{r{0ª#jtp|`
jkuz¤u¤j¨preR{|jtcn}m²syxHednqp|sv© ª#`bjk`Ð{prnyp{O®°|svc
½
n}pcsycedup
»
Î*_ `¤edul¤{jkubYN×p"¤Ù{+®°syrclbmonsyube
`¤eRÍz{0p|`¤n}p
´O
¶0PQ
R
¼X
¶0PQ
R ¸
¶0Z
R Z
LÓ{|symkye{0p|`beÏ#hz±¯³
O
¶0PQ
R
姃
´V
¶0PQ
L ¸
·\[
L
R
¾!´^]q¼V
¶0PQ
_ ¼ O
¶0PQ
_
¼X
¶0PQ
_
¸W`
]ab[
L
R X
¶0PQ
R
`dc
_ M
N×®
¾
jo{mkjtuben}#pr`bedup|`be5{svmtlzprjtsvuws}®²n}«HsqyeÏ#hz±¯³(`¤nv{n}uÑedzxbmtjojtp=®°sv|clbmonn}u
~
pr`beeèvln}mkj¨p
¹ ´°»¼|½ ¸EÂeO
¶0PQ
¶
Âef ´O
¶0PQ
¶
¸9^jteRm~ {#p|`bedmknv{|{|jkRn}màÕbeR^ubcn}uhg1ny¯®°syrclbmonbÎ
uwpr`bes}pr`bed=`¤n}u ~ ¤jt® ¾ nyu ~ p|`bes^e¬7jkedup{#s}®E©(ny|e l¤{p¯Ú/jtx¤{r`bjtp|Ôdsyup|jku^lbsyl{9p|`beRuÑp|`be
Ï#hz±¯³{p|jkmtmv`¤ny{nlbubjoèvl¤e²{svmtlbp|jksyu+«blzppr`beE²±¯³`¤nv{àubscsv|eEn#dmknv{|{|jkRn}m}{|symklzp|jksyu/{|s p|`¤n}p/syu¤eE`¤ny{
prs¯dsyu¤{|j~ edvedubeRrnymtjkÔde ~ {svmtlzprjtsvu¤{RÎHNºun={|edrjkeR{sy®bx¤n}xHed{jkubjtp|jonqpre ~ «^2AtvC^syube9syu{j~ ed{à^jo{|dsv{|j¨p
{|symklzprjtsvu¤{=nyu ~ csyre|eeRuvprmtjkubAiC!nyu ~ At+Csvubedsyu¤{|j~ ed{{svmtlbp|jksyu¤{=jtuhzsy«Hsymkedw{x¤nve{ ®°sy¯p|`be
0±=³Ün}u ~ xbrsqye{ p|`nqp9p|`be+xbrsy«¤ny«bjkmtjo{prjk=jtupredrxb|edprn}p|jksyuvjtveduny«sqve=redcn}jku¤{²p|rlbeyÎ =u^ª n[y^jku
«Hs}pr`Ìpr`beR{|en}xbxbrsvnv`beR{RU{jku¤ep|`¤eds^e¬7jkedupr{+ny|e5Újkx¤{|`¤j¨prÔ5dsyup|jku^lbsyl¤{RHp|`beny|Ívsq*x¤|sze{|{
V
ª#j¨pr`*jkuz¤ubjtp|e{jkcn}mUsyxHednqprsy9©&jo{ n ~ jt§Hl{jksyu*xb|szdeR{r{²ª#`bjo`*{rnqprjk{¤e{0nhz±¯³Ènyu ~ {|ssvube¯cn[
l{eOp|`¤ej¤sqª
V
¶0PQ
R
nyu ~ jtpr{=ny{r{szjonqpre ~ {prs^`ny{p|joORn}molbmkl¤{+Î
Nºuwpr`bjk{=x¤n}xHed=ª0edsyu¤{|j~ ed=n{|edcjt¥¦eRmtmkjtxbp|jo1{^ccep|rjo1{esyu ~ sv~ ed ~ jt§HeR|eRup|jon}m/svxeRrn}p|sv ©
Öʪ#`bjo`jo{²ª#rj¨p|p|eRujku ~ jkyeR|vedu¤de#®°syrchkvubs5ubsvu ~ edvedubeRrnv5syu ~ j¨prjtsvujo{0nv{|{|lbce ~ Ø ª#jtp|`ceny{|lb¥
ny«bmkes^e¬7jkedup{dÎ!_ `bedu
V
jo{En{|^cc5edp|rjk#Ìn}rÍysqxbrsze{|{!ª#`bjo` ~ s^eR{Eubscsyre{|n}p|jo{®°5nhz±¯³Î
_ `¤edre=ezjo{p{²u¤scsyre¯n<j¤sqªÜredxbreR{|edup|jkub
V
n}u ~ p|`beOl¤{|l¤n}mU{prsz`¤ny{p|jo=Rn}molbmkl¤{9syrreR{|xsvu ~ jkub
prshb±=³Ù{9`¤ny{0prs«He¯redx¤mknve
~
«pr`be+dn}mol¤mtl¤{9ny{r{|s^djkn}p|e
~
p|s{l`nx¤|sze{|{0«^ÕblbÍ^l¤{|`bjkcnÖÊ{|ede
AfC°ØÎ
¢
exbrsqyepr`¤nqpp|`ben}«Hsqye²±¯³`¤nv{nÑlbu¤jkèlbe{|symklzprjtsvuÓ¹Ðjtu¿{svc5e7±¯jk|jo`bmkep5{|x¤nve7n}u
~
pr`be{prs^`ny{p|jojkuvpredrxbreprn}p|jksyuÑjo{Oyjkyeduny{¯¹
´Ê»¼½
¸ÂlO
¶
m
¼#n
Q
abo
M
Uv¼
ª#`beR|e n Q
jo{=pr`be5mkn[ªsy®Ep|`be
x¤|sze{|{
V
{prn}|p|jkub®°rsyc
½
n}u ~ O jo{pr`be{svmtlzprjtsvus}®²n*Ï#hz±=³£ÖÊnyu¤n}mksyvsyl¤{#p|s7pr`¤nqp1svu¤{|j~ eR|e ~
ny«sqveØsyupDÄ ¼ Ç aÈ»
F0M ¢ enymk{|sË{|symkyenÌvedubeRrnym9Ï#hz±¯³ vjtve*nËsycx¤ny|jo{|syuÓpr`bedsv|eRc n}u ~ l¤{e
pr`bedc£jkusy ~ ed²p|s{|symkyen{prs^`ny{p|josvuvpr|svmbxbrsy«bmkedc£ny{r{|s^djkn}p|e ~ prs1p|`¤e¯ny|Ívsqxbrs^deR{r{
V
n}u ~
prse{pn}«bmkjk{|`Ñpr`bemkjkubÍÑ«edpª0eReduÌp|`bjo{¯dsyup|rsymxbrsy«bmkedc nyu ~ p|`¤e5dsyr|e{xHsyu ~ jtubrq=n}cjkm¨prsyu9snydsy«bj
Ï9eRmtmkcn}ueRèl¤nqprjtsvuÎ
_ `be#xn}xHedEjo{ syrvnyubjkÔde ~ ny{®°svmtmksqª{dÎtNºu7h^eRp|jksyu¤{0¯nyu ~ >+ª0e l¤{|en}un}mkvprjkRn}mzcepr`bs ~ {!prs{svmtve
pr`ben}«HsqyeO²±¯³Üª#|jtppreduwjkuqny|jonqprjtsvu¤n}m®°syrc
[ L
m
´r´
¹ ¶ ¼|µ ¶u
¸
·bv^´
¹ ¶ ¼ u¶
¸¸,`
» Â [ L
m
´¦¾!´°»¼
M¼ ¹ ¶ ¼rÀ
¹ ¶ Á¸
¼ u¶
¸`
»·(´
É ¼ u L ¸ ¼
ª#`¤edre
u
jk{1n*preR{p1®°lbu¤prjtsvuÎqeR|e v ~
edubsyp|eR{Opr`beedu¤edryny{r{|s^djkn}p|e
~
p|spr`besyxHednqprsy1© M !=s}p|e
pr`¤nqp À ¹ ¶ ~
s^eR{ ubsyp#ezjk{pjku*svlb#{jtp|l¤n}p|jksyuz«¤lzpª0e1ny|e+n}«bmkeOp|syjkye+n5ª0en}Í
~
ed¤ubjtp|jksyuws}®
À ¹ ¶Á M
syredsqyeR ª0e¯xbrsqye p|`nqp9p|`beOn}«Hsqyeª9eRnyÍeèvlnqp|jksyujk{0eRèlbjkqn}mkedup²ª#jtp|`7pr`beOc5jkm
~
eRèl¤n}p|jksyu7jku©
{|edu{e
¹
´Ê»¼
¸EÂ
n
L
¶ É
´
¸ · [ L
¶ n R
¶º¾!´-Uv¼
M¼¹ R
¼rÀ
¹ R Á¸
´
¸ ` U M ÖØ
¢ e7ny{r{|lbcepr`¤nqp
¾
jk{1Ú/jtx{|`bjtp|Ôdsyup|jku^lbsyl¤{+ª#jtp|`CreR{|xep+p|sÑjtpr{+mony{pqn}rjkny«bmkeyn}u ~ {rnqprjk{¤e{On
csvubs}prsyubjojtp7svu ~ jtp|jksyuwª#j¨pr`w|e{xHeRp9prs¹xM
Nºuwh^eRp|jksyu¤{0fnyu ~ 1ª9e=ed^p|edu ~ p|sp|jkc5ed¥E{x¤nve#®°l¤u¤prjtsvu¤{0p|`beO|eRxbreR{|eduprn}p|jksyu5p|`beRsyredc£yjkyeRu
jku AfC¦ÎEÑsv|e+xbreRdjk{|edmkª9e1xb|sqve=pr`¤nqp#jt®¹Ìjk{9pr`be{svmtlbp|jksyus}®p|`beny«sqve¯eèvlnqp|jksyu*p|`¤edu
¹
´-U0·Ð»¼WV
¶ ¸ a ¹
´Uv¼V
m
¸ÃÂ
K
[ ¶
m
´U9·E]q¼V
_ ¸ `
´]
¸ a
ÖIyØ
a [ ¶
m
¾!´-U0·b]q¼WV
_ ¼ ¹
´U9·E]q¼V
_ ¸
¼
¹
´U9·E]q¼V
_
¸¸W`
] M
_ `be=cnyprjtu¤vn}mkeR{
ª#`bjo`nyxbxHeRn}²jtupr`be=n}«Hsqye ®°sv|clbmon1n}re p|`bes^sv ~ jku¤n}p|e cn}|p|jkubvnymte{
jkup|rs ~ l¤e ~ «^7Õbl¤Íl{`bjkcnb
jk{#nyeRubedn}mkjkÔRnqprjtsvus}®
Q nyu ~
!
¹wjo{9nyeRubedn}mkjkÔRnqprjtsvu7s}®
À
¹UÁ M
!=s}pren}mo{s+pr`¤nqp²jkup|`¤e=mony{r{jodnymb{jtp|lnqp|jksyuÑÖIÚjkx¤{|`¤j¨prÔ=svuvprjtu^lbsvl¤{ s^ed¬djteRupr{Ø `"iÂ(Á `dcWª#`¤edre
c£jo{#nÏ9|sqª#ubjon}u*cs}p|jksyu/βh^s5p|`beny«sqve=®°sv|clbmkn5jo{9p|`benyu¤n}mksyvlbeOs}®/pr`be{prn}u ~ n} ~ Ï#hz±¯³Î
Nºuwh^eRp|jksyu7fª9e=x¤|sqve p|`¤n}p+Ö¦yØ `¤sym~ {Elbu ~ ed
n$#
ª#`¤edre&% jk{²p|`beOÚeR«e{vlbe=ceRnv{l¤|eÖʪ#`bjo`
redx¤|e{eRuvp{1pr`bewjtu^qn}rjknyupceRnv{lbre7®°sv5p|`bewxbrszeR{r{
V
Ønyu ~ jkuÈhzeRprjtsvuÈ̪0ew{p|redubyp|`beRuÐp|`¤jk{
reR{|lbmtpjtupr`be{eRu¤{|e+p|`¤n}p#p|`beeèl¤n}mkj¨p`bsym~ {#lbu ~ edeny`
n Q
ª#j¨pr`
½')(+*R¼
ª#`bedre
(
jk{n{|epsy®
u^lbmkmdnyx¤nydj¨pvÎ
Nºuh^ep|jksyu-,ª9e+edcxbmksq7p|`¤en}«HsqyeO|e{lbmtpr{=n}u ~ n{pn}u ~ n} ~ bze ~ xHsyjkup=n}rylbceRuvp prsxb|sqve
edzjk{p|eRu¤e1n}u ~ lbu¤jkèlbeRubeR{r{ ®°sy pr`be{|symklzp|jksyu¤{9sy®pr`be1yeRubedn}màÏ#hz±¯³
O ¶ Â/. ·\[
L
¶
¾!´Uv¼0¼ O R
¼X
R Á ´^V
R
¸|¸` U ab[
L
¶ X R M`"
R ¼
ª#`¤edren}vnyjtu*p|`¤e1®°lbu¤p|jksyu
¾
ubede ~ ubs}p«HeÚjkx¤{r`bjtp|Ôdsyup|jku^lbsyl¤{ ª#jtp|`ÑreR{|xHeRp#prs 21bsyu¤mt*c5svubs}¥
prsyubjojtp@{lz¬7deR{RÎ8ÖI_ `be ~ n}prn/.Ünyu ~
¾5´Uv¼0¼
Ä ¼
ĸwn}reÐny{|lbce ~ prs(«eRmtsvubÝp|sL©43
´65
¸ ¼
|e{xÎ
© 387
5
1|©
´Ä ¼ ǯ¸9
¼;:-<>=
MØE_ `be#eRèl¤nymtjtp`bsvm~ {!®°sy²nymtm
»?'
DÄ ¼ Ç F¼ nyu ~ n Q
ao
MU
®°sy²edyeR|
½-'!(+*
M
!=s}prep|`¤n}ppr`beedzdedxzprjtsvu¤n}m9{edp
(
cn[Ë«e`bsv{|eduÓjku ~ eRxeRu ~ eduprmtCs}®@. M!_ `be7jkup|eR|e{p5s}®{l`
n|e{lbmtpjo{1pr`be®°symkmksqª#jtubBAsvube7cn[Ë`¤ss{epr`be7edzdedxzprjtsvu¤n}m9{edp
(
{|l¤`Cp|`¤n}pR!jt®#p|`be7x¤|sze{|{
V
{prnyp{1®°rsyc nÑxHsyjkuvp5s}®
(+*
!j¨p5|eRcn}jku¤{nymtª n[z{1jtuÐp|`bjo{{|epÎh^s (+*
redxbreR{|edup{nÑu¤n}p|lbn}m
{prn}p|eO{x¤nve=®°sy9p|`be+ny|ÍvsqxbrszeR{r{
V M Nºu{svce¯{|edu¤{|e=pr`bjo{0ª9syl¤m
~
redx¤mknvepr`beOmknvÍ5sy®àp|`be j¤sqª1Î
¯{1nwsyu{eèvl¤edu¤de5svube5¤u ~ {1nved{jksyus}®9p|`be{|symklzp|jksyu˪#`bjo`C{rnqprjk{¤e{=pr`becjtm~ eRèl¤n}p|jksyuÝÖØ
xHsyjkupª#jk{|eyÎ
Õjku¤n}mkmtvyjkuhzeRprjtsvu)C=ª9e xb|sqve9nOsvcx¤n}rjk{|syu|e{l¤m¨pEn}u ~ l¤{|e j¨pEjku5sy ~ edp|s1{|symkye n+{prs^`ny{p|jo
dsyup|rsymUxbrsy«¤mteRcÎ
>@ #$dd68"1 d#
jkyeRuÓnw«svlbu ~ e ~ ceny{|lbrny«bmke®°l¤u¤prjtsvuÐÁ S I
K
I
K I
¼
ÁÜÂ 7Á
9ÑÖ#ª#jtp|`ÓlbxbxHedjtu ~ e
Â
=v¼
MMM
¼
^n}u ~ mtsqª9edEjku ~ ed5Â
=y¼
MMM
¼
Ø!ª9e ~ ed¤ube
o SÂßÁàÁ= 7o P
9}ª#`beR|e o P
Â
Á
Á
nyu ~ u¤s}p|eÑÖI{eRepr`bexbxHedu ~ j¨¤Ø¯pr`¤nqp+p|`¤edre5edzjk{pr{+nceRny{|lbn}«bmke®°l¤u¤prjtsvu S'I
K I
I K
{|l¤`*p|`¤n}p
Á Á*ÂÜÁ
y¼
ÁbÁÂ(Á
¼
Á
Â
bÁ Â = M
_ `be1cn}p|rj¨7edm~ jo{ bze ~ p|`brsyl¤y`*p|`be1x¤nyxeRRÎ Õ¤lbpr`bed=jtup|rs ~ l¤de¯pr`be1«bjkmtjkuben} ®°syrc
v´
¹
¼
¸EÂ [6o
P
´°½
¸ µ ¹
µH½
´°½
¸
µ
µH½
´°½
¸W`
½¼
¹
¼)'! #"
* ´ IKO¸
¼
ª#`¤edre
"
* ´I K ¸9jo{ pr`be{xnye+sy®jkuz¤ubjtp|eRmt
~
jt§HeR|eRuvprjkny«bmke+®°lbu¤prjtsvu¤{#ª#jtp|`Ñsvcx¤nyp={lbx¤xsvpn}u
~
pr`besvu^yeduprjtsvus}®!{l¤c5cn}p|jksyuwsyu|eRxenqpre ~ jtu ~ jkdeR{#jo{ jtu®°svrdeyÎ ¢ enymk{|s ~ e¤u¤e
v ´¹
¼$
¸EÂ
´¹
¼
¸ · v^´
¹
¼$
¸ ¼
ª#`¤edre
´ t¼
¸9jo{9p|`¤eOl{l¤nym{|Rn}mon} xbrs ~ l¤p#jkuÑ© Î ¢ e+l¤{|e v^´
¹U¸0®°sv
v^´
¹ ¼
¹U¸9n}u ~ v ´
¹U¸0®°sv
v ´¹ ¼
¹à¸M
¢ e i* 46 p|`¤n}p
´%
"
* ´I K ¸
¼v
¸5jo{7mksv{rn}«¤mteCÖÊ{|ede AfC®°sv ~ epn}jkmte ~¿~ ed¤ubjtp|jksyu¤{n}u ~ ubs}pnqprjtsvuØÎ
hzsyce*vedubeRrnym9d|jtp|eR|jonjkcxs{jkubCsvu ~ jtp|jksyu¤{5syu
o
jtuÈsy~ eRpr`¤nqp v
«HeÑmksv{rn}«bmke*ny|e*yjkyeRu&jku
hzeRprjtsvuB>¤Îtsy®AfC AOe{|{|eduprjknymtmkÌeRpn}jkuÌx¤nyprjknym ~ edrjk[n}p|jkye{¯sy®
o
`¤n[yep|sw«HemtszdnymtmkjkuË© sy
o
`ny{9p|svedrj¨®°*nmtszdnymàedmkmtjkxzprjkdj¨pny{r{lbcxzprjtsvuβÑsv|e+xbreRdjk{|edmkybn{|lz¬7jkedup=dsyu ~ jtp|jksyujo{ p|`¤n}p#®°sy
eny`
½-'
I K pr`bedre+jk{={svc5e ] <
Ä{|l¤`pr`¤nqpsvube+s}®p|`be+®°svmtmksqª#jkubpª9ssvu
~
jtp|jksyu¤{ `bsvm
~
{A
jØ)'&
)(*
Q ' © ´c _
´°½
¸¸
¼ = Æ
¼,+
Æ
¼
jkjoØ _ `¤edreOedzjk{pr{{|syce#-
<
Ä{|l¤`p|`¤n}p oU´
z¸/.0-21®°sv#edvedr ' c _
´°½
¸"M
_ `be ~ svcnyjtu¿s}®#pr`be®°sv|c
´3
"
* ´I K ¸
¼#v
¸Ö¯j¦ÎevÎCp|`bewmksv{|lbres}® "
* ´I K
¸ª#jtp|`È|e{xHeRpp|s
v
زjo{ ~ edu¤s}p|e ~ «^54
¼
pr`be1ny{r{szdjkn}p|e ~ {eRc5jkyrsyl¤xjo{
´n/¶
¸
¶,6
m ¼
¥Eª#`bjk`*jk{#n5{|ccepr|jo+n}rÍysq^jon}u
{|edcjkyrsylbxp|`¤n}pn ~ cjtpr{9p|`¤eÚed«HeR{|yl¤eOceny{|lb|e8%ny{#nyu*jku^qn}rjknyuvp ceRnv{lbreyÎ!_ `be1jkuz¤ubjtp|e{jkcn}m
svxeRrn}p|svs}® pr`bjk{O{|edcjtv|svlbxÑjo{ ~ eRubs}pre ~ «^
´;Ì´
©0¸
¼
©9¸Î ¢ e5|edn}mkmpr`¤nqp
Å ¹ ' ©
¼ÅH»@<
Ä
¼n ¶¹ '
Ì´
©0¸8794 nyu ~ ¤jt® ¹ ' ´
©0¸
¼$ '
4pr`bedu v^´
¹
¼
¸²Â a´
©E¹
¼$
¸M¤@vedrl¤{|e®°l¤m/jkubeRèl¤nymtjtpjo{ÖÊ{|ede
Ú/edccnyÎ>bÎ>jku AfCØ
v´n/¶
¹à¸9Æ
=
: » ¹ ¼ ÅU»<
Ä ¼ ¹ ' © M Ö>vØ
¢ e1jtupr|s ~ l¤deOp|`be{|x¤nyde+s}®®°lbu¤prjtsvu¤{
L Â ; u SDÄ ¼
Ç)FHGhI K
I=< u¶
'
4W®°syn}mkcsv{p#eRny`
»¼9[
L
m
v´
u¶
¼
u¶¸`
»?>A@ܼ
» u ¶
jk{ ~ j¨§Uedredup|jon}«¤mte1jku© n}u ~ » µ¶ u ¶
jk{#© a
syuprjtu^lbsvl¤{9syu DÄ ¼ Ç)F3B
¼
ª#`¤jk`p|l¤|u¤{sylzp#prs«e1pr`ben}x¤xb|svxbrjkn}p|e1{xnye+sy®preR{p®°lbu¤p|jksyu¤{RÎ0±¯edubsyp|e1«^ L Pm
pr`be{|x¤nydeOsy®
®°l¤u¤prjtsvu¤{
u 'C
LÐ{|l¤`pr`¤nqp u m
ÂßÄbÎ Õbsv
u 'C
LCª0e
~
e¤ube
u L SÂEDF$GIH
¶0Z
L u¶
·\[
L
m
v´u¶
¸W`
»KJML
N
nyu
~
ª0e
~
eRubs}pre1«4Wpr`besvc5x¤mtedp|jksyus}® LÓª#jtp|`wreR{|xep p|s L M
*â Ðz ¹
' 4 , "$"%42 +:i +3
»¼
¹ ¶ ' 4
¼
L
m
v^´
¹ ¶
¸W`
» >9@ܼ
$
¹ L
ÂEDFG H
¶0Z
L ¹
¶2 · [ L
m
v´
¹ ¶
¸W`
»KJ
> @ M
¹
' 4 $'
u ' 4 , "$
L
m ¹ ¶`
»'
4 $'
v D [ L
m ¹ ¶ ` »¼ u J Â [ L
m
v^´¹ ¶ ¼ u ¸W` » M
, r, 4 $ h ""+ B1^ 1 # h* $'+-$1¹
' ©
7#DÄ ¼ Ç)FHGhI
K 9 *,#
^0¹ ¶ ' 4"J%428 +: "3 »!' DÄ ¼ Ç F ¼
L
m
v^´¹ ¶ ¸W` » > @ <$ Y*, ^9¹ ¶ ' ©
"r$3
» ' DÄ ¼
Ç)F$' , 646
» ¹ ¶ ©
a
$$*,* 6$ DÄ ¼
Ç)F"!#' $"%% %"3 &''# 42$)(
$,32*,** $'+-$ ^96+ #(+*, +$'
´¹
¸T + $' ,"3
¹ P¶ Â [
m -¹
¶/.
R ` U ¼ » ' DÄ ¼ Ç)F
^ 0* $'+$ * $' "^ <$, (%1 "$)(12* '+% 5 "$,$9" ¹ $ %(d +8$, ":% DÄ ¼ Ç
·32
F
*,h^/¹
¶'
44"T%42= +:i +3
» ' DÇ ¼ Ç
·52
F768"T 4'%
-¹ ¶ Â(¹
L
¶
"
» <
Ç9'!
!=s}p|ep|`¤n}p "
* ´ DÄ ¼ Ç)F GhI
K
¸ jo{#ubsypubeRdeR{r{|ny|jkmt ~ edu¤{|e1jku 4ª#jtp|`ÑreR{|xep#prs M L ¼
«¤lzp=jkuwn
ª9eRnyÍyeR9{|edu¤{|e+yjkyeRu*jkup|`be+®°svmtmksqª#jtu¤mkedccnbÎ
:0â Ð;<9= 5 +:i +35¹
' 4 ¼
^ + Tr+ ("*, +$' O¹ ' "
* ´DÄ ¼
Ç)FxG I K ¸ ¼ 'MѼ
*,#
^ L
m v ´¹ ¶ a ¹ ¶ ¸ ` » Ä,M
> ææ@? 0syu{j~ ed!pr`be{rdn}mon}!xbrs ~ l¤p v ´¹
¼$
¸EÂ L
m v ´¹
¶¼$q¶
¸ ` »
n}u ~ mtedpBAß«e pr`bemksv{|lbre9sy®
"
* ´ DÄ ¼
Ç)FG5I K ¸ ª#jtp|`reR{|xHeRp0p|s1pr`be=u¤syrc nv{|{|szjonqp|e ~ p|s v
M^h^jku¤e
v
jo{²dmts{|ny«bmtev
v
jo{²dmts{|ny«bmke
nymk{|s7Ö°j¦ÎeyÎ jt®¹
Ä1jku© ´ DÄ ¼ =
F,G2I K ¸Enyu
~ ´¹ ¸
jk{ 9n}l¤`^5ª#j¨pr`7|e{xHeRpEp|sp|`be¯ubsyrc jtu
~
l¤e
~
«^
v ¼
pr`bedu¹
Ä1ª#jtp|`7reR{|xHeRp²p|s1pr`¤nqp9ubsyrc7Ø {s+pr`¤nqpCAEDFÂTS;R¹
' © ´ DÄ ¼ =
FGTI K ¸)S}¹
¶?'
4 ¼
®°sy¯n}mkc5s{p edvedr
»
nyu ~ L
m
v^´
¹ ¶¸ ` » >9@
BdM
Ú/ep+l¤{Oxb|sqve1p|`nqpA ÂGFYM N×p+{|lz¬7e{prs`¤eRÍp|`¤n}pOjt®0H
' F nyu ~ HJIKAÒÖ°jkup|`¤e5{|edu¤{|esy®
v
Ø#pr`beduLH ÂWÄ,MU¯{|{|lbcep|`¤n}p L
m v ´H
¶¼
¶¸ ` » ÂWÄ
¼ºÅ
' "
* ´ DÄ ¼ Ç)FxG9I
K
¸Mà#edx¤mknvjkub
¶
«^
M ¶ ¶ ª#jtp|`
M ' "
* DÄ ¼ Ç F/ª0esyumkl ~ e1pr`¤nqp¯®°sy¯nymtcs{p=eRyeR|
»¼v
´H
¶¼N
¸ Â@Ä
¼Å@N+'M
"
* ´I K
¸M
hzjtu¤deKH
¶ '
4pr`bjk{jkedm~ {OH
¶
Â(Ä,M
Õjku¤n}mkmtv¤{|jtu¤deP4GDFpr`be1xbrs^s}®jo{dsycxbmkep|evÎRQ
Õ¤sy
¾ ' © ´ DÄ ¼
Ç)FtG I K
¸nyu ~ É ' © ´I K
¸#ª9edsyu¤{|j~ edpr`be®°svmtmksqª#jkub7x¤nyrny«svmtjo1eRèl¤n}p|jksyuÑsvu
DÄ ¼ Ç)FHGhI
K
ª#jtp|`predrc5jku¤nymsyu ~ j¨prjtsvuÑnqp p|jkce1ÇpS
´Êµ¶·
©9¸º¹
·¿¾
姀
¼
¹'LÌÂ ÉM Ö°fØ
Nºup|`beRny{|e o . -21
¼
ª#j¨pr`-
< Ä ¼
p|`be5pr|enqp|ceRuvp1sy®Ep|`bjo{1eèvlnqp|jksyujk{+dmknv{|{|jkRn}m#ÖÊ{|edeJAC/C°ØÎ5Nºu
ª#`nqp®°svmtmksqª{#ª9e{mkjtv`p|mk*cs ~ jt®°pr`bemony{r{|jkRn}m/ny|vlbcedupr{#{|sny{#p|sdsqyed pr`be ~ eRyeRubednqpredny{|eyÎ
_ `¤e+{|edcjkyrsylbx
´n¶
¸
¶,6
m ¼
ª#`bs{eOezjo{predu¤de1jk{#nv{|{|lbce ~ zª#jtmkm«He1l¤{e ~ jkuÑn}ueR{r{eRup|jon}mઠn[yÎ
â !" "Êæ#! %$'& . * $'+$¹
' 4 r%% # Ë{p|rsyubÑ{|symklzp|jksyupr (+*,-$ 6)( 0
´É
¼r¾
¸ » ¹ ¶
ÂL¹
´°»¼
¸=© a
! + +$,-'+% 6$ DÄ ¼ ǯ¸
¼²µ ¶¹ ¶ ' © ´ DÄ ¼
Ç)FxGYI
K ¸ ¼ ¹ ¶ 'Ì´
©0¸
"%46i) ": +3
»?'
DÄ ¼ Ç)F
¼
$', T (+*,%- 8$6%( T%r%42= +:i +3x + !
.* $ "-$¹
' 4 %% rª9eRn}Í{|symklzp|jksyub"8 (+*,-$56)( 5^ "%% $ (r +%-$r,%
[ L
m
´´
¹
¶¼rµv¶u ¸ · v^´
¹
¶r¼ u ¶
¸|¸`
» Â [ L
m
´I¾[¶¼ u ¶
¸,`
»·(´
É ¼ u
L!¸ ÖIyØ
" ": +3 u '!
L Pm !
ubedn}u{|ede« ~ jk|ep=vedrj¨Hdnqprjtsvup|`¤n}pOn}u^{pr|svub{|symklzp|jksyuwjo{¯nª9eRnyÍ{|symklzp|jksyu/Î ¢ eu¤s}p|e
nymk{|sp|`¤n}p#p|`be1redmonqprjtsvu ~ ed¤ubjkubn5ª0en}Í{|symklzp|jksyujo{ eRèlbjk[nymteRup#p|s
[ L
¶+*
´´
¹ ¶
¼|µ
¶u ¸ · v^´
¹ ¶¼
u¶ ¸|¸ ` » Â [ L
¶+*
´¦¾
¶ ¼ u¶ ¸ `
»/·(´
É ¼ u L ¸ aÈ´
¹ ¶+*
¼
u¶+* ¸ ¼
Ö;,vØ
®°sv=eRyedr
» m ' DÄ ¼ Ç Fn}u ~ edvedr
u '
LEàny{¯syu¤ednyu ~ jk|ep|mkÑ`beRÍUÎ=Ï9eRmtsqª@ª9eyjkye{lz¬7djteRuvp
dsyu ~ j¨prjtsvu¤{9®°sy pr`be1ezjo{predu¤de+nyu ~ lbubjoèvl¤edube{|{#sy®n{p|rsyub{|symklzp|jksyuÎ
> æ-,²æ " "Êæ#! /. ;/É
' © ´I K ¸ , "$
» n L ¶ É 7© a
#$,-$,*,* h$ DÄ ¼ Ç)F
¼ © 10
! + "$ "% 5$ DÄ ¼
ǯ¸8$
µ¶n
L ¶ ÉÈÂ
a © n L ¶
ÉM
32x +
¾
SDÄ ¼
Ç)F G I K I B%* $ "-$p*,# ,
»
¾[¶
B! "# "$ "% B$Ý©
$'
» µ¶¾[¶
1© a
$$*,* 2$ DÄ ¼ Ç F!5 +$ ^ * $'+$ H
´°»¼|½
¸1ÂTS L
¶ n R
¶¾
R
´Ê½
¸ ` U
© a
! "# "$ "% 5$BDÄ ¼ Ç)F=$'
µ ¶H
´Ê»¼½
¸²Â
a n L ¶¾ L
´Ê½
¸
·\[
L
¶ n R ¶ µ R ¾ R
´Ê½
¸ ` U M
79 : "&
» µ ¶H
´Ê»¼½
¸© a
#$,-$,*,* $BDÄ ¼
Ç)F0M
5.#* 46 8^ É ' © ´I K ¸
¼²¾
+/+ ", 3, $ $'r + $'
¹
´°»¼½
¸EÂ n L ¶ É
´°½
¸ · [ L
¶ n R
¶¾
R
´°½
¸ ` U M
, "$*¹ 6-$)(2+%* $B" 6)( !
> ææ@?
jØ+_ `beny{r{eRprjtsvuÌjo{1ndsyu¤{|eRèlbeRu¤evn}csyu¤*syp|`beR+p|`bjkubv{Rsy®²p|`¤e®Ênyp+pr`¤nqp n¶
É '
´
©0¸0®°sv#eRny`
»<
ÄM
jkjoØ Nºuwsv
~
eR9p|s7svcxblzp|eOpr`be
~
edrjk[n}p|jkye µ¶
Hsyube+ª#rj¨preR{
H
¶/.
_ a H ¶ Â [ L ¶ _
m
nI¾[¶/.
_
.
` a [
L
¶
m
nʾ[¶/.
`
 [ L ¶
m
nb´I¾[¶/.
_
.'aо[¶/.
¸ a [ L ¶
L ¶ _ nI¾[¶/.
_
.
`M
_ `¤edusyube ~ e ~ l¤deR{9p|`nqp
=
] ´H
¶/.
_ a H ¶¸ [ L
¶ n R
¶µ
R ¾ R ` U a n L
¶¾
L M
Ú/ep#l¤{ ubsqª(x¤|sqve=jkjtjØÎEÏ9jØ0©
n R
¶¾
R Â µ R
´n
R
¶¾
R ¸ a9n
R
¶µ
R ¾ R
n}u ~ ®°lb|p|`beRR^«^7jØ0nyu ~ jkjØ
©²¹
¶ Â © n L ¶ É · [ L
¶ © n R
¶¾
R ` U
 aµ ¶n L ¶É aÓ¾ ¶ · n L ¶ ¾ L aE[
L
¶ n R ¶ µ R ¾ R ` U  aµ ¶¹ ¶ aC¾ ¶ M
Q
> æ-,²æ
"
"Êæ#!
.#* 46 2^
¾ ' © ´ DÄ ¼ Ç FtG9I K ¸$ *É
' © ´I K ¸ ! +$b^ r #(+*,-$
6)( ,i6* $("*, 6+%* -$*¹
' 4 $' 7(:i +$3+3
¹
´°»¼½
¸EÂ n L ¶ É
´°½
¸ · [ L
¶ n R
¶¾
R
´°½
¸ ` U M Ö6C}Ø
79 : "& ^ <+%* -$9+-+ 8^ "%% $ (J +%$
¹ ¶2
· : [ L
¶
v0´¹ R
¸,` U Â : [ L
¶
´I¾
R ¼¹ R ¸ ` U ·= É ¼ÄÆ » ÆÈÇ Ö Ø
¹ L Æ v L
É
·=d¾
N
m P L
M ÖvØ
> ææ@?=ubjkèlbeRubeR{r{RÎ Ú/ep=¹Â Ta
Hª#`beR|e
¼
H '
4£n}re+pª0sª9eRnyÍ*{|symklzp|jksyu{#s}®ÖIvØÎ#_ `beRu
¹{rnqprjk{¤e{
[ L
¶+*
´´
¹
¶¼|µ¶
u ¸ · v^´
¹
¶|¼
u ¶
¸|¸ ` » Â a´
¹ ¶* ¼u ¶* ¸ ¼ ÅH»
m
.¿Ä
¼ u ' L M
´
¸
±¯e¤u¤e
¹
´Ê»¼½
¸²Â
=2 [
m ¹
¶/.
R
´°½
¸W`
U
