• Aucun résultat trouvé

An elementary introduction to Malliavin calculus

N/A
N/A
Protected

Academic year: 2021

Partager "An elementary introduction to Malliavin calculus"

Copied!
53
0
0

Texte intégral

(1)

HAL Id: inria-00071868

https://hal.inria.fr/inria-00071868

Submitted on 23 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents

An elementary introduction to Malliavin calculus

Vlad Bally

To cite this version:

Vlad Bally. An elementary introduction to Malliavin calculus. [Research Report] RR-4718, INRIA.

2003. �inria-00071868�

(2)

ISRN INRIA/RR--4718--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

An elementary introduction to Malliavin calculus

Vlad Bally

N° 4718

Fevrier 2003

(3)
(4)

)+*-,/.102,/*3*34

57698:2;=<?>A@CBD:E9FHGJIKBMLON+;ILQPRIKBM:2BHSTGUITBDLQN

V ;!STWCSXIT8Y:2;YSZLO:2P9FM;[R;YS

\^]KLU_`;I abGJIK6Rc

d GUP9P/LQ]TI V ;=]T;eZf69;]fZf69;gNih<CjCkelm>on;pC]TBD;] qQrQrQs!>o<utPvGQwQ;eS

xzy{Y|}J~9Q|Q€‚ ;7wQBDpQ;7GUN?BDNOIK]TL V EƒZ„ITBDLQNITLabGQFMFDBHG…puBDNZYGUFHZE9FDEvS†‡LOFMFDLJˆBMNvw‰IK69;7N9LUIK;YSLU†ƒ†‡LQE9]FD;YZITE9]K;YS

IK6vGJIŠ^wuG…pQ;‚BMNIT69;=ˆ'LQ]K‹CBMN9wwQ]KLQE9PzLU†ŒIK69;=]T;eSX;eGU]fZf6IT;eGU:abGUIT6Rc+BMNbŽZ„IKLQ/;]qUrOr9kQ‘5769;eSX;N9LUIK;YS

ZLQNuIKGQBMNIT6v]T;Y;‚IKLQP9BHZSY’

kQ‘“ N;FD;:2;NuIKGQ]TWP9]K;YST;NuIKGUITBDLQNLU†a”GUFDFMBHG…pCBMNŒ•S V BM–i;]K;NuITBHGUF/LQP/;]fGJIKLQ]fS^GUN V LQ†—IT69;‚BMNuIK;wQ]fGJIKBMLON

CWzPvGQ]XIfS †‡LO]T:E9FHGˆ6vBDZf6]T;YP9]K;YST;NuIKS˜IT6v;!Z;NuIK]KGQFiITLCLQF—BDNIT69BHS7IT6v;LQ]KWQ‘

qR‘5769;  BM;YN9;]Zf6vGQLOS V ;YZLQ:2PƒLuSXBMITBDLQNmGUN V IT69; V BM–/;Y]T;YNuITBHGUFLQP/;]fGJITLO]KS‰LU†˜a”GUFDFMBHG…pCBMN™ZGUFHZEvFMEvS

GQS7LOPƒ;Y]KGUITLQ]fS'LQNIT69;  BD;Nv;] STPvGQZ;Q‘

s9‘5769;šGUPvP9FMBHZGUITBDLQNmLU† IK69BHS!ZGQFDZE9FMEƒS‰IKL+IT69;SXITE V W”LU†˜IT69; V ;YNvSTB›I`WbLQ†˜IK69;2FDG…ˆœLU†7G V B›–iEvSTBMLON

P9]KLCZ;YSKS‘5769BHS'ˆ'GOS^IK69;‚BDN9BMITBHGUF—GUPvP9FMBHZGUITBDLQNšLQ†abGQFMFDBDG…pCBDNŒ•S˜ZYGUFHZE9FDEvS žGQN V P9]KLJpCBV ;YS'G!P9]KLQR

GU9BDFDBDSXITBHZ=P9]KLuLQ†LQ†Ÿ LO]T:GQN V ;]e•S˜6CWCP/LUIT6v;YSTBDSY‘

5769;mGUBD: LQ†IK69;YST;bN9LQIT;eSšBHSITL¡;Y:2P96vGQSTBHSzSTLQ:2;¢vGQSTBHZ¢BV ;YGQSzGUN V IK;YZf69N9BHZSš]T;YFDGUIT; V IKL¡IT6vBDS

IK69;LO]TW¢GQN V NvLUIITL+P9]KLJpuBV ;?GšIK;[CIƒLCLQ‹i‘576v;?/LCLQ‹¢LQ†'£ EƒGUFHGU]TIY¤i†‡LQ]=;[RGQ:2P9FM;O¤—BDS=GQN”;[9Z;FDFM;YNOI

NvLQ:2LQwO]KGQP96CWšLQN¢IK69BHS IKLQP9BHZU‘@CLšˆ';FDBMpO;LOERISTLQ:2;IK;YZf69N9BHZGQFP/LQBDNuIKS¥‡ˆ';ST;N V IKLšIK69;PƒGUP/;]fS LQ]

/LCLQ‹RSˆ69;Y]T;!ZLQ:2P9FD;IK;!P9]KLuLQ†¦S:G…Wz/;†‡LQEvN Vv§ GQN V+V LšN9LQI IT]K;YGUI P9]KLQ9FD;:SBDNbGUFDFwO;N9;Y]KGQFMBMI`WšLQ]

EvN

V

;]7IK69;:2LQ]K;‚wO;N9;Y]KGQF—GQSKSXE9:2PRIKBMLONvSY‘



;=69LQP/;=IT6ƒGJI7IT69BHS Zf69LOBDZ;‚Pv]TLJpCB

V

;YS7G2SXLQ†¨IP9]T;eSX;YNuIKGJIKBMLON

LQ†žIT69;2STE9R_`;YZIˆ6vBDZf6bP/;]K:2B›IfS‚ITLzEvN

V

;]fSXIKGUN

V

ƒ;IXIK;]IT69;?LQ9_`;YZ„IfS‚GQN

V

IK69;B

V

;eGQSZLQ:2BDN9wšLONmGQS

ˆ';FDFŒGQS'IK69;P/LOSKSXBD9FD;=GQP9P9FDBDZYGJIKBMLONvS‘

©¢ªu«i¬K­® }J¯{Q€ abGUFDFMBHG…pCBMN•S°'GQFDZE9FDEvS¤  BD;Nv;]Zf6ƒGULOS V ;YZLQ:2PƒLuSXBMITBDLQN¤Š±NOIK;wO]KGUITBDLQN²CW³PvGQ]XIfS¤

Ÿ‰LQ]K:GUN V ;Y]Y•S˜IT69;YLQ]K;:+‘

(5)

{ € Ž‚N P9]K;YST;NuIK;E9Nv;BDNuIT]KL V EvZ„IKBMLON GQE³ZYGUFHZE9F V ;+abGQFMFDBDG…pCBDN³STE9BDpQ;NuIFD;YS?N9LUIK;YS V •E9N9;

STE9BMIT; V ;uEvGJIK]T;[RP/LOSYSOEv;‚_•GUB†¦GQB›I V GQNvSFM;ZG V ]K; V E™wQ]KLQE9P/; V ;IK]KG…pJGQBMF V ;2F3•uE9BDP/;2a”GJIT69c

;YNLRZITLQv]T;!qUrOr9kO‘бFHSZLONOIKBM;YNuIIT]KLQBHS'STEJ_`;IKSY’

kQ‘ N9;P9]eSX;YNuIKGJIKBMLONYFY:?;YNuIKGUBD]K; V ;YSLOPY]KGUIT;Ev]KS V BM–]K;NuITBD;FHS V E¢ZGQFDZE9F V ;a”GUFDFMBHG…pCBMN;I V ;

FDG?†‡LO]T:E9FD; V •BDNuIwO]KGUITBDLQNPvGU]PvGQ]XIKBM;uE9B—;eS`I F-•BDNvSXIT]KE9:2;NuI‰Z;YNuIT]fGUF V ;!Z;IXIK;=IT6YLQ]KBM;O‘

qR‘G V eZLO:?P/LOSTBMITBDLQN ;N Zf6vGQLOS V ;  BM;YN9;];IFHG+P9]YST;NuIKGUITBDLQN V ;YS!LQP]fGJIT;YE9]fS V B›–]K;NuIKBM;YFDS

SXEv] Z;I ;eSXPƒGQZ;O‘

s9‘^•GUP9PvFMBHZGUITBDLQN V E¡°'GUFHZE9F V ;?abGUFDFMBHG…pCBMNšF-•ITE V ; V ;FDG V ;NvSTBMI V ;FHGšFDLQB V •E9N9; V BM–/EvSTBDLQNŒ‘

°‚•;eS`I˜F3•GUP9PvFMBHZGUITBDLQNBDN9B›IKBDGQFM; V ;‰Z;IXIT;‰IT6YLQ]KBM; uE9B/G V LQNvN E9N9; V :2LQNƒS`IK]KGUITBDLQN2P9]KLQvGQ9BMFDBHS`IK;

V EIT6YLQ]K8:2; V ;!Ÿ LO]T:GQN V ;]e‘

;š9ERI V ;zZ;eSN9LUIK;YSN•;eS`IPƒGQS V ; V LQN9Nv;]!E9Nv;šP9]YST;NuIKGUITBDLQN ZLQ:2P9FD8IK; V ;šFHGIT6YLQ]KBM;:GQBDS

_`EƒS`IK; V ;˜:2;ITIT]K;7;NpCBV ;NvZ;˜FD;YSLQ9_`;IKSP9]KBHZBDPvGUER[!;IžFM;eSBV Y;˜;IIK;YZf69NvB!uE9;eS;Y:2P9FMLJW"YSY‘бFRW!G V 8`_XG

V •;[RZ;FDFM;YNuIT;YS:2LON9LQwO]KGUcv;YSSTE9]žFM; STEJ_`;IY¤CZLQ:2:2;Z;YFMFD; V ; £ EƒGUFHGU]TIPƒGU]#[C;Y:2P9FM;O‘žŽ‚NG V LONvZ7FDGQBDSKS

V ;Z$QI?Z;]TIKGQBMNƒS P/LQBDNOIfSIK]T8eS IK;YZf69N9B%uE9;YS‚;ILQNbN•GPƒGQS‰†‡LQ]fZ;:2;YNOI V LQN9N&FD;YS‚]T;eSXE9FMIKGUIKS‚SXLOEvS FD;YS

6CWCP/LUIT6&YST;YSŒFD;YSP9FDEvSw'N&]fGUFD;YSY‘Ž‚N!;YSTPƒ8Y]T;(OEv;˜Z;eZBuG P/;]K:?BMI V ; V LON9N9;]E9N9;˜P9]eSX;YNuIKGJIKBMLON!GUFDFYw'Y;

uE9B—P/;]K:2;I V ;:2BD;ER[+ZLO:?Pv]T;YN V ]K;‚FD;YS7LOR_`;IfS7;I FD;YSBV ;eSGUNvSTB)uE9;FD;YS7P/LOSKSXBD9FD;YSGUPvP9FMBHZGUITBDLQNvSY‘

* ® |e{ ¬ '+ { € °'GUFHZE9F V ;¢abGQFMFDBHG…puBDNŒ¤-,.eZLQ:2P/LOSTB›IKBMLON³;YN Zf6ƒGULOS V ;  BD;N9;Y]Y¤'бNuIYwQ]fGJITBDLQN³PvGU]

PƒGU]TITBD;YSY¤9576LO]T8Y:?; V ;!Ÿ‰LQ]K:2GQN V ;Y]Y‘

(6)

+ ,

!

a”GUFDFMBHG…pCBMN¡ZGQFDZE9FDEvS!ˆ7GQS!ZLQNvZ;BDpQ; V BMN IK69;zWO;YGU]fS.-/102GQN V BMN IK69;zWO;YGQ]KS435/1062GQN V87 /502G¢6CE9wO;

GQ:2LQE9NuI!LU†7ˆ'LQ]K‹b6vGQS!ƒ;Y;N V LON9;BMN IK69BHScv;YFV ‘ŠI/;YZLO:2;YS!GQN GQNvGUFDWRSXBHS=LQN IK69;  BD;N9;Y]STPvGQZ;

GQN V SX;YpQ;Y]KGQF:?LON9LQwO]KGQP96vS?LQN IK69BDSSXEvR_`;YZIzGU]K;bG…p…GQBMFHGUvFM;+N9LJˆ7G V G…WRSY’¢£ EvGQFDGQ]XI"9Dk%:%; ¤‰Ž‚‹RSX;YN V GUF

9Dk<;-¤Rб‹O; V G  GUIKGUNƒGU/;.9DkYr=;‘^5769;:GUBDNGQP9P9FDBDZYGJITBDLQNzLU†a”GUFDFMBHG…pCBMNzZYGUFHZE9FDEvS'ˆ7GQS ITL?wQBDpQ;=SXE(>šZBM;YNOI

ZLQN V B›IKBMLONvS!BDN¡LQ] V ;]!IT6vGUIIT6v;šFDG…ˆ LQ†Gb]KGQN V LQ: pJGU]KBDGQ9FM;z6vGQS!G”SX:2LCLUIK6 V ;NvSTBMI`WmˆBMIT6 ]K;YSTP/;YZ„I

IKLŒ;Yƒ;YwQE9;O•S‰:2;YGQSTE9]K;GUN V ITLzwOBMpO;ƒLOE9N V S‰†‡LQ]‰IT69BHS V ;NvSTBMI`WGQN V BMIKS V ;]KBMpJGUITBDpQ;YSY‘7бNb69BHS‰BDN9BMITBHGUF

PƒGUP/;]fSabGUFDFMBHG…pCBMN™EvST; V IT6v;zGUvSTLQFDERIK;2ZLQNuITBDNCE9BMI`W”Z]KB›IK;]KBMLONmBDN¡LQ] V ;Y]‚IKL¢P9]KLJpQ;?IT6vGUI!E9N V ;Y]Ÿ LO]X

:GQN V ;]e•S‰ZLON V BMITBDLQN¢IK69;FHG…ˆ LU† G V B›–iEvSTBMLONbP9]KLRZ;YSKS 6ƒGQS‰GSX:2LCLUIK6 V ;NƒSXBMI`W¢GUN V BDN¢IK69BDS‚ˆ7G…W69;

wuG…pQ;šG¢Pv]TLOvGU9BDFDBDSXITBHZ2P9]KLuLQ†LU†‰Ÿ LO]T:GQN V ;]e•S=IT69;YLQ]K;:+‘b“†¨IK;]Kˆ'GQ] V S!P/;LQPvFM;šEƒSX; V IT69BHS?ZGUFHZEvFMEvS

BDNbpJGU]KBDLQEvS STBMITEvGUITBDLQNvS‰]T;YFDGUIT; V ˆBMIT6 @uITLRZf6vGOS`IKBDZ@?BADC40E2GQN V STLšLQNŒ‘‚5769;YST;FHGQSXI‰WQ;eGU]fS abGQFMFDBDG…pCBDN

ZYGUFHZE9FDEvS†‡LQEvN

V

Nv;ˆ³GQP9P9FDBDZYGJIKBMLONvSBMN?P9]KLQvGQ9BMFDBHS`IKBDZ˜NCE9:2;]KBDZYGUFC:2;IT6vL

V

S¤O;YSKSX;YNOIKBDGQFMFDW=BMN?IK69;'cv;F

V

LQ†

:GUIT69;Y:2GUITBHZGQFicvNvGUNƒZ;Q‘ž5769;eSX;GQP9P9FDBDZYGJITBDLQNƒS'GQ]T;uE9BMI

V

B›–i;]K;NuI†‡]KLQ: IK69;P9]K;pCBDLQEvS'LQNv;YS'ƒ;eZGQEvSX;

IK69;BMNuIT;YwQ]fGJIKBMLONuW2PvGQ]XIfSž†‡LO]T:E9FDGBMNabGUFDFMBHG…pCBMNZGUFHZEvFMEvS˜BHS ;Y:?PvFMLJWO;

V

BDNšLQ]

V

;Y]^ITL?P9]KL

V

EvZ;SXLO:2;

;[RP9FMBHZBMI ˆ';BDwQ6uIKS'ˆ69BHZf6¢ZLQ:2;=LQNBDNNvLQNFMBDN9;YGQ] GUFDwQLO]TBMIT69:SY‘

“'IFDBHS`IZLQNvSTBV ;]K; V †‡]TLO: IT69;P/LQBDNuILU†'pCBM;YˆœLQ†'P/;LOP9FM;ˆ'LQ]K‹CBMNvwBDN¡ZLQNƒZ]K;IT;GQP9P9FDBM; V :GUIT6R

;Y:GJITBHZSY¤^abGQFMFDBDGQBMpCN ZGUFHZEvFMEvS?GUPvPƒ;eGU]fSGOSGm]KGUIT69;Y]STLQP96vBDSXITBHZGUIT; V GUN V IT;YZf6vN9BDZYGUF'IT6v;LQ]KW ˆ69BHZf6

]K;uE9BD]K;YS GQN¢BM:2P/LQ]TIKGUNuIBMNCpQ;eS`IK:2;NuIKSY¤ƒGQN V IT69BHS‚:2G…W/; V BHSTZLQE9]fGUwOBMNvwv‘D‘7576v;GUBD: LU†žIT6v;YST;!N9LUIK;YS

BHS‰ITLzwOBMpO;GUN”;FD;:2;NuIKGQ]TWBDNOIK]TL V EƒZ„ITBDLQN+IKLšIK69BDS‚IKLQP9BHZU‘=@CBMNƒZ;!IK69;?P9]TBDLQ]KBMI`WBHS IKLšIK69;;eGQSTWGOZZ;eSTS

ˆ';GUFDFMLJˆ LQE9]fST;FM†ITL2/;]KGUIT69;Y]BMN9†‡LQ]K:2GQF—LQN¢STLQ:2;=PƒLOBMNuIKSY¤CITLšG…pOLQBV IKLuL2IK;YZf69NvBDZYGUF—P9]KLCLU†¦SY¤CITL2FMBDpQ;

LOERI‚STLQ:2; V ;YpQ;YFMLOP9:2;NuIKS‰LU†IT69;IT69;YLQ]KWzˆ69BHZf6”GQ]T;=IKLuLz69;YG…pCW+GUN V ¤/GJIFMBHS`I‰†‡LQ] IK69;:2LO:?;YNuIY¤iGU]K;

NvLUI V BD]K;YZ„IKFMWEvST; V BDNšGUP9P9FDBHZGJIKBMLONvSY‘  ;IT]KWITLST;I EvPšG:2BMN9BD:GUFvIT;[CI ˆ6vBDZf6wOBMpO;YS^GQN?BV ;eG!GU/LQERI

IK69;vGOSXBHZLQR_`;eZ„IKS7GQN V BDSKSXEv;YS7BDNabGQFMFDBHGUBDpuN+ZGQFDZE9FDEvS7BMN+LO] V ;Y]'ITL269;FDPIT69;]K;YG V ;]'IKL?EvN V ;]fSXIKGUN V

IK69;?N9;ˆgFDBMIT;]fGJIKE9]K;ZLONvZ;Y]TNvBMN9wGUP9P9FDBHZGJIKBMLONvS‚LQN”LON9;6ƒGUN V GUN V ITLGOZZ; V ;ITL:EvZf6m:?LO]T;?ZLO:

PvFM;IT; IK;[CIKS¥¦GOS^IK69;‰:2LON9LQwO]KGQP96vS#uE9LQIT; V GU/LJpQ; § LQN2IT6v;‚LQIT69;Y] 6vGQN V ‘ž5769;‰PƒGUP/;]'BDS LQ]KwOGUNvBFY; V GQS

†‡LOFMFDLJˆ SY‘!бN @R;YZ„IKBMLON³k?ˆ';wOBMpO;?GQN™GQvS`IK]KGOZ„IBMNuIT;YwQ]fGJIKBMLON”CW¢PvGU]TIKS‚†‡LO]T:E9FDGGUN

V

BDNupO;YSXITBDwOGUIT;?B›Ie•S

ZLQNvST;uE9;YNvZ;eS†‡LO]‰IK69;SXITE

V

W¢LU†žIT6v;

V

;YNvSTB›I`W+LU†žIK69;?FDG…ˆ LQ† Gš]fGUN

V

LO: pJGQ]TBHGU9FD;Q¤iZLO:2P9ERIKGUITBDLQNbLQ†

ZLQN V B›IKBMLONvGUFŒ;[RPƒ;eZ„IKGUITBDLQNvS‚GUN V ST;NƒSXBMITBDpCB›I`WˆB›IK6b]T;eSXP/;YZI ITLG2PvGU]fGU:2;IK;]e‘'576v;YST;GU]K;;YSKSX;YNuITBHGUFDFMW

SXIKGQN V GQ] V V BHS`IK]TBD9ERIKBMLON IT69;YLQ]KW”]K;YGOSXLON9BMNvwOS=BDN G¢P9]KLQvGQ9BMFDBHS`IKBDZ?†‡]fGU:2;Q‘+бN @C;eZ„IKBMLON q+ˆ˜;zP9]T;eSX;YNuI

a”GUFDFMBHG…pCBMNŒ•S‰ZYGUFHZE9FDEvS‰B›IfSX;YF›†`‘  ;?S`IfGU]TI‰ˆBMIT6mSTBD:?PvFM;†‡E9NvZ„IKBMLONvGUFHS‚ˆ69BDZf6bGQ]T;]KGQN V LQ: pJGU]KBDGQ9FD;YS LQ†

IK69;†‡LO]T:HGJILKNMPO.Q RTSUU6US'OWVX*Y[ZR \^ˆ69;]K;OW]R!I_^`M ]abZ

V X

\cd^`M ]

V X

\^GQN V KBDS˜G!ST:?LCLQIT62†‡E9NvZITBDLQNŒ‘^@CL

GSXBD:2P9FD;‰†‡E9NƒZ„ITBDLQNƒGUFiBDS_`EvSXIGUNGQwQwO]T;YwOGJIK;LU†BDNvZ]T;Y:?;YNuIKS˜LU†ŒIK69;fe']KLJˆN9BHGUNš:2LUIKBMLON`^ GUN V IT69BHS'BDS

ZLQ69;Y]T;YNuI ˆB›IK6¢P9]TLOvGUvBMFDBDSXITBHZ=NCE9:2;]KBDZYGUFŒ:2;IT69L V S  IT6vBMN9‹†‡LQ]‰BMNvSXIKGQNvZ;!ITLIK69;4g^E9FD;]STZf6v;:2;=†‡LQ]

G V B›–iEvSTBMLON¢P9]TLRZ;YSKS‘˜ŠI ITE9]KNvSLOERI IK6vGJI‚GUI IT6v;!FM;YpQ;YFŒLU†žSXBD:2P9FD;†‡E9NvZITBDLQNvGQFDS‚abGUFDFMBHG…pCBMN•S V BM–i;]K;NR

IKBDGQFLOPƒ;Y]KGUITLO]KS GQ]T; OEvBSTBM:2BDFDGQ] IKLzEvSTEvGUF V BM–/;Y]T;YNuITBHGUFLQP/;]fGJITLO]KS LU†ihjVXkU—5769;?SXPvGOZBHGUF†¦GQZ„Iˆ69BHZf6

wOBMpO;YS˜GQZZ;YSKS^IKLGQNzBDNRcvN9BMIT; V BD:?;YNvSTBMLONvGUFiZGQFDZE9FMEƒS BHS˜IT6vGUIlOW]R ImOWVn]R

aoZop

O

V]6aoZ

R

aoZ

UC576vBDS˜Pƒ;Y]T:2BMIKS˜ITL

;Y:/; VdqBr MhjV XTs

hB\^BDNuITL qBr MhjV Xt*u%s

hB\U9@CL?LQNv;‰LQ9IKGUBDNvS7GZf6ƒGUBDNzLU†cvN9BMIT; V BM:2;YNvSXBDLQNƒGUF—SXPƒGQZ;eS¤

GQN V abGQFMFDBHG…puBDNŒ•S V ;Y]TBDpJGJIKBMpO;‰LQP/;]fGJIKLQ]fS¤Qˆ69BDZf6GU]K; V ;cvN9; V LQNIK69; cƒN9B›IK; V BM:2;NƒSXBDLQNvGQF/STPvGOZ;YSY¤CGU]K;

BDN V ;YPƒ;YN V ;YNOI=LU† IT69; V BD:?;YNvSTBMLONŒ‘5769BDSGUFDFDLJˆ S IKLz;[CIK;N V IT69;eSX;?LQP/;]fGJIKLQ]fS!¥‡EvSTBDN9wdvwV!N9LO]T:S § LON

BDNRcvNvB›IK; V BD:?;YNvSTBMLONvGUFSTPvGOZ;YSY‘ “ ‹O;WPvGU]TI BMNIK69;!;[CIK;NvSTBMLON+P9]TLRZ; V E9]K;BHSP9FHG…WQ; V CWšIK69; V EvGQFMBMI`W

/;I`ˆ';;YN!IK69;abGUFDFMBHG…pCBMN V ;]KBMpJGUITBDpQ;xAœGQN V IK69; @C‹QLO]TLO69L V BMNuIT;YwQ]fGUF(y7ƒ;eZGQEvSX;˜IT6v; V EvGUFDB›I`W]K;FHGJIKBMLON

(7)

wOEvGU]fGUNuIKBM;eSIT6vGUI!IT69;šI`ˆ˜L¢LOPƒ;Y]KGUITLQ]fSGU]K;ZFDLOSKGU9FD;Q‘+£ ;[CIˆ˜;

V

BDSKZEƒSTS=IK69;š]K;FHGJIKBMLON™/;I`ˆ';;YN IT69;

V

B›–i;]K;NuITBHGUFLQP/;]fGJITLO]KSGQN

V

IT69;Zf6vGULuS7;[RPvGUNƒSXBDLQNLQN+IK69;



BD;N9;Y]‰SXPƒGQZ;O‘ ŠI IKE9]KNvSLQE9I IT6vGUI IT6v;W

6ƒG…pQ;!GšpO;]KWzN9BHZ;?GUN

V

;[RP9FDBDZB›I;[RP9]K;YSKSXBDLQN+LQNbIT69;



;YBMN9;Y]Zf6ƒGULOST;YSGQN

V

IK69BDS‚P/;]K:?BMIKS‰ITLP9]K;YZBHST;

IK69;BD] V LO:2GQBMNƒSžBMN?IK;]K:2SLQ†ƒIK69; STPƒ;Y; V LU†iZLQNCpO;]KwQ;NƒZ;'LU†vIK69;‰Zf6vGQLOS;[RPvGQNvSTBMLONST;]KBM;eS‘D‘nBDNvGUFDFDW!ˆ˜;

Pv]TLJpO;˜abGQFMFDBDG…pCBDNŒ•SBDNuIT;YwQ]fGJITBDLQNuWPvGU]TIKS†‡LQ]K:EvFDGGUN V wQBDpQ;˜IT6v;7ZLQ]K;YSTPƒLON V BDN9w‚GQvSTLQFDERIT;ZLONuITBDNuEvB›I`W

Z]TBMIT;Y]TBDLQN‘бN¢@C;eZ„ITBDLQN¢s?ˆ';wOBMpO;IT69;!GQP9P9FDBDZYGJITBDLQNƒS˜LQ†a”GUFDFMBHG…pCBMNZGUFHZEvFMEvS'IKL V BM–/EvSTBDLQN+P9]KLCZ;YSKS‘

(Œ^ !2+( Jž

576v;?Z;NuIT]fGUFITLCLOFBDNmIT69;abGQFMFDBDG…pCBDN”ZGQFDZE9FDEvSBDSGQN”BDNOIK;wO]KGUITBDLQNbCW¢PvGQ]XIfS‰†‡LO]T:E9FHG9‘!5769BHS=BDSˆ6CW

ˆ';=wQBDpQ;=BDNIK69BHS SX;eZ„IKBMLON+GQNGQvSXIT]fGQZ„I GQP9P9]KLOGQZf6zITL2IK69;BMNuIK;wQ]fGJIKBMLONCWšPƒGU]TIKS'†‡LQ]K:EvFDG2GUN V IKL2IT69;

GQP9P9FDBDZYGJIKBMLONvS?LU† IK69BDS†‡LQ]K:E9FHG¢IKLmIK69;¢S`IKE V W LQ†‰IK69; V ;NvSTBMI`W LU†‚IT69;+FHG…ˆLU†=G”]fGUN V LQ: pJGU]KBDGQ9FD;

GQN V IKL™ZLQ:2P9ERIfGJITBDLQN LU†‚ZLQN V BMITBDLQNƒGUF';[RPƒ;eZ„IfGJITBDLQNƒS‘ “ FDF'IT69BHS?]T;YP9]T;eSX;YNuI?SXIKGUN V GU] V †¦GQZIKS?BMN IT69;

V BDSXIT]KBMvERITBDLQNIK69;LO]TW9ERI7ˆ˜;‰P9]T;eSX;YNuI^IT69;Y: BMNzGP9]KLQvGQ9BMFDBHS`IKBDZ7FHGUN9wOEvGUwO;ˆ69BHZf6šBDS'GUPvP9]TLOP9]KBDGUIT;'†‡LQ]

LOE9]GUBD:S‘žn9LO]SXBD:2P9FMBHZBMI`Wš]K;YGOSXLONvS ˆ';P9]T;eSX;YNuI˜cv]fSXI˜IK69;=LQN9; V BD:2;NvSTBMLONvGUF—ZYGQST;GQN V IK69;N+ˆ';‚wOBMpO;

IK69;;[CIK;NvSTBMLONzIKL?IT69;:E9F›IKB›V BD:2;NvSTBMLONvGUFi†‡]fGU:2;O‘

!"$#&%'"$ (

)+* ª ¯ ª-, {/.‡| « ®10 | * ª +3~ ­ Œ;IM32fSGxSn?4\ƒ;GP9]TLOvGUvBMFDB›I`WbSTPvGOZ;2GUN V FM;I4GxS547682:9 h ƒ;

BDNuIT;YwQ]fGU9FD;=]KGQN V LQ: pJGU]KBDGQ9FM;eS‘  ;SKG…WšIT6ƒGJI IK69;!BDNOIK;wO]KGUITBDLQN+CWzPvGQ]XIfS†‡LQ]K:EvFDG<; ?MG s 44\769LQFV S

IK]TEv;BM†IT69;Y]T;;[CBHSXIKSGUNBDNOIK;wO]KGQ9FM;]fGUN V LO:pJGU]KBDGQ9FD;>= MPG s 44\STEvZf6IT6vGUI

;5?MPG

s

44\ CDM3?A@MG4\44\wI C`MB?bMG4\= MG

s

44\ƒ\S UCD?FE q r

G

MhB\ ¥`k §

ˆ6v;]K; qBr

G

MPhB\'BDS7IK69;!STPvGQZ;LQ†BDNRcvN9BMIT;YFMW V BM–/;Y]T;YNOIKBDGQ9FD;‚†‡EvNvZ„IKBMLONvSˆBMIT6¢ZLO:?PƒGQZ„I STE9P9P/LQ]TIY‘

a¢LO]T;YLJpQ;Y]Y¤ˆ˜;STG…WmIT6ƒGJIIK69;BMNuIT;YwQ]fGJIKBMLON¡uW PvGQ]XIfS†‡LO]T:E9FDGH;5?JI(MG

s

44\!6vLQF

V

S!IK]TEv;zBM† IT69;Y]T;

;[RBDSXIKSGQNBDNuIT;wO]KGQ9FD;‚]fGUN

V

LQ:pJGU]KBHGU9FD;K=LI*MPG

s

44\STEvZf6IT6vGUI

; ?MIMG s 44\ CDMB?8NPO5QMPG4\R44\wImCDMB?bMPG4\R=<IMPG s 44\ƒ\S UC8?FE q r

G

MPhj\ ¥3q §

£‰LUIK;IT6vGUIS;5?MPG

s

44\—ZLQBDNvZB

V

;eSiˆB›IK6T;5?

Z

MPG

s

44\—GQN V = MPK

s

44\ IU=

Z

MG

s

44\UYa¢LO]T;YLJpQ;Y]Y¤BM†V;5?MPG

s

44\

GQN V ; ?MG

s = MG

s

44\ƒ\'69LOF

V

IT]KE9;IT69;YNW; ? V MG

s

44\'69LQF

V

S˜IT]KE9;ˆB›IK6F=

V

MG

s

44\iIX= MPG s = MPG

s

44\\U

“‰N¢GUNvGQFMLOwQE9;eS'GOSTST;]TITBDLQN69LQF

V

S'IK]TE9;=†‡LO]69BDwQ69;Y]LQ]

V

;]

V

;Y]TBDpJGJIKBMpO;YSY‘576vBDSFDB

V

SEvS'IKL

V

;cvNv;T= I MRY%\

CWš]K;YZE9]K]T;YNvZ;O’M=

Q

MPG4\iIZY1S5=<I

abZ

MG4\wIX= MPG s =<I(MG4\\U

5769;ˆ˜;YBMwO6OI= MPG s 44\7BDN; ?MG s 44\BDSNvLUI E9N9B%uE9;Q’^†‡LQ]‰GQNuW h STEvZf6IT6vGUI CDMB?bMG4\ƒhB\ IL/?LQN9;

:G…WEvST;T= MPG s 44\

p

hgGQS7ˆ';FDF3‘бN+NCE9:2;]KBDZYGUF—:2;IK69L V S'IT6vBDSPvFDG…WRS7GUNBM:2P/LQ]TIKGQNOIPvGQ]XI/;YZYGUEvST;Q¤

BM†žLQN¢ˆ7GUNuIKS'IKLzZLO:2P9ERIT;.CDMB?bMG4\= MG s 44\ƒ\‰EvSTBMN9wšGabLQNuIT;?°'GQ]TFDL2:2;IT6vL V IK69;N”LQN9;ˆ'LQE9FV FMBD‹Q;

IKL!ˆ'LQ]K‹ˆBMIT6zGˆ';BDwQ6uI ˆ6vBDZf6šwOBMpO;YSž:2BDN9BD:2GQFƒpJGQ]TBHGUNvZ;!¥¦ST;;+[G&\^]-_B_ S`Y`aƒGUN V [G&\b]!_B_Sbcda§ ‘£ LUIK;‰GUFHSTL

IK6vGJIBMNmLQ] V ;]‰ITLPƒ;Y]X†‡LO]T:GzabLQNuIT;š°'GU]KFMLGUFDwQLQ]KBMIT69: LQN9;6vGQS‰ITLSXBD:EvFDGUIT;.G GUN V =&MG s 44\UiбN

STLQ:2;‚PvGU]TITBHZE9FHGU]'ZGOSX;eSe= MPG s 44\ :G…W?/;=ZLQ:2P9E9IT; VV BD]T;eZ„ITFDWQ¤CEvSTBDN9wSTLQ:2;‚G V 69LRZ :2;IK69L V e'ERI

a”GUFDFMBHG…pCBMNŒ•S ZYGUFHZE9FDEvS˜wQBDpQ;eS˜GSXWRSXIT69;Y:2GUITBHZ‰GOZZ;eSTS ITLIT69;ZLQ:2P9ERIfGJITBDLQNzLU†—IT6vBDS'ˆ';BDwQ6uIY‘ž5˜WuPvBDZYGUFDFMW

(8)

GQwQwO]T;YwOGJIK;!LU†

V

BM–/;Y]T;YNOIKBDGQFžLQP/;]fGJIKLQ]fS¥¦BMN abGUFDFDBDG…pCBDNŒ•S=ST;NvST;

§

GOZ„ITBDN9w+LQN GxUŒ5769;eSX;uEvGUNuIKB›IKBM;eSGU]K;

GQFDSTL]K;FHGJIK;

V

ITLSXLO:2;SXITLRZf6vGOS`IKBDZ‰;uEvGJIKBMLONvS7GUN

V

STLLQNv;:G…WEvST;=SXLO:?;=GUPvP9]TL…[RBD:GJITBDLQNƒS^LU†IT69;eSX;

; uEvGJIKBMLONvS'BMN+LQ]

V

;Y]˜IKLP9]KL

V

EvZ;ZLONvZ]K;IK;GUFDwQLQ]KBMIT69:SY‘

;IzEvSšwQBDpQ;¢G¡SXBD:2P9FD;b;[9GU:2P9FD;Q‘ 5GQ‹Q; G I KNMOW\2GQN V 4 I MPO!\2ˆ69;Y]T;"KkS³GU]K;bSXLO:2;

V B›–i;]K;NuITBHGUvFM;†‡E9NvZITBDLQNvS‰GUN V O BHS G2Z;YNOIK;]K; V wOGQEvSTSTBHGUNz]fGUN V LQ: pJGU]KBHGU9FD;=LU†pJGQ]TBHGUNvZ;TU/5769;N

CDMK

0

MO!\ MPO!\ƒ\iI_CDMKNMPO!\MTMOW\

O

c

0

MO!\ƒ\ƒ\ ¥¦s §

STL<;5?MPG s 44\'69LOFV S'IT]KE9;ˆB›IK6F= MG s 44\ I MPO!\ 8c*0MPO!\U

5769;Pv]TLCLU†/BDSžG

V

BD]K;YZ„I^GQP9P9FDBDZYGJITBDLQN?LU†ƒIT69; SXIKGQN

V

GQ]

V

BMNuIT;YwQ]fGJIKBMLONCWPvGQ]XIfS¤JvERI^BMN?IK69;P9]K;YST;NvZ;

LQ†IK69;wOGQEvSTSTBHGUN V ;NvSTB›I`WoMk\ I Z

V

Mc

V \ 6

C`MPK

0

MPO!\ MPO!\ƒ\ I K 0

Mk\ MT\ bMk\!""I c# KNMk\ M

0

Mk\oMk\

p

Mk\

0

MT\ƒ\!"

I c KNMk\M

0

Mk\

p

Mk\

0MT\

bMk\

\oMT\$!% I_CDMPKNMO!\M MPO!\

O

c

0

MPO!\\ƒ\

abGQFMFDBHG…puBDN¢ZGQFDZE9FMEƒSP9]KL

V

EvZ;eS7IT69;ˆ˜;YBMwO6OIfS= MPG

s

44\'†‡LO]‰GFHGU]KwQ;ZFHGQSKSLU†]fGUN

V

LO: pJGU]KBHGU9FD;YS'

¥3s

§

]K;Pv]T;eSX;YNOIfSIT6v;‰SXBD:2P9FD;YSXI ;[9GU:2P9FD; LQ†/IK69BHS uE9BDN

V

9E9I IT6vBDS BDS N9LUI^IK69;‚STE9R_`;eZ„I LQ†iIT69BHS ST;YZITBDLQNŒ‘

Ÿ‰;]K;=ˆ˜;=wQBDpQ;STLQ:2;ZLONvSX; uE9;NvZ;YS'LU†IT6v;=GQƒLJpO;=P9]TLOPƒ;Y]XI`WO‘

&˜ª ~('*)

,+%+) -

G

) PE).w)

;5?MPG

s

Y=\U0/

- -`21 3

G

E)`"4) #( #'P*)

1 - n)5+n !-76o4')$81@!'=)(.T-W1 )P`3B-@21 E)981$= (4

oMT\iI C`MRY:

";r=<

MG4\= MG s Y%\ƒ\U ¥‡< §

\^]KLCLU†`‘b5769;š†‡LQ]K:2GQF GU]KwQEv:?;YNuI!BDS!IT69;z†‡LQFDFMLJˆBDN9wv’?SXBDNvZ;`y

Q

M>\@I@?BA-Y:

Q ;r0<

M>\LQNv;;:2P9FDLJWQ;YS

; ?MG s Y%\˜BDNLO] V ;]'ITLLORIKGQBMN

CDMPy

Q

MPG cT\ƒ\iI_CDM?BA-Y":

Q ; r=<

MG cCk\\iImC`MRY:

Q ; r=<

MGLcCk\R=

Z

MG s Y%\ƒ\ I_CDMY:

; r0<

MPG4\R= MPG s Y=\ƒ\U

бNLQ] V ;]ITLFM;IIK69BDS]K;YGOSXLON9BMNvw ]TBDwQLO]TLOEvS—LON9; 6vGOSŒIKL‚]K;wOE9FHGU]KBFY;žIT69; ,‚BM]fGQZ^†‡E9NvZ„IKBMLONŒ‘ž@CL‰ˆ'; IfGU‹O;

G=P/LOSTBMITBDpQ;7†‡E9NvZITBDLQN ? E qBr

G

MhB\ˆBMIT6IK69;‰STE9P9P/LQ]TI^;uEvGUFvITL [cKY1S Y aƒGQN V SXEvZf62IK6vGJIED&?bM>(\!">DIY

GQN V †‡LQ]^;eGQZf6yF /=ˆ'; V ;cƒN9;?

M>\ I_y Y Z`?oM>(y Y Z\UOabLQ]K;LJpQ;Y]ˆ'; V ;cvN9;HG

IKL!ƒ;7IK69; P9]KBD:?BMITBDpQ;

LQ†>?

wQBDpQ;N uWIG

M>\DIJD

A

Y r ?

MK*\$!LK GUN V ˆ˜;+ZLONvS`IK]TEƒZ„ISXLO:?;]fGUN V LO: p…GQ]TBHGUvFM;eSNM

LU†‚FHG…ˆ

?

M>\$!%>zGUN V ˆ6vBDZf6+GU]K;=BMN V ;P/;N V ;YNuI LU† G2‘^n9LQ];eGQZf6 K E qBr

G

MhB\'ˆ';=6vG…pQ;

CDMPKNMPG4\ƒ\iIPORQRS

TjQ

C@KNMPG_cCM

\ƒ\U ¥P: §



;!ZLO:?PvERIT;N9LJˆ

C@KNMPG_cCM

\\ I @ KNMU`cV(\?

MV*\!"VB! ?XW G Y Z

MUT\ I Y KNMK*\?

MUDcCK*\$!LKL!5?W G Y Z MUT\

I KNMK*\CDM3?

MGLcCK*\ƒ\!%KWIX KNMZK \ƒCDM5G

0

MG cCK*\ƒ\!%K

I KNMK*\CDMG

MG c(K \= MG s Y%\ƒ\!%KU

(9)

5769;˜GU/LJpQ;ž]K;FHGJITBDLQNIKLQwOGUIT69;Y]—ˆB›IK6š¥P:

§

wQEvGQ]KGQNuIT;;eS/IT6vGUIŒIT6v;^FDG…ˆ LQ†*G BHSGUƒSXLOFMERIK;FDW‚ZLQNuITBDNCE9LQEvS

ˆBMIT6 ]T;eSXP/;YZIITLbIT69; Œ;/;YSTwQEv;z:2;YGOSXE9]K;Q‘mŽ‚N IT69;LUIT6v;]6ƒGUN V G

M>\ 9 Y:

";r=<

M>\;[9Z;P9I!†‡LQ]

>I /SuSTL G

MPG c K*\$9 Y:

Q ;r=<

MG c K*\NI Y:

;r=<

MG4\S? cF]kU25UO576v;NzEƒSXBDN9w ;/;YSTwQE9;eS

V

LO:2BMNvGUIT;

V

ZLQNCpQ;Y]TwO;NvZ; IK69;LO]T;Y:ˆ˜;=PvGOSTS'IKL?IT69;FDBM:2BMI BMNIK69;!GU/LJpQ;]T;YFDGUITBDLQN+GUN V ˆ';LQRIfGUBDN

C`MPKNMG4\\iI* KNMK*\CDMY:

; r=<

MG4\= MG s Y%\\$!LKU

5769;?GU/LJpQ;BDNuIT;wO]KGQF]K;P9]K;YST;NuIfGJITBDLQNIT6v;LQ]K;:wOBMpO;YS STLQ:2;!:2LQ]K;!BDNR†‡LQ]K:GJIKBMLON¢LQN+IT6v; V ;YNvSXBMI`W

F6

+¦~9}d.¦|

«ž‘ ‚SXBDN9wŒ;Yƒ;eSXwOE9;Q•S V

LQ:2BDNvGJIK;

V

ZLONupO;]KwQ;YNvZ;IK69;LO]T;Y: LQN9;=:G…W;YGOSXBDFDWšZf69;YZf‹

IK6vGJIH9 oMT\ IJCDMY:

;r0<

MPG4\R= MPG

s

Y=\ƒ\‰BDSGšZLONuITBDNuEvLQEvS †‡E9NvZ„IKBMLONŒ‘



;ST6vGQFMFSX;Y;!BDN¢IT69;?SX; uE9;F

IK6vGJIe¤UBM†vLQN9;'6vGOSSTLQ:2;':?LO]T;˜BMNuIK;wQ]fGJIKBMLON!CWPvGU]TIKS†‡LQ]K:EvFDGOSŒIK69;N?LQN9;˜:G…WP9]TLJpO;žIT6ƒGJIIT69; V ;YNvSXBMI`W

BHS V BM–/;Y]T;YNuITBHGU9FD;Q‘

® , ¯{ @CEvP9PƒLuSX; IT6vGUI = MPG s Y%\žBDS˜SuEvGQ]T;BDNuIT;wO]KGQ9FD;Q‘5769;NŒ¤uEvSTBMNvw?°˜69;vBDST69;pi•S^BDN9;uEvGQF›

BMI`W

oMT\ ?MPGIk\ = MG s Y%\

V U

бNPvGU]TITBHZEvFDGQ] ORQS T r

oMk\iI_/GQN V IK69;ZLONupO;]KwQ;YNvZ;]fGJIK;‚BHS7ZLQNuIK]TLOFMFD; V CW2IK69;IKGUBDFHS˜LQ†ŒIT6v;FHG…ˆ

LQ†wGxUŒn9LO]‚;[9GU:2P9FD;BM†wG 6vGOS‚cvN9BMIT;?:?LO:2;NuIKSLU†˜LQ]V ;] ”IK69BDSwOBMpO;YS oMT\

U—бNmSTBDwQN9BMcƒZGQNuI

;[9GU:2P9FD;YSY¤OGQS V BM–/EƒSXBDLQNšPv]TLRZ;eSTST;YSY¤UIK69; IfGUBDFDS 6vG…pQ;‰;pQ;YN;[RP/LQN9;YNOIKBDGQFƒ]fGJIK;Q‘ž@CLIK69;‚Pv]TLO9FM;Y:œLQ†—IT69;

EvP9Pƒ;Y]˜/LQEvN V S˜†‡LQ]˜IT69; V ;YNvSXBMI`W2BDS˜]fGJIT6v;]7SXBD:2P9FM;?¥¦GUI˜IK69;ZLQNuIT]fGU]KWQ¤QIT69;P9]KLQ9FD;: LQ†ŒFMLJˆ';]˜ƒLOE9N V S

BHSž:EvZf6:2LQ]K; Zf6vGUFDFM;YN9wQBDN9w § ‘5769; GU/LJpQ;7†‡LO]T:E9FDG=wOBMpO;YSžG!ZLQNuIT]KLQFC†‡LO] 9mUQбNLO]V ;]žIKLLQRIfGUBDN

STBD:?BDFHGU]/LQE9N V S'†‡LQ] W9 c LON9;=6vGQS'IKL;:2P9FDLJWIT6v;=†‡LQ]K:E9FHG

oMT\iI_CDMRY

Y r ; <

MG4\= MG

s

Y%\ƒ\U ¥< §

5769BHSBHS7LQRIfGUBDN9; V BDNzIK69;!SKGU:2;=ˆ'G…WzGOS=¥‡< § EƒSXBDN9w?IT69;Pv]TBD:2B›IKBMpO;G

M>\ I c D r

A ? MZK \!%KU

 ;wQL2N9LJˆ †‡E9]XIK69;] GQN V IT]K;YGUI'IK69;P9]KLQ9FD;: LU†IK69; V ;Y]TBDpJGJITBDpQ;eS7LU†IT69; V ;YNvSTB›I`W†‡EvNvZ„IKBMLONŒ‘

&˜ª ~ ) ,+%+)f- ;5?MPG s = ]

MG4\ƒ\S Im/S U6UU6S -6=)! / - -B1 )P E) !)

" P%46Wk

]<

MT\iI MƒcKY%\

]

CDMY#

";r=<

MG4\=

]aoZ

MG4\\ ¥-j §

$ } ®Œ®10K‘#Œ;INIZY1U  ; V ;cvNv;&%

Mk\wI D

Y r G

M>\$!%>kS9STLIT6vGUI'%f00

I ?

S9GUN V ˆ';ZLQ:2;vGQZf‹

IKL?IT69;P9]KLCLU†LU†FM;Y:?:GkQ‘  ;=EvST;=I`ˆBDZ;‚IK69;BDNOIK;wO]KGUITBDLQNzCWPvGU]TIKS'†‡LO]T:E9FDG?GUN V LORIKGQBMN

CDM3?

MG c(K \\ I CDMG

MPGLc(K*\R= MPG s Y%\\ ImCDM(%

MG c(K \= MG s = MG s Y%\\ƒ\U

@CBDNvZ;=OQRS

TjQ

%

MG c(K \ I MGLcCK*\

a

ˆ˜;=LORIKGQBMN

CDMPKNMPG4\ƒ\iI KNMZK \ƒCDMƒMPG_c(K \a = MG s = MG s Y%\\ƒ\$!LK

(10)

bMK*\ ImCDMMG c(K \ a = MG s = MG

s

Y%\ƒ\\U

5769;PvFM;eGQSKGUNuIP/LQBDNuIžBMNIT69BHSžN9;Yˆ BDNuIT;wO]KGQFC]T;YP9]T;eSX;YNuIKGJIKBMLON!LU†/IT6v; V ;NvSTB›I`WBDSIK6vGJI K+9 MPG c K*\

a

BHS V BM–/;Y]T;YNuITBHGU9FD;Q‘^5GU‹CBDN9w V ;]KBMpJGJIKBMpO;YS7BDNIT69;!GQƒLJpO;‰†‡LO]T:E9FDGwQBDpQ;eS

0

MK*\ IJc CDMY:

; r=<

MG4\= MG s = MG s Y%\ƒ\\ I c C`MRY:

; r=<

MG4\=

V

MPG4\ƒ\

GQN V IT6v;”P9]KLCLU†BDSZLO:2P9FM;IT; V †‡LQ] I YO‘gбN1LO]V ;Y]šIKL V ;YGQF‚ˆBMIT6 69BMwO69;] V ;]KBDp…GUITBDpQ;eSLQN9;

;Y:2P9FMLJWO;YS':2LQ]K;=BMNuIT;YwQ]fGJIKBMLONzCWšPvGQ]XIfS7BMN+LQ] V ;Y]˜IKL2LQRIfGUBDN

oMK*\ ImC`MM

]

MPG cCK*\

a = ]abZ

MPG4\ƒ\

ˆ69;Y]T;7M

]

BHSGUN ^IKBM:2;YS V B›–i;]K;NuITBHGUvFM;=†‡E9NƒZ„ITBDLQN¢STEvZf6IT6vGUI M ]<

]

MT\iIY":

Q ;r=<

Mk\„‘

;I EvS:GQ‹Q;=SXLO:?;=]K;:GU]K‹RS‘



GUIKGUNƒGU/;

V

;pQ;YFMLOPvSBMN 9qUr=;‰¥3SX;Y;GUFHSXL9Mker=;

§ G V

BDSXIT]KBMvERITBDLQNmIT69;YLQ]KW¢LQNmIT69;



BD;Nv;]!STPvGQZ;

]K;FHGJIK;

V

IKL‰abGUFDFDBDG…pCBDN!ZGUFHZEvFMEvSY‘Ž‚Nv;^LU†RIK69;˜Z;NuIT]fGUFOB

V

;eGQSBMNIK69BHSŒIK69;LO]TWBDSIT6v;˜STL‰ZGQFMFD;

V

P9]KLRZ;

V

E9]K;

LQ†±P9EvST69BMNvwzvGOZf‹¢@9Zf6uˆ7GU]TIF

V

BDSXIT]KBD9ERITBDLQNƒSˆ69BHZf6”;eSTST;NuIKBDGQFMFDWZLQBDNvZB

V

;eS‰ˆBMIT6mIT69;IT]KBHZf‹EƒSX;

V

BDN

LOE9]P9]KLCLU†`’LQN9;;Y:2P9FMLJWO;YS7G2Z;Y]XIfGUBDNNCE9:ƒ;Y] LU†BMNuIT;YwQ]fGJIKBMLONCWšPvGQ]XIfS7BMN+LO] V ;Y]'ITL]K;wQEvFDGQ]TB6F;‚IT69;

V BDSXIT]KBMvERITBDLQNvSY¤Cˆ69BHZf6ZLQ]K;YSTPƒLON V S^BDNLQE9]7P9]KLCLU†ŒIKLIK69;=]T;YP9]T;eSX;YNuIKGJIKBMLONCW:?;eGUNvS'LQ†IT6v;‰†‡E9NƒZ„ITBDLQN

M ]

ˆ69BHZf6+BDS^ITBD:?;eS V BM–/;Y]T;YNuITBHGU9FD;Q‘

5769;‚BMNuIK;wQ]fGUFv]T;YP9]T;eSX;YNuIKGJIKBMLON?†‡LQ]K:E9FHGš¥-j § P/;]K:?BMIKS ITLLQRIfGUBDNE9P9P/;]'ƒLOE9N V S˜LU†—IK69; V ;]KBMpJGJ

IKBMpO;YS=LQ†˜IK69; V ;NƒSXBMI`W oUŒŠ±N PvGU]TITBHZE9FHGU]e¤STE9P9P/LOST;?IT6vGUI@G 6vGOS=cvN9BMIT;:2LO:?;YNuIKSLQ†7GQNuW”LQ] V ;]GUN V

IK6vGJI ;5?MPG s = ]

MG4\ƒ\‰69LOFV S‰IT]KE9;†‡LQ];pQ;Y]TW E GUN V = ]

MPG4\‰GQ]T;SuEvGU]K;!BDNuIT;wO]KGQ9FD;Q‘‰5769;YN mBDS

BDNRcvNvB›IK;FDW V B›–i;]K;NuITBHGUvFM;zGQN V

]< MT\ ?MGPF k\ `=

]

MG4\ V

†‡LO];YpQ;]KW E U@CL

FE SRIK69;@RZf6Cˆ7GU]TIF=SXPvGOZ;=LU†]fGUP9BV FDW V ;YZ]T;eGQSTBMN9w†‡E9NvZITBDLQNvSY‘

5769; GU/LJpQ;7FD;:2:GSX6vLJˆ SIT6vGUI^IT69;Y]T;BHS^GQN?BDNuITBD:GJIT;]K;FHGJIKBMLON¥uEvGQSTB9;uE9BDpJGUFD;NvZ; § /;I`ˆ';;YN

IK69;2BMNuIK;wQ]fGJIKBMLONbCWbPvGQ]XIfS‰†‡LO]T:E9FHGGQN V IK69;;[RBHS`IK;NvZ;LQ†'GwOLuL V V ;NvSTBMI`W¢LU† IK69;2FDG…ˆ LU† GxUбN

†¦GOZ„Ie¤^SXE9PvPƒLuSX;zIT6vGUIWG bMT\$!% GQN V BDS V BM–/;Y]T;YNOIKBDGQ9FD;GUN V 0MG4\!BDS?BDNOIK;wO]KGQ9FM;O‘”5769;YNŒ¤†‡LQ]

;YpQ;Y]TW`KHE qBr

G

MhB\

CDMPK 0 MG4\\ I K 0Mk\oMT\$!% I c KNMT\ 0Mk\$!% I c KNMT\

0Mk\

oMT\

Y#

Q <

MT\ bMk\!"

I c C`MPKNMG4\

0MG4\

bMG4\

Y#

Q <

MPG4\ƒ\U

@CL ; ?MGxS`Y=\š6vLQFV SzˆBMIT6= MPG s Y%\I c @

<

<

Y

Q <

MG4\ E v Z ¥‡/;YZGQEvST; 0MG4\ E v Z MB2l\\U

ŠIK;]fGJIKBMN9w=IK69BHS^ˆ'; LQRIfGUBDN2IT69; †‡LQFDFMLJˆBDN9w!Zf6vGQBMN2LU†—BD:2P9FMBHZGUITBDLQNvSY’J;5?MPGxS5=<I

aoZ

MG4\\69LOFV S^IT]KE9;

BHSIKBM:2;eS V BM–/;Y]T;YNuITBHGU9FD;!GUN V I < MPG4\E"vxZM32l\ ;5?MPGxS5=<I

Y[Z

MPG4\ƒ\ 69LQFV S7IK]TE9;O‘a¢LQ]K;LJpO;]7LQN9;

6ƒGQS=<I(MG4\ I McKY=\

I

NPO5Q

<

<

Y#

Q <

MPG4\ EdvxZ=U

(11)

®1,

¯M.‡|/.

®1,

~&+

ª ª

U|e~R| .

®A,

{

5769;¢ZLO:2P9ERIKGUITBDLQN LU†‚ZLQN

V

B›IKBMLONvGUF;[CP/;YZIKGUITBDLQNvS?BHSZ]KEvZBHGUF˜†‡LQ]

STLQFDpCBMNvw™NCE9:2;Y]TBHZGQFMFDW Z;Y]XIfGUBDN²N9LQN FDBMNv;YGU]Pv]TLO9FM;Y:SCW

V

;eSTZ;N

V

BDN9w G ,‚WCNvGU:2BHZGUF‰\^]TLOwQ]fGU:2BDN9w

GQFMwOLQ]KB›IK69:+‘ d;YZ;NuITFDW ST;pO;]fGUF‰GQERIT6vLQ]fSb¥¦ST;; [G&\^]-_ƒS`Y _haS/[G&\b]-__Bc aSUU6U\!;Y:?PvFMLJWO;

V

SXLO:2;¢†‡LQ]K:E9FHGQS

ƒGQST; V LQNmabGUFDFMBHG…pCBMNbZYGUFHZE9FDEvS IT;YZf6vN9BDZYS‰BDNbLO] V ;Y]ITLZLO:?PvERIT;?ZLQN V B›IKBMLONvGUF;[RP/;YZ„IfGJIKBMLONvS‘‚бN¢IT6vBDS

ST;YZITBDLQNˆ˜;=wOBMpO;‚IK69;!GUƒS`IK]KGOZ„I'†‡LQ]K:LU†IT69BHS7†‡LO]T:E9FDGv‘

&˜ª ~ 6 G T 4 4' Z1i@'T1 $%46) ) -- ; ?MG s Y%\ k ; ?MG s 44\ -

! / -

CDMB4GJI k\ I

CDMRY":

;r=<

MPG4\R= MPG

s

44\ƒ\

CDMY:

";

r=<

MG4\R=&MG s Y=\ƒ\

¥¦l §

1 - - n$P -j -` - 8,-(H-k")'5dE)`( 1 - - 5TWk E)

(

$ } ®Œ®10K‘ ;INMMk\ V ;YSTBMwON9;;eS=IT6v;IT;Y]T: BMN IK69;š]KBMwO6OI!6ƒGUN V STBV ;LU†7IT69;GU/LJpQ;2;uEvGQFMBMI`WQ‘  ;

6ƒG…pQ;šIKL™Zf6v;YZf‹ IT6ƒGJI?†‡LQ]2;pO;]KW8K:E qBr

G

MhB\LQN9;6vGQS!CDMKNMG4\44\`I CDMKNMG4\$MMG4\\U ‚SXBDN9w™IT69;

]K;wOE9FHGU]KBFeGJITBDLQNš†‡E9NvZ„IKBMLONvS7†‡]KLQ: IK69;P9]KLCLU†LQ†;:2:G+kˆ';=ˆ]TBMIT;

C`MPKNMG4\44\ I C`M4 OQRS

TjQ

KNMZK \5?

MG c(K \!%K*\ I OQRS

TjQ

KNMK*\CDMB4T?

MG c(K \\$!LK

I ORQS

TjQ

KNMK*\CDMG

MPG_c(K \= MG

s

44\ƒ\!%K.I KNMZK \ƒCDMRY":

Q ;r=<

MPG_c(K \= MG

s

44\ƒ\!%K

I KNMK*\MMZK \oMZK \!%K@ImCDMPKNMPG4\ MMPG4\ƒ\

GQN

V

IT69;P9]KLCLU†BDS ZLQ:2P9FD;IK;

V ‘

)+*

ª { ª-, { .¦| . D.¦| « } ® y+ª бN :2GQNCW²GQP9P9FDBDZYGJIKBMLONvSLQN9;™ZLQNƒSXBV ;]fS OEƒGUNuITBMITBD;YSLU†IT69;m†‡LQ]K:

C`MB?bMG

\ƒ\iˆ6v;]K; G

BDSG'†¦GU:2BMFDWLU†C]KGQN V LQ: pJGU]KBDGQ9FM;eSiBMN V ;[R; V LONG'cvN9BMIT; V BD:2;NvSTBDLQNvGQFQPvGQ]KGQ:?;IT;Y]

oU—“gI`WCP9BHZGQF;[9GU:2P9FD;BHSfG

I

ˆ69BDZf6mBDS=G V B›–iEvSTBMLON”P9]KLRZ;eSTSS`IfGU]TITBDN9wš†‡]KLQ: oU—бN”LQ] V ;Y] ITL

SXITE V WIT6v;ST;NvSTBMITBDpuBMI`WšLQ†ŒIT69BHSuEvGUNuIKB›I`WˆBMIT6]T;eSXP/;YZI˜IKL?IT69;=PvGQ]KGQ:?;IT;Y] +LQNv;‚6vGOS˜ITL2P9]KLJpQ; IK6vGJI

9 C ?oMPG \=BHS V BM–i;]K;NuITBHGU9FD;GUN V IKL¢;pJGUFDEvGJIK;2IT69; V ;]KBMpJGJIKBMpO;Q‘š5769;Y]T;šGQ]T;?I`ˆ'L+ˆ'G…WRSITL+IfGU‹O;

IK69BHSP9]KLQ9FD;:+’EvSTBMN9wG?PvGUIT6Cˆ'G…WRS7GUPvP9]TLuGQZf6LQ]GUN¢GUPvP9]TLuGQZf6šBDN+FDG…ˆ‘

5769;mPvGJIK6Cˆ'G…WRSzGQP9P9]KLOGOZf6²STE9P9P/LOST;YSšIT6vGUI 9 G Mw\šBHS V B›–i;]K;NuITBHGUvFM;¢†‡LO]GQFM:2LuS`Iz;YpQ;Y]TW

¥¦GQN V IT69BHSBHS2IT69;bZYGQST;†‡LQ] Z9

Mw\†‡LQ];[9GU:2P9FD; § GUN V ?1BDS V B›–i;]K;NuITBHGUvFM;bGQFDSTLv‘ 5769;YN

?

CDMB?bMG \ƒ\I|CDM3? 0MPG \$?

G \U e'ERIIT69BHSGQP9P9]KLOGOZf6¡9]KGQ‹Q;eS V LJˆN BM†? BDSN9LQI V BM–/;Y]T;YNuITBHGU9FD;Q‘

576v;šSX;eZLQN V GQP9P9]KLOGQZf6™LJpQ;Y]KZLQ:2;YS‰IT6vBDS V B>šZE9F›I`WmEvSTBMN9w+IK69;šST:?LCLQIT69N9;eSTS!LU†'IT69; V ;YNvSTB›I`WmLU†7IT69;

FHG…ˆ LQ†fG U˜@CLv¤'BDN³IK69BHSšGUP9Pv]TLuGQZf6 LON9;¢GQSKSXEv:?;eS?IT6vGUI`G @ M>\$!%> GQN V 9 M>\?BDS V B›†¨

†‡;Y]T;YNuITBHGU9FD;†‡LO];YGOZf6>TU—5769;N ?

CDM3?oMPG \\fI D ?bM>\?

M>\$!%>I D ?oM>\$?

O M>\ M>\$!%> I

C`MB?bMG \$?

O MPG4\ƒ\UC@CLO:2;ITBD:2;YS˜;N9wOBMN9;Y;]fS^ZGQFMF ?

O MG4\žIT69;‚STZLQ]K;7†‡E9NvZ„IKBMLON”¥¦ST;;^GQSOEv; F

“‰vG

V

9DkYt;

§

‘De'ERI!LQ†7ZLQE9]fST;IK69BDSGUP9P9]KLOGOZf6”ˆ'LQ]K‹RS‰ˆ69;YN LQN9;2‹CN9LJˆ SIT6v;

V

;NvSTBMI`WbLU†'IT69;FHG…ˆ LQ†

Références

Documents relatifs

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe -