Genetic genealogical models in rare event analysis
33
0
0
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Genetic Genealogical Models in Rare Event Analysis Frédéric Cérou , Pierre Del Moral , François Le Gland , Pascal Lezaud. N˚5878 Avril 2006. ISSN 0249-6399. ISRN INRIA/RR--5878--FR+ENG. Systèmes numériques. apport de recherche.
(3)
(4)
(5) !#"$ %& '( )+*-,/.0,/*-132546,/*87:9 ; < 13=/*8*-=5>=/?A@B7:*DCE? ; )+*-CEF G/7E1IHJ=LKM?ICEF . ;< CEH-2/CE?NJ=/O/CE9 . PRQTSVUXW8Y6`cZ-giS\hjZa[RU]^klY(SV_amn`cb bed/]^Z-S f oqp m^mEgr`XUAsTZN`cZ8tvu^Z8`ctvu^Zw[xzyr{/|}{L~k R`Xbeizr ~i$m pr Z8S ∗. †. ‡. §. [n0s 8a}^Zar[^rZ eg bZlt mnT`XY(Z-SVgRZ8s^[zUZab[grUX`u^Z-bSjS Ucbp Y `XUXbt Ucbe[Z p$ Z8ta[^ Z SXUcS+betqgrUjQ/` mEZ`XZb[/Z8UXrZ8Za` [/p UvtS Ucb[ `c bSVbem [ p `XUXbeb[t em^ZluRSXQTQTSVSjbUctaZaS YS[ns(p [^ grZaUj`cbUXgzu^`XY p [ eQTp SVbS8¡ p Z$ `cZamnp `XZ-SVZ8[zUlUXu^Z(sTbSjUc`Xbe¢^p]TUcbgz$[5p gr p `c rg}p mn`XgTt Z-SXSlp u^bUVUcb[ ` `XZ U ` p ZaUlbe[MUXZ8`XYS grp p p¦¥ Z8Q/[nY p [R§R¨ p tY(gTsTZabe[©m p Ucu©SXm p taZr¡ p& £5Zªp SXu^g}«u^g}Ucu^Z8SXZ¢^` p¤[np tvu^bep [ m pp `XUXbt eZ&Y(gTsTZ8eS `s^p Z8`cScZt `cZabr¢EZ8Z8[/s¦UcS be[¬p S\m^`XZ8ZaReb gzp ]nSSNUcu^gzZ`XsT TbS SjUct `Xp be¢^[]T¢UcbZgz[&]nSVgrZ- sLUcu^UXgMZ m^Z-Sj`cUcgTbt Y Z8Scp SUcZ(b[UXu^Ucu^ZbeSqm^`X`XgzZ ¢ bep Y6¢nbZzeb®¡ UjQLgiUXu^Z&t gr`c`cZ8SXmgz[nsTbe[ ¯ª°/±:²c³´ }µ0r be[/UXZa` p t UXbe[ m p `XUXbt eZ6SXQTSjUcZaYS8¶·` p `XZ$ZazZa[/UcS8¶ ¥ Z8QR[^Y p [T¸¹¨ p tY6gTsTZ8eS8¶ Za[^ZaUXbt p ®¸ gr`cb®Ucu^YSa¶ Za[^Z p eg bt p nUc`XZ8Z8S º »¼X½¿¾8À\¼Á0Âe½¿ÃIÄ Å ðåaæ äí}â ñvÆ·ònèéÇDê íiÈÊÉ+伿ÃèËê®éÏÌæôÍÏè Îóë¹ÑDãDË·ÍϼIÌvÐvæXà3½É×já¹ÑDæVÒ·Ðvß8ÍÏîEÒ3ÖDÈÓêõÐvì ÂÔÊéÏÙiÔ ÕèX×VïaÍ^ÉIæö ÖD÷Vç·ÑDø8ÑièÛ Ìé¹ÂÍùêeÒIúëV×j¾-Øû ì è üÙ ê®ü-éÏÕ6æ ýVö-Ú·ó Þ-âÛ â Ü0èäVÂÝiå ã ÞT¼Xþ ì ÖDê½¹ßDèØDÂ×jêäÍÑDíÒIÍ3Ç-ÌæV×AÍÂr3þrÑ-Ðvê®õÍ3ÀÌvõàϾ8×jì ×á¹èÒI¹ÂÉÞTêä Ðvâí ßDØð¼X¾8ã-â À\äVå ¼X¼VæXÂé¹ã-½êeäVÿVçåa¾DæVèæ ½ôé¹ÑDÂçêÍÏëXÌ è¾8Íé¹ì IÂèÐvêê®ëXõé3¾8æVõì½Aèê®× æXé3¿í$æVÞ ½îEó ãDêì è ÂÝéÏ®íièX ïaæX÷8æ½ Þ þäí}ÂéÏäìzè íiëXåÂêäþrí& äXë¹þaã-Ä-è ¼I½IëXÂê Eêeë ÿX¾Dæ íièÑ-ì Í3Ì ½¿Íꮽ03Ðvä õõ× V¹Þé¹êÐvåßDþ ؽ¿Ù  ÕæVÀÒIÇ-½ × º ÖDÍÏÌ» Ñi×VÐvߪÚ:ÅÌõó ÑDõÍÏÌvÈÊà3ÉI×jÉ3á¹ÈÊÌÒ·ß6ßaÖDÖDõßDØ-Ùi×V×jÍÍ ÏÒ3ÝaÇDÆ×Nø8ÑDùÍÏüÌvü-àÏý¿×jø8ájÒúù 꽿ü-ÂeÞ é¹¿ê jí ¾-¹ä  æ¿é å À\èÂêä íNþzëVêeæVíië¿æ iæ¿ë¹ã-íiäì ä¿ï ÑDÍ3Ì Í3Ðvõõ× Ü ÐvÏÙiÝ ÌÍIÐcû Ò3ÌÛ0ÈÊÍ3Ü × ÞzÚÐv0õÈÊ×jÑDÖDÖDØDÉ·Ìß-ØDß × :Þ ×V ÐvßDÖDÈ ÔÊÈÊ×jnÖ/Û ÞDÈÊú áj×ü Þi ÝaùÌÑDÜ ÇDÈÓÐÛ0Û ßa:Ò3ÝAÈÊÑiÚ ÌÔÊjÈÊØDÉX×Þ r^Þ Ðv}ÍÏáÍIÐv:ß-ÐváV×ÔÊÍÏÌÉI×ÞDü -ýVü \Û ÈÊáj×ÞiÚ jØD× lù-Þ }Í3ÐvßDáj× Ú:Ü ×jÏßaÝ Ò3ÍÏ×Nû Ø Û0nÜ Ò3ÖDØDÞz×jÚÉ\ÐvØ-õ×lÑDÔÓÖDÐÉ·Û0ØDÐ × -:È Ð×VcÐvÒ3ÖDÈÊÌÔÊÈÊß ×jÖ/ÞDjú Í3ÈÊ×jü ßD ßDù×Ü Þ÷ Û0Ð Û ×V:ßaÝAÖD× Ú jØDØD× ÌrÖ}Þ Ðv}ÍÏØÍIÐvß-:áV×j× ÔÊÈÊß/Þ üü -Þú-ýVü Æ 0Ý Ú jØD× DÞ }Í3ÐvßDáj×
(6) . . . . . . . . . ( 21 PO@. .Q @5 9R N >+=+Z-[VF0<+J+Z*H\[@J*]*^;B_GIV?0A0=LBM< 0( ( 21 ‡ ( /( Cb \S c § 65 -7#) V8 . Z*=*f+J0G+>+H A0=0I JLBM< ∗. †. ( /( 0(. .
(7) #"$ &. % '. !. -.. S!.. 5. 3 *). 4. 3 *). 4. . . (. . *)+. -,. 4. 65. -7 98 VU W. .;:+<+=*>+=*<@?0ACBED=*< F G H?-I+<@?KJLBM< X Y5 -7 8. 4 \S. 65 . -7 98. .YZ*=*` Z*J0I+>+H?\<@?0a0JLBM< Y16U) 3 33 4e R N e R . 0T . d1. Unité de recherche INRIA Rennes IRISA, Campus universitaire de Beaulieu, 35042 RENNES Cedex (France) Téléphone : 02 99 84 71 00 - International : +33 2 99 84 71 00 Télécopie : 02 99 84 71 71 - International : +33 2 99 84 71 71. ..
(8) . 0& & ¦
(9) & V% j #"
(10) $ « . ' . gz]nSm^`c_8SXZa[/Ucgr[nS$s [nS6t ZaU `VUcbetaZ]n[ gz`XbUXu^Y(Zs^Zm `VUcbeta]^Z-SZ8[b[/UcZa` tUcbgz[ ^s ZUjQRmZ _8[^_ Ucbed/]^Z&Z U$]n[ Y(gTsTWaeZ p _a[^_ p eg p bd/]^ZmEgr]^`$p Z8 SVUXbeY6Z8`$]n[^Z&ta p p SXSXZs _az_a[^Z8Y6Z8[/UcS$p ` p `XZ-S mns^`XZ g}zZa[ `Xp z[zg}U$sTd/Z ]^b p UXm^UXuRZ8QRb[/SXUbdz]n]n[Z_aZazUl_asT[^Z Z8Y( p Za[[/pUe`QTSV`cZNZAsTtaZ-grSY(`XY(_-SVZAZ p ]^] [& ¡Y6 gTsTgz]nW8SZNZ sT RZ mn`XbeZaY(QRgr[^[nY S\ [Rp §R¨ gzbtAs s ]^[¤[nm^Sô`XgT Z8taSXZ8m ScSV]ntaZS s^Z8Sô£MUX` p p hjZ8t UXgzb`cZ8S8p ¡ gr]nSY(gr[/UX`cgr[nSt gzY6Y(p Z8[zU\t Z-SY(gTsTWaeZ8SsTZ¢^` p [ntvu^¥ Z8Y6Z8[/Up sTZAm pp `XUXbt p ]^eZ8SsT_8ta`Xp bUcS `s p p `c[nZ8SlS8¶ sTp Z8beSq[nSXUXb·` p dz ]np Z ] & p mngz`X_-bt sT_8]¤s^Zam^[/`cUcgTSt mEZ8ScZaSX]n]nrS\Za[/s U p [n SUX`ctaZZN]T`XUc_ bebbeSVY6_-SqZz¡mEgr]^`lZ8SVUXbeY6Z8`A p m^`cgr¢ p ¢^bebUX_s! _8r_a[nZaY(Za[/UcS "w´ - ² # SXQTSjUcWaY(Z8SsTZAm p `XUXbt ]^eZ8SôZa[be[/UXZ8` p tUXbegr[¶r_az_a[^Z8Y(Za[/UcSô` p `XZ-Sa¶zY6gTsTW8Z-Ss^Z ¥ Z8QR[^Y p [T¸ ¨ p ti¶ p gz`XbUXunY6Z-S _a[n_ UXbd/]^Z8S8¶ p `c¢^`cZ8S _8[^_ p g bd/]^Z8S .
(11)
(12) "!#$%'&)(*+-,#/.* 01 . 2. $ & ô ¢Z p tagr[/UXbe[R]^gr]S¹§/UXbeY6Z(SVUX`cgr[ ¤£Mp `c rg}ªmn`XgTt Z-SXSlU p Rb[ p ]^Z-S be[SXgrY(Z 5 Z U f gzbSVXu¤SV=U p Uc{XZ SVm ,p tt Z ≥S0}¡ ¥ gr` p6 berZ8[U p ` Z U76gr`cZa·SXZ U B ⊂ S Z sTZ
(13) 8n[nZNUXu^ZNu^bUVUcb[ UXbeY6Z 3. 4. t. S6UXu^Z98n`cSVUUXbeY(ZMu^Z8[ UXu^Zªm^`cgTt Z-SXS u^bUcS ¡ Z U]nS ScSV]^Y(Z¤Ucu U u S eY(gzSVUSV]^`cZaeQ `cp b u/U\tagr[/UX¡@be[R?¹]^[¦gr]Y Sa¶/[RZaQ Uqm^beY(mnb®bUct Z8sUXbeUXgr` [np S8hj¶:Z-tUXUXu^gzZX`XbeSXZ8Z S'U : o;)Bb5<S5<UXu^= 5 ¶ZAp :3[nSVs]^mEUcZau p ` p U eBZarZ8bStaSVZag/U p SVZ-gis X¡->\Scu^t p Za[ p `lY(ZluZ p SVz]nZA` Ucu ¢^p ZU = p pp p p X B ]n[ntUcb∈gz[ Bφ sT
(14) Z 8[^Z8p sgr[ p S ¶^bI¡ Zrp ¡ TB = inf{t ≥ 0 : Xt ∈ B} ,. TB. ¹[6Ucu^beSôt p SXZr¶rZbe p ScSV]^Y(Z\Ucu p U φ beS]^m^mEZa`ôSXZaY(b®§Rt gz[/UXbe[/]ngr]nS8¶}u^btvu6Z8[nSX]^`XZ-S0UXu p U B beSôt egzSXZ8s¡ Ucu^ZSVZaUqgi o;)55 UX` bp hjZ-tUcgr[n`cZabrZ-SZ8`be[ SX]nStatabZa[nZ8s^snZSN TZ8UXsg S) SXg¤UXu U ¢R¥ Qgr` Ip ¡ [/Q `XZ Z p p e be[zp UcQTZaS`c U p p rIZ(Ucu^ZNZ&tabgre[RsTrZ8Z8[^[/UXgibeUcgrZ [ ¢RinfQ D(I, `cZ p tvu¤UXu^Z(sTZ-SVbe`XZ-sªU p ` Z U B B¡ ?¿U Y p Qu p m^mEZa[5∅ =UXu p ∞UY(gzSVUlgrp UXTunZ$`c=Z p ∞bDC p UXbegrX[nSAgi X [^Z8rZ8`l`cZ p tvu Ucm u^Z`XUXSXbZ t U]^ B`\E¡ gr>\[nu^ZZt grgz]n`Xe`cs&Z8SXmbe rgzZ[nsTUcgbe[ Z8 SVUX` bep Y `cZNUXZaZzZaUX[/unUZm^dz`X] gz¢ [/p UX¢nbUXbbeeZ8b®UcS bZ-S p `cZNZ RUX`cZaY(ZaeQsTbFt ]^UqUcg p [ p eQGCaZz¡-?¹[ p p p p :IHrD¡ H = p [s 5 p (X , 0 ≤ t ≤ T | T ≤ T ) , P(T ≤ T ) unZa`cZ T bS\ZabUXu^Z8` p sTZ UcZa`cY6be[^bSjUcbe't 8[^b®UcZNUXbeY(Zr¶ • p § p eY(gzSVU6SX]^`XZ8¶TJQ b3¡ Zr8¡ [^b®UcZªSjUcgrm^m^beb[ UXu UXbeY(Zr¶gz`(b[nSVU p [nt Z&UXunZunb®UXUXbe[ UXbeY6Z¤gr p `cZ8ta]^`c`XZ8[zU • 6grP`cZa·SXZ U R⊂S T =T p [ns P(T < ∞) = 1 . T = inf{t ≥ 0 : X ∈ R} >\unZNSXZ8tagr[nst p SVZ tag}rZa`vSôUXu^ZNUj g KVsT] p M LNSXb®Uc] p Ucbgz[nSa¡ PR]^m^mEgzSXZlUXu^Z SVU p UcZlSXm p taZ S = A ∪ R beS\sTZ-t gzY6mEgzSXZ8sb[/UXg$Ujg$SXZam p ` p UXZA`cZ bgz[nS A p [ns R ¡ • >\u^Zm^`XgTtaZ8ScS b[/Ucg p U p ` ZaU XB Sj⊂U p A`XUcS ¢EbZ [ gzA`XZp Z [n RsbUXbe[ Z Ap [z¡<U ?¹[(UXgUcu^Z8SVbeSUXbet Y gzp[zUXUcZZ RUXU unZUXu^mnZl`Xgzt ¢gr[p ¢^sTbeb®eUcb®bUjgzQN[ gip nUXsTu^beSVZUXZa`c[/be¢^UX]T` pUXbe[grt [AZ:IUcHrbY(¡DH Z = `XZ8m^`cZ8SXZa[/UcSUXunZ p gi0UXu^Z m^`cgTt Z-SXSbe[&Ucu^bOS N¢ p ebSjUcbePt Nq`cZ bY(Zr¡ PR]^¡ m^>\mEu^gzSXZ Z6m^UX`cu^gTZt Z8SVScU S p UXZZ8SXmrpgrtaerZ Z-SSb[&=UXBu^Z ∪`cZ CbgzbeS [ s^Z8t gzY(unbemtvu¤g/SVt Z-gzsM[zUbe[zUcbe[ngS Ujgªt grSXeZaeZ8m tp Uc`bgzp [&UXZgr`cR Z N u bgz[n`vSsBgz¢npSjU [ns ¸ • Q C t eZ8*S N$`XZ8m^`XZ-SVZ8[/UXZ8sLX¢RQ p SX]^¢nSXZ U R ⊂ C ¡ >\Cu^Zm p `XUXbt eZ6bpS Rbeep Z8s p S SXgRgr[ p SNbUpZa[/UcZa`vSUXu^p Z NQLu p UX`cuns5Zgrm^¢n`cSVgrU ¢ p p t ¢^ePZ beN$bUjQ5SVZaU giRZ @¡ Rb?¹UX[Lbe[ UX u^UcbSu^ZtagrmE[/gTUXtv Z r /Z UU$UXungiZ(grUj¢nSVg¤U p d/t ] eZ8p S [/UXCbUXbeZ8SS bUX:Iu^HrgzD¡ ]TH =U`c¢EZaZam^be[`cZ8 SXZa R[/beUNeZ8`Xs Z-SVmEp [nZ8t s¦UXbeUXru^Z Z ¸ sTbSjUc`Xbe¢^]TUcbgz[grUcu^Z m^`cgRtaZ8ScS\u^btvuSX]ntataZaZ-s^SUXgZ-SXt p mEZl`cgrY«UXu^bS`cZ begr[¡ B = {x ∈ S : φ(x) ≥ λB } .. ?. B. B. t. B. B. R. R. Ü Ü ß-T ÷. 3X *X. t. R.
(15) $% $%
(16) <R < +. ?¹[ e:UXunZSVZ-dz]nZaI¶ bee·¢Z gr0tagr]^`vSVZN]^[n /[ng}[·¶T¢^]TUq[^Z8rZ8`VUcu^ZaeZ8ScS p ScSV]^Y(Z-sUcg¢Z VS UX`cbtUXeQp mEgzSXb®UcbzZr¡ P(T ≤ T ) Z8[ >\be[^u^Z8Z6Za`cZ8b[ SVUX beY m^`cp grUc¢nbgzZ8[5Y(giS8ô¡ UX?¹u^[ªZ-SV[nZ(Z Ujd/] grp `c[/ UXbUXp Z-[ S p p QT`cSXbebeSXSZ8SASX]nb[Ltvu Y p p S[RbQª[ `Xp Z-sTSVZ pp [n`vtvtau Z8s&p `XUXZZ8pZ-St gzSV]Y6tvY$u ]^p [^Sbtbe[5p UXm^begruR[¤QTSVSVbQTtaSVS UXZap Y[s S [nSVUXZ ]UjsTbgrZ-`cS $Xm^`cUcgT` t p Z-sTSXb®SXUcbb[ gz [ phjgree¢nQ&S8-¡`cZa?¹m^[`cZ8UcSXu^Zab[/SUvtaS\grUX[/u^UX ZZ$ RUôZ8[ UXun UcZ8uMSXZAgiUjSXg$Za`cd/R] betap Z[zUct b®Z8Uc[zbZ-UcS#Za`v:ISqHr¡DbeH [ = p `X[ªZ8m^gz`cZ8mSXZ8Za[[/}Uôta`cg/ZamESVZ-Z8s¤tUcdzbz]nZaZae]^Q(Z8UXb[u^Z mn`Xgz¢ p ¢^beeb®UjQgr0¢n] EZ8`V¸¿g}rZ8` g}qS p [nsUXu^Zs^beSVUX`cb¢n]TUXbegr[ªgr Ucu^Zd/]^Za]^Z8b[ m^`XgTtaZ8ScS\b[¤Ucu^beSAg}rZ8` ng} `cZ bPRY(Z8Zzr¡Z8` T[/]nY6Z8`Xbt ^Y(Z Ucu^gTs^Su zZ\¢Z8Za[m^`cgrmEgzSXZ8s6be[$Ucu^Zqeb®UcZa` Uc]^`XZ\UcgZ8SVUXbeY UcZ\UXu^ZqZ8[/UX` [ntaZ mn`Xgz¢ p ¢^beeb®pUjQb[/Ucg p ` p p `cZSVZaU8¡ Z$`Xp ZaZa`qUcu^Z6`XZ p sTZ8`UXgUXu^Z$Z ^t Z8eZap [/Ulm p mEZa` p u^btvuMt gz[zU p bep [nS p mn`XZ-t bSVZ`XZ8RbZ8 gr[&UXu^Z-SVZNY(Z Ucu^gTs^S p SZae p S p sTZ U p beZ-sebeSVUgi0`XZaZa`cZa[ntaZ8S8¡ ¥ gr`UXunZNtagr[RrZ8[^beZa[ntaZ grUcu^Z `XZ p sTZ8`\Z m^`cZ8SXZa[/U\u^Za`cZ p UXZ8` p ¢n`XbeZ +sTZ8Sct `cbemTUXbegr[gi UXu^ZNUjg(Z p sTbe[ bes^Z p S8¡ ` p `cZr>\E¡ u^>\RZ u^8beSq`cSVm^U+`Xgrgz[^¢ Z\p b¢nS+be¢ beSVp UXSXbZ8t sp grmn[m^`Xtvg u p p tv[ u& begi[ UX Z8UX[ªu^Z`XZ-`cd/Z ]^Z8b`c`XZ8Z8S[nt UcZu^Z m^`c8ngr[¢ sTp b¢^[ b ebUjgiQ SVUXglu^ZUXu `Xp b Uu/` Uqp `ctvZu pSXZ [ Uv S Z ¢giZ- t grY(Y(Z Z-p S SV]^Z-`cSXZrS ¡ >\unbeS+SVUXZ8m6bS ]nSX] p eeQsTgr[nZ]SVbe[ p ` Zs^ZaRb p UXbegr[nS UcZ8tvu^[^bd/]^Z8S8¡ kq[^grUXu^Z8`+Y6gz`XZ\m^uRQTSXbet p p m^mn`Xg p tvu ¢EtagrZ [ngzSX`XbSjZ+UvbSqU be`c[5Z p SVtvmnu^Z8bUVSEUcUcb[ u^ Zô` UXpu^`cZ6Z0SjU U p p ` Uc Z$Z U8SX¡<m >\p tau^ZbeS b[/SVUcmng bUVp UcbSX[ Z8 d/]^SjU Z8pi[n t Z+ZbeSgrô¢ pSV]nSXZ8¢TsN¸IegrZaz[ Zap SmnUc`Xu^Z-Zt bSVm Zp mn`VUcu/beQTtaSXZbet [^p ZarZ-s^s^Z8SqSct Uc`cgbmTm Ucbp gzSX[ S gr UXu^Z$Zargz]^UXbegr[¤gi+Ucu^Zm^`cgTt Z-SXS¢EZ UjZaZ8[ªZ tvuªZ8rZa Z sTbe[ UcgUXunZ` `XZSXZ U-¡ >\u^Z[nZ RUSVUXZ8mªbeS Ucg$be[/UX`cgTsT]nt Z p SXQTSjUcZaYgi m p `XUXbt eZ8SZazgre/be[ p be[Ucu^bSeZazp ZasTZ- t grY(mEgzSXb®Ucpbgz[giUXu^ZNSjU p UcZr¶Rb[&u^btvu Z p tvum p `XUXbt eZ¢^` p [ntvu^Z-S p SSXg/gz[ p S\b®UqZ8[zUcZa`vS\b[/Ucg p unb u^Za`\eZarZ83¡ >\u^Zm^]^`cmg/SVZ gr Ucu^Zm^`cZ8SXZa[/U `XUXbt eZNbeSqUXgtagr[^[^Z-tUUcu^ZY]n®UcbeZazZa SXm^eb®UXUXbe[ UXZ-tvu^[^bd/]^Z8SbUXu Ucu^Z¢^` p [tvu^b[ Mp [nsbe[/UXZa` p t UXbe[ mp p `VUcbetaZSXQTSjUcZaYS p m^m^`cg RbeY p Ucbgz[nSNgi ¥ Z8QR[^Y p [R§T¨ p t(sTbSjUc`Xbe¢^]T¸ Ucbgz[nSSVUX]sTbZ-sb[ m^`XZ8Rbgz]nS p `XUXbt eZ8S8A¡ >\u^bSgr`c 5u p S¢EZaZa[¬be[ n]nZa[ntaZ8s¬¢RQLUXu^ZUXu^`cZaZ`cZ Z8`XZ8[nt Z-s m p mE lZa`v]^S `Ê|/gr¢T¶hj{ Z- tp Uc[bzs Zqbe Sô¿Uj¡ gigrs¡ b`vSjUZAm^`Xgzmg/SVZ Z8Q/[nY [R§R¨ t\`XZ8m^`cZ8SXZa[/U UXbegr[(giUcu^Zld/] [zUcb®¸ UcbZ-#S :IHrD¡ H = ¡ >\u^Z Z8[^Za` p besTZ p ¢E¥Zau^be[nsgz]^`tagr[nSVUX`c]np$t UX¥ begr[bSUcp g$t gz[np SXbesTZ8`Ucu^ZAeZazp Zata`Xg/SXSXbe[ £Mp p `c rg} grtvu p UXbeu^[Zbeta[&grm`c`Xp Z-UXSVu¤mEgrSVm[psTt b[Z p [SVU s p UXp Z8SXSASXgTt bb ep UXtaZ-grs[/U p bUXb[ u&p UXu^·Z be[TSXm^gzeb®`XUXY UXbep [ Uc bgzgi[¤·grUcu^[¤ZUXSVunU Zp Uc Zp SXQm pUXu^t ZrZ$-¡ m^>\`XgTu^taZ Z8SctaS\gr[nm t p p ScUXSVZ8Z-[ Sp Z Ucp bgztv[u e[ ZazUXZa]n/` ¢ZanY(gr`cZ ZZ8S[ [/UX8nZaZa`cbes[ UjQRbemE[/UXZ glUXZau^[^EZ ZaUX8nb[tqp m p `V[nUcbestaU Z p ` m^Zam^U0`c`g p T`cbZôY SXZ UXU-be¡gr[(6 pgiSXZ8Ucsu^Zgr[s^Z8Ucu^SXbb`cS Z8Y6sgTd/sT] Z8[zbe[Ucb® UcbZ-#S Z:Ibe[zHrUcD¡ H `XgT¡ sT]t grZ `cpZ be[/p UXZ8`XZ-p SjUcb[ p Q$Z p SVgSXu^g} u^g}©UXu^p Z Za[nZ p p eg bet p ^SVUX`cp ]ntUc]^`cZqgi:UXunZAm p `VUcbetaZ pp UôZ p tvuZ8rZ8= t p £ [¢Z ]SVZ-sUc@g 8n[ns&UXu^Zs^beSVUX`cb¢n]TUXbegr[gi0UXu^Z mn`XgTt Z-SXS\sT]n`Xbe[ b®UvSZ ^ta]^`cSXbegr[nSUXg6Ucu^Z ` p `XZlU p ` Z U-¡ u^Z8[UXu^Z&SjU UXZSXm t ZbSSVmnbUVUcZ8sbe[zUcg eZarZ8eS UXu^Zm `XUXbt eZ6Y(gTsTZ8ôZazgrerZ(be[zUcg SVUXZamSa¡ kUlUXbeY(Z n = 0 UXu^p Z6SjU p `Xp UAb®Ucu N be[nsTZ8mZ8[nsTZam[/UNt grmnbZ-Sqgi+Ucu^p Zmn`XgTt Z-SXS X ¡ >\u^Z$m p `XUXbt emZ8SAu^betvu Z8mn[/`XgTUXZasT`]nUXtau^Z8SZ`c[¤Z8ta g ]^:`c`XSXZ8m^[z`cUb[ SXZ ¡ U ?¿RUXunp Z `cZ Ru^bgreeZ-Z s SVQTp SV[nUXsZaY be[nbeSVSU p R[/beUceZ8Q sp UXunm Z p `XUX bt greZ`cbUXu u^p YRb[ bSq SV`XUXZgrp mntvmu^Z-Z-ssM ¡ Uc Au^UXSZ u^8nZ8`X`vSjUbSXeZlZaz¢RZaQ tagr[nSVUX`c]nUXt u^UXbeZ p gr[¦giôm Ucu^`VUcZbeta¢^Z-be`VSUcube[¬p Uc[nu^s Z sT8Z `cp SVUXU$ueZa`cz]^ZaeZ6ZarZ(gzzgrZ¢TU p t8be[ t gzN`csTbem[ p p `VUcUcgMbetaUXZ-unS Zp U S UcY(u^@ZZ8n`c]^`vSjeZUeZaS zZaUXIun¡(Z&km^U `cgTUct bY(Z-SXZ S p p `cZ RbeeZ8s p [ns¤be[nSjU p[/UXeQ m `VUcbetaZ n ¡ = q1Z8`XZ Nb[ªm `XUXbt eZ8SqunbetvuMZ8[/UXZa`AUXu^Z$`XZ-t ]^`c`cZa[/UlSXZ U i p z p p X u p /be[ `XgzY `XZ p Uctvu^u^bSZ-s¢^Uc`cu^beZ ZSVsTZ-Z-t SXgzta[n`XsbemTeUXZabezgrZa[#m^`cZLgTsTSX]nZaZ5taZ8UXS u p [¤U g Uc:u^ZMSXm^`cgzb[ `XY((Zap R`&[sm p SX`Xg6UXbgrt [·eZL¡ Sctvp u^Z8Y6ZMgrpeg}qSp UXu^Zp S Y(Z SXm^eb®UX¥ UXbe[ SjUc` p UXZ bZ-S p SUcu^Z¤gr[^Z¤sTbSXta]nScSVZ-sbep![ ¿¡ >\u^Z¤[^Za«p Y p UXu^Z8Y p UXbt p Y(gTsTZ8eS6m^`cZ8SXZa[/p UXZ-s B. Û0Ü (. ( .
(17) y.
(18) "!#$%'&)(*+-,#/.* 01 . unZa`cZ eg}qS Ucgt be¢^` UcZb®Ucum^`cZ8t bSXbgz[$Ucu^Z SVQRY(mTUXgrUXbt¢EZau Rbegr`grEUcu^bSm `VUcbetaZUcZ8tvu^[nbed/]^Z-S S Ucu^ZSXbp C8Z gi0UXu^ZSXp QTSjUcZaYp SUXZ8[ns^SqUXgbe[ 8n[^bUjQr¡E?¹[ p p s^sTbUXbegr[ p [s&pbe[ªt gz[zUc` p SjU\Ucgp m^`XZ8Rbgz]nS p `XUXbt eZ8S\gzp [ Ucu^ZaZA[^Z SX]^e¢Tg hjZ-bett U+Ucu^SjUcZ `X¥]Z8tUXQR]n[^`XY Zp gr[R§TUc¨ u^pZ8tSXZp [SXm^p ebQRUVSXUcbbS [ b[m m p `XUXUXubt eSXZm p Y(taZgRs^m^Za`cZ8SaSX¡ Za[/ UXZ-Zs$unZabe`côZ SXp u^ Ucg}Zaw` p Ucu eg}qUS UXu^UcgZSjZ8UcY(]nsTm^Q$b`cbUXt u^Z Y(Z SXp ]^`XZ- S p ScSVgTt b UcZ8sb®Ucu&Ucu^Z t gz`X`cZ8SXmgz[n sTbe[ p u^beSVUXgz`Xbt :m^`XgTtaZ8ScSVZ-Stagr[RrZ8` Z Sp pZ c U ( g X U ^ u p p p p p N →∞ s^beSVUX`cb¢n]TUXbegr[gi0UXu^Z ungreZm p UXu¤m^`cgTt Z-SXS¢EZ UjZaZa[¤Z p tvueZazZaSa¡ kA[ZaY(m^be`cbet nY(ZaUXu^gTst eZ-s DH ¶ H2:t [ eSXg¢EZA]nSXZ8s(Ucg$tagrY(m^]TUcZq` `XZqUX` [nSXbZ8[/U ZUc8u^rZ(Z8[zU UvS ` p [nZ Us¢EUXZ u^Z6gzp `Xm^Z`cgrt ¢gzpY(¢^bbe[ bUjQ¢ p grtv ª` p Uc`Xg ZZ8r`XZ8Z-[zt Uv]^Sq`c`Xbe[5Z8[/SVU UXZ SXp Z sTU-¡Q¤p ?¿SjUU p p UXZz¶S [nsTgiZ8UlrZ8gz[^gzem^Q&mUcZ-u^sLZUXmngM`Xgzt ¢gzY(p ¢^m^bep e]Tb®UjUXQ&Z(UXUcgu^p Z`cZ ` p tvUcu Z gr+g/Sjp UAm p tv rZaUcS\Ucu^`cgr] u p [^Z p Ujgr`c pbe[ p SVUXZ p sTQz§RSVU p UXZ `cZ bep Y(Zr¡ ¥ `cgrY p Y p Ucu^ZaY p UXbt p mgzb[/UAp gr RbeZa ¶ Ucu^beS$bSZ-dz]nb p eZa[/U$UXgMUcu^Z` p t UXbegr[grUXbeY(ZUcu p U$UXu^Z&UX` p hjZ-tUXgz`XQ¦SVmEZa[s^Sbe[ p m p `XUXbt ]^ p ` SXZ U ¶ SXQRY(mTUXgrUXbt eeQ S ¶m^`cg}RbesTZ-s©Z SXSX]^Y(Z¤UXu UUXunZLSXQRSVUXZ8Y bSZ8` gRs^betr¡ ?¹[ gUczu^`cs^Z(ZaB` ` `XUcZ6gMp Za¢zZ Za[/p U8¢^¶·eZUXu^UcZ g5p ¢E gigzUXp u`XbUXSXtunbeYY→]^SV m^p +∞ UcebZUcSUXu^UXunZZ(SVQTUX` SVUXZ8hjZ8Y!t UXgzgz`X[beZ8UXSp u^t Z&`cgzgzSc[SVbe [ UXbepY(UXu^Z Zp eZa[nzsZaSX@S ZaZN ]^mR`cZ8 d/]^`cZ8sn[zS U$ N /: bSXZab®UVUvUcS b[ UXg tag/SVZ8`ôp UcgUcu^ZN` p `XZlZazpZa[/ U = ¶ p [nst p [ntaZaEUXungzSXZpta`Xg/SXSXb[ N s^g}[/ p `vs^S*N}¶iZ ^t Zam^Ugz[^Zlgrp ·UXunZaY t p eZ-s Ucgzu^[^ZZY gr p SjUcZa`$ UX ` p hjZ-¡ tUX?¿gzU6`XQz]n¡SXZ8PTsg5¢RUcQu^Z&m^` Y t p UXbb[ UXbegrm^[^]^[^`cZ8m`cg/S8SV¶0Z&¢^gi]TqUUcUcu^u^bebS S$p Y( ZagrUX`cu^b®gTUcu^s¬Y `cZ8bed/S(]^d/be`X]^Z-bSUXZ¤SXgrsTY(b :ZZa`cY Za[/U6Ucu^Za`XYgzY UcbeUXt u^Z p p p p p m^tatam^gr`c]^g [/ TUbY ¢Rp QUcbUXgzun[nZSm^`cu^ZaRbetvbugz]np S`cZNgr[^`cY giUQzebeZ SXUY¤¡Z8PRg¶]n[nsT]^Za`v`XSVUXUXu^gRZ8gT`qs¡ gz£ `X &gr`ct Zagrg}]nzeZas`ô¢EUXu^Z bS[¤Y(Z Z RUcUXu^Z8gT[ns&SXbbgzS\[[^grgr UUcU u^p Z zZa[gz`XbY([/UXZag ` p mp p `XUXbt eZNSctvu^Z8Y6Zgz`qt g}rZ8`Xbe[ p ¡ p k SVu^gz`VUsTZ8Sct `cbm^UXbegr[ grUcu^Z5m mEZa`beS Sgreg}qS8¡wPRZ8t UXbegr[ ¬gi UXunbeS&m mEZa`SVZaUcS&gr]TU&UXu^Z ¥Ucu^ZaZQR[^t Ygz`Xp `c[TZ8SX§RmE¨ grp [ntsTbe`X[ Z8m^`XZ-Z8SV[^Z8Z [/UcU bep t Uc¸3bUjgzQR[MmEZ giôb[/UXp UXu^Z8Z` d/t] Ucbp [ [/p Ucb®mUcbZ-`VS Ucbe:ItaHrZ ¡DH =SXQT¡6SjPRUcZaZ8Y t UXbegrm^[ m^2&`cg m^ T`cbg}Y RbesTUcbZ-[ SqpUcY(u^Z(gTsTsTZaZ8ISc¡t `cPRbZ-m^tUXUcbegrbgz[5[ gr be[/UXUXZ-`cs$gTsT]nbUXt u6Z-S Ucu^p Zm m^p `cUcZauT6R¸¿b gzp ]n]nS Z8s6Z8[^b[/Z UcUcZabet` p t Ucbgr[ `c b®Ucp m u^p Y¤`XUX¡ b t eZ\b[ p SXQTeSjQzUcZa¶/YPRZ-S+tY(UXbegrgT[sTZay^p sTgrZ `+UcS+u^Zqb®u^Ucbu p SjUcgr`c[R be]^t pY(TZ8m^`X`cbgTt t TZ-SXZ S p Y(ScSXgRm^taeZrb®¸ ¶ ¢p p SXZ8s¤gr[¤Ucu^Z l`c[nSVUXZ8b[T ¸
(19) qunZ8[/¢EZ8p tv &m^`cgRtaZ8ScSa¡¥ Sjp UcbY p Ucbgz[Mgi+ Z RbUAp UcbY(Zgr`lp s^b :]nSXbgz[nSAp tagr[/pUX`cgreZ-s ¢RQmEgiUcZa[/UXb p S p `cZlSX] r Z8SVUXZ8s p SVg¶/SXbe[nt ZUXunZl p tv grZ p t U\t p t ]^ p Ucbgz[nS8¶/Z8rZ8[b0SXgrY(ZAu^Z8]^`cbeSVUXbtaS Y p Q¢EZ p mnm^beZ8s·¡ &\' ! . } & n \" ?¹[$m^` p tUcbetaZUcu^Z\m^`XgTtaZ8ScS X ¶}¢EZ gz`XZRbeSXb®Ucb[ R gr` Z8[zUcZa`cb[ b[/UcgqUXunZ\s^Z8SXb`cZ8sSXZ U B ¶rm p ScSVZ-S·Ucu^`cgr] u p sTZ-t `cZ p SVbe[ SVZ-dz]nZa[ntaZgr0tag/SVZ-s&SXZ UvS B=B ⊂ B ⊂ ··· ⊂ B ⊂ B . >\unZqm p ` p Y(ZaUXZa` m p [ns(UXunZlSXZ8d/]^Za[t Zqgr:eZazZa:SVZaUcSsTZamEZa[nsgr[Ucu^ZAm^`cgr¢nZ8Y p Uôu p [s¡ Ztvu^gRgzSXZ Ucu^ZZ8rZaTSXZ UvS·Ucgl¢EZ[nZ8SVUXZ8sUXglZa[nSX]^`cZôUcu p U0UXunZm^`cgTt Z-SXS X t p [^[^grU0Z8[zUcZa` B ¢Zagr`cZRbSVbUXbe[ B ¡ >\unZ6tvu^gzbetaZgrUXu^Z6`cZ8t ]n`X`cZa[/USXZ U sTZamEZa[s^SNgr[5UXu^Z6[ Uc]^`cZgiôUXu^Z(]^[ns^Za`cQRbe[ m^`cgTt Z-SXS ¡ >g UcRg6bSV]UXu^p ZNeb C8UjZ9gUcu^N Z8sTSXZ(] Z8NArZabe[z+UcSXZaZ `cUvm^SN`cZ U Z(UXm^begr`cgr[nRmES\gzmnSX`XZ$Z-SVunZ8Za[z`cUcZ Z8p s UcbZa[`UXUXununZ Z$bUj[/p UX`cg gTsTN sT]n]t UXp be grN$[·t ¡ gz[nSVUX`c]ntUcbgz[n SNt gr`c`cZ8SXmgzX[nsTbe[ p p ?¹[UXu^Z¢ p ebSjUcbet(`cZ bY(Z(Ucu^ZsTZ8ta`XZ p SXb[ SXZ8d/]^Z8[nt Z B = B ⊂ B ⊂ · · · ⊂ B ⊂ B • `XZ8m^`cZ8SXZa[/UcSUXunZ m^uRQRSXbt p eZazZaSUXu^Z mn`XgTt Z-SXS [^ZaZ-s^SUXgm p ScS\¢Zagr`cZbUq`cZ p tvunZ8S B ¡ X m+1. m. 1. 0. n. m+1. Ü Ü ß-T ÷. 3X *X. n−1. m. 1. 0.
(20) . $% $%
(21) <R < +. ¹[6UXu^Zt p SVZ\gr p m p `XUXbt eZ X ZazgreRb[ be[ p mEgRtv zZ U C ginUcu^ZSjU p UcZ\SXm p t Z S tagr[/U p b[nb[ Nu p `vs gr¢nSVU p t eZ8S*NNUcu^Z6SXZ8d/]^Z8[nt Z B = B ⊂ B ⊂ · · · ⊂ B ⊂ B `cZam^`cZ8SXZa[/UcSqUXu^Z6Z TbUeZazZa SVZaUcS\Ucu^Z m^`cgTt Z8ScS[^ZaZ-s^SUXg`cZ p tvuUcg Z Ugz]TUgi C ¢EZ gz`XZN¢EZaZabe[ RbeeZ8s&¢RQ p [gr¢SjU p t eZr¡ > gNt m^UX]^`cZôUcu^Z¢Z8u Rbgz` gr ¢EZ UjZaZ8[Ucu^ZsTb :Za`cZa[/U Z8rZ8eS ZNb[/p Uc`XgTsT]ntaZUXu^ZsTpbSct `cZ UXZNZ8XrZa[/UV§RsT`cbzZa[SjUcgRtvu p SVUXbtlSXZ8d/]^Z8[nBt Z = B ⊂ B ⊂ · · · ⊂ B ⊂ B bUXu E = [ D([t , t ], S) X = (X , T ∧ T ≤ t ≤ T ∧ T) ∈ E gz` p [RQ 1 ≤ n ≤ m + 1 ¶TunZa`cZ T `cZam^`cZ8SXZa[/UvSôUcu^BZ 8n`vSjUUcbY(Z X `XZ p tvu^Z-S B ¶RUcu p UqbS •. ?. m+1. m. 1. 0. m+1. n. t. n−1. m. 0. n. 1. 0. 00. t0 ≤t00. n. n. Z bUXuMZ RUXUcu^ZaZ6[ns t grp [RezZaUc[/u^UXZNbegrUX[` p infhjZ-tUX∅gz=`XbeZ8∞S X¡¢RkQ UA0Ucu^SVbe]SltvmEugrUcbe[zu Up UqUcZ$u^[^ZaQ ZaZ-p s¤`cZ UXgsTZ
(22) Za8n[n[nsTZ8g}s¤ grE[&UXunbZUXu p u^grσe¸Z p `c Z Zap ¢^` ebep [^¡ Zr¥¡ be `cSVZU s^Zaun[^begrtvuMUXZ¢RZ$Q taX˜gr[nUXSXu^bsTZ6Zat `AgzUX`Xu^`cZ8Z SXσmgz¸ p [n sT beZa[ ¢^ ` p(Z R UcZ8Za[^[nZasT` Z-p s¤UcZ8UXs¤` p ¢RhjQZ-tUXUcu^grZ`cQrPR¡ z>\gr`cu^grZ8u^Q gTsp `XY(ZUXZ u^UX`cZ8b[5ti¡BZae>\ZaY(u^ZaZ8[L[zUgiZ taD(R, gr[nSXbsTS)Za`l¶:UXgru^[ Z mn`XgTsT]nt U$SXm p taZ E˜ = D(R, S) × R¯ × R¯ Za[ns^g}Z-sb®Ucu¬UXu^Z&m^`cgRs^]ntU σ¸ p Za¢^` p ¡ ¥ b[ p eQ¦UXg ˜ ¡PRg¦Zu p zZ s, t) ∈ E bep Y[RQ¦¢EZ8Z8snsTZ8Y(Z8s Za[/EU Xbe[ ∈E˜ Ebe[L¶SXsT]nZ tv8nu [^pZ8s grp [QUcp u [p U be[/p UXZae`cUcu^p Z([s,SjU p t][n¶s p `cZ sªp SX]^SX[ngTtt Ucb bp gzUc[ Z p (SX,˜ : sup, u rZ gRgTs Y(Z p SX]^` p ¢^bebUjQ¤m^`XgzmZ8`VUcbZ-Sa¡ ZsTZa[ngiUXZ6¢RQ B (E) Ucu^Z(Y(Z p SX]^` p ¢^eZ¢Egr]n[nsTZ8s5inf,]^[n. t . UX. be= gr[nSp `cgr Y E :Ïgr`Z-dz]nb p eZa[/UcQbUcSqbY pi Zlbe[ E˜ = be[zUcg R ¡ giUXbt ZNUcu p U b® ¶TUcu^Za[ X = {X } p [ns X = X 6∈ B ¶ • T <T b® ≤ T < T ¶TUXunZa[ X = (X , T ≤ t ≤ T ) p [ns X = X 6∈ B ¶ • T 8n[ p eQr¶Eb® ≤ T ¶^UXu^Z8[ X = (X , T ≤ t ≤ T ) `XZ8m^`XZ-SVZ8[/UcSUXunZm p UXuªgi X ¢EZ UjZaZ8[ • UXu^ZSX]nt8t Z8ScTSXbzZl eZazZaS B p [ns B ¶ p [ns X = X ∈ B ¡ b p [ns&gr[^eQb T ≤ T -¡ 6Qt gz[nSVUX`c]ntUcbgz[&Z p SVg6[^giUcbetaZNUXu p U ; gz[nSXZ8d/]^Za[/UcQz¶ X ∈B Tn = inf{t ≥ 0 : Xt ∈ Bn }. +. +. b. n−1. n. n−1. n. n. T. n. t. n−1. n. t. n−1. n−1. Tn ∧T. p [nsgr`Z p tvu. Tn ∧T. n. T. Tn ∧T. T. n. n. Tn ∧T. n. n. Tn. n. n. T0 = 0 ≤ T1 ≤ · · · ≤ Tm ≤ Tm+1 = TB. p [ns X = {X } 6⊂ B ¶Rgr` p p ≥ n) . ¥ `XgzYUXunZ8SXZgz¢nSXZa`c p UXbegr[Z t p [ p UXZa`c[ p UXberZaeQsTZ8n[^ZNUXunZNUXbeY6Z-S T ¢RQUcu^Z b[sT]ntUcbzZNgr`cY]^ p n. (Tn−1 > T ) ⇒ (Tp > T. p. T. p. n. bUXu¤Ucu^Z$tagr[RrZ8[/UXbegr[ SVg(Ucu U bZabUXunZa` gz`qb®ôSVU `VUcb[ b[ UcbY(Z T Ucu^Z m^`cgTt Z8ScinfS[^∅Zaz=Za`∞`cZ p tvu^Z8S p B T¢EZ >gz`XTZUXbeY(Z T ¡ T Z p e>SXg6Tgr¢nSXZa`crZlUXpu p U Tn = inf{t ≥ Tn−1 : Xt ∈ Bn } n. n−1. n−1. Bn−1. pU. n. (TB ≤ T ) ⇔ (Tm+1 ≤ T ) ⇔ (T1 ≤ T, · · · , Tm+1 ≤ T ). Û0Ü (. ( .
(23) |.
(24) "!#$%'&)(*+-,#/.* 01 . 6Q©UXunZSVUX`cgr[ `X zg} m^`cgrmEZa`XUjQ Ztvu^Z8tv ©Ucu UUXu^ZSVUXgTtvu SVUXbt5SXZ8d/]^Z8[nt Z gz`XYS p [ E ¸I p e]^£MZ-sp £Mp `c rg}¦tvu p be[·¡ l[^Z p QLUXp gtvu^Z8tv ¦u^Z pUcu^Za`6Ucu^Z&m p UXu u (Xp S6SV,]·ta·t ·Z8,Z8XsTZ-sUXg ) `cZ p tvuUXu^Z(sTZ8SXbe`XZ-s n¸3UcuMZ8rZ8bSqUcg&t gz[nSXb®beUcsTu Z8`qUcu^ZmEgi¢RUcQ Za[/UXb p ]^[nt UXbegr[nS g gr[ E s^Z
(25) 8n[^Z-s¤gr`Z p tvu 0. m+1. n. x = (xt , t0 ≤ t ≤ t00 ) ∈ D([t0 , t00 ], S). ¹[Ucu^beS\[ngiU p UXbegr[·¶TZ u p zZAgz`Z p tvu ?. bI¡ Zr¡. t0 ≤ t00. gn (x) = 1(x. t00. ∈ Bn ). n. (Tn ≤ T ) ⇔ (T1 ≤ T, · · · , Tn ≤ T ) ⇔ (g1 (X1 ) = 1, · · · , gn (Xn ) = 1) ,. 1(T ≤ T ) = n. p [ns. n Y. gk (Xk ) ,. k=0. f (Xn ) 1(T ≤ T ) = f (Xt , Tn−1 ≤ t ≤ Tn ) 1(T ≤ T ) . n n. ¥ gr` p UcZa`m^]^`cmEgzSXZr¶TZbe[/UX`cgTsT]nt ZUcu^ZNgreg}be[ [^giU p Ucbgz[. (X0 , · · · , Xn ) = (X0 , (Xt , 0 ≤ t ≤ T1 ∧ T ), · · · , (Xt , Tn−1 ∧ T ≤ t ≤ Tn ∧ T )). ¹[/Uc`XgTsT]ntab[ Ucu^Z ¥ Z8QR[^Y p [R§T¨ p tsTbeSVUX`cbe¢^]TUXbegr[ η sTZ
(26) 8[^Z8s¢RQ Y γ (f ) b®Ucu η (f ) = γ (f ) = E(f (X ) g (X )) , γ (1) gzZa` QRp[^[/Y Q$¢[Tgz§R]^¨ [nsTtlZ-s`cZaY(m^`cZ Z8p SXSVZa]^[/`U p ¢nUXbegrZ\[]^gr[ tUcUXu^begrZ[ d/f] s^[/
(27) Z Uc8nb®Uc[^bZ-Z-Ss: grHr[ ¡DH E¡ ¶/Z p `cZq[^g} p ¢nZUcgSVU p UcZUXunZqgreeg}b[ ¥ p p p p = ´ #3 $ +. n
(28) $# . f ∈ B (E) °R´ ° " # # ° ° #
(29) °/± ·¯ ^ ( = [Xt , 0 ≤ t ≤ Tn ∧ T ] .. ?. n. n. n. n. n. n. n. k. k. k=0. . b. E(f (Xn ) E(f (Xt , Tn−1 ≤ t ≤ Tn ) | Tn ≤ T ) =. n Y. gp (Xp )). p=0. E(. n Y. gp (Xp )). p=0. P(Tn ≤ T ) = E(. Ü Ü ß-T ÷. 3X *X. n Y. k=0. gk (Xk )) = γn (1) .. =. γn (f ) = ηn (f ) , γn (1).
(30) {. $% $%
(31) <R < +.
(32) $O+.. f ∈ Bb (E n+1 ). ( E(f (X0 , · · · , Xn ). E(f ([Xt , 0 ≤ t ≤ Tn ]) | Tn ≤ T ) =. E(. \unZNSVUX` p b u/UXgr`c p `cs(gz`XY$]^ p. n Y. n Y. gp (Xp )). p=0. gp (Xp )). p=0. >. P[Tn ≤ T ] =. n Y. P[Tk ≤ T | Tk−1 ≤ T ] ,. mn`XungTbesTtvu©]nt SVUAu^gig}+q`cSZ u^SVg}gz[ Uc¢^u^eZ&QSXzY Za`cQ¬eSV¢^Y ]Tp Ue[^m^giUl`XgzUX¢ gRp g¢nSXbeY b®UjQe grt gzp [nsT` bp UX`cbeZgr[ Z8r·Zam^[/`cU(grt¢ p [ ¢^be¢bUXZªbeZ8sTS8Z-¶nt Z grY(tvuªmEgzgiSX+Z8s¬u^beb[/tvUXuMgLt UXgzu^`VZ ¸ `cZ8SXmEgr[nsTbe[ UXgp UXu^Z p Uc` p [nSXb®Ucbp gz[¢EZ UjZaZ8[¤Ujg(Z8rp Za[/UvSa¶Et p [¤¢EZp `cZ8t g}zZa`vp s`cgrY UXu^Zp UXu^ZZae®§/ R[^g}[ bsTZ8[zUcb®UjQ k=0. γn (1) =. n Y. ηk−1 (gk ) ,. [nsbee/mn`Xg}RbsTZUcu^Z¢ SXbeSgr`0UXunZZ FtabZ8[/U0[R]^Y(Za`cbt mnm^`Xg TbeY UXbegr[ be[$UXZa`cYSgr [$b[/UcZa` tUcb[ m¢Rp Qp `XUcUXu^bt Z eZNSXQTgzSj`XUcbZaUXY¤unY ¡ >\u^S\Z-SVp ZZaeta3¡gr[ns^b®Ucbgz[ p m^`cgr¢ p ¢^bebUXbeZ8S p p `cZp [ngiUq R[^g}p [b[ p s^ p [nt Zz¶ p [p s p `XZNeZ p p `X[^Z-s p p ! & &a ?¹[(m^`cZaRbegr]nS+SVUX]sTbZ-S Ê|T¶r { ZsTZ-SVb [ p t gzeZ8t UXbegr[6gi:¢^` p [ntvu^be[ p [ns$be[zUcZa` p t UXbe[ m p `XUXbt eZSVQTSVUXZaYS m^m^`cg TbY Ucb[ Y(gTsTZaSlgr`SXgre/be[ Za[^Z8` 0ta SXSgr Z8QR[^Y [R§T¨ tY(gTsTZ8eS8¡ >\unZ8SXZ6m `XUXbt eZ Ucp Z8tvu^[nbed/]^Z-pSt p [ ¢ZN]nSXZ8sUXg(SXgrerZlUXu^Zp&gr `cY]n p p Zlm^p `cZ8SXZa[/UcZ8¥ sbe[ >\p u^Zagz`XZ8p Y HrD¡ Hr¡ BZ 8n`vSVUgTt ]p S\gr[ p SXbeY6mnZNY]^U p Ucbgz[}SXZaeZ8t UXbegr[ Z8[^Z Ucbet p gz`XbUXunY¡ k=0. .
(33) . > gsTZ8Sct `cbe¢ZNUcu^beSqm p `XUXbt eZ p m^m^`cg TbY p UXbe[ Y(gTsTZ8·Z'8n`vSVUq`XZ-t p e:UXu p UUXunZ ¥ ZaQR[^Y p [R§R¨ p tlsTbSVUX`cb®¸ ¢n]TUXbegr[ ng} η ∈ P(E) sTZ
(34) 8n[nZ8s&¢/Q Y γ (f ) b®Ucu η (f ) = γ (f ) = E(f (X ) g (X )) γ (1) bSSXgre]TUXbegr[gi0UXu^ZNgzeg}be[ Y(Z p SX]^`XZN p e]^Z-s&sTQR[ p Y(bet p ·SXQRSVUXZ8Y : 2^D¡ H = η =Φ (η ) n. n−1. n. n. n. n. n. p. p. p=0. n+1. n+1. n. Û0Ü (. ( .
(35) .
(36) "!#$%'&)(*+-,#/.* 01 . \unZY p m^m^be[ S Φ `XgzYUXu^ZSXZ Uqgr Y(Z p SV]n`XZ-S >. n+1. be[/UXg. P(E). p `XZ sTZ8n[^Z-s&¢RQ. Pn (E) = {η ∈ P(E) , η(gn ) > 0}. Φn+1 (η)(dx0 ) = (Ψn (η) Kn+1 )(dx0 ) =. Z. Ψn (η)(dx) Kn+1 (x, dx0 ). \unZ `X zg} rZ8`X[^Z8eS `cZam^`cZ8SXZa[/UUcu^Z `c rg}¬Uc` p [nSVbUXbegr[Sgi Ucu^Ztvu p be[ X ¡ >\u^Z ]nmEs p UX£Mbe[ p Y ¢Rp Qm^m^Ucu^be[ ZN S gr`cΨYK]n p (x,`cZdxs^Z
(37) 8n) [^Z-s `XgzY P (E)£5pbe[zUcg P (E) p [ns¬gz` p [RQ η ∈ P (E) p [ns p f ∈ B (E) >. E. 0. n. n. n. n. n. n. b. \uR]nS\Z SVZ8ZUXu p UUXu^Z `cZ8ta]^`cSXbegr[ :2^¡DH = be[/zgrerZ-SôUjgSVZ8m p ` p UcZSXZaeZ8t UXbegr[ Y]TU p UXbegr[Uc` p [nSVbUXbegr[S :2^¡Ê = η ∈ P(E) −−−−−−→ ηb = Ψ (η ) ∈ P(E) −−−−−−→ η = ηb K ∈ P(E) ?¿U+bS p SXgtagr[RrZ8[zUcbZ8[/U Ucg`cZ8t p e/UXu p U+UXu^)Z 8n[^bUXZ p [ns$mg/SVbUXberZ\Y(Z p SV]^`cZ8S γ gz[ E t p [$¢EZ\Z Tm^`cZ8ScSVZ-s be[UXZ8`XYS\gi UXu^Z g} {η , · · · , η } ¶T]SVbe[ UXu^Z Z p SXbeQtvu^Z-tv rZ-sgr`cY]n p Ψn (η)(f ) = η(f gn )/η(gn ). >.
(38) . . n. n. n. n. n+1. n. n+1. n. 0. n. n−1 Y. γn (f ) = ηn (f ). ¹[Ucu^Z8SXZN[^giU p UXbegr[S\ZN`cZ p s^beQgr¢SVZ8`XzZqUXu p U ?. p [ns. ηp (gp ). p=0. γn (gn ) = P(Tn ≤ T ). rg>\unUXZ u^ZZa[^grZa`cUXY bt&:Uj2^Q/¡DmEH Z= beNS§RUXu^m Zp `X£MUXbt p e`cZ rg}SXQTLSjUctvZau Y p be[ p SXξSXgTt =b p Uc(ξZ8s,· ·bUX·u , ξp [ ) p U ¢p Sj RUcbe` [ p tU pY(e]^Z pZ-SSX]^p `cU$Z&Z p p tvu]^Z-UXs beY(mnZ `XgTnt Z-beSX[ S UcUcu^` Zª[nm^SXb®`cUcgTbgzsT[n]nS tU`XSVZU p sTUXZZM8n[^SXm Z-sp taZ8SSqEgreg}q∪S8{∆} ¡ ¥ gz` p u^[/Z8Q¤`XZ t gz∆[ 8 Sj U ]^p ` [np s^UXbeSgr[ grx` p= t (xZ8Y(,Z ·UX·Z8·`XQ , xgr`)ta∈gFE[ mESXgr]nbe[/tvuªU8¡ Ucu ?¿p UcUS p p p ZNSXZ U 1 X δx ∈ P (E) ηbn (f ) = Ψn (ηn )(f ) = E(f (Xt , Tn−1 ≤ t ≤ Tn ) | Tn ≤ T ) N. n. 1 n. N n. 1. N. N. N. N. i. n. i=1. P(ξn+1 ∈ dy | ξn = x) =. Ü Ü ß-T ÷. 3X *X. N Y. p=1. Φn+1 (. N 1 X δ i )(dy p ) N i=1 x. ^¡. : 2 2 =.
(39) - H. $% $%
(40) <R < +. unZa`cZ bS [¬b[ 8[^b®UcZ8SXbY ô[^Z8b u/¢Egr`cu^gRgTs5gr nu Za[UXdyu^Z=SXQTSjdyUcZaY ×p ·`X·`cbe·r×Z8Sdybe[¤SXgrY(Z p t gz[ 8 ]^` p UXbegr[ p ξ = x SV]tvu&Ucu p U y = (y , · · · , y 1. N. 1. n. N. ) ∈ EN. N 1 X δ i 6∈ Pn (E) N i=1 x. ¡. Ucu^Zªm p `VUcbetaZ p gz`XbUXunY
(41) beSSVUXgzm^mEZ8s p [ns©ZªSVZaU ξ = ∆ ¡ >\unZªb[^bUXb p lSVQTSVUXZaY giNm p `XUXbt eZ8S tagr[nSXbSjUvSbe[ be[nsTZ8mZ8[nsTZa[/U¤` p [nsTgzY p `cb p ¢^eZ8Sb®Ucu t gzY(Y6gz[# p η = ξ = (ξ , · · · , ξ ) ¡)>\unZSV]^mEZaN 5 p (X ) = 5 p (X ) Ucu^Z m p ` p Y6ZaUXZ8` N bS\UXu^ZSXbDCaZNgi UX`vu^ScZt `cSXbQTmTSjU UcZaiY=S 1,p [n· s&· · Uc,u^NZ m^`c`cZaZ8mnt `XbZ-SXSVbgzZ8[[zUvgiS0UXUXu^unZZ p p ¢ Z8gr·`cgibUX u^UXY¤unZ ¡ m p `XUXbt eZ p [ns Z RUZMs^Z8Sct `cb¢EZ¤b[ Y(gr`cZªsTZ U beeS(Ucu^Z Z8[^Z UcbetZ8rgre]TUcbgz[ grlUcu^Zªm UcuT¸¿m `XUXbt eZ8S8¡#kUUXu^Z UcbY(¸I Z e]^nZ-s&= ` 0 [UXsTu^grZY be[^ bUXb `cp b ¢^tagrZ-[ S 8 ]^` p b®UXUcbeuMgrp [¦t grt Y(gz[nY(SVbgrSV[ªUcS b[ N ¡Ab[PTsTbZa[nmEtaZaZ[ns^ZaZ [/u U p r[nZ sLbp sTZa[/UXbtp p eQ5gr` sTbSVUX¸`cb¢^eY(]^UXgzZ8SVs U p p p p ξ p η p g (u) = 1 η p S Z8rZ8`XQ u ∈ S ¶zZAY p Q(sTbSXt p `vs6UXu^ZNSVZ8Z-tUcbgz[ p UôUcbY(Z n = 0 p [nsSVZaU ξb = ξ gr`Z p tvu 1 ≤ i ≤ N ¡ ?¿0Z]nSXZUXu^Z t gz[RrZa[/Ucbgz[ T = Tb = 0 p [nsb® ZNSXZ U T = Tb = 0 Z[^grUXbt ZlUXu p UUcu^ZSXb[ Z SVU p UXZ-S ξ p [ns ξb t p [¢Z `cbUVUXZ8[¤b[&UXu^Z m p UXu^¸3gz`XY p [ns ξb = ξb (0) = (ξb (t) , Tb ≤ t ≤ Tb ) ξ = ξ (0) = (ξ (t) , T ≤ t ≤ T ) >\u^Z UlUcbY(Z bSlsTZ 8n[^Z-s SAgreeg}qSaB¡ ?¿ m ZUXu^SVZa¸ImU ξ`V Ucbeta-Z =T ∆´ ¡ Aa}Uc u^ Za`c Ï be SX´ Z sT]^ξb`cb[ → Y$ξ ]TU p UXp begr[·¶ b[sT(nZamEZa+[ns^1)Za[/UXeQ5grZ p tvu¦p giUXunZa`-¶·Z p tvu¦SXξbZaeZ8=t UXZ8∆s p p 1 0. 0. n+1. N 0. 0. 0. 0. i 0. 0. 0. i 0. i −1. i 0. i 0. i −1. i 0. 0. i 0. i 0. i 0. i 0. i 0. i −1. i 0. i 0. n. i 0. i 0. i −1. i 0. n+1. n. n+1. ξbni = (ξbni (t) , Tbn−,i ≤ t ≤ Tbn+,i ). cUZ8brY(gzZ zZ8(nSN+` p [n1)sTgzSXY(g$UcQ u p p U tatagr`vsTb[ UcgªUXunZ £Mp `c rg}¤Uc` p [nSXb®Ucbgz[. Kn+1. gi\UXunZ £Mp `c rg}Mtvu p be[. Xn+1. pU. −,i +,i i i ξn+1 = (ξn+1 (t) , Tn+1 ≤ t ≤ Tn+1 ). bS p ` p [nsTgzY« p `cb p ¢^ZbUXuMsTbSjUc`Xbe¢^]TUcbgz[ K (ξb , dx ) ¡ i Z$?¹[&VS U gip UX`VunU Za`ôp UX` grp `vhjs^Z-S8t¶iUcUcgru^`cQZ p `c grgrY `cb®Ucξbu^Y p UA gRUXbeZ8Y(SZ ebe rTZqUcu^bSô=¢ETZ bUjZaZ8¶ [p [nSjUcsªZamneZ S Unb®UNp Za[zs grenr+Z` 1p ¡ [¥ sTgzgr`ôY(Z epQ tvup S m p p `Vt UcgrbetamRQ Z gi UXu^Zm^`XgTtaZ8ScS {X , s ≥ T } ¶T]^[/UXbe·UXunZSjUcgrm^m^be[ UXbeY(Z T ¶^u^btvu {ξ bS\ZabUXun(s)Za` , s ≥ T } n+1. −,i n+1. i n. i n+1. −,i n+1. s. i n. 0. +,i n −,i n+1. i +,n+1. +,i −,i i Tn+1 = inf {t ≥ Tn+1 : ξn+1 (t) ∈ Bn+1 ∪ R},. be[ªt p SVZgi p `XZ-t ]^`c`XZ8[/U\SXZ UUcg6¢EZ p rgzbes^Z8s¶Rgr`. +,i −,i i Tn+1 = T ∧ inf {t ≥ Tn+1 : ξn+1 (t) ∈ Bn+1 },. be[ªt p SVZgi p sTZ UcZa`cY6be[^bSjUcbet'8[ p :UcbY(Zr¶sTZamEZa[ns^b[ gz[Ucu^Z m^`cgr¢^eZaY p Uqu p [ns¡ Û0Ü (. ( .
(42)
(43) "!#$%'&)(*+-,#/.* 01 . HH. >\u^Z beS sTZ
(44) 8[^Z8s p SEgreg}qS8¡ ¥ `cgrYBUXu^Zôm^`XZ8Rbgz]nS:Y]TU p Ucbgz[ Uc` p [nSXb®Ucbgz [&° # ° ZNr gr ¢T´ U p bea[ } N m Ïp Uc´ uT ¸¿m ξp `XUXbt e→Z ξb n+1. n+1. −,i +,i i i ξn+1 = (ξn+1 (t) , Tn+1 ≤ t ≤ Tn+1 ). [nsUXu^ZNgrUXu^Z8`gz[^Z8S l[nQSXgrY(ZAgrUcu^Z8SXZlm `XUXbt eZlu zZlSX]ntataZaZ-sTZ8sUXg6`XZ tvuUcg6sTZ-SVbe`cZ8sSXZ U u p rZ6 p beZ-s¡ ZsTZa[ngip UXZ¢RQ I p UXu^Z p ¢Z8eSgi\UXu^p Zm p `XUXbt eZ8S u p /be[ BSX]ntataZap Z-sTZ8sLUXg5`XZ p tvu¦UXu^Z ¸3UcueZazZa (n + 1) n+1. N n+1. ¿ I ?. +,i N i In+1 = {i = 1, · · · , N : ξn+1 (Tn+1 ) ∈ Bn+1 }. N n+1. =∅. UXu^Z8[¤[^gr[^Z grUcu^Z m p `XUXbt eZ8S\u p rZNSV]nt8t Z8Z8sTZ-sUcg`XZ p tvuUcu^ZsTZ-SVbe`XZ-seZazZaI¡PRb[t Z N In+1 = ∅ ⇐⇒. N N 1 X 1 X i gn+1 (ξn+1 ) = 0 ⇐⇒ δ i 6∈ Pn+1 (E) N i=1 N i=1 ξn+1. ZSVZ8Z\UXu Ub[6Ucu^beSôSVbUX] UXbegr[6UXunZ gr`cbUXu^Y bSSVUXgzm^mZ-s [ns ¡ AUcu^Za`cbSVZUXunZqSVZ8Z-tUXbegr[ cU ` p [nSXb®Ucbgz[p grTUXu^Z N ¸Im p `Xp UXbt eZY(gRs^Zap S : 2n¡ 2 =p [sS: 2^¡Êy =·p `cZp sTZ
(45) 8n[nξbZ8s p S=gr∆eeg}qSa¡ ?¹[NUcu^ZQ8`cSVU SXb®Uc] p Ucbgz[ UcY$u^]TZNU SVQTUXbeSVgrUXZa[ Y ` ξb[ns^grY«= ( ξb`cb ¢^, e·Z8·S · , ξb ) tagr[nSXbeSVUcSbe[ N be[nsTZ8mZ8[nsTZa[/U : bzZa[Ucu^Zlm p SVU]^[/UXbeEUXu^Zl p SjU p = p p p n+1. n+1. 1 n+1. bUXuªtagrY(Y(gr[¤sTbeSVUX`cbe¢^]TUXbegr[. Ψn+1 (. N n+1. −,i +,i i i ξbn+1 = (ξbn+1 (t) , Tbn+1 ≤ t ≤ Tbn+1 ). N N i X ) gn+1 (ξn+1 1 X 1 δξ i ) = δξ i = N N N i=1 n+1 |I n+1 X n+1 | i=1 j gn+1 (ξn+1 ). X. N i∈In+1. δ. −,i +,i i (ξn+1 (t) , Tn+1 ≤ t ≤ Tn+1 ). ¹[ SXbY(m^eZgr`vs^S8¶aZsT` p 5UXu^Z8Y ]n[^b®gz`XY(eQ p Y6gz[ UXu^ZSV]ntaZ8ScSj]nimnbZ-t Z8S·gi/UX` p hjZ8t UXgz`XbeZ8S {ξ ¡ I } j=1. ?. N n+1. . . . i n+1 , i. ∈. klS Y(Za[/UXbegr[^Z-s p ¢g}zZôUcu^Zqtvu^gzbetaZgrUcu^Z N ¸Im p `XUXbt eZ p m^m^`cg RbeY p Ucb[ Y(gTsTZaTgr : 2n¡H = beS [^grU]^[^bd/]^Zr¡ 6Z8g} ¶RZmn`Xgzmg/SVZ p [ p UXZ8`X[ p UXberZNSXtvunZaY(ZNu^betvu¤t gz[/U p be[nSbe[¤SXgrY(Z SXZa[nSXZNZ-SXS` p [nsTgzY([^Z8ScS I¡ >\unZA rZ8Q(besTZ bSUcg[^grUXbt ZAUXu p UUcu^ZA]^m:s p UXbe[ Y p m^m^be[ Ψ : P (E) → P (E) t p [¢EZA`cZa`cb®UXUXZa[ be[UXu^ZNgzeg}p be[ gz`XY Z : 2^¡ = Ψ (η)(dx ) = (η S (η))(dx ) = η(dx) S (η)(x, dx ) , bUXuUcu^Zt gzeZ8t UXbegr[gi £5p `c rg}6UX` p [nSXbUXbegr[ rZa`c[^Z8eS S (η)(x, dx ) gr[ E sT
(46) Z 8n[nZ8s¢RQ n. n. 0. n. 0. n. n. n. 0. E. n. 0. Sn (η)(x, dx0 ) = (1 − gn (x)) Ψn (η)(dx0 ) + gn (x) δx (dx0 ) ,. Ü Ü ß-T ÷. 3X *X.
(47) D H. $% $%
(48) <R < +. unZa`cZ p [ns&u^Za`cZ. gn (x) = 1(g (x) = 1) = 1(x ∈ g −1 (1)) , n n. gn−1 (1). gn−1 (1). ¹[sTZaZ-s ?. SjU p [ns^Sgz`\UXu^ZSXZ Ugr0m p UXunS\be[ E Za[/UXZ8`Xbe[ UXu^Z eZazZa B ¶RUXu p UqbS n. 0. 00. 00. = {x ∈ E : gn (x) = 1} = {x ∈ D([t , t ], S) , t ≤ t : xt00 ∈ Bn } .. 0. 0. (η Sn (η))(dx ) = Ψn (η)(dx ) (1 − η(gn )) +. unZa[ntaZ. 0. Z. E. η(dx) gn (x) δx (dx0 ) ,. gz` p [RQ¢Egr]n[nsTZ8s¤Y(Z p SX]^` p ¢^eZl]n[ntUcbgz[ f sTZ8n[^Z-s&gz[ E ¶nu^btvu¤m^`Xg}zZ8S :2^¡ = ¡ ?¹[¤UXu^bS[^grU p Ucbgz[·¶ :2^¡DH = t p [&¢EZ `cZa`cb®UXUXZa[ p S (η Sn (η))(f ) = Ψn (η)(f ) (1 − η(gn )) + η(f gn ) = Ψn (η)(f ) ,. bUXuUcu^Zt gzY6mEgzSXbUXZ £Mp `c rg}6Uc` p [nSVbUXbegr[ rZ8`X[nZa K. sTZ
(49) 8n[nZ8s&¢/Q. ηn+1 = ηn Kn+1 (ηn ) , n+1 (η). Kn+1 (η)(x, dx0 ) = (Sn (η) Kn+1 )(x, dx0 ) =. Z. Sn (η)(x, dx00 ) Kn+1 (x00 , dx0 ). \unZ UXZ8`X[ UXberZ ¸¿m `XUXbt eZY(gTsTZa p ScSXgRtab p UXZ-s¤b®UcuMUXunbeS[^Z8wsTZ8Sct `cbemTUXbegr[MbSsTZ
(50) 8n[nZ8s p SA¢EZ gz`XZ ¢RQ`cZap mn p t be[ p :2^¡ 2 N= ¢RQ p Y : 2^¡Êy = 1 X P(ξ ∈ dy | ξ = x) = K ( δx )(x , dy ) N 6QªsTZ 8n[^bUXbegr[5gr Φ p [s K (η) Zu p zZgr` p [RQªt gz[ 8 ]^` p UXbegr[ x = (x , · · · , x ) ∈ E bUXu 1 X δ ∈ P (E) >. N. n+1. n. N. i. n+1. p=1. n+1. E. p. p. i=1. 1. n+1. N. N. N. N. xi. n. i=1. Φn+1 (. N N X gn (xi ) 1 X δxi )(dv) = Kn+1 (xi , dv) N N i=1 X i=1 gn (xj ). ¹[¤Y$]ntvuUcu^ZS p Y(ZN p QZ'8n[ns&UXu p U ?. Kn+1 (. j=1. N N 1 X 1 X δ xi ) = S n ( δ i ) Kn+1 N i=1 N i=1 x. Û0Ü (. ( .
(51)
(52) "!#$%'&)(*+-,#/.* 01 . H2. bUXuUcu^ZSVZ8Z-tUcbgz[Uc` p [nSVbUXbegr[ Sn (. unZa`cZ. N N 1 X 1 X δxi )(xp , dv) = (1 − gn (xp )) Ψn ( δ i )(dv) + gn (xp ) δxp (dv) N i=1 N i=1 x. Ψn (. N N X gn (xi ) 1 X δ xi δ xi ) = N N i=1 X i=1 j gn (x ). \uR]nS8¶TZSXZaZUXu U\UXu^ZNUc` [nSXb®Ucbgz[ SXZam p ` p UXZ Z8[^Z Ucbetlp UjQRmEZ Y6Z-p tvu p [nbeSXYξS >. giUcu^ZNgr`cY(Za` £Mp `X zg}6Y(gTsTZ8eS\SXm^bUcSq]^m&be[/UXg6Ujg j=1. n. → ξn+1. .
(53) . ξn ∈ E N ∪ {∆} −−−−−−→ ξbn = (ξbni )1≤i≤N ∈ E N ∪ {∆} −−−−−−→ ξn+1 ∈ E N ∪ {∆}. Qtagr[nSVUX`c]nt UXbegr[ZN[^grUXbt ZUXu p U 6. QsTZ
(54) 8[^b®Ucbgz[giUcu^Zm UXu ]nZ8s `X zg}5tvu p b[ p ]^Z-sm p `XUXbt eZ8S p p £Mp. p [ns ξb = ∆ Ucu^beS Z8[^Z UcbetY(gRs^Zatagr[nSXbSjUvSbe[ N ¸¿m p Ucu. ξn = ∆ =⇒ ∀p ≥ n ξp = ∆. 6. Xn. p. ξni. = (ξni (t) , Tn−,i ≤ t ≤ Tn+,i ) ∈ D([Tn−,i , Tn+,i ], S). ξbni. = (ξbni (t) , Tbn−,i ≤ t ≤ Tbn+,i ) ∈ D([Tbn−,i , Tbn+,i ], S) .. \unZ` p [sTgrY UcbY(Z ¸¿m p b`vS (T , T ) p [s (Tb , Tb ) `cZamn`XZ-SVZ8[zUUXunZ#8`cSVU p [ns& p SjU\UcbY(ZNgi0UXu^Z tagr`c`XZ-SVmEgr[sTb[ m p UXunS8¡ ?¹[UXunZ p UXZa`c[ p UXberZY6gTsTZ8 : 2n¡ y = Z p tvum p `VUcbetaZ >. −,i n. +,i n. −,i n. +,i n. −,i +,i i i ξbn+1 = (ξbn+1 (t) , Tbn+1 ≤ t ≤ Tbn+1 ). bSS p Y(m^eZ8s p tatagr`vsTbe[ Ucg(UXu^ZSXZaeZ8t UXbegr[¤sTbSjUc`Xbe¢^]TUcbgz[ Sn+1 (. N 1 X δ j )(ξ i , dv) N j=1 ξn+1 n+1. = (1 −. =1. i gn+1 (ξn+1 )). +,i i (ξn+1 (Tn+1 ). Ü Ü ß-T ÷. 3X *X. N 1 X i Ψn ( δ j )(dv) + gn+1 (ξn+1 ) δξ i (dv) N j=1 ξn+1 n+1. 6∈ Bn+1 ). Ψn (. N 1 X δ j )(dv) + 1 i δ i (dv) +,i (ξn+1 (Tn+1 ) ∈ Bn+1 ) ξn+1 N j=1 ξn+1.
(55) H. $% $%
(56) <R < +. £ gr`cZlm^`cZ8tabeSXZaeQZNu p rZ +,i i i i ξn+1 (Tn+1 ) ∈ Bn+1 =⇒ ξbn+1 = ξn+1 .. ?¹[5UXu^Z6grmnmg/SVbUXZ6Z6u p zZ u^Za[LUXu^Z6m p `VUcbetaZ$u p S[^grUNSX]ntataZaZ-sTZ8s¤Ucg&`cZ p tvu Ucu^Z (n + 1)§/UXu¤Z8rZ83¡ ?¹[UXu^ξ bSqt (Tp SXZ ξb ) 6∈ bBStvu^gzSXZa[` p [nsTgzY(Q p [ns]n[^b®gz`XY(eQbe[UXu^ZSXZ U +,i n+1 i n+1. i n+1. n+1. j j +,j j N {ξn+1 : ξn+1 (Tn+1 ) ∈ Bn+1 } = {ξn+1 : j ∈ In+1 },. gr p e:m p `XUXbt eZAu p Rbe[ SX]ntataZaZ-sTZ8s(Ucg$Z8[/UXZa`be[zUcg B ¡ ?¹[giUcu^Za`gr`vs^SZ p tvum p `VUcbetaZlu^btvusTgRZ8S [neZagizUZa·ZaSX[/m^UcZaeb®`UvS\be[/b[/UXg¤UXg(UcUju^Z g((ng+SVmn1)`Xbe§z[ Ucu¬S8¡ Z8rZ8beS /beeZ-s p [nsbe[nSVU p [/UcQ p sTb :Za`cZa[/Um p `VUcbetaZb[Ucu^Z B ZsTZa[ngiUXZ ¢RQ τ UXu^Z eb®Za UXbeY(Zgi UXu^Z N ¸ Z8[^Z UcbetY(gTsTZa n+1. n+1. N. τ N = inf{n ≥ 0 :. cU¥ u^grZ `NZ p ¸¿tvm up UX`XbeUXY(bt ZeZnY6<gTsTτZ8 ZNsTZa[^grUXZ¢RQ N. giUcZlUcu p U . p [ns. ηbnN. UXu^ZNm p `VUcbetaZsTZa[SVbUjQm^`cg8neZ8S p ScSVgTtab p UXZ8s(bUXu. p [s ηb = Ψ (η ) . UXunZ N §/m p `VUcbetaZ p m^mn`Xg TbeY p UXbe[ Y(Z p SX]^`XZ-S γ p ScSXgRtab p UXZ-sªb®Ucu ¢RQ. ¥s^Z
(57) gr8n`N[^Z Z-p s&tvuMgrUc` bY([RZ Q n < τ p f ∈ B (E) b. p [ns. N 1 X δ i N i=1 ξn. ηnN =. N. ηnN. N 1 X δ i 6∈ Pn (E)} . N i=1 ξn. N n. n. N n. N n. γnN (f ) = ηnN (f ). n−1 Y. p `cZ. γn. ηpN (gp ) .. p=0. γnN (gn ) =. n Y. ηpN (gp ) =. p=0. ηbnN = Ψn (ηnN ) =. n Y |IpN | , N p=1. 1 X δ(ξ i (t) , T −,i ≤ t ≤ T +,i ) . |InN | n n n N. u\p unSNZ ¢Ep ZaSXZaQR[Y6SVm^UXUX]gisTUcbebZ-tNs¢EZab[u p Y /bep gr[R` QLp SNgz`X T→S8¡ ∞ Zgi `cZ UXunZ8Z`NbUX[/u^UcZZa` `Xp Z tp UcsTb[ Z8 `lmUcgªp `XUXUXunbt ZeZ SV]^Y(`cgRrZ8s^QMZam p Z mZ8u ` p zÊ| Zt begr[L[UcSju^Uc`XZ]nt t UXp Z-SXs Z gr SVUX`cbtUXeQmEgzSXb®UcbzZ mgrUXZ8[zUcb p eS g p [s ¶Ey·gz`[^gr[¤[^Z zp UcbzZNmgrUXZa[/Ucb p eS8¡ ¥ gz`\UXu^Ztagr[RrZ8[^beZa[ntaZ Z-grSjUcUcbYu^Z p UX`XZ8Z S8p ¶ sTpZ8[`\s ZNp unp]ztZNUX] tvp unUXgzbegrSXZa[[`cUcZ8g(SX]^m^®U-`X¶^Z-SVbe[Z8[/UcUu^UcZ u^pZ [ bepY6QTm SXp betSUqgrgr00` pSXgr`cZlY(Z8ZNrZZ8 T[zmEUvSagr¡ [^Z8[zUcb p p [ns L ¸IY(Z p [Za`c`Xgz` >. i∈In. n. p. Û0Ü (. ( .
(58) Dy.
(59) "!#$%'&)(*+-,#/.* 01 . . °R´ ° . $A .. N ≥1. 0 ≤ n ≤ m+1. +$0
(60) D
(61) 0 S
(62) .
(63) $ . p≥1. H. . $%A 1 0 " Q A + 0 +. cn.
(64) . P(τ N ≤ n) ≤ cn exp (−N/cn ) . γnN (gn ). $% * . E(γnN (gn ) 1. (. (τ N > n). ) = P(Tn ≤ T ). √ (E| γnN (gn ) 1(τ N > n) − P(Tn ≤ T ) |p )1/p ≤ ap bn / N ,. . +
(65) $A .. . .
(66) $ S Q B + 0 + a < ∞ +/.@ B" +$% @ $ p -
(67) $ S Q+ 0 + b < ∞ p /. +
(68) * " @ +$% @
(69) $ n
(70) $ . 0 0
(71) f n∈ B (E) kf k ≤ 1 b. . √ (E| ηbnN (f ) 1(τ N > n) − E(f (Xt , Tn−1 ≤ t ≤ Tn ) | Tn ≤ T ) |p )1/p ≤ ap bn / N .. Z(be]SjUc` UXZUcu^Z6bY(m tUNgiUcu^beS SVQRY(mTUXgrUXbttagr[RrZ8` Za[ntaZ UXu^Z8gr`cZaY¢RQªtvu^gzSXb[ SXgrY(Zm `V¸ Ucbeta]^ p `qUcZ8SVUA]^p [tUXbegr[Sa¡ ¥ grp`lZ p tvu u > p 0 ZsTZ8n[^ZUXu^Z]^[ntUcbgz[ f gz[ E ¢RQSXZ UXUXbe[ gz`AZ p p tvu b®Ucu s ≤ t ¶ x = (x , s ≤ r ≤ t) ∈ D([s, t], S) b® : 2^¡ = 1 b® |t − s| ≤ u f (x) = 0 |t − s| > u ?¹[Ucu^bS[^grU p Ucbgz[ u → Ψ (η )(f ) bS UXunZ`XZ8m p `XUXbUXbegr[¦]^[tUXbegr[ giUXu^Z&b[/UXZ8`VUcbY(Z T − T ¢EZ UjZaZ8[Ujgt gz[nSVZ-t ]TUcbzZNZ8rZaS B p [s B ¶RUXu p UqbS (u). r. (u). n. n. (u). n−1. n. n−1. n. \unZm p `VUcbetaZ p m^m^`cg TbY p UXbegr[¤gi+Ucu^beSd/] p [/Ucb®UjQ¤bSqUcu^Z$mn`Xgzmgz`VUcbgz[ªgim p UXunSlu p Rb[ m p ScSXZ8s`cgrY UXg be[&>Uc¸3bUjY(QRmEZ Zu`c¡ Z8SX]^®U$gr[¬Ucu^ZZ8`X`cgr`n]tUX] UXbegr[Sa¡ Z U6]nS 8`cSVUbe[zUc`XgTsT]t ZUcu^Zgzeg}b[ B g} B p 5 [ngiU p UXbegr[ pJ;)5 Ψn (ηn )(f (u) ) = P(Tn − Tn−1 ≤ u | Tn ≤ R). >. n−1. n. an =. p [ns. n X p=0. bn =. E [∆np−1,p (Tp , XTp ) 1Tp ≤T − 1]2 |Tp−1 ≤ T ,. n X. E 1Tp ≤T [∆np,p (Tp , XTp ) − 1]2 |Tp−1 ≤ T ,. bUXuUcu^ZN]^[nt UXbegr[nS ∆ sTZ
(72) 8[^Z8s¢RQ p=0. n p,q. ∆np,q (t, x) =. Ü Ü ß-T ÷. 3X *X. P(Tn ≤ T |Tq = t, XTq = x) . P(Tn ≤ T |Tp ≤ T ).
(73) - H. . $% $%
(74) <R < +. °R´ ° . $O+.. . . Q +
(75) $ . (*$0
(76) . 0≤n≤m √ N N Wn+1 = N 1τ N <n γn+1 (1) − P(Tn ≤ T ). +(
(77) $P
(78) B J1 0 $0 A G 0 . N (0, σn2 ). . . Z Za[sUcu^bSSVZ-tUXbegr[bUXu p m^uRQTSVbt p :s^beSct ]nScSXbgz[&gz[Ucu^ZNUXZ8`XYS a ¶ p [s b ¡ ´ô´ Ï ´ $O+.S @ $0 (O
(79)
(80) $ σn2 = P(Tn ≤ T )2 (an − bn ).. n. an − b n. n. n X. 1 = −1 P(Tp ≤ T |Tp−1 ≤ T ) p=0 # " 2 n X P(Tn ≤ T |Tp , XTp ) + − 1 |Tp ≤ T E P(Tn ≤ T |Tp ≤ T ) p=0 1 − P(Tp ≤ T |Tp−1 ≤ T ) . × P(Tp ≤ T |Tp−1 ≤ T ). ¥ be `c´·SVUX´ eQrc¶TZNgr¢nSXZa`crZlUXu p U. E ∆np−1,p (Tp , XTp ) | Tp ≤ T = E. UXunZN p tUUcu p U Z t gz[nt e]nsTZUXu p U. =.
(81) lSVbe[. P(Tn ≤ T |Tp , XTp ) | Tp ≤ T P(Tn ≤ T |Tp−1 ≤ T ). . P(Tn ≤ T |Tp ≤ T ) . P(Tn ≤ T |Tp−1 ≤ T ). q ≥ p =⇒ P(Tq ≤ T , Tp ≤ T ) = P(Tq ≤ T ) E ∆np−1,p (Tp , XTp ) | Tp ≤ T =. ¹[¤Y$]ntvuUcu^ZS p Y(ZN p Qz¶/ZNgr¢nSXZa`crZlUXu p U ?. . E fp (Tp , XTp ) 1Tp ≤T | Tp−1 ≤ T E 1Tp ≤T | Tp−1 ≤ T. gz` p [RQY(Z p SV]n` p ¢^Zl]^[nt UXbegr[ f gz[ p. (R+ × S). ^¡Ó|. : 2 =. 1 . P(Tp ≤ T |Tp−1 ≤ T ). = E fp (Tp , XTp ) | Tp ≤ T. ¡Q>\u^beS\QRbeZas^SUXu p U. E fp (Tp , XTp ) 1Tp ≤T | Tp−1 ≤ T. = E fp (Tp , XTp ) | Tp ≤ T × P(Tp ≤ T |Tp−1 ≤ T ).. Û0Ü (. ( .
(82) }|.
(83) "!#$%'&)(*+-,#/.* 01 .
(84) lSVbe[ : 2n¡Ê| = ¶TZB8n[ns&UXu. H. pU. E ∆np−1,p (Tp , XTp ) 1Tp ≤T |Tp−1 ≤ T. = E ∆np−1,p (Tp , XTp )|Tp ≤ T × P(Tp ≤ T |Tp−1 ≤ T ) = 1.. ¥ `XgzYUXunZ p ¢Eg}rZgr¢nSXZa`c p UXbegr[Sa¶RZ p `X`cberZ p U
(85) lSVbe[. E [∆np−1,p (Tp , XTp ) 1Tp ≤T − 1]2 |Tp−1 ≤ T. = E [∆np−1,p (Tp , XTp )]2 |Tp ≤ T × P(Tp ≤ T |Tp−1 ≤ T ) − 1.. pi/p be[ :2^¡Ó| = ¶nZNZa[s&]nmbUXuUcu^ZNgreg}be[ gr`cY]^ p E [∆np−1,p (Tp , XTp ) 1Tp ≤T − 1]2 |Tp−1 ≤ T ". = E. Z RU8¶nZ SXZaZUXu. ∆np−1,p (Tp , XTp ). E ∆np−1,p (Tp , XTp ) |Tp ≤ T. pU. #2. . |Tp ≤ T ×. 1 − 1. P(Tp ≤ T |Tp−1 ≤ T ). E [∆np−1,p (Tp , XTp ) 1Tp ≤T − 1]2 |Tp−1 ≤ T . =. an. =. 1 −1 P(Tp ≤ T |Tp−1 ≤ T ) " #2 ∆np−1,p (Tp , XTp ) 1 − 1 |Tp ≤ T × +E P(Tp ≤ T |Tp−1 ≤ T ) E ∆np−1,p (Tp , XTp ) |Tp ≤ T. 1 −1 P(Tp ≤ T |Tp−1 ≤ T ) p=0 " #2 n X ∆np−1,p (Tp , XTp ) 1 n − 1 |Tp ≤ T . + E P(T ≤ T |T ≤ T ) E ∆ ) |T ≤ T (T , X p p−1 p p T p p−1,p p=0. n X. g6U p rZUXu^ZB8[ p ·SVUXZ8m·¶^Zgz¢nSXZa`crZlUXu p U >. E. Ü Ü ß-T ÷. 3X *X. . ∆np−1,p (Tp , XTp ) ∆np−1,p (Tp , XTp ). |Tp ≤ T. . = =. P(Tn ≤ T |Tp , XTp ) E P(Tn ≤ T |Tp , XTp ) |Tp ≤ T . P(Tn ≤ T |Tp , XTp ) = ∆np,p (Tp , XTp ). P(Tn ≤ T |Tp ≤ T ).
(86) -{ H. $% $%
(87) <R < +. \unbeS\Z8[ns^S\Ucu^Z m^`cg/gr grUXunZ m^`Xgzmg/SVbUXbegr[¡ g} ZNZ Tm^ be[Ucu^ZY(Z [^b[ grUcu^beSqm^`Xgzmg/SVbUXbegr[¡-?¿ ≤ T |T , X ) sTgRZ8S[^grUqsTZamEZa[ns gz[ (T , X ) bzZap [ (T ≤ Tp ) ¶/b3¡ Zr ¡+sTgRZ8S[^grUsTZ8mZ8[nsgz[Ucu^P(T Zlunb®UXUXbe[ UcbY(Z p [nsmgzb[/U\giUXunZleZazZa SXZ U B ¶RUXu^Z8[ >. n. p. Tp. p. Tp. n. p. [nsb·UXunbeSu^grs^Sôgr` p [RQ XSp begr[. ". E. P(Tn ≤ T |Tp , XTp ) −1 P(Tn ≤ T |Tp ≤ T ). p = 0, 1, . . . , n. σn2 =. n X. 2. |Tp ≤ T. #. =0. ¶zUcu^Za[UXu^Z p SXQRY6m^UXgiUcbetA p `cb p [nt Zq`cZ8s^]nt Z-S+UXg$UXunZAZ Tm^`cZ8SV¸. 1 −1 P(Tp ≤ T |Tp−1 ≤ T ). . S bzZa[be[ DH8 I¡-?jsTZ eQr¶/UXunZleZazZa:SVZaU SVungr]^s¢EZtvungzSXZa[SX]ntvuUcu U ≤ T |T , X ) sTgRZ8S n[p giUô sTZ8mZ8[ns(gr[ (T , Xp ) berZ8[ (T ≤BT ) ¡ rZ8[(b®:UXunbeSôbeSt eZ p `ceQ]^[^p `cZ p P(T bSjUcbetgz`Y(gzSVUm^` p tUcbet p mn`Xgz¢^Z8YSa¶RUcu^beS\gz¢nSXZa`c p UXbegr[ berZ8S p [&be[nSXb u/U\gz[¤u^g} Ucg(tvung/g/SVZlUXunZ Z8rZa·SXZ UvSa¡ >\unZ Z8[^Z Ucbetm `XUXbt eZ m^m^`cg TbY UXbe[ Y(gRs^Zas^Z8Sct `cb¢EZ8sbe[UXu^Z&m^`cZaRbgz]nSSXZ8t UXbegr[¬t [¬¢Z&b[/UXZ8`V¸ mn`cZ `XZatvU up UcUcZ8u^s Zp sTS Z8SXp be`Xp ¢^Z-bes`VUceu Zazp Zap [ s¬[sTsZ pbUAUXu¬sTp ]^m m^p e`VbtUcbetaUcZ8ZS\Y(be[ªgTsTSVgzZ8Y639¡ ZN>\ g u^:ZSXm^m `cpb[ `XUXbSt eZu^sTZabe[¤Z8S$bUqb®u^bb®U6UvS\sTUXgRunZ8beSSe[^ZagizU6pZaISV¡ ]n lt8t [^Z8ZNZ8s¦ UXQg Ucgp Y(gRs^ZaUcu^Z Z8[^Z p g btp p nUc`XZ8Z p [ns&UXp unZbe[^Z gr p [nt Z-SjUcgr`vS\gi 0UXu^Zm p `XUXbt eZ8S p eberZ p UASXgrY(Z berpZ8[ s p UXZNbS\UXgtagr[nSXbes^Za`\UXunZSjUcgRtvu p SVUXbtlSXZ8d/]^Z8[nt Z p=0. p. p. Tp. n. p. Tp. n. z§ UcbY(Z-S Uc?¿UN` p b[nSASX[^b®UcgrbgzU[nsTS bFQt ]^UUX`cg¤grY tvu^EZ8tv &eb UX[/u UXp g U EY gz`XYS p UcbY(Z6b[^ungrY(g Za[^gz]nS £Mp `X zg}tvu p b[ªbUXu M£ p `c rg} Yn = (X0 , · · · , Xn ) ∈ En = E × · · · × E {z } | (n + 1) n. 5. n+1. n. n+1. Qn+1 (x0 , · · · , xn , dx00 , · · · , dx0n , dx0n+1 ) = δ(x , · · · , x ) (dx00 , · · · , dx0n ) Kn+1 (x0n , dx0n+1 ) 0 n. Z U h ¢EZNUXu^Z Y p m^m^be[ `XgzY n. En. b[/Ucg. [0, ∞). sTZ8n[^Z8s¢RQ. hn (x0 , · · · , xn ) = gn (xn ). Û0Ü (. ( .
(88) -.
(89) "!#$%'&)(*+-,#/.* 01 . ¹[Ucu^beS\[ngiU p UXbegr[Z u p rZlgr` p [RQ ?. E(fn (Yn ). fn ∈ Bb (En ). n Y. H. UXu^Z ¥ ZaQR[^Y p [T§R¨ p tl`cZam^`cZ8SXZa[/U p UXbegr[. hp (Yp )). p=0. µ bn (fn ) =. E(. n Y. hp (Yp )). p=0. = E(fn (X0 , (Xt , 0 ≤ t ≤ T1 ), · · · , (Xt , Tn−1 ≤ t ≤ Tn )) | Tn ≤ T ).
(90) lSVbe[ Ucu^ZS Y(Zeb[nZ8Sqgr+`XZ SXgr[^be[ S ¢g}zZNUXu^Z ¸Im `XUXbt eZ m^m^`cg TbY UXbe[ Y6gTsTZ8 SXSXgTt b UXZ-s bUXu UXu^Z-SVZ ¥ p ZaQR[^Y p [R§R¨ p t+sTp beSVUX`cbe¢^]TUXbep gr[S·p bS przp be[ pN Z8[^Z p Ucbet p gzp `XbUXu^Y b®Ucp uY ]^U p Ucbgz[ UXp ` p [SVbUXp begr[nS [nsmgrUXZa[/Ucb ]n[ntUcbgz[nS ¡ AZa`cZAUcu^Zlm p UXu^¸Im p `VUcbetaZ p UUXbeY(Z n U p zZA p e]^Z-Sbe[ E p [sUXunZaQ Q t p [p ¢Z `cbUVUXZ8[ p p Sgreeg}qS h p [ns ζb = (ξb , · · · , ξb ) ∈ E ζ = (ξ , · · · , ξ ) bUXugz`Z p tvu 0 ≤ p ≤ n p [s ξb = (ξb (t) , Tb ≤ t ≤ Tb ) ∈ E ξ = (ξ (t) , T ≤t≤T ) >\unZNSXZaeZ8t UXbegr[&UX` p [nSXbUXbegr[¤t gr[SVbSjUvS\b[¤` p [ns^grY(QSXZaeZ8t UXbe[ p m p UcuT¸¹SVZ-dz]nZa[ntaZ = E(fn ([Xt , 0 ≤ t ≤ Tn ]) | Tn ≤ T ). n. n. i n. i p,n. i p,n. i 0,n. i p−1,n. mn`Xgzmgz`VUcbgz[ p eQ6Ucgb®UvSR8^UX[nZ8ScS. n. i n,n. i p,n. i n. i 0,n. i p,n. i p,n. i n,n. i p−1,n. n. i p,n. i i ζni = (ξ0,n , · · · , ξn,n ). i i i hn (ξ0,n , · · · , ξn,n ) = gn (ξn,n ). \unZY$]TU p UXbegr[ SjU pi Z¤tagr[nSXbSjUvS$be[ Z RUXZ8[nsTbe[ UXu^ZMSXZaeZ8t UXZ-s m p UXunS p tatagr`vsTb[ Ucg p [ Z8Z8Y(Za[/U p `XQ ¸3Uc` p [nSXb®Ucbgz[·¶RUXu p UqbeS K >. n+1. i i i i i i i ζn+1 = ((ξ0,n+1 , · · · , ξn,n+1 ), ξn+1,n+1 ) = ((ξb0,n , · · · , ξbn,n ), ξn+1,n+1 ) ∈ En+1 = En × E. cUu unp ZaU`cZUcu^ξZ Zazgre]TUXbegrbS[ p p Sc` SVp gT[tasTb p grUXY Z8s p `cb®Ucb p u¢^UXZu^Z Z8b®Uc[nu5s mp gz b[/KUcSqgi(Ucξbu^Z m ,p ·)UXun¡'S 6Q p SVbeY(m^Z p ` ]^Y(Z8[zUlZSXZaZ p [s ξb = (ξb , · · · , ξb ) ∈ E ξ = (ξ , · · · , ξ ) mtagrp beUX[nu^t ¸Ibm sTp Z6`VUcbetab®UcZ u5£5Ucu^p Z `c r g}Z8[^Z tvUcu betp b[ªp Y(gzgT`XbsTUXZ8uneY(SqSNUXu^sTZ$Z-ZaSXtar`Xgzbe¢E]^Z8UXsMbegr[ªbe[¬be[¤PRZ8UXt beUXY6begrZ$J[ gi2^ ¡ UXun Z$Zt gzt `Xgz`c[nZ8t SXmEe]ngrsT[nZ6sTbeUX[ u (p UN Z8UX[^u^ZZ6p gzg `X Y(bt Zap ` i n+1,n+1. n+1. n. Ü Ü ß-T ÷. 3X *X. 1 n,n. N n,n. n. i n,n. 1 n,n. N n,n.
(91) r. $% $%
(92) <R < +. Uc`XZ8Z8S8¡ ¥ gr`+Z p tvuUcbY(Z n < τ ZsTZa[^grUXZ\¢RQ µ p [ns µb Ucu^Zm p `VUcbetaZ\sTZa[SVbUjQm^`cg8neZ8S p SXSXgTt b p UXZ-s bUXuUcu^Z p [ntaZ8SVUXgz`ebe[^Z8S\gr Ucu^beS Za[nZ p eg bet p UX`cZaZN¢ p SVZ-s p gr`cb®Ucu^Y p [s µb = 1 X δ(ξ , · · · , ξ ) 1 X µ = δ(ξ , · · · , ξ ) N |I | bUXu N. N n. N n. N. N n. i 0,n. i=1. N n. i n,n. N n. i 0,n. N i∈In. i n,n. i i InN = {1 ≤ i ≤ N : ξn,n (Tn,n ) ∈ Bn }. \tagrun[/Z UXZ p RSVUNQRY(grmTSVUXUXgr`cUXbebt tAUXe¢EQMZau mEp gz/SXbeb®grUcb`zZ6gi mE giZaUc[^ZaZ [/p UXeb g S bet p [^sMUc`XZ8]^Zl`X¢ UXunp ZaSXZ8`s sTpZ8r Z8grgz`cb®m^Ucmu^Z-Y s5u be[p S ¢E+ZaZagr[` Sj[^Uc]ngzsT[LbeZ8[^s&Z b[UXbe{rZ(b[&gr[^UXZ-u^SaZ ¡ p p zp ?¹[¤gz]^`qt gz[zUcZ RUUXunZ m p UcuT¸IzZa`vSVbegr[gi0UXunZ L ¸¿Y(Z p [&Z8`X`cgr`Z-SjUcbY p UcZ8S\m^`cZ8SXZa[/UcZ8sb[Ucu^Zagz`XZ8Y 6t p [ ¢EZSVU p UcZ8s p Sgzeg}qSa¡ °R´ ° $9 . p ≥ 1 0 ≤ n ≤ m + 1 +.J
(93) 0 f ∈ B (E ) ( >. p. kf k ≤ 1. n. . b. n. √ p 1/p (E| µ bN ≤ a p bn / N n (fn ) 1(τ N > n) − E(fn ([Xt , 0 ≤ t ≤ Tn ]) | Tn ≤ T ) | ). . . .
(94) $"
(95) @ Q+ 9 + 0 + a < ∞ /. +$% @ $ 0 + b < ∞ p @ %$ S $ n n. . . p. . J
(96) @ Q+ . ¥ greg}be[ Ucu^ZAgz¢nSXZa`c p UXbegr[nS berZa[(Ucu^ZZa[nsgrUXu^Zlm^`cZaRbgz]nSôSXZ8t UXbegr[eZ U]nStvu^gRgzSXZ p tagreZ-tUXbegr[gr UcbY(Z-S u > 0 ¶e¡¡e¡¶ u > 0 ¡ 5 Z U f ¶ u = (u , · · · , u ) ¶^¢EZNUXunZNUXZ8SVU]^[nt UXbegr[¤gr[ E sTZ
(97) 8[^Z8s¢RQ 1. bUXu. (u) n. n. 1. n. n. fn(u) (x0 , · · · , xn ) = f (u1 ) (x1 ) · · · f (un ) (xn ). f (up ). sTZ
Documents relatifs
2) Le développement local c'est-à-dire les expériences d'intervention en développement économique communautaire, en économie sociale et en économie solidaire mises
La définition générale du concept d'évaluation formative des apprentissages retenue pour cette recherche constitue une synthèse découlant de la recension d'un ensemble
Effets et pouvoirs de la voix dans la philosophie et la littérature antiques Paris, Classiques Garnier, 2021 (Kaïnon - Anthropologie de la pensée ancienne, 19), ISBN :
relève de traités, et que les États membres sont des États au sens du droit international public, c’est à ce juste titre que J. Fischer peut qualifier l’Union actuelle de
Afin de saisir le déploiement de ce discours, j'analyserai dans un premier temps les représentations liées au temps dans le roman, en m'arrêtant d'abord sur toutes celles qui,
Cette recherche, de type qualitatif, consiste en une exploration de la littérature dont les buts sont de décrire selon divers auteurs, différe ntes expériences
For each sam- pling point is also given the value output of the merging process with the difference between the SH degree 100 expansions subtracted, the index corresponding to the
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des
