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December 2019
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>?!3OD !"4F¬.,% I 3!QP+"#.#.4 +F I X%(+3UE+2JE%(3%(-.4 ' 3!"#3KE\-'>?'%>?+Ò&DP+3*'.,%¬X%(.#"#$& >?!3 "#E>?!"4+0G+E "# "#!",%(. 6ÎK*MTKKÏ ]0G$ I %( "40G ].=?+E!6{)% I +0G+E"#E>?+3« I ? I ' 83!"4\-'6 3!' "#+ I %& 6[L%('X !"·>?':% I [++Ò1' D ' ¬ $1 O0G+0G I %(3X% I $+3!"4!"4\E'UP+ !"V.4 I ' 3O ¬.=?'O% I [++Ò1%(.43!' 3 .402%(3 I [$1QV' 3!"4+3:ÎKKK% I !"4/F Ñ KÏ \E'>?'&3+>?+ +Ò2ÎKÎKÏÔF2.,%;9v I "4$+$]Ð:$+$+3X%(.4 Ç>?'¬.,%("4 I %(. I ' .4+3>?!36QP$+${¸ I KM#M#M GD !"4G-]' Z%(' !3;m I !"4 !' D DPKX]\E'j< ' Øm G ÂDP+"4Ó%(0G$+3!" I %("#Ó $+!"4+ K?Õ *' I %(D "#X%(. /5" I [+.#"#+OU?+*3 X%DL%(3!!" I "#DL%(!"4]%>K%(E·.,% I %(X%!!3D [%( I $K02%("4UDV' 3 S3X%("4XS¸L I %(.4Ue'/DP' 3DL%SDV+3*3Á *3"# 3+JE%(3Â!' 3Á.,%/J?!!"4Â O.=?+E3 I ' 3!"#3RF:' &!!3X%($+JK"4 I ' >?+3!!' 3 3!"4X\E'&» Ê ´!¿U©¡§¢-© ¼s $O!' 3Á.=?+-KM-N I %ÁSv"4+ES?v\-'$6DL%(3 .,% I [-' I ' 3UML@.#' \-'/.46J?Æ /.,%2!DV$ I ' .,%(!"4 D 3¹+$1%('*) %( JK.4(HµX%) U I v3X%("4+06Q .,%(Q .4+0G+- I O' I "V OX$ I ' 3!"#$ \-' "q{)*D .#"4\E'O.4O $+>?+.4D DV+0G+ER!DV I X% I ' .,%("#3 R02%(3 I [$ $+3!"#>?$ I !X%($6%(' I ' 3O ÎK] +3! "4W+3 %( $&%('*);NX%( O "4eM Ã .3!2F {)vD .#"4\-'+3/.423 .42\-'¬Ç'+-6.4602%(![$+02%(!"4\E'1%( I +' "#>?+3 ' 2$+!' 2!X%(!"4!!"4\-' I ' 3 ®QP' 3 I *' "#' S?+3! .4+' 3/$+>?.#' !"4 3$+DV$+:\E'j< "#.1UK%(!"42S< ' GJK3X%(]0Q 3 Ç!+S< ' RD "4W I 0G L%("4/KS?+*3OXD !"4 »§.4ÑUÏ !3{)*+0]D .4e¼ >?+*'Á!3D I [W+3UKDV+3 j?'*3X%*q>?+*'!3DQPZ02%(3 I [$2+.#.4ODP+' >?+- I *' "#3F1.,%=%("#.#.#"#6 O$+X%(Q .#"4X+0G+E 3$+D ' $ "# !!3!' I !"#Q .4/V»§'Ç!' 3 I & $+>K%!X%(+' 36´ ´+ ¿E´¬°4´{ÌU¡=´{¥?¼&M*¬ +3! "4+3RDV"#Ej?$M f !f 1b?" ''RQV3!+3U*DP' 3.=K%(E6« Ä +0]D!*"4 I 3+ »s]Re< "#E$+3DL%(3{)*+0]D .4O%(' I ' 3Á Á% I !"s\-'jK%(.4+' 3¸ )*$»§DL%(3]{)*+0]D .4KF I [L%\E' I (. !' 3 I 3 ?¼:M I ' 3O!"#0' .#!"#D .#"4$/DL%(3 Î1"#*"#>v"4$¬DL%( '>?+3!3RD .#'.4"#.43X%("402%(![$+02%(!"4\-'O\-' " I *' "4X+E F=%("#3 I +!: +3! "4W+3:[EÅvDP![WX1»m3! $+.4"#JK$6*' I 0]DV3!+0G+E&3$+.jS< ' % I !"4 ,¼ N $+D "#: 1 I %(3X% I W+3]"#0]D .#"4!KP.=?1DV' 3&D .#'!"4+' 3¬3X%("4 ' S?.,%(!"#.#"#$:*' Î

02%(3 I [$+RZDL% /%+%( I »§[L%('!"4W+3:' QL%("4!"4W+3e¼DL%(3{) +0]D .4KM 5®%("4R"#.S>K%'R=%(.#.4"#3Rm%("#3¬' DP+'Z ¬02%(![$+02%(!"4\-' %(' !3X%>K%("#. ,

+3X%1$ ' 'ZD .#' fLdU_*d %§fqg q« 9-" ' ++0Q .4:¸ "Á*.4:0Q 36S
!#"%$&'( !*)+" ,-.0/!

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21

43

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2

$ I [L%(-!"#.#.4Ó X%("#.#.4 ØDV' 3!3X%D 3+*3DV' 31!DL% I .=%(.4+' 3%(p % P 9 < # " Á . U T < ( % K J # " 6 S < ' ( % ? J ] V D ' / 3 4 . 6 + $ 4 . ! 4 " D 3 $ ! 4 " + E ! 4 " + # . 4 . ; I p E |Ω| = np v>?"#R\-'O.=?:-Q "4+Â\-' 3!X%(E¸ "-JK"#JE%(-\E'KM (n = 20.000.000, p = 2.000) 1 +{) +0]D .4G >?"4"#Z*'ZD 3$ I $ +E DP+' X+3!>-"#3 :0Gv W+.4/%(']Ç!+'Z :.4ZM 9v"4+E + +'*)8+-!"4+3 %U>? I »m%('.4s3X%(U%("4] ¼ ' 1$+D 3+' >?1+3X%G' n p >0 p?& %(.43 p E = {0, 1, 2, . . . , n − 1} Ω I %(3*"#L%(. n !¬' +p+06Q .4]¸ " I %(3*"#L%(. + +'*)+E!"4+3 "#*m$+3!"4+' 36F I ["4!"# %('[L%H 5 X%(3 E] +'*)'/+X+06Q .4 + n +p.4/\E'q + > 0 .=?6 E |F | = p |G| = q Ω F %GG %( I I %

4D

GD

JE

9E

{(F, G)| |F | = p ; |G| = q}

|Ω| =

n p

n q

5768596 :FC ? 4<>C?

A 8C >C

+.#.4:\-' P Ä hAcfqa·_ a&%/%d !' 3 Ω +3X%' 6%(D D .#" I %(!"4 : Ω 7→ [0, 1] P (ω) = 1 ω∈Ω % + X 3 1 % { $ ¸ 4 " / L D ( % 3 · h ( c L f A a _ & a % / % 5 (#·gvgbv`b-gPd d Ä A A = ω∈A (ω) Ä I ' D .4 (Ω, ) X+3X%]%(D DP+.4$ bUhA_21?bhAc(fLaA_ a%/% ' + I 3 bUhA_21?b 5Ab hAc(fLaA_ a%/%d Ä .=?%UÅE%(E' D 3QL%(Q "#.#"#$ ÂE' .#.4¬!R%(D DV+.4$ #·hAhfLcd ' D 3D 3!"4$+$ \-' "j!R>?$+3!"¸L$+' DP"#-O*' !' D DV3! *"#:>-3X%("4 hAc(b #·b ·cbv`bvgPd!h" #" 9-" + EÎ $+>?$++0G+- 8*"4!DVK SK%( I [Ø!" '%A + E;*"4sÇ!"#E®×.4+' 33$+' "4ÔDL%(3 +(A 2.=
D

5768596 I< )

*

,+# -(-

(ω) =

; < ;<>=@?BA8<>C

1 ∀ω ; |Ω|

A=

»ÑKM#Ñe¼

|A| |Ω|

1!DL% I $+D 3+' >?"#-!3-*' "#G%('DL%(3X%(JK3X%(D [/. " . " .®$+X%(E]!' D DVK$06' "41 G.,% D 3QL%(Q "#.#"#$ ' "m3!0GG I %(. I ' .4+3O.,%D 3QL%(Q "#.#"#$ O$+>?$++0G+E&!' "#>K%(E/« _ .,%G0]0G: O +'*) $!R' Z0' .#!"#D .46 a O D 3+0]"4+3 +O +3! "4+30GO ¬.=
-D GD 4D GD

3

n

5

! /B$ , $ !*$ )$

43

5 7698

8

;:<=5

,-!* . , ! >:?8

@8A<=5

!

0]0G1.4+' 3&0G&.=< "#*"4\E'+-:DL%q.4Ì¯¥{¡s¯CB{°#´{³G%(.4$U%("#36E:+83$U%(.#"#$ ED¢ ¨ s¡m¢ ³ D .#' D 3 $ I "4$+0G+-«

576 685

FG,HEI J+ K I

¢¯? ´{°§°4´{¥¯ÂÌ(¯¥¡m¯B{°4´1¯° Ë ¯K =¡§¥´9M2Ì(¯°#´+º-¥³¿?¯¢ ³:ºv¢ ´{¢ ³´+ÉNB+°#´ L

E

º-¢V´¯{?q°²¡ ¨ ¯? s¡s¢¿E´

Ω

¿?¯¢ ³

E

9-" ' ;>VMT%*M+ ' ¤']++0Q .4Z .=?$++0G+- ! E {ω ∈ Ω | X(ω) ∈ Γ} = X −1 (Γ) ' >?+-X $:DL%(3.4RD 3QLΓ %(Q "##. "4 M [X ∈ Γ] @' "4\-' !¸ "=.=K%(.4+' 3\-'¬DP+' D 3+*3 ! R.#' "%('X!"V¸ "ÁM Ω

X(Ω)

X

576 6 I< ; < ;<>=@?BA8<>C + ${¸ "4DL%(3 3+D 3$+E+-.4A3$!' .#X%( _-D >K%(3!",%(Q .4%(.4$U%("#3 %("#!" % 3 I [$ODL%(3R.4&D 3+0]"4+3O+R.4: X I X$ ' /%(' !3¬>VMTX %*M "#E(i,$+j) 3=X%(-i ;6X! (i, j) = j S = X +X S(3, 4) = 7 « aGD Í ${¸ "#O :.,%0G¹+0G:m% >K%(3!",%(Q .4 %(.4$U%("#3&F>%(.4+' 3 %( p E

*

,+# -(-

1

2

1

2

1

2

!" SDVK+3X% M9-".=? ω = (x1 , x2 ...., xi , ..., xp ) Xi (ω) = xi (i ∈ {1, 2, ..., p}) ω I 0]0G' 6.#"463 $1 p 0G&3 I '+"#.#.#"4DL%(3&' ®+\E'¹++' 3U Xi (ω) 3+D 3$+-+3X%Â.4 i − eme 0 Ù

I +!:.#"4!G»s+Rj-"*'Z!'Ç!+/*"#!X%(.#.4R' LD %(3!!"#!"4' 3 + 3+D 3$+EX%(-3DP I !"#>?+0G+-6.=%(3!",%(Q .4&%(.4$UE%(1"#3E O2\-'¬.=? I %(!+E!"4X+3E Z1 (ω) = ]{i |Xi (ω) ∈ E1 } ;

Z2 (ω) = ]{i |Xi (ω) ∈ E2 }

»ÑKMTÎ?¼

N.#.4 E F>K%(.4+' 3%(O.=?$+3!"¸L+-O.=< "4 +-!"#$ {0, 1, 2, ..., p} Z 1 + Z2 = p 1 7O'1Ç!+'Z*'Z.4 .,%10]0G6 $+.4$+0G+EO¸JK' 3X%(E&%(R.4:!"#3X%(J?

4D GD 576 6 5

ω

(F, G) → ](F ∩ G)

)

<>C

! I

9v"4+-

+

< C -

' ++0Q .4 + ' Â'Á++0Q .4 .=< "#*" I %(+' 3R +3X%/DL%(3 ${¸ "#!"4Â.,%¬m I !"4 E F ${¸ "4/DL%(3 F

E 1F : E → {0, 1}

1F (x) = 1

!"

x∈F

1F (x) = 0

!"

x ∈ Fc

-9 ' 3]' ¤!DL% I Z ;D 3QL%(Q "#.#"#$ .=< "#*" I %(+' 3S< ' ¤$+>?$++0G+E !] I ' ;>K%(3!",%(Q .4 %(.4$U%("#38>K%H A .4+' 3' !"#.#"4X%(!"4 DV+3!0G+Z ®3+0]D .,% I +3Z.4DV$+3X%(!"4QV-.4$+ Z' 3Â.4 ++06Q .4 .4+' 3%( »§"#E+3X I !"4{0, 1}3$+' "4 + I M#M#M¼DL%(3O I %(. I ' .4 %(.#J?$+Q 3!"4\-'O3*"#L%("#3M @%(3O{)*+0]D .42 1A∩B = 1A .1B = 1A ∧ 1B ; 1A∪B = 1A + 1B − 1A .1B = 1A ∨ 1B (= 1A + 1B si A ∩ B = φ) 1A∪B∪C = 1A + 1B + 1C − (1A .1B + 1A 1C + 1B 1C ) + 1A 1B 1C

+.,%&Ç'!!"¸L¬.4RX%(!"4

A ∩ B = AB

+

A ∪ B = A + B si A ∩ B = φ

b#' h·c(fLhAc;%Ed c(b?dUbvg6%§c ÑKM 9v"4+-

X

' ¬>PMT%*MLF>K%(.4+' 3%(

GD

E

+

' ZX'++06Q .4

Γ

E

M

8%.=
1[X∈Γ] = 1Γ ◦ X

DL%(3 7 "#!"RV%(¬.=VMT%*M3$+.#.4 ;» ¼R' $+-' 0G$+3X%(!"4® È .4 E&*"4!!"# I Î*M 9v"# X a1 , a2 , ..., an X(Ω) ai + .=?$++0G+E M %]!' "#G» ¼ I !!"#!':' /DL%(3!!"#!"4 +R.=
[X = ai ]

Ai

X=

Ω

i=n X i=1

a i 1 Ai

43

»ÑKM -¼

% $+0G!!3X%(!"4¤ I ]ÎDP"#-]!"#0]0G$*",%(«·"#.R!' G] 2>?$+3!"¸L+3G\-'.4s I !"4¸JK' 3X%(EÂ%('*) 8 Î 0G+06Q 3] ®»ÑKM ?¼/+1 8»ÑKM -¼6D 3+ +-].,%Z0G¹+0G>%(.4+' 31+' DV"#E M '6"#->-"#1.4.4 I +' 3 "#{)*DP$+3!"#0G+-$FRD 3+*3\E'+.4\-'A0]"#E' DV' 3A$ I 3!"#3.,% $+0G!!3X%(!"4 I 0]ωD .4W+ Á02%( "4W+3RF U
576 6 :FC ?

4<

7 I

bg%,d %§fPg P¡s

X

G<

-I

BA < 7A

K +# K

< ><>A8A8<

ºv¢P´6Ì¯¥{¡s¯B+°#´1¯° Ë ¯K =K¡¥´9MGÌ(¯°#´+º-¥³6¥ Ë {´ °§°#´³ =¢2*(³´+¥¯

[X] =

X

»ÑKMTÙ?¼

X(ω) (ω)

1 %(D DV+.4\-'+.4\E'D 3D 3!"4$+$ X+E!"4+.#.4O ¬.=
Ä Uh -c_g61?b 5(#Ag %sg'5&%1?_*dUb#·c 1 Í % [1A ] = A \-'+.4\-':"#R.=?$++0G+E A Ä ^ %s g -_ c+%d 1 -9 " X + Y XE Î>VMT%*M 3O+!" α + β - Î06Q 3 3$+.4/ %

«

[αX + βY ] = α [X] + β [Y ]

!/DVK!"#!"#>?Z· !DV$+3X%( I 2!6DPK!"#!"#>?+6j?{)*2L%(.43 [φ(X)] ≥ φ( [X]) h /§ b 1 O+' 3!6+ I 3]' s"4/F.=

1

GD

[Z1 ] =

p X

[Yi ] =

1

p X 1

[Xi ∈ E1 ] =

p X n1 1

n

=p

n1 n

\-'L%(%DVK$ ]E1 = n1 , ]E2 = n2 Ä jbE+c 1 %1?b 1 9v"4+- A, B, C !3 "4 $+>?$++0G+E&S< ' !DL% I / ¬D 3QL%(Q "#.#"#$ Ω 0GE!3+3 .=

A+

B+

C − ( [A ∩ B] +

Ð:$+$+3X%(.#"4+3 F' Z0Q 36\-'+. I \E'/S?$++0G+-M

576 6

[A ∪ B ∪ C] =

- I

<

BA8< 7A

K

K +#

[A ∩ C] +

[B ∩ C]) +

[A ∩ B ∩ C]

<

P¡s º-¢P´ Ì(¯¥¡m¯B{°4´:¯° Ë K¯ =K¡¥´ MÌ¯°4´{ºv¥X³:¿?¯¢ ³ ¢qKºE³ ¯E ´{°§°#´{¥!¢ ³&°K¡£oK¢¿K¡§ ¯º-³X³¡³+ºvÌ(´{¢q ¿K¡,³+ s¥¡KB{º* s¡s¢?ª ¿E´ X°: ¯ q¥!B!¯B+¡°²¡§ Ë ¿ ¶¢¡=´:³{º-¥:°À ´{¢ ³´+ÉNB+°#E ´¶¢L¡ *¯K¥ X ΠX Ë X(Ω)

ΠX (k) =

[X = k]

∀k ∈ X(Ω)

´{º¾ÂÌ,¯¯E¿É2´+ m =¯¢L Á°¯É{É2´6°¡³U´{¥!¢L ¿¡§ µ´{³ Ë +º-¡m¿¡³{ s¥{¡JB+º Ë ´{³

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1

Π

{0, 1}

Π(0) = 1 − α , et Π(1) = α \E':.=v"4+E& %1D 3QL%(Q "#.#"#$ Π

${¸ "#3O!' 3 : eK%(3!",%(Q .4O%(.4$U%("#3 ${¸ "4 F6.,%¸*'DL%(3X%(JK3X%(D [ . " "" ER$\-' "4*4" H !3!"#Q '$« I L[ % I ' S
H

GD

)

n

1

GD

Î*M /§f<%ja&%§g·fq` %§_ /b M O!%(.=VMT%*M "#E!3v*' "#O+$. " " Π Z1 6' /SK%(.4+' 3/%(:.=?$++0G+- !:.=
k!(p−k)! n1

n2

Π(k) =

[Z1 = k] =

p! n1 n2 ( )k ( )p−k k!(p − k)! n n

Í %(D DP+.#.4+3X%¤°¡ B+¡¢qKÉ¡m¯°4´Z¿E´]*¯K¥¯É+ s¥´³ (α, p) ¿ µ´{¥{É1¡¢ Ë ´*¯¥ {0, 1, 2, ..., p} Ë Π(k) =

7 "#!"

Z1

»§3!DjM

Z2

(α ∈]0, 1[

°#¯;P¥B¯B{¡§° ¡s µ´

p ∈ N)

p! αk (1 − α)p−k k!(p − k)!

¼!' "#O' ¬.4"jQ "#0]",%(.46 ¬DL%(3X%(0GW+!31»

H

n1 n ,p

¼¬j»§3DjM»

n2 n ,p

Π

³{º-¥

¼!¼:M

D

6 .=VMT%*M -fq` ?dUc;% #·b X(F, G) = ](F ∩ G) X D 3+Â>%(.4+' 3 %(.=< "#E+3!>K%(.#.4 ! " ( % D L D ( % ! 3 ! 4 " + E F I + "#E+3!>K%(.#.4E% I = [(p + q − n)+ , p ∧ q] k

*M /§f<% :PijhbEc

]{(F, G)|X(F, G) = k} =

63!:\E'

n p

[X = k] = H(n, p, q, k) =

p k

p k

n−p q−k

n q

n−p q−k

,

»ÑKM ?¼

DV3!$/DL%(3 $6DL%(3 ! %(D DV+.4$/.4"·[EÅvDP+3!J?$0G$+!3!"4\-' % D 3QL%(Q "#.#"#$ ] Π I Π(k) = H(n, p, q, k) /DL%(3X%(0GW+!3 M n, p, q M jbE+c 1 %1?b »s DP' 3!3X%8' !"#.#"4X+3 »m%?¼ÂB¬$+3!"¸L+3]DL%(3]' I %(. I ' .D ' 3+0G+-%(.#J?$+Q 3!"4\E' \E' P p, q, k) = 1 k H(n, .=< "4 +-!"#$ + / + $ E J ( % 4 . + : 3 4 . v 4 " + E 6 ¸JK' 3X%(-6%(¬.46Î I I (1 + x)p (1 + x)n−p = (1 + x)n xq 0G+0Q 3¬¼&M »§Qq¼ %(O.=?"#3 E' 0G$+3KOJE%(JKL%(EM k % I L%("4XX%( I / :.,%1.4"·S< ' :>%(3!",%(Q .4%(.4$U%("#3/DV+3!0G+O I %(. I ' .4+3 ZDP$+3X%( I GLR0G0G+EX% mE!"4ZJ?$+$+3X%(!3!" I + I M#M#M D .#'RJ?$+$+3X%(.4+0G+- %.4/3$!' .#X%(O!' "#>K%(E/«

H

GD

3

4D

(c fLhf! %d %§fqg%1 P¡=´{¢L ºv¢P´GÌ ¯;MZÌ(¯°#´+º-¥³¿?¯¢ ³ ´+ º-¢P´Â¯Eq° ¡ ¨ ¯K s¡m¢ ¿E´ ¿E¯¢ ³ ´³+ ¯°#¥³]º-¢P´GÌ¯¥{¡m¯B{°4´ R φ(X) ¯° Ë ¯K =¡§¥X ´¥ Ë ´+°°4´1¿?¢L °À ´³µ Ë ¥!¯¢ ¨ ´]´{³{ E¿?¢¢ φË ´ ¯¥/°#¯D¥{É1º-°4´ E

[φ(X)] =

X

k∈X(Ω)

φ(k)ΠX (k).

»ÑKM )K¼

Ë ÉG¢ ³{ s¥!¯K s¡m¢«vO3$!' .#X%( $ I ' .4O"#0]0G$*",%(+0G+- .=-"4 +-1E!".=
X

Φ(X) =

k∈X(Ω)

Φ(k)1{X=k}

+ :.,%.#"#$U%(3!"#$ : .=
H

X

X

[X] =

kΠX (k) ,

[X r ] =

k∈X(Ω)

@%(3O{)*+0]D .4 !"

X

X

k r ΠX (k) ,

k∈X(Ω)

!' "#O' ¬.4"jQ "#0]",%(.46 ¬DL%(3X%(0GW+!3

[sX ] =

X

sk ΠX (k)

»ÑKM ¼

k∈X(Ω)

(α, p)

L %

k=p X p αk (1 − α)p−k . k [X] = k k=0

I %(. I ' .*'ZX I ;0G+06Q 36' Z{) +3 I " I I .,%"4\E'¬!' 3R.,%m3!06' .4/*'Q "# 0G»s $+>?+.4D DP+3 p (x + a) 3+!3' >?]%("#!"S< ' $+3!"#>?+3]D ' "4 =%("#3 %(&.=K%(.4+' a3= !31' − >?$/ +G. " " DP' 3O.=
α=

n

jbEc+1 %1?b ÑKM %(. I ' .4+3 %(.4¬0G¹+0G/D 3!"#6*.4¬0G0G+E&SK%(3!",%( I / Z1 Î*M %(O.=?"#3]0GE!3$.=< "4 +-!"#$ ¢P¿¡ ¨ ¯K s¡sK¢ ³®« Ã .RU

H

GD

p(1 + x)

DV.#Åv 0G

576 6

x k+1 (1 + x)p−1 (1 + x)n−p = (1 + x)n−1

- I

=

0

-

(k + 1)

? A< <

BA8<>C 7A

C K

$

K +#

x

<>C

v9 "4+- + +'*)Ø>%(3!",%(Q .4G%(.4$U%("#3 ${¸ "4G!' 31.4Â0G¹+0GZ!DL% I ; X1 : Ω 7→ E1 X2 : Ω 7→ E2 D 3QL%(Q "#.#"#$ M :%(.43¬' >K%(3!",%(Q .4G%(.4$U%("#3GFÂ>%(.4+' 3 @AKGPDP(Ω, ' 3¬' ) %(R.=K%(3!",%(Q .46%(.4$U%("#3 "#.+3$!' .#/\-']« X Ä %1.4" ΠX *' I ' D .4 (X1 , X2 ) !O' ¬D 3QL%(Q "#.#"#$6!' 3.=K%(3!",%(Q .46%(.4$U%("#363$+.#.4/+

[f (X1 , X2 )] =

X

x1 ∈F1 ;x2 ∈F2

f (x1 , x2 )ΠX (x1 , x2 )

%U>? I 4. I %RDL%(3!!" I ' .#"4+3&!' "#>K%(E Q +E'O+ I ["4!"4%(E I ->?+L%(Q .4+0G+E¬.,%mE!"4 f Ú

Ä -9 " A1 ⊂ F1

[X1 ∈ A1 ] = ΠX1 A1 = ΠX (A1 × F2 ) =

Ä 9v"# φ1 : E1 7→ R %

[φ1 (X1 )] =

X

X

ΠX (x1 , x2 )

x1 ∈A1 ;x2 ∈F2

φ1 (x1 )ΠX (x1 , x2 )

x1 ∈F1 ;x2 ∈F2

-O%(D DP+.4$¬°#¡³ ÉG¯¥m©¡¢P¯°#´{³:*' I ' D .4 "#.V3!' .#: Î .4"S I L[ % I ' / R +'*)Â>%(3!",%(Q .4 DL%(3 {)*+0]D .4G DV"#EO\-' "SD 3$ I W +E&\E'j
X

ΠX (x1 , x2 )

x2 ∈F2

¢¢P´Á ´+ºv P ¯(³¬¿ Ë ¿º-¡§¥´:°#¯1°#¡A¿À ºv¢ ¨ ºeq°4´6¿E´¬°¯³U´{ºv°#´ ¨ ¢¢q¯K¡,³³+¯¢ ¨ ´:¿E´&³U´³ °#¡³OÉG¯¥m©¡¢P¯°#´{³{M ¢Ì(´+¥X³U´{É2´{¢L @A' 3 U%("# I 3 Z3$L$ I [ "#3X%F.=%(-6« jbEc+1 %1?b

Í ;DPKX Ω = {1, 2, 3, 4, 5, 6}2 , X1 (ω) = i , X2 (ω) = j si ω = (i, j) M Í ;06' "# Ω :Î/D 3QL%(Q "#.#"#6« !R.,%]D 3QL%(Q "#.#"#$/' "m3!0G/!' 3 *X%(*"4O\-' G< !R.,%]D 3QL%(Q "#.#"#$/' "m3!0G/!' 3R.,%]*",%(J?L%(.4 Ω Ñe¼0]DL%(3+3O.4.4"4R02%(3!JK"#L%(.4Ω *' I ' D .4 +.4Z\-' !0' " ' 2< (X , X ) Ω DP1' >K%("#2HµZD 3$+>?"#3 I /3$!' .#X%(OX%( I %(. I ' . Î?¼0]DL%(3+3 + 0 2 [X1 + X22 ] [X12 + X22 ] ?¼!0]DL%(3+3 + 0 [(X1 + X2 )2 ] [(X1 + X2 )2 ] Gv"4 +0]0G+E:' ]{)*DP${H 3!"4+ I &!3W*" S$+3+E¬*'Ç+ ¬Î6 $2\E' "V!0G- $+.#"4$ DL%(; 3 .4 I !"4 $+3+3R' I +R%( JK.4 {)*DP$+3!"#0G+-X%(. DV+3!0G+O I 0]D 3+*36DV' 3\-'" I OÎ10Gv W+.4 +E L%("4X%( I 6%('*);0G¹+0G.4"4R02%(3!JK"#L%(.4eM

!' J?$+$+3X%(.#"4X%(!"4;=% I "#.4 O.,% ^ _8gfLd %§fqg 5Ab 5·b0 _ c;%_ a&/§b _ /'E_*dUf!%§c(b n− uplet (X , X , ..., X ) 1 2 n !"4 I ' D .4 .4O%%(D X%(!"4 \-' "jU< "#0]DVK+E E.,%("4X$O%('Z.4 I + ' 36M Zg b jb-`h&/§b/1&/§_ /§f!%Á` #/d %§g·fq` %§_ /b/" 9*%(.4:*"#31v'O%>?O!' D DPKX$&\-'&!3:%(J?&D 3$$+.4 I H 3X%(. I I +3!L%("#.4 I ' 31 $+.4 I !"4D 3$"4 +E!"4+.#.4Z»§"#.Á3!X%("#\E'ÂÎ I %(*"4%(¼ DV+' U X%( I [L%( J?+3 S?+ "#3ÑUÙ&Ç!' 3O+8%(3!3!"4W+36+OU< "#-$+3+3¬%('ZD 3+0]"4+3O' 36qF] %(*"4%(:Y+R!' 3!' F6.4+' 3$+.4 I +' 3R\-' "Ps3!0G+-' DL%(3!!"#!"4 &.,%/DVD ' .,%(!"4 qI , E2 , ...Eq E DVK+3X% + .=
Zj = card{i |Xi ∈ Ej }

(1 ≤ j ≤ q)

» 3+D 3$+E2.4106Q 32S
ÑU

Z = (Z1 , Z2 , ..., Zq )

!&=% I "#.41+¬X+3X%2.,%("4X$]%('8.4 I +' 32».,%.4" V!¬Q "#0]",%(.4] I %(. I ' . ¬.4"4¬02%(3!JK"#L%(.4/ DL%(3X%(0GW+!31» nj ¼!¼ "#E$+3XZ'R02%("#-+L%(E:F1.,%.4" I
Z

,p

ΠZ (k1 , k2 , ..., kq ) =

[Z1 = k1 ; Z2 = k2 ; ....; Zq = kq ]

\-'L%( M k + k + ...k = p 1.=?1/ 2 I F I q %(. I ' .4+3:.4 I %(3*"#L%(. +3!0G%( + I M#M#MY@A' 3m%("#3 N I +.,%· 3+02%(3\-'S
p k1

p − k1 k2

Í Q "4+-O I ¸L%(.4+0G+E1«

p − k 1 − k2 k3

ΠZ (k1 , k2 , ..., kq ) =

....

p − k1 − k2 − ... − kq−1 kq

=

p! . k1 !k2 !....kq !

N p! n1 n2 nq = ( )k1 ( )k2 ....( )kq 1[k1 +k2 +....+kq =p] p n k1 !k2 !....kq ! n n n

+!:.4" ¬D 3QL%(Q "#.#"#$!' 3 !O%(D DV+.4$:.4"j0' .#!"#0]",%(.4 ¬DL%(3X%(0GW+!3 F = {0, 1, 2, ..., p}q : ¼ M Z 3 + 2 0 ( % 3 E \ ' + X 3 G % E \ ' : ! ' D DP3!R.4/'++0Q .4/!!3!" I O « » n1 n2 nq Í n

,

n

, ...,

n

, p, q

F

{k ∈ F | k1 + k2 + .... + kq = p}

( )+"J (0$ &'

1 %(D DV+.4&Q 3!"4W+>?+0G+E:.4¬ ${¸ "#!"4:+-!"4+.#.4:+&\-'+.4\-' 3$' .#X%(&m% I "#.4¬ E¬'& +3

DL%R.4 $+0G!3X%(!"4

g'5>EhÁb-g65·_g61?b5 (#·g·b o& _` &m%(0]"#.#.4:¸ "4

%//b5

-g·b-` b-gPd$1 S?$++0G+-&*"#:"# $+DP+%(-6!"

(A1 , A2 , ..., An )

[

\

Aj ] =

j∈J

g'5>EhÁb-g65·_g61?b5Ab A 5 b#' 9v"4+-

C

+

D

Y

∀J ⊂ {1, 2, ..., n}

%

Aj

j∈J

1 /§_2Ub$1

+'*) I .,%S?$++0G+- *"#3X%]\E' + ER"# $+DV+%(E¬!" C D ∀C ∈ C ∀D ∈ D

[C ∩ D] =

C D.

6>VMT%*M +3X%2*"#"# $+DV+%(E]S< ' I .,% !"A.4 I .,% XE¬"# $+DP+*H D {Y ∈ Γ; Γ ⊂ E} et D %(EM Y

ÑKÑ

g'5>EhÁb-g65·_g61?b5 (#·g·b &o_` %//b5·b!*_ c;%§_ a/§bG_ /'E_*dUf!%§c(b 9v"4+-» ¼' &=%(0]"#.#.4&¸ "4: ¬>%(3!",%(Q .4 %(.4$U%("#3 Z*"#\-' I +!&m%(0]"#.#.4:!R"# $+DV+*H %(E!" Xi : Ω 7→ Ei

∀Γ1 ⊂ E1 , ∀Γ2 ⊂ E2 ..., ∀Γn ⊂ En ,

43

[X1 ∈ Γ1 ; X2 ∈ Γ2 ; ...Xn ∈ Γn ] =

[X1 ∈ Γ1 ] [X2 ∈ Γ2 ]... [Xn ∈ Γn ]

@A' 3\E'j< "#.S+"#R%("#!"= "#.S!' GR\E'&.4$+JE%(.#"#$D 3$ I $ +E "4+E>?$+3!"¸L$\-'L%(Â.4X'+X+06Q .4 Γi -3*' "# F1 RDV"#EM 9-" XE %(D D .#" I %(!"41+&!" !!"#!'6' /m%(0]"#.#.4/"# $+DP+%(-Â"#.+ (X1 , X2 , ...Xn ) I i 7→ Fi ! :f0Gi ¹+: 0GE6 ¬.,%m%(0]"#.#.4n (f1 (X1 ), f2 (X2 ), ..., fn (Xn ))

g'5>EhÁb-g65·_g61?b bEdGh·c(f95%d 5·b /f<%21 9v"4+- + ÎZ!DL% I /D 3QL%(Q "#.#"4$ %(D DV+.#.4+3X%Öq¥!B!¯B+¡°²¡§ Ë q¥e¿º-¡s : + (F1 , Π1 ) (F2 , Π2 ) Π 1 Π2 \-'.=?+0G+EOΠÎ6'F+1×+06 Q .4R 2 9-"RL02%("#-Γ+1LetΓ %(EÂ !&' I ' D .4] >K%(3!",%(Q .4/%(.4$UF%(1etF "#32G Π(Γ1 × Γ2)-=:"#Π1 (Γ $+DP1+)× %(Π-2(Γ/2!)" X = (X1 , X2 ) X1 etX2 M ++' .4+0G+- !" B' I !X%(+Ò ΠX I =e\-Π'X1× %(ΠX.4 2I %S< ' I ' D .4 >VMT%*MU"# $+DV+%(Ee.4j.4"4j02%(3!JK"#L%(.4 $++3!0]"#+- .,%.4"*' I ' D .4 02%("4R' .4R>VMT%*M :-DL%R"# $+DV+%(EM#M#M

UdUb-g61?b b?d #Ag6%1 %dØb-g /§f<% 5(#Agb #%dUb 5·b _ c;%_ a&/§b 8_ /'E_*dUf!%§c(b %sg65 -hb-g65·_gPdUb/ #% 5&%dUc;% a&# -b #" 9v"4+- ' ++06Q .4&¸ "S+ ' .4"S &D 3QL%(Q "#.#"#$/!' 3 .,%D 3DVK!"#!"4Z\E' "S!' "#R%DP' 3RQ*Ç!+.,% E µ E I !!3!' I !"48S< ' /!' "# (X1 , X2 , ....., Xn ) ¬>K%(3!",%(Q .4 %(.4$U%("#3 "# $+DP+%(- + $\E' "4*"4!!3!"#Q '$ .4" M 7 I + V+6*DVK µ Ä Ω = En Ä Xi (ω) = xi " ω = (x1 , x2 , ...., xi , ..., xn ) Ä = µ × µ × ... × µ = µn Í %]%(.43R.4/3$!' .#X%(O!' "#>K%(EO -.,%] $+0G!!3X%(!"4vm% I "#.42 !R.,%("4$/%('Z.4 I +' 3M '%

c(fLhf! %d %§fqg ÑK= M !R.=< ' 4" \E'6D 3QL%(Q #" .#"#$6!' 3 X'.,%\E'+.#.4:.4O>PMT%*M XE"# $+DV+%(E& ¬.4" Ω X1 , X2 , ..., Xn µ Î*= M !R.,%1D 3 QL%(Q "#.#"#$/' " m3!0G/!' 3 "j+O+' .4+0G+E !" !R.,%1.4"S' "m3!0G6!' 3 M

Ω

µ

E

jb-`h&/§bG1#·g·b _ #dUc(b _ hAh·c(f91+:·b 5Ab /§_ /§f!%a%sg·fq` %_ /§b 9v"#e» ¼G' ' "#8 $+>?$++0G+-Z"# $+DV+%(E ®0G¹+0G®D 3QL%(Q "#.#"#$ DVK A1 , A 2 , . . . , A p p α Pi=p M '%(.#.4>?"#3\E' ' "#Â' ®.4"¬Q "#0]",%(.4 ®DL%(3X%(0GW+!3 MN >?+ 3!!' Z = .=< "# $+DV+i=1 %( 1I A6i %1+ S+/ !" Z !' ¬DL%(3!!"4 I %(3*"#L%(. α, p Ai I {1, 2, ..., p} k

{

\

i∈I

Ai

\

i∈I c

Aci }

k

= α (1 − α)

p−k

,

ÑUÎ

[Z = k] =

p k

αk (1 − α)p−k

GD

H

m%("#\-' %*0G+!&0G¹+0G&.4"P\-' .,%/>VMT%*M *';%(J?1»s{)*+0]D .4 a ¼{EjK%(3!",%(Q .4%(.4$U%("#3 E.4%(D D .#" I %(!"4 = Ep I v3 $ SPMT%*MSXE6 I "# $+DV+%(XEi+/!' "#>?+-6' 1.4" -& I "# $+DV+%(E: D 3QL%(Q "#.#"#$ n1 I \-' "·DV+3!0G+ ' "m3!0G' 3 M ¬$+>?$++0G+E n SV∈MT%*EM 1 ] P M Z1 = 1 i [Xi ∈E1 ]

Uh Ec(_g61?b5 (#·gÓh·c(f95#%d5Ab _ c;%§_ a/§b 2_ /-_*dUf!%§c(b %sg65 -hb-g65A_gPdUb $1 9v"4+- + XE:Î2>K%(3!",%(Q .4:%(.4$U%("#3¬"# $+DP+%(-6F2>%(.4+' 3:%( + +'*) s I !"4 X Y E f g E' 0G$+3!"4\-'&!' 3 %(.43

E

[f (X)g(Y )] =

[f (X)] [g(Y )].

NZDL%(3!!" I ' .#"4+3 !" + -3$+.#.46 M X Y [XY ] = [X] [Y ] 7 D 3WO%>?"#3 3+02%(3\E'$/\-'j< "#.j!' GO ¬0G-!3+3O.,%] +'*)*"4W+0G6%X+3!!"4 *DPKX

43

[XY ] =

X x,y

xyΠZ (x, y) =

X

xyΠX (x)ΠY (y) = (

x,y

Í Z+3X%]\-':!"S.=< ' / Î1>PMT%*M I +E!3$2

X

x

xΠx (x))(

X

Z = (X, Y )

yΠY )(y) =

%

[X] [Y ]

y

[XY ] = 0

:_ c+%§_g61?b 5(#Agb efq` ` b 5Ab "#_"#c"%sg'5>Ehbvg'5A_gPdUb 9v"# ' &= %(0]"#.#.4/ >VMT%*M 3UM "# $+DP+%(- DPK M (X1 , X2 , ..., Xn ) n S = X1 + X2 + ... + Xn Í %]%(.43 V ar(S) = V ar(X1 ) + V ar(X2 ) + ... + V ar(Xn ).

XE I +E!3$ .,%2¸*' I %(. I ' .·>K%G+' "# " X/3X%(0GW++S+&%("4$+0G+-:%(' I % È.4O>VMT%*M Í Xi I 0]D :+-'*'DV"#ED 3$ I $ +-M < [--Å DV![W/S< "# $+DV+%( I !R" I "j+-!"4+.#.42M jbEc+1 %1?bG1 N $*' "#3:' Z' >?U%(' 0Gv : I (% . I ' . ¬.,%]>%(3!",%( I /S< ' /.4"jQ "#0]",%(.4GM

Ñ

I 0]D 3 $+[+!"4¤%(D D 3(m*"4; I I [L%(D "#!3;!"#*"4!DP+%(Q .4DV' 36.,%!' "# ' I 0]0G+ I +3 DL%(3' X3 %(D VD +. R!"4 $+.4$+0G+EX%("#3M

( <

3

54: 5A<

0

:

"%$ , . )+/)

:

%:D 3QL%(Q "#.#"#$ I *"#!"4*H D 3QL%(Q #" .#"4$ %U>? I 9v"4+E + +'*)2$+>?$++0G+ES< ' Â!DL% I +.#.4 A B \-'L%( »sZ*"#&%('!" +X % I [L%(E U¼(Ω, +3X%Z)*DL%(3O ${¸B "#>!"40 [A|B] A B B

GE

E

[A|B] =

Í %1$+>v"4 +0]0G+E .4$+JE%(.#"#$

(A + A0 |B) =

(A|B) +

AB B

(A0 |B) ;

[Ω|B] = 1 ;

[φ|B] = 0.

% m I !"4 S?+.#.4/D 3QL%(Q "#.#"#$!' 3 Ω B B (A) = B +'*)D 3D 3!"4$+$ !' "#>%(-O- I *m3!0G F.=< "4 $:"#-!' "#!"#>?6\-':.=? X% I [L%(- \-'j< "#.SD .4+' 6!$+JE%(.4F;Ñe¼¬M Ä $+>?$++0G+-;"#E>K%(3!",%(EÂDL%(3 I I [L%( J?+0G+-Z ZD 3QL%(Q "#.#"#$®EÂ{) % I +0G+-;.4$+>?$++0G+- »§.,%®D 3QL%(Q "#.#"#$ "# $+DV+%(EÂ «R¡§°Ì¯;P°#´{ºvÌ¡§¥!]0Gv*"¸L$ZDL%(3].,% I L%("4%( I Z .=?$++0G+-«V°4´ µ´{BÉ L³´³{ ¨ ºvÌ(´{¥+ /*02%("4RA¬.=?$++0G+E&"# $+DV+%(E« À ¯¡ ºvG ¢ B{¥!´+°¯¢®³U´{¥{ÌU¡¼¬M

jbEc+1 %1?b $1 ÑKM Í *"4!DVKÂ I %(3!G%("#" I !!"#!'$ «.,%D 3+0]"4W+3!Q .,%( I [!' 3]]ÎZm% I ·.,%8 I "#3]' 3¬X/ÎG=% I :+:.,%!3"4!"4W+0G]DVKW ' m% I 1Q .,%( I [G+¬' m% I "#3 ' %(0]"!"#3]%(' [L%%(3;.=< ' :S?'0GE!3:' : ¬=% I :«*+.#.4:!Q .,%( I [ \-'+.#.4:! .,%]D 3QL%(Q "#.#"#$6\-'¬.=v"4*' R I 0]D 0]"#.#.4DV+3 %(!+"#E8S< ' I +3!X%("#02%(.,%*"4 M Í ×*"4!DVKDV' 3G.,% $ I +.4+3ÂS< ' Ó!\E' "j

H

Ñ+

Ä %(D D .#"4\E'$6F' /DP+3 /02%(.,% G*"#. :' Z3$' .#X%(DPK"#!"%>? I ' ¬D 3QL%(Q "#.#"#$6 0, 9 Ä %(D D .#"4\E'$6F' "#*"#>-"4*'ZQ "4+ DP3!X%(-6 "#.j!RDVK!"#!"%>? I .,%1D 3QL%(Q "#.#"#$ 0, 1 Í Â!"#3:%('[L%%(32' Â"#*"#>-"4*'%( I +! DPD ' .,%(!"4 "#.V!Á$ODVK!"#!" \-'+.#.4 .,%6D 3QL%(Q "#.#"#$ \-'j< "#."#R02%(.,% Â»§3$+DVG«SÑ vÑKÑUÎ?¼ * M v"4$O%(0G$+3!" I %("#;!DVK!$ +>K%(E!3"4DV3! I .4K*"4!"#0' .,%(ERÎ I [W+>v3 +6' I %*"#.#.,% I ' ;Ç!'+' 32j$+>v"4 +0]0G+E $!"#3+'*) ]JE%(JK+3/.,% I %*"#.#.,% I I ["4!"#%(' [L%X%(3' DV3!&M ? .=v3&%(.43' & Î6DP3! I ["4!"4]+m%("#+ØX3! 0GE!3+31' I [W+>v3 "#."#*"4\-';%(.43]%(' I %(*"4%(]\E'j< "#.DP+' 1+ I 3 0Gv*"¸L+3 I [")]U< "#.*.4R $!"#3¬M?+.#' "H I "q%H=H="#. "#E$+3¹+FO.4=%("#3 /»§3$+DV:«' " U< "#. $!"#JK.=?"#!' 3e¼¬M

+6859685

J+ K

I

II

< A8A <

- I

<

BA < 7A

K +#

<

9v"# ' G>%(3!",%(Q .4G%(.4$U%("#32FÂ>K%(.4+' 3/%( ,. %8°K¡ ¨ ¢q¿¡s s¡sK¢¢P´{°§°#´ \E'L%( +3X%® X B DL%(3O ${¸ X"#!"4*.,%.4" \-'L%( !0' " :.,E%D 3 QL%(Q "#.#"# $ %1 ΠX/B I X

Ω

ΠX/B (x) =

B [X

B

= x] =

[X = x |B] =

[X = x ; B] B

∀x ∈ X(Ω)

58E!3O' 3R' {)*+0]D .46\E'¬.4 I *"#!"4 +0G+- DL%(3O' Z$+>?$++0G+E&0G-*"¸L:.,%.4"·S< ' :>PMT%*M jb-`h&/§b Í ®Ç!' n m"4GF ' 8Ç!+' ;D "#.4Â'=% I ;Q ",%("4$ .4 k I ' D¤!13+D 3$+E$;DL%(3]' Â>VMT%*M Xk .,%;!' "#G 1>VMT%*M» ¼ !/!' D DVK$1"# $+DV+%(EKMA9-" +3!' .#.#">?$+3!"¸q%(E Pp [Xk 3=+D 1] 3$=+-α;.40Q 3 ÂD "#.4Xk01QP≤ $Gk%(' ≤ I n' 3] D 3+0]"4+3 I ' D ' p ≤ n Sp = k=1 Xk p X%U>?O\-'¬.,%.4" !RQ "#0]",%(.4 :DL%(3X%(0GW+!3 M Sp (α, p) 'R02%("#-+L%(E&' %(' !3/+-!"4+3 + {) %(0]"#O.,%.4" \-'L%( M Í 8%

H

[Sp = k | Sn = q] =

q≤n

[Sp = k ; Sn = q] = [Sn = q]

Sp

[Sp = k ; Sn − Sp = q − k] = [Sn = q]

[Sn = q]

[Sp = k] [Sn−p = q − k] [Sn = q]

% +3! "4W+3$+JE%(.#"#$/D 3e>?+L%(E¬ I 6\-' !O"# $+DP+%(- + %10G¹+0G6.4"\E' M O ] Sn − S p Sp Sn−p %I (. I ' .S=% I "#.4:0GE!3%(.43\-'

[Sp = k | Sn = q] =

NZDL%(3!!" I ' .#"4+3*DP' 3

H

p=1

[X1 = 1 | Sn = q] =

p k

n q

n−p q−k

[S1 = 1 | Sn = q] =

»mÎ*M#Ñe¼

((p + q − n)+ ≤ k ≤ p ∧ q)

q , n

[X1 = 0 | Sn = q] = 1 −

q n

»mÎ*MTÎ?¼

%(.4 I *"#!"4 +0G+E:DL%(3O.=?$++0G+- .,%].4"jQ "#0]",%(.4 ! +>?+E'.,%].4"[-Å-DV+3!J?$(H n = q] 0G$+!3!"4\-']3+ I E!3$;%('DL%(3X%(JK3X%(D [» . " "[S /\E'L%(-F;S+.#p.4]3!$G' ].4"Á 0+3!' .#.#" " ¼ I +.#.4G 02%("4/%>? I ' ®DL%(3X%(0GW+!3]*" S$+3+E]M Í 3+02%(3\-'+3X%;\E'X.41¬.4"4 I *"#!"4 +.#.4:!3' >?$¬1 $+DP+ +- D .#' α

ÑUÙ

+68596 : ;< 4<

,

+

9v"4+-

G<>=@<

K+# II

I

+'*)Z+E!"4+3

I

+

< A = BA

- I

K

-

J+ I

= 7A < % K

p > 0 et α1 , α2 , ..., αq Z*"# \E'j< ' :>VMT%*M qDVK q reels > 0 H = {k ∈ {0, 1, , ..., p}q | k0 + k1 + k2 + ... + kq = p} Í F¬>K%(.4+' 3%( ! 0 ' # . ! # " ] 0 , " ( % 4 .

O L D ( % X 3 ( % G 0 +!3 Z = (Z1 , Z2 , ...Zq ) {0, 1, 2, ..., p}q . " " ¼!"j+.#.4/%*0G+ODV' 3R.4"j.,%D 3QL%(Q "#.#"#$!' 3 *DV3!$:DL%(3 ${¸ "4(α :DL1 ,%(α3 2 , ..., αq , p, q) q {0, 1, ..., p} H

[Z1 = k1 ; Z2 = k2 ; ...Zq = kq ] =

p! k αk1 αk2 ...αq q k1 !k2 !...kq ! 1 2

»I

∀k ∈ H

9v"# ' +.#.4>%(3!",%(Q .4R%(.4$U%("#3 0GE!3+3\-'.,%O.4" I *"#!"4 +.#.4 : » ¼·\-'L%( Z Z1 , Z2 , ..., Zq−1 [Zq = j] !R.,%1 .4"S0' .#!"#0]",%(.4 /DL%(3X%(0GW+!3» n1 n2 ¼ & M nq−1

+68596 :FC ? )

4<

7 I

<>A A <

K+ K I I I

n−nq , n−nq , ..., n−nq

G<

-I

A< A

K

,p − j ,q − 1

< #<>A A <

+# K

v9 "4+E ' />K%(3!",%(Q .4%(.4$U%("#33$+.#.4+ ' $+>?$++0G+-: /D 3QL%(Q "#.#"#$ -' .#.4 .=?+0G+EFG.,%2D 3QL%(Q "#.#"#$ »§"#E!3v*' "#G%(' $+Q ' ¬ I I [L%(D "#!3e¼P+3X%2$ ' [X|B] [X] B ++3X%G%(D DV+.4$/!DV$+3X%( I I *"#B!"4 +.#.4 \-'L%( M

B

»mÎ*M ?¼

[X1B ] B

[X|B] =

: $+0G-!3%("4$+0G+- !"S.=
B [X]

=

X

ai

B Ai

=

X 1 1 1 X a i 1 Ai ] = ai [1Ai 1B ] = [1B [X1B ] . B B B i

i

i

I %(. I ' .D 3X%(!"4\-' ;.=K%(3!",%(Q .4Z%(.4$U%("#3ZU

[X|B] =

X

»mÎ*M -¼

kΠX/B (k)

k∈X(Ω)

jb-`h&/§b $1

+>?+%('*) .4"4 I *"#!"4 +.#.413+ I E!3$G%('DL%(3X%(JK3X%(D [ÂD 3 $ I $ +E] $DL%(3.46s3!0' .4 »mÎ*M#Ñe¼+6»mÎ*MTÎ?¼ ZQ !"4+E +8%(D D #. "4\E'L%(-»mÎ*M -¼

1

[Sp | Sn = q] =

»§B"#3 F I /'Ç+R.=
+68596 : ;< 4< H %(R.=

pq , n

. " "

¼

[X1 | Sn = q] =

q n

,

» Ç!+ /Î] $¼& +$ +3!0]"#+3 .,%1.4" I *"#!"4 +.#.4 \E'L%( X1 [S = q] +Z $*' "#3 \E' q

(2 ≤ q ≤ 12 , S = X1 + X2 )

[X1 |S = q] =

Ñ

2

»mÎ*MTÙ?¼

+68596

<>=@< <

J+ K I I

0 I

I

+

#? <

,+ I

I

4PMT%*M + ER"# +$ DV+%(E DP' R 3 \-'j< "#.j+Z"#%("#!"= Í Z*"#3X%1\-' "#.V=%(' O+R"#.!' XG\-' "#R"#B $+DV+%(E& X 1B I \-' "jU
43

[X = k | B] =

[X = k] ∀k.

»mÎ*M ?¼ ! I ' I *"#!"4;$ I %("#3O+' 2X%(E¬%!' 3X%(-Á.=< "# $+DV+%( I & + M 2+3X%\E'»mÎ*M ?¼ X B Í +E!3X%4 »mÎ*M )K¼

43

ΠX/B = ΠX

[X|B] =

+68596 : ;< 4<

[X]

,

Í .,%( I m"4]' ¤ $ .4]3$' .#X%(G!' II !"m]-G$ X1 , X2 , ..., X8 +].=
+68596 : ;< 4< ,

×%(D DV+.#.4 v9 "4+- !3"4G>VMT%*M 3UM"# +$ DV+%(E8 .,%8.4"O X1 , X 2 , X 3 S = X 1 + X2 + X3 µ X3 ! $6DL%(3 0GE!3+3&\E'¬.,%].4" I *"#!"4 +.#.4 -\ 'L%( S

[X1 = i; X2 = j]

»mÎ*MTÚ?¼

[S = k | X1 = i ; X2 = j] = µ(k − i − j);

+ $*' "#3:.=
»mÎ*M#ÑU?¼

[S | X1 = i ; X2 = j] = i + j + [X3 ].

58E!3+3»mÎ*M#ÑU?¼R*"#3 I +0G+E»§"=MYKMLX%( I (% . I ' .4+3O.,%.4" I *"#!"4 +.#.46 ¼&M S

+68596 : ;< 4< ,

Í !' D DVK&.=?$+3¸L$:DP' 3R' />PMT%*M 3UM*D 3+L%(-&Î>K%(.4+' 3 0GE!3+3&\E' X !O%(.43"# $+DV+*H %(E M 3' >?+3&' I E!3{Hµ{) +0]D .40GE!3X%(-¬\-' I +.,%Gj%(.4+' 3M 3

43

+6 685

(

?' >?+E2$+$Z%(0G+$ÂF I !"4 $+3+32 ]{)*D 3!"4G ;.,%m3!0G È !O' />VMT%*M3$+.#.4+ ' />VMT%*ML\-'+. I \E'F]>K%(.4+' 3 %(O' ++0Q .4¸ " MqN F J?$+[X|Z $+3X%(.j=.4:i]3$' .#XX%(O $+DP+ I :\E' "SZ'&%(0GW+6F1 ${¸ "#3O' :s I !"4 DL%(3 i

φ:F →R

[X|Z = i] = φ(i)

Ñ )

»mÎ*M#ÑKÑe¼

» Í ZDVK+3X%**DL%(3 I ->?+E!"4j !" ¼{M ¬02=%( 0"4W+36D [Z .#'R=X3 %(i]02= %0$K Ã .j!O%(.43L%(!' 3+ . ¬DVK+3Uφ(i)

[X|Z] = φ(Z)

E

»mÎ*M#ÑUÎ?¼

@' "4\-'Z.4 I 0G+06Q 38 »mÎ*M#ÑUÎ?¼G!G' >VMT%*M\-' ">K%(' O» I ?+' Â"#X%(E%('*)Â{) +0]D .4*'ÂDL%(3X%(JK3X%(D [ " .]DP' 3RU< [L%(Q "#!'+3OF I +!& X%(!"4M Z=i $+JE%(.#"#$»mÎ*M#Ñe¼:j»mÎ*MTÎ?¼¬j»mÎ*MTÙ?¼¬»mÎ*M ¼ +>v"4+ +E¬%("#!"S3!DV I !"#>?+0G+E

E 1

[Sp = k|Sn ] =

p k

n−p Sn − k n Sn

Sn Sn ; [X1 = 0|Sn ] = 1 − n n Sn Sn [Sp |Sn ] = p [X1 |Sn ] = ; n n [S|X1 + X2 ] = X1 + X2 + 21

[X1 = 1|Sn ] =

%(' 3X%$\-'.,%>PMT%*M I *"#!"4 L%(E DP+' ¬D 3+*3]&>K%(.4+' 3¬%(¬' ®DL% I D 3v*' "# I

+6 6

G<

GI

C

[S|X1 , X2 ] =

+# + I

c(fLhf! %d %§fqg

G< :

[S|Z] = X1 + X2 + [X3 ]

[X|Z]

!R.=< ' 4" \E'/>PMT%*M 3 >?$+3!"¸q%(E .4OÎ1D 3D !3 "4$+$ !' "#>%(-6« Y KÑ M !' &m I ! "4 $++3!0]"# "4! Y Z Î*M [Y ψ(Z)] = [Xψ(Z)] ∀ψ : E → R Ã .·! I .,%("#3 S?$+3!"¸L6.,%]D 3D 3!"4$+$ÂÑKM Ë ÉG¢ ³{ s¥!¯K s¡m¢ .4$+JE%(.#"#$

[X|Z]

[φ(Z)ψ(Z)] =

X i

φ(i)ψ(i) [Z = i] =

1

X

[X1[Z=i] ]ψ(i) =

i

[X

X i

1[Z=i] ψ(i)] =

H

0GE!3+-6\-' >?$+3!"¸L]Î*M O$ I "#D 3v\-'+0G+EUS"# ' >VMT%*M 3UML>?$+3!"¸q%(-2ÑKMP+:Î*M I +!1 +3! "4W+3 Y Y = h(Z) D 3D 3!"4$+$$ I 3!"#6DV' 3 L"jQ "4+8\-' I *' "#:F].=
Ñ

+6 6

9? C

)

7

K+# I

KÑ M [aX + bX 0 |Z] = a [X|Z] + b [X 0 |Z] ∀a ∈ R ∀b ∈ R Î*M X ≥ 0 =⇒ [X|Z] ≥ 0 * M [ [X|Z]] = [X] M 9-" + ER"# $+DV+%(E X Z [X|Z] Ù*M 9v"# ' &m I !"4Z-' 0G$+3!4" \E'!' 3 % ψ:F →R

F

»mÎ*M#Ñ?¼

[Xψ(Z)|Z] = ψ(Z) [X|Z].

NZDL%(3!!" I ' .#"4+3

[ψ(Z)|Z] = ψ(Z).

*M6»mÎ*M#Ñ?¼DP+' O¹+!3/J?$+$+3X%(.#"4$ I 0]0G6' #" % I "4:F1.,%s I !"4 H : Ω×F →R ! "S.=%(3!",%(Q .46%(.4$U%("#3/3$+.#.4 % Y = H(., Z)

K(., i) =

[H(., i)|Z]

»mÎ*M#Ñ+-¼

[Y |Z] = K(., Z)

DPKX ' :>VMT%]F>%(.4+' 3 %( "# $+DV+%(E + E Z h:E×F →R M %]%(.43 v(i) = [h(X, i)] (i ∈ F ) Í

)vM 9v"4+-

[h(X, Z)|Z] = v(Z)

»mÎ*M#ÑUÙ?¼

$-` fqg6dUc(_*d %fPg' ÑKM+OÎ*ML$+X%(EO$+>v"4 +EUDL%OF" 8%

[φ(Z)] =

X

φ(i) [Z = i] =

X

[X1Z=i ] =

[X] .

i

i∈Z(Ω)

MR@A' 3Z0G-!3+38Ù*M#R 3+02%(3\-'+3X%¤\-' DV"#EMD 3U>v"4+E I \-' %1φ(i) I = [X] ∀i ∈ Z(Ω) M [Xψ(Z)|Z = i] = ψ(i)φ(i) [Xψ(Z)|Z] = ψ(Z)φ(Z) = ψ(Z) [X|Z] 58E!3O+*¸ .,%s3!0' .42»mÎ*M#Ñ+-¼{M@AK DP' 3 I +.,% 8%]%(.43 u(i, j) =

S
[Y |Z = i] =

[Y |Z] = u(Z, Z) = K(., Z)

[H(., i)|Z = j]

[H(., i)|Z = i] = u(i, i)

M %m3!06' .42»mÎ*M#ÑUÙ?¼+3$!' .#6%("4$+0G+-« !"S.=
.,%m I !"4 >?$+3!"¸L6%(.43 +>?+3!!' ¬.=< [-Å-DV![W/S< "# $+DV+%( I 2M u u(i, j) = v(i) ∀i ∀j ( % ' X 3 Z % G $ \ ' ] 4 . : V > T M * % M \ ' Á " E : m I !"46 $++3!0]"# "4! Ç'+-6' 3 .4GDL%(3!!" I ' .#"4+3 Í U Z %(R.4 I *"#!"4 +0G+- DL%(3 M N.#.4 E I %(3X% I $+3!"4$DL%(3.,%D 3D 3!"4$+$!' "#>K%(EG« \-'+.4\E'6"# .=? ?.,% I L%("4X%( I Z +-!3X%4 I +.#.4 M Ã E!' "#!"#>?+0G+-UvDP+' 3+D 3$+E+3 I +.,% ω Z(ω) U (ω) I 0]0G:.4 I -!3X%("#3/ ¬.=< "# $+DP+%( I «*!"= +ZV+U*' ¬>VMT%*M V !R"# $+DV+%(E Z *.,% I L%("4XX%( I j%(.4+' 3DPK"#Q .4 M Z(ω) V (ω) ÑUÚ

+6 6 : ;< 4<

,

I

G<>=@< < ? <

K+# II

I

+

,+

+ K I

+

7A PMT%*M 3¬!' 3 + .4 9-' D DVK (ω) > 0 ∀ω '+X+06Q .46 s3!0G$6 s I !"4|Ω| =$+n+3!0]"# "4!O V DVK+3X%]+*¸ M Ω VZ V Z |Z(Ω)| = p +3 .,%]*"#0G+"48 6.== M [XY ] VZ 0GE!3+3 \-' Î*M 9v"# !R.,%D 3(Ç! I !"4+' I "4*"4+ / !' 3 M X ∈V [X|Z] X VZ *M Í 2 .4X'!DL% I R s3!0G$ >PMT%*M 3UM?\-' "-Á m I !"4% GÁ M '+.#.4 !X% RZ *"#0G+"4 658E!3+3&\E' 3R.,%]D 3V(Ç I !"4 +' I "4*"4+ !' 3 Z$/DL%(3R.=
3

Y =

[X] + r(Z − [Z])

\-'L%(Z%DVK$ r = cov(X,Z) var(Z) M58E!3+3&X%( I %(. I ' .4R.=< "#$+JE%(.#"#$ M [(X − [X|Z])2 ] ≤ [(X − R[X|Z])2 ] ! " Ù*M58E!3+3&\E' 3

[X|Z] = R[X|Z]

8

,-! "%$ , 6

>:

|Z(Ω)| = 2

0!

:54:5 <

'Â' D DPK+3%( I 8 DL%(3X%(JK3X%(D [\-' .4®!' D VD 3! ( !;$+JE%(./F M7>K%(E I 0]0G+ I +3 Ω M Ã E!3v*' "4 OS(% -G« "#E+3!3J? 'O' 3R.,%1 $+PD +%( I +

Z

[X|Z]

_ ced %d %§fqg×b-g L bvg'5Ac - b hA_ c A # gb! _ c+%§_ a&/b_ /'E_*dUf!%§c(b " !!"#!' 9v"# ' >VMT%*M >K%(.4+' 3:%( .,%2m%(0]"#.#.4¸ "41S?$++0G+E Z E {[Z = k] , k ∈ Z(Ω)} I ' GDL%(3!!"#!"4¸ "42 %(D DV+.4$GDL%(3!!"#!"4Ø+ J?+*3$2DL%(3 +.#.4G+3X%$ M +>?+%(.431F;.,% PZ ${¸ "#!"4 ¬.=
1

X

[X|Z] =

Ak

c k 1 Ak

k∈Z(Ω)

Í ®>?"#:!' 3 I +!6m3!06' .4]\-' + J?+*32M 7 #" !"=

[X|Z]

1 $+DP+

[X|Z] =

Z

\-'DL%(3:.=< "#E+3!0G$*",%("#3G .,%DL%(3!!"#!"4\E'j
[X|Z + 1] =

[X|Z 3 ]

D ' "4\-'.4]DL%(3!!"#!"4Â+ J?+*3$2DL%(3 E2"4 +E!"4\-' +¤3+>K%( I [K!" !F®>K%(.4+' 3 Z, Z + 1, Z 3 Z %( + 2 -*" S$+3+EeM %/>?$+3!"#X%(Q .4&!"4;"#E$+3%(E¬! I I +.#.4& +1} [X|Z] I *"#{−1 !"4 ,+0G +E¬DL%(3R' :DL%(3!![X|Z "#!"4 ]Z! I *' "# %('*) ${¸ "#!"4&!' "#>%(-/«

/EaAcbb-g

Lbvg'5Ac -b hA_ c #Agbh·_ cd %d %§fqg " Ä " P ' DL%(3!!"#!"4 Ω q®%(D DV+.#.4+3X%8¯° © B{¥´´{¢v©E´{¢q¿¥ Ë ´:*¯¥ P +&.=?$++0G+E m3!0G$6 O3$+' "4 SK%(.4+E/F I +.#.4: &.=
ÎK

3+.,%(!"#>?+0G+-;%('*)¤DV$+3X%(!"4GQV-.4$+ \E';j
F1 etF2 ∈ F =⇒ F1 ∪ F2 , F1 ∩ F2 , F1c ∈ F

+3X%2$+JE%(.4+0G+E&\E' %(D DL%(3!!"4+ +-¬F Í Ω et φ +3E&%(D DP+.4$¯K =É2´³6 ¬.=?$+3!"¸L1%("4$+0G+E:\-' I +! P P0 P D 3D 3!"4$+$:$\E' "#>K%(.4+E .=< "# I .#'!"4 M Ã .S+!%("#!"j!"SP+R+' .4+0G+-O!"V' O%(0G: 0 0 F ⊃F F %( M F

+6 685 A < < 9 < )

I

I

>< ?

< 7= A8A8< <

/- I

4

BA8<>C 7A

C K

2

K +# K

<>C

9v"# ' >PMT%*MVF2>%(.4+' 3:%( M T< %(.#J?W+Q 3 + J?+*3$DL%(3 +3X%*LDL%(3¬ ${¸ "#!"4jV.=?F(Z) "#3O\E' PZ

F(Z) = {[Z ∈ Γ] ;

9-"

Γ ⊂ E}

R' I ' D 4. *' &%(0G %(D DL%(3!!"4+E¬F D ' "4\-'jK%(3!",%(Q .4·%(.4$U%("#3«U"# Í Z1 , Z2 , . . . , Z N ' /!' "#: />PMT%*M .=
PZ

Fn

+6 6 )

BA8<>C 7A

C K

$

K +#

F(Z1 , Z2 , . . . , Zn )

<>C F =@<>C

F1 ⊂ F 2 ⊂ . . . ⊂ F N -

BA8<>C

9v"# ' DL%(3!!"#!"4]+ J?+*3X%(E' R%(.#J?W+Q 3 ' >PMT%*M +3X% *"# ¦oÉG´{³ºv¥¯B+°#´O!"-+.#.4! I !X%(E F !' 3 I [L%\EP'6%(0G »§"=MYKM!' 3 I [L%\-'/$+>?$++0G+F-& ¼{M Y F P R3+02%(3\-'O' "#>%(-'OX+3E' !"#.4 Ä Y etY 0 - Î1>VMT%*M 3UM F H=0G' 3X%(Q .4O>?$+3!"¸q%(- [Y 1F ] = [Y 0 1F ] ∀F ∈ F L%(.43 Y = Y 0 M Ä Z$+>?$++0G+E F ∈ F !"j++' .4+0G+- !" 1F F H=0G' 3X%(Q 4. :_ c+%§_ a&/§b G_ /'E_*dUf<%§cb

F(Z)

`b #·c_ a&/§b

'%(.#.420GE!3+3\-' I ZXEG' 2"#0]D .4+0G+EÂ.4]>PMT%*MÁ\E';'Â%>?;%(D DP+.4$/Ç~'\-'j< " I " 02%( "4W+3:DV+' $+.4$+JE%(E s I !"4 $++3!0]"# "4!O 1M Z cfqhf % d %§fq@ g 1 9v"# ' :>VMT%*M 3UM .4 +'*)D 3D 3!"4$+$O' "#>%(-O-O$\E' "#>K%(.4+E Y ÑKM ! H=0G' 3X%(Q .42M Y F(Z) Î*M Ã .j{)*"4!¬' ¬m 3!!"4ZE' 0G$+3!"4\-' !' 3 +.#.4/\E' M φ E Y = φ(Z) Á9-"ÁÎ*MS!¬>?$+3!"¸L$K Y ! I !X%(EG' 3:.4/%(0G/ F(Z) SS?$+3!"¸LÎ*M#%>? I p P c1 , c2 , ..., cp Y = p1 ci 1{Z=zi } Y »=Á!M4»ÑKM ?¼!¼ p

E

φ=

E

1 ci 1{zi }

Î*Ñ

+6 6 )

)

J+ K I I

0 I

<>=@< ? I

+

< 7A <@6

- I

M Í %(D DV+.#.4+3X% $g %,d %§fPg?1 9v"4+- ' >PMT%*M 3UMP+ =.
F

[X|F]

[X|F] =

X

[X|B]1B

B∈P

Í 6+3X% \-' [X|F(Z)] !·$+JE%(.4FR.,%R>VMT%*M [X|Z] "#-!3-*' "#+ " " .qMB" I "v02%("#E+L%(-.4jD 3D 3!"4$+$ 4. RD .#'R"#0]DV3!X%(E& :.=? I 3!"4+.·»s *"#0G+!"4 ¼· A>%(3!",%(Q .4Á%(.4$U%("#3 3$+.#.4 {¸ "#/%(.43¬V' G%(D D .#" I %(!"4.#"#$U%("#3]+&DVK!"#!"#>?|Ω| 1 %(¬.41'¬!DL% I X 7→ [X|F] V »s 6*"#0G+!"4 ¼m3!0G$¬DL%(3.4R>VMT%*M 3UM H=0G!' 3X%(Q .46M |P| F Î*M g·b q_ /%d 5·b 1?fqg Pb '%d 9v"# ' 6%(D D .#" I %(!"4 I E>?{) %1.=< "#$+JE%(.#"#$ u : R 7→ R »mÎ*M#Ñ ?¼ [u(X)|F] ≥ u( [X|F])

NZDL%(3!!" I ' .#"4+3

»mÎ*M#Ñ )K¼ »mÎ*M#Ñ ¼ [X 2 |F] ≥ [X|F]2 . M %/>%(3!",%(Q .4&%(.4$U%("#3¸JK' 3X%(-%('D 3+0]"4+30G+0Q 3 »mÎ*M#Ñ?¼!$+JE%(.4 Ë ÉG¢ ³+ s¥¯? s¡s¢9v"# B∈P 1+3!0]"#.,%& $+0G!!3X%(!"42+G%(D D .#"4\-'L%(E ' 3 F +.4X I 0G+0Q 3OF .=< "#$+BJE%(.#"#$[u(X)|B] I E>?{)*"#$6F1.,%D 3QL%(Q "#.#"#$ u("#-[X|B]) !3-*' "#+ " .

[|X||F] ≥ | [X|F]|

B

*M Zg·b

!;.=< ' "4\E'>PMT%*M 3UM>?$+3!"¸q%(- .4 ÎD 3D 3!"4$+$ 1?_ c(2 _ 1Kd Ec;%U_*d %§fqg &µfPg'5A_`bvgPdU_ /§b 1 [X|F] ' #" >%(-6« Qq¼ %?¼ ! H=0G!' 3X%(Q .4 Y F ∀F ∈ F, [X1F ] = [Y 1F ] %("4X%(E%('Â.4 I +' 3.4&X"#Â & $+0GE!3+3R.,%6DL%(3!!"4¬*"#3 I K-''QV3!+3F60GE!3+3\-'j< ' >VMT%*M 3UM X%(!"4=%("4X%(E %('*);DV"#E %?¼+OQq¼!R$ I X%("#3+0G+- $+JE%(.46% .=?+3!Y!'Z /%?¼{-"#.j{)*"4!&' m%(0]"#.#.4: : I %(.,%("#3 +.#.4:\-' [X|F] P (cA ; A ∈ P) Y = A∈P cA 1A Qq¼& %(D D .4\-'$6F' Z$+>?$++0G+E DP+3!0G+R I %(. I ' .4+3 %+Z S+:« cB [Y 1B ] = cB B I I ? I F G F ⊃G \-'+.4\-'"#.,%>VMT%*M 3UM X

43

[X|F|G] =

[X|G] ;

[ [X|F]] =

[X]

»mÎ*M#ÑUÚ?¼

Ã .'/!' G3X%; 10G-!3+3/.,%ÂD 3+0]"4W+3G$+JE%(.#"#$ .4¬D 3+0]"4+3+/ I ®0G+06Q 3 I +.#.4{H I " S\E' '&%(D DP+.#.4+3O3!DV I !"#>?+0G+E $+X%(- =H 0G!' 3X%(Q .4U' O3+>-"4+- F0GE!3+3 .4$+JE%(.#"#$ U et V

[U 1G ] =

ÎKÎ

G

[V 1G ] ∀G ∈ G.

5®%("4

.,% +'*)*"4W+0G®$+JE%(.#"# >?+L%(-Z*'Ó=%("#;\E' M .4;X%(!"4 "" . " RG ∈%FLD %(3

1 ++[U1' ] =I [$\-[X|F]1 '+ I D 3X]%(!="4\-'[X1 I ]®3$!' .#X%(«%>? I

G

G

G

{) +0]D .4

Ù*M

7 %

Y

b Ud

F

`b 4#c(_ a/§b

Ë ÉG¢ ³+ s¥¯? s¡s¢

H

[X|Z1 , Z2 , . . . , ZN |Z1 , Z2 ] = [Y |F] = Y

$"#JKL%(EODL%(3

X

»mÎ*MTÎK?¼

[XY |F] = Y [X|F].

.4 +'*)0G+0Q 3O 2»ÑU?¼¬*'&%>?&F0GE!3+3 \-'

U etV

[X|Z1 , Z2 ]

D .#'OJ?$+$+3X%(.4+0G+EUP\E'+.4\-'"#R.,%1>VMT%*M 3

[U 1B ] =

[V 1B ] ∀B ∈ F.

B'/.,%("4X%(E I %(. I ' .4+36.41D 3+0]"4+3/0G+06Q 3K/\E' [V 1B ] = [(Y 1B ) [X|F]] = [XY 1B ] + >?+3!!' *M Qj<¼¬M B" I "·02%("#E+L%(-&' 6J?$+$+3X%(.#"4X%(!"4®' >?+E&' !"#.#"4$6 6.=%(- « + -Á +'*)G>PMT%*M-F¬>%(.4+' 3Á3!DV I !"#>?%( + K'2!Á' RmE!"42-' 0G$+3!"4\E' E F X ' YD DPK ' 3 H=0G!' 3X%(Q .4 !"S.=
Y F

H(ω, y) =

[f (X, y)|F](ω)

»mÎ*MTÎ*Ñe¼

[f (X, Y )|F](ω) = H(ω, Y (ω))

5-!3 ' &SVMT%*M*¸JK' 3X%(E/%('8 I Z0G+0Q 3 Â»mÎ*MTÎ*Ñe¼/»s\-' Ë ÉG¢ ³+ s¥¯? s¡s¢ '+3R' "#>%(-.=< 'X%(J?¬ D 3QL%(Q "#.#"4!/ ¼ H=0G!' 3X%(Q .4 I ["4"4DV' 3 I +.,% ' Ø%(0G · $!"#JKDL%(3 +6DL%(3 .46H(., >K%(.4+Y' 3);» I F!X%(E¼/D 3!"463!DP I !"#>?+0G+-1DL%(3 B F y !' 3 Z>?"#U+Z 3+JE%(3%(-c X%1 ${¸ "#!"4jq\E' ! $+JE%(.46F.,% I !X%(- Y et [f (X, y)|F] !' 3 +.#.4/ I B H=0G' 3X%(Q .4 +Z' !3GDV+' O$ I 3!"#3G« H(., Y ) c B F

[1B H(., Y )] =

I 6\E' "S0G-!3/.=Ehbv' g 5A_gPdUb 5Ab X F D 3$ I "4X+0G+E«

H

[1B H(., y)] =

.,%>%(3!",%(Q .42%(.4$U%("#3

)

[X|F] =

[1B f (X, y)] =

[X|F]

[1B f (X, Y )],

13$*' "#6FÂ' I !X%(- D .#'

[X]

%( I I %8.,%s3!0' .4»mÎ*MTÎ*Ñe¼2 "#0]D .#"¸L8%("#!"«Á!" .=
+6 6 H

[1B [f (X, y)|F]] =

8C

J+ K I I

! I

<>A A <>C

[f (X, Y )|F] = h(Y )

»mÎ*MTÎKÎ?¼ %*+ »mÎ*MTÎ ? ¼

%( I :DL%(3X%(JK3X%(D [ $!"#JK+3X%]' :>%(3!",%(Q .4%(.4$U%("#36F1>K%(.4+' 3%( M X E ÑKMK¢q¿¡s s¡s¢L¢P´{É2´{¢L ¯¥-º ¢V´¯K° ©B{¥!´ ¬9v"# ' %(.#J?W+Q 3»s+ J?+*3$ DL%(32' DL%(3!!"#!"4 ¼ !" F P ZDVK+3X% x∈E

ΠX (x|F) =

[{X = x}|F] =

[1{X=x} |F]

» "4+ +3G\-'j< "#.RUVMT%*M DVKW + -.4OÎ1D 3D 3!"4$+$ !' "#>K%(E3' .#X%(E"#0]0G$*",%(+0G+-¬F I 6\E'¬'O%U>?>v' Π + X (x|F) " "

Î

m» %?¼ ! H=0G!' 3X%(Q .4+ ∀x ∈ E ΠX (x|F) F ≥0 P »§Qq¼ ∀ω ∈ Ω Π (x|F) = [1 |F] = 1 @A' 3¬Å;>?"#3 D #. ' I .,%("#3ULx∈E &D 3eX>-"4"#3+0G+- Ω .,%]s I !"4® ÎG>%(3!",%(Q .4& ${¸ "4DL%(3 Q(x, ω) = .4D 3D 3!"4$+$»m%?¼+»§Qq¼e%(-G« ΠX (x|F)(ω) Q ! H=0G!' 3X%(Q .4+ »m%?¼ ∀x ∈ E ω 7→ Q(x, ω) F ≥0 »§Qq¼ !R' :D 3QL%(Q "#.#"#$6' 3 ∀ω ∈ Ω x 7→ Q(x, ω) E +!m I !"4 q\-'6.=%(3!",%(Q .4G%(.4$U%("#3GF>K%(.4+' 3/%( ' D DPK+3X% I 0]0G8S< [L%(Q "#!' 8\-' M Z% °#¡ ¨ K¢q¿¡s s¡s¢L¢P´{°§°#´8¿E´ +º ¯¢q¿ +3XF%DL%(3 Z(Ω) = F X Z ${¸ "#!"4 MS+&Q*Ç!+& I I 0]0G]%(' DV"#E]ÑKMqD 3$ I $ +E: E¬'&3+D 3+ ΠX (.|F(Z)) .4X%(!"4U' ¬m I !"4 Î>%(3!",%(Q .4»)V ¼ 02%("4 S?']Π%(X,Z) >?+Ò;X(x, %(1 ' Â $~Ç!F >-'%( >?!3RÇ+' X]«

[f (X, Z)] =

X

ΠZ (z)

z∈F

X

f (x, z)q(x, z)

x∈E

Í DP+' :3+!3' >?+3 I +!]s3!0' .4]DL%(3:' ]>?"41D .#'6%(Q!!3X%("# » I G\E'&Ç!>?' I +"#.#.4G =%("#3 DV' 3Á>?'[L%(Q "#!'+3R%('202%( "4+0G+ER !DV$+3X%( I I *"#!"4 +.#.4¼ I v'Z+ »mÎ*MTÎKÎ?¼\E'¬.=< "# $+DV+%( I 6S< ' I ' D .42» ¼ X, Z G>PMT%*M 3UMj+-!3X%("#L%("#.=?+3 ' I -!3{Hµ{)*+0]D .4e¼ "#.A{)*"4!1+[X|Z] ®3+>%(= I [1[X] ' I *"#!"4$ I XX%("#31+¬' 2X%(E]S< "# $+DV+%( I + +3!0GO ¬.4" I *"#!"4 +.#.4 %.4:3$!' .#X%(O' "#>%(-6« ´ ³¢L ¡§¢q¿ Ë ´{¢q¿?¯K¢L µ´³:³¡´ ³U´{º-°4´{É2´{¢L Á³¡ X Z ΠX (.|Z) = ΠX +!&$+JE%(.#"#$&!"#JK "¸LO\-'.,%:m%(0]"#.#.4 D 3QL%(Q "#.#"#$"# {)*$ODL%(3 ¸JK' 3X%(ER%('2D 3+0]"4+30G+06Q 3¬! +;3$U%(.#"#$¬3$*' "#/F.,%+' .4&.4"V02%(3!JK"#L%(.4: M %1 $+0G!3X%(!Ω"4Z I ¬3$' .#X%(U !3WR!"#0]D .4K ! X .,%("4X$6%('.4 I +' 36M

43

+6 6 : ;< 4< )

,

ÑKM= fPg'5&%d %§fqgAgbv`b-gPd05 (#·g·b/§f!%` #6/d %sg·fq` %§_ /§b " 9v"4+E + +'*);+-!"4+3 p q 0 et α1 , α2 , ..., αq ZDVK > Z *"#O\-'j< ' :>PMT%*M Í q q reels > 0 F>KH %(.4+=' 3{k j%(∈{0, 1, , ..., p} | qk0+S06k1' .#+!"#k20]+",%(....4+ kDLq %(=3X%(p} 0G+!3 Z = (Z1 , Z2 , ...Zq ) »I . " " ¼!"j+.#.4/%*0G+ DP' 3R.4".,{0, %D 1, 32, QL%(..., Q "#.#"#p} $6!' 3 (α${¸1 , α"4:2 ,DL..., %(3 αq , p, q) q *DP3!$/DL%(3

{0, 1, ..., p}

[Z1 = k1 ; Z2 = k2 ; ...Zq = kq ] =

p! k αk1 αk2 ...αq q k1 !k2 !...kq ! 1 2

H

∀k ∈ H

' /+.#.4>%(3!",%(Q .4%(.4$U%("#3 0GE!3+3:\E'/.,%].4" I *"#!"4 +.#.4 Â» ¼R\-'L%( Z1 , Z2 , ..., Z q−1 !R.,%.4"j06' .#!"#0]",%(.4 :DL%(3X%(0GW+!3» n1 n2 ¼ +3X%G\-' nq−1 Zq .46!' D DP3! I +!:.4" I *"#!"4 +.#.4O%(.4n−n $U%(q "#,3n−n 2M q , ..., n−nq , p − Zq , q − 1 9v"#

Z

Î(

Î*M 7S_g6:d %dUcb 9v"# ' !' "#O >K%(3!",%(Q .4%(.4$U%("#33$+.#.4U?"# $+DV+%(Ee-\-' "H n *"4!3!"#Q '$ ZDVK(X 1 , X2 , ..., Xn ) Sn = X1 + X2 + ...Xn 0GE!3+3 \-' »m%?¼Z9v"# f : R 7→ R [X1 f (Sn )] = [X2 f (Sn )] = ... = [Xn f (Sn )] »§Qq¼;N $*' "#36\-' Sn

+6 6 )

:·_ %§g·b

[X1 |Sn ] =

>C =

,+#

5·b

_ c+3Lf

8< G<>C

II

n

G<=

-I

< A

+# K

<

1O+>?+3X%(D "4 +0G+-!' 3Á.,%6!"4

Ê ¯K¡¢P´¬¿-´8¯¥(Ì]\-' "V%6$+$ $+!'*"4$:%('ÂD 3+0]"4+3R+0G!!3M

H

%(

I ¬DL%(3X%(JK3X%([2LZ $!"#J?+3X%1DL%(3 Ä E ' ++06Q .4/¸ " I 0]DP3!X%(- k $+.4$+0G+-O\-':'O%(D DV+.#.4+3´³µ*¯ ¨ ´G¿À Ë =¯K §³6 ¬.,% I [L%("#2M Ä P = (pij ) ' /02%(!3!" I I %(3!3$6S-"4+ Â·»§"=MY pij ≥ 0 ∀i, j + Pj pij = 1 ∀i) !" f P R' ¬s I "4 E' 0G$+3!"4\-'6!' 3 DPK+3X% !' "# >VE MT%*MvF6>%(.4+' 3%(Pf (i) =+3X%%(jD pDPij+f.4$(j) ¨{Ê ¯K¡¢P´6¿E´®¯¥ Ì¿E´¬É]¯? s¥¡ ¨ ´/¿-´ (X0 , X1 , X2 , · · · , XN ) E s¥!¯¢ ³¡s s¡m¢ !"j\-'+.4\-'/"#R.,%m I !"4ZE' 0G$+3!"4\-' !' 3 P

f

[f (Xn+1 )|Fn ] = P f (Xn )

E

∀n ∈ {0, 1, 2, · · · , (N − 1)}

.4 I +' 3:>?$+3!"¸L+3X%SFÂ!"#!3]S?+ "#36«

[Xn+1 = j|X0 = i0 ; X1 = i1 ; · · · ; Xn = i] =

[Xn+1 = j|Xn = i] = pij

*_ dUc+%1?b 5·b dUc_g6 %d %§fqg 5(#Agb ` _ c41;:·b _ /'E_*dUf<%c(b 9v"4+E ' .4"E D 3QL%(Q "#.#"#$!' 3 d Á!' D DP3!j¸ " + ' Á!' "# >VMT%*M"# $+DV+%(E $\-' "4*"4!µ!3!"#Q '$O :.4" F1>K%(.4+' 3R%( d M Í ZDVK F ((Yn ) ; n ≥ 1) µ

R

X0 = 0

,

Xn =

i=n X

Yi

i=1

∀n ≥ 1

´{³{ ¯°K¥X³:ºv¢P´ {¨ Ê ¯¡§¢P´¿E´8¯¥(ÌG¿E´6ÉG¯K s¥{¡ ¨ ´¿E´ s¥!¯¢ ³{¡§ s¡m¢ Pij = µ(j − i) (Xn ) $+0G-!3 I + .,% 8%; +Z>?+3!!' (2.23)

H

[f (Xn+1 )|Fn ] =

[f (Xn + Yn+1 )|Fn ] =

X

k∈F

f (Xn + k)µ(k) = P f (Xn )

P P + $*' "#O"#0]0G${H \-'L%( 8%1DPKX$ *",%(+0G+- .4¬3$' P .#X(% f (i)%( = I k∈F $2M f (i + k)µ(k) = j∈F +i f (j)µ(j − i)

ÎKÙ

+ 9v"# ' ;!DL% I & OD 3 QL%(Q "#.#"#$G»¸ " ¼ Â' D DPK+3X%%(' I I [L%(D "#!3¬\-'O.4&!' D DV3! ! $+JE%(.·(Ω, % ) +3X%1' 0Q 3 6+E!"4+3&!!3!" I +0G+E DPK"#!"·¸ ) $:%(R' :.,%]' "#]M Ω N

>, $ +6859685 )

(5 84:

!%! . -!*$( $ " -)! *"0, )+/ ! ! BC

;:5A<

54:

:

>:

5

5 698

FG,HEI J+ K I

' 3 MO!DL% I %(D DP+.#.4+3X%O¶° s¥¯? s¡s¢!' 3 ' O' "# I 3"4X%(- S
Ä Í Z*"#3X%G\E'j< ' /' "# (X0 , X1 , X2 , ..., XN ) ¬>PMT%*M !¯?¿?¯P Ë ´F I +!¬¸.#!3X%(!"4 !" H=0G' 3X%(Q .4GM ∀i ∈ {0, 1, 2, ..., N }, Xi

Fi

Ä Í 8*"#3X%2\-' I +!!' "#6Rq¥ Ë ÌU¡³¡KB{°4´1!" ∀i ∈ {0, 1, 2, ..., N } , Xi ! Fi−1 H=0G!' 3X%(Q .4 »s+ I ->?{H L%(-O\-' ¼{M F−1 = F0

Ä v9 "#/» X0 , X1 , X2 , ..., XN ¼' &' "#& >VMT%*MPM Í %(D DP+.#.4+3X%&¶° s¥!¯K s¡sK¢Z¢q¯K sºv¥!´+°°4´:+ J?+*3$¬DL%(3 I +! »§'%>?>-'G+ "" .\E'j< "#.PU? I .,% I ->?+E!"4 %/!' "#?» ¼!%%(D $&% I +!R¸.#!3X%(!"4ÂX%(*"4\-'.,%6!' "#» Xn Xn−1 X−1 = ¼&YRD 3$+>-"4!"#Q .4KM Í ZDP+' I !3!' "#3F.=VX MT%*nM "4+-¬FG>%(.4+' 3&%(&.4/0G¹+0G!DL% I Xn +O 'O' /!' "#2» ¼SVMT%*M 3UM E φn : E n+1 7→ R ! H=0G!' 3X%(Q .4q»s+.#.4 ! I X%(E&!' 3 I [L%\E'¬%(0G n ¼RE Yn = φn (X0 , X1 , X2 , ...., Xn ) X3!:\E':.,%1' "# !O%%(D $F n %(R.4 I %'.4>VMT%*M XE3$+.#.4*.4!' "# Fn (Yn ) XE X%(n"#!"j !' "#O%%(D $M Sn = X0 + X1 + X2 + .... + Xn et Πn = X0 X1 X2 ...Xn

Î

$,

9v"#»

+6 685 )

$ !

:5

43

Ω, (Fn ),

=8

¼' Z!DL% I :¸.#!3$GM

BC

FG,HEI J+ K I

'*"#31\E' I +!' "#!' 9v"#» ¼¬' Â!' "#Â%%(D $;F .,%;¸.#!3X%(!"4 Mn ; 0 ≤ n ≤ N H=02 %(3!!"# JE(% .4Â»s'ZD .#'R Q 3!4" W+>?+0G+- ' :02%(3!!"# JE%(.4e¼{q!"j+.#.4(F ¬>?n$+)3!"¸L/.4O$+JE%(.#"#$

(Fn )

[Mn+1 |Fn ] = Mn

»*M#Ñe¼

∀n ∈ {0, 1, 2, ..., N − 1}.

'*"#3O\-' I ?$+3!"¸L:.4R"#$+JE%(.#"#$ »§3!DjM ¼

[Mn+1 |Fn ] ≤ Mn

[Mn+1 |Fn ] ≥ Mn

9-".4 I E{)*] I 0]DP3!1DL%: ¸.#!3X%(!"4¤P"#.+3X%;'Hµ+-+*'\E'.=K%("#.#.4]%(¬.,%2¸.#!3X%(!"4 L%(!' 3+ .#.4KM

+6 6 )

<>=

%

<>C ? 9?

>C

,+#

v9 "# ' /02%(3!!"# JE%(.4 (Mn ) ÑKM 9-" +DL%(3!!" I ' .#"4+3 n ≤ n + p ≤ N , [Mn+p |Fn ] = Mn Mn = [MN |Fn ] Î*M 9-" ! I E>?{) R» ¼!1' '02%(3!!"# JE%(.4 I
*M

I 0Q "#L%("4 .#"#$U%("#3/S< ' :m%(0]"#.#.4¬¸ "46 :02%(3!!"# JE%(.4O!R' /02%(3!!"# JE%(.42M 3+!"4+*3X%G\-' M / $+DP+ZDL% [Mn ] = [M0 ] ∀n [M ] n :D 3+0]"4+3ODV"#E3$' .#¬"#0]0G$*",%(+0G+E: Gn»mÎ*M#ÑUÚ?¼ %1+ S+

[Mn+2 |Fn ] =

[Mn+2 |Fn+1 |Fn ] =

[Mn+1 |Fn ] = Mn ,

S?{)v"#$Â»mÎ*M#Ñ?¼ %1+ZS+

.4 !3"4!"4W+0G¬!$+>-"4 +- "L>?' %>?+Ò¬DL%Á0G-!3+3R.4O\-'L%(!3!"4W+0G1 I ?' %]$ I [L%(D DV$ "#.!RD .#'O\E'¬+0]D S< Å3+0G$*"4+3M

+6 6 )

[φ(Mn+1 )|Fn ] ≥ φ( [Mn+1 |Fn ]) = φ(Mn ),

)

<>= < C < ;<>=@?BA8<>C %

¼/' ' "# >VMT%*M 3UM ÑKM 7Sfq``b 5·b "#_"cC"=1?b-gPdUcEb %sg65 -hb-g65·_gPdUb1 9v"# » X0 , X1 , X2 , ..., XN Ç! I +-!3$R"# $+DV+%(E DVK Sn = X0 + X1 + X2 + ... + Xn ; Fn = F(X0 , X1 , X2 , ..., Xn ) *"4O\E' !R' :02%(3!!"# JE%(.4 + V+/D ' "4\E' !R"# $+DV+%(E (Sn )

Xn+1

Fn

[Sn+1 |Fn ] = X0 + X1 + ... + Xn + [Xn+1 |Fn ] = Sn + [Xn+1 ] = Sn

Î )

-9 ' D DVK /D .#'.4>PMT%*M $\-' "4*"4!!3!"#Q '$ +.#.4&E&%(.43O' O.,%10G¹+0G/>K%(3!",%( I 6\-'¬' X n %(D DV+.#.4+3 2 %( I +!:"#!'L%(!"4 2 R%('X!"V' ¬02%(3!!"# JE%(.4 I 0]0G:.4¬D 3' >? Sn − (n + 1) σ 2 .4/DP+!"# I %(. I ' σ.' "#>%(-6«

2 [Sn+1 |Fn ] =

2 [(Sn + Xn+1 )2 |Fn ] = Sn2 + 2Sn [Xn+1 ] + [Xn+1 ] = Sn2 + σ 2

S

2 [Sn+1 − (n + 2)σ 2 |Fn ] = Sn2 − (n + 1)σ 2

Î*M _ cd %§g L_ /§b8hAc(f95#%d 1 9v"#Â» ¼1' G!' "# G>VMT%*MSDPK"#!"#>?2S"# $+DV+%(E T0 , T1 , T2 , ..., TN SP=MT%*M T0 T1M TN2 Tn

[Mn+1 |Fn ] = T0 T1 ...Tn [Tn+1 ] = Mn

Ã . Å%¬SK%(E.,%]D 3D 3!"4$+$6 ¬02%(3!!"# JE%(.4 '&%(.#.4$+!'*"4+3 +'*)S

,-$ 0! , )! (0 $ ,

<

:5

$ !

=8

9-' 3 .=-"4"#Q .4 n ) .,%8X%(!"4×!' "#>K%(E «!" !]' Z!' "# >VMT%*M 3UM Ã E!3v*' "42S
Xn

∆Xn = Xn − Xn−1 .

Í DP+' 13 I !!"#!'+3Z» Xn ¼ I L%("4%(E1.,% ' "# ;1% II 3"4+0G+-]+1%>K%(.4+' 3"# "#!",%(.4®«AØ% 02%( "4W+3/$+>v"4 + E Xn = X 0 +

k=n X

∆Xk .

»*MTÎ?¼

k=1

Í Z+3X%]\-'j< ' /!' "#/%%(D $ (Mn ) !R' :02%(3!!"# JE%(.4!"S+O+' .4+0G+E&!"

[∆Mn |Fn−1 ] = 0 ∀n ≥ 1

Î

»*M ?¼

+6 685 )

)

FG,HEI J+ K I

Í %(D DV+.#.4+3X%1!3X%(m3!0G$/ (Mn ) DL%(3 (Un ) +O.=
k=n X

»*M -¼

Uk ∆Mk

k=1

! I %(3X% I +3!"4X$/DL%(3R.4R3+.,%(!"4« Ã .S3$!' .#/ I /\-' "S>-"4+-OS
bEb3 Xb-g65

Zg

(U ∗ M )0 = U0 M0

»*MTÙ?¼

∆(U ∗ M )n = Un ∆Mn ∀n ≥ 1

qf gPdUb _ c+/§f Ã 02%(JK"#\-'>?'PÇ!' "4+Ò% D "#.4'1=% I %(' I %!"#1«>?'JE%(JK+Ò/ÑÏ8!"*.4 I ' DG!D "#.4+ADV+3 +Ò ÑÏØ%(R.4 I % I -!3X%("#3 DV' 3O%(3!3!"#>?+3 % I ¬3$!' .#X%(O%(R' Z+->-"#3 +0G+- 3X%(D DV+.,%(E I +.#' "S< ' :X%(.#.4 Ç+'*)®**"4D .#' R\E'&>?'0]"4+Ò:' "4\-'+0G+E/ÑÏ!' 3D "#.4 >?'JE%(JK+Ò:' s"4>?!3&0]"4:+ I % G!' II W2j+:.,%;DV+3 +Ò2+ I %/S?'6*"4DPK+Ò2F>?!32%(3!3!"#>?$G%(/.4:Ç+' S< ' /0]0G6SVMT%*M*"# $+DP+%(- $\-' "4*"4!!3!"#Q '$O /.4" Π \-'¬'+3» Xn ; 1 ≤ n ≤ N ¼ 3+D 3$X+E+3X%].4¬JE%("#D 3 I ' 3$:DL%(3R.4¬*H="4+0G I ' DX%(*"4O\E' +3X%.,% Xn \-'L%(E!"#$6S?'O*"4!DVK+3+Ò6F I +"#X%(E6M@' "4\E'¬.4M On>P=MT%*M M0 +XE1 +I +X E!23$+U... + Xn !R' 02%(3!!"# JE%(.4;4» I VMT%*M $+JE%(!"#>?2 +\-':.=?.#' !"48 ¬>?!3¬s3!!' 6!R' /!' 3!02%(3!!"# JE%(.4M Xn 9-"j>?' ¹+' ZDV+']Ç!'+' 3LÎ I [K>?'ODL%(3X%("#!3E&"#!' D DV3!X%(Q .4« /DL%DV' >?"#3OJE%(JK+3 S?K0]"4%2ÑÏ\E'L%(>?'Á>?'+-+Ò&+Â>?+"#&S?$+ "4+E+G"#-!3-*' "4X%(-' R' "#6» ¼ 06Q 33$+.4Á !!"#$ F ¸JK' 3+3Á>?K 0]"41»§' 60]"4X6$+JE%(!"#>?1!"#JK "¸q%(-&\-'6>?'¬%>?+Ò1DL%(3!"4$!' 3U=% nI e¼6MP0]0G+E/ $++3!0]"#+3+Ò1>?'&>?!3 0]"4 /9-"V>?'¹+ I 0]0G&' .4 0G Kv>?'+3+Ò¬"# '+ I $¬DL%(3.=?.#' !"4Z*'6Ç!+'»mF63! I 0]0G¬ >K%¬.4>?"#3%(\-'+.4\E'.#"#JK02%("4Á.,%("4X.,F I +! \E'!"42DV' 3Á.40G0G+-¼+>?!3O0]"4X F¬.=< "#!X%(- +3X%m I !"4 Q+3!>K%(!"4O\-'¬>?' %(' 3+Ò:=%("#Ç'\-'j
NØ+3!0G102%(![$+02%(!"4\-'Z I +.,%X2!3X%*' "#]DL%(3.4m%("#1\E'Â>?!3Â0]"4 1%(.4$U%("#3+1\E'.,% ! ' "# !6%%(D $MS58%(.#[+' 3+'+0G+-S.46*"#3 I +' 36 I %!"#2EU :nDL%/*"4DPK$F%*0G+!!3 j< "#0]DV(U 3!n&) \-'+.#.4&!' "#&%%(D $1«-"#.P' G3X%("#¬-!"P+.P$+X%("#.4 I % v O0]"4X+3 6 ) DV' 3+0]DP I [+3' ]0]"#.#.#"4 §3X%( I =% I +3!X%("# F I [L%\E' I ' D ]U>?n'="#10 0]DP(1 KX{X +3Xn%:=1} −I 1 {X n$ =−1} I "4 +3 >?!3 0]"4X %('*H="4+0G I ' D+Â*"4DPKX%(-' "4\E'+0G+- "#*m3!02%(!"4s' 3! "4DL%(3.4ÁÇ+'Ç~'X\E'jK%D .#' U¼ %!!3+"#-O n −I 1» ¼F1¹+!3: X+' .4+0G+E&%%(D $/02%("4q¥ Ë ÌU¡³K¡ B{°4´2M Un

43

E

E

7ÅE%(-O0]"4$: I +!:m% I j*>?!3¬m3!!' 6%('Z+0]D +>?+E'2L!"S.=v"4+E&\E' n U0 = 1 M0 + U1 X1 + U2 X2 + ... + Un Xn = UO M0 + U1 ∆M1 + U2 ∆M2 + ... + Un ∆Mn .

ÎKÚ

B'O%U>?+Ò63 I E'.,%!3X%(m3!0G$/ :.,%02%(3!!"# JE%(.4 DL%(3O.4:D 3 I X!'D 3+>-"4"#Q .4 M 3X%(n$+)JK"4:JE%(JKL%(E\E'G.=K%("# I '6 ].=?"#3 DL%$/ E' "#¬F1+3 +O$+!'*"4+3& $+3!"4 S< ' JK3X%(0Q 3 I ' D ' /+.#.4/!3X%($+JK"46! %(D DV+.4$:%(.4 .,%( JE%(J? I ' 3X%(-R' {02%(3!!"# JE%(.4 ]*\E'j< "#.P=%(' ROJE%(3 +3 I *m*3/%>? I .,%!"4Z 02%(3!!"# JE%(.4 02%(![$+02%(!" I "4+]M 6![$3W+0G\-' "!' "#&!O $DP$+3X%(-«"#.0GE!3\E'j?K:0]"4/]DP+' :!3X%(!s3!0G+3.,%Â02%(3!!"# JE%(.42 ]QL% (Mn ) +' G'!02%(3!!"# JE%(.4!!3!" I » I /\-' "j+3X%("#RQ "4+%(JK3$U%(Q .4*D ' "4\E'6R!DV$+3X%( I Rm3!0G+3X%("4+EO' 6!' "#:!!3!" I +0G+E I 3"4X%(-e¼:M

E

E

Efqc v`b 9-' 3A' !DL% I Á¸.#!3$¬» ¼j"4+-» ¼V' 02%(3!!"# JE%(.4¬»§3DjMe' !' 3A'' X'H=02%(3!!"# JE%(.4e¼ Ω , (Fn ) , Mn +Â» ¼:' !' "#2D 3$+>-"4"#Q .4»§3!DjM' !' "#2D 3$+>-"4"#Q .4GDVK!"#!"#>?e¼ !%(.43' 202%(3!!"# JE%(.4 (U ∗ M )n »§3!U DjM n' :' 3O'' /'H=02%(3!!"# JE%(.4e¼{M 3! ' %(' I % O02%(3!!"# JE%(.4&+ DPKX M 0 = (U ∗ M ) ®% Ë ÉG¢³{ s¥!¯K s¡sK¢ n n Sv"4!"#Q "#.#"#$/ 0 :

∆Mn = Un ∆Mn

(Un )

+6 6 )

)

-

A ? 9?

K

K +

[∆Mn0 |Fn−1 ] = Un [∆Mn |Fn−1 ] = 0

< <#? >C <

I

?

+# +# I

C BA8< K

v9 "4+-2» ¼R' 02%(3!!"# JE%(.4» ¼ X%G¸.#!3X%(!"4L%(!' 3+.#.4 *"#3X%2\-' DVKH 0 = M0 , M1 , ..., MN Fn (Mn ) W .,%Âq¥!q¥{¡ Ë Ë ¿E´¥´µq¥ Ë ³U´{¢L =¯K s¡m¢®q¥ Ë ÌU¡³¡KB{°4´!"=A\-'+.4\E'"#.,%» ¼µH=02%(3!!"# JE%(.4 j"#.{)v"4!2' Fn (Zn ) +.4\-' M I !X%(- +' D 3 I X!'RD 3$+>v"4!"#Q .4 c Un (Zn ) = c + (U ∗ M )n ∀n

'&3+!3' >?+3 I +!D 3D 3!"4$+$1%(&.4&D 3 I [L%("# I [L%(D "#!3:' .40 ÉG¯¥ ¨{Ê*Ë ³ ¨ É q°4´+ §³ DV' 3.=< "#!X%(EO'%(.#.4!"#0]D .4+0G+E' +02%( +3ÁU< "#.{)*"4!R +.#.402%(3!!"# JE%(.4 .43$!' .#X%(\E' " >?E!' "#>v3R0GE!3+-:E\-'O%(!3 I %*3&*"4 I 3+Á"#.P+2{)v"46 I +3!&K02%("4!3WÁDV+'q» I E!3X%("#3+0G+E %('*);0G- W+.4&F+0]D I E!"#-'ÈZQVU%(' I ' D :02%(3!!"# JE%(.4s%(0G+-X%(.4ODVKW +-.,%1DjM 3UM DjM¼{M 9v"# ' !' "# >PMT%*M 3UMÁ"# $+DP+%(-$\-' "4*"4!!3!"#Q '$® I +E!3$ DVK + (1 , 2 , ..., N ) = 0 0 %(D DV+.4 .,%¸.#!3X%(!"4L%(!' 3+.#.46+ J?+*3$6DL%(3 I +!/!' "# I !"4 $+3O.,%102%(3!!"# JE%(.4 (Fn ) ME@·' 3$+>-"#+3.4Á!3!"#>-",%(.#"#$ K'Á' D DPK+3 Â $+J?$+$+3$6M ¸.#!3X%(!"4 Mn = (0 + 1 + ... + n ) 1 L%(!' 3+.#.4G+ J?+*3$2DL%(3].4]!' "# + XE]"4 +E!"4\-'U I 0] 0G DV+' 1.4;>?$+3!"¸L+32F8!"#!3 (Mn ) (n ) S
DPKXW .,%D 3D 3!"4$+$/ ¬3+D 3$X+EX%(!"4ZD 3$+>v"4!"#Q .4:!"S+R+' .4+0G+E !" D 3+ (c fLhf! %d %§fqg 1 (Mn ) 1 >K%(.4+' 3%(' Z++0Q .4/3$*' "# F2ÎDP"#-6M !' D DVK¬' 6S%(.4+' 3 Ë ÉG¢³{ s¥!¯K s¡sK¢×«9v"4+E a et b > 0 1 %( +R0GE!3&\-' DVKW /%(.43O.,%DjM 3UM DjM DVK

{−a, b}

(Mn )

p=

[1 = b] , q = 1 − p =

[1 = −a].

@' "4\E' ! I +-!3$K SV− MT%*M (1 3UM − p)a H=0G!' p3X= %(Q .4a+b >?+,3!"¸qq%(= - a+b Fn Fn [Fn |Fn−1 ] = 0 0G-!3+3 \-'j< "#.¬{)*"4! H=0G!' 3X%(Q .4®+.#.4®\E' S
Fn−1

Fn = γ n n

K

"" "#.{)v"4!' 2s I !"4E' 0G$+3!"4\-'

' 3 n+1 » I %(3!",%(Q .4]3$+.#.4{¼+.#.4\E' MÁ0]0G !1"# $+DV+%(E n+1 ) .=
\-'L%( ®%]DVK$ fn (0 ,1 ,...,n−1 ,b) I +.,%2 $+0GE!3!3%X+3!!"4MV!"4 $+3O02%("#*H b %(D D .#"4\E'L%(- I \E' "6D 3$ I W F +L%(-' 02%(3!!"#γ nJE%(=.4 R"#.>-"4+- (N ) F = ∆N n n n Pn M Ã .Á3!2!"#0]D .4+0G+-GF3+02%(3\-'+3\-' \-' " H=0G!' 3X%(Q .48!Nn' = N0 F0 N0 + 1 γk k I X%(E/3$+.#.4KM @%&F.,%13$ I "#D 3v\E' DVK 2 2 ML0]0G6.,%1!' "# [1 ] = [k ] = σ 2 Á m ! 3 G 0 $

V > T M * % E M # " + $ V D + %(E+ I +E!3$: Pn 2 2 !' (2k − σ 2 ; k = 1, 2, ..., N ) −σ ) 02%(3!!"# JE%(.46E' .#.4F.=v1"4!("#Q k.4: +.#.4/\-' (γn )

k=n X k=1

(2k − σ 2 ) =

k=n X

γk k

∀n ∈ {1, 2, ..., N }

k=1

D ' "4\E' 2 +.,%Z+-!3X%4S 0 I S
n (ω) 6= 0 ∀ω

Fn−1

n

2n − σ 2 = n

[

2n − σ 2 |Fn−1 ] = n

[

2n − σ 2 ]= n

[

21 − σ 2 ] 1

7 D DV+.,%(E .,% I !X%(E¸JK' 3X%(EG%(' +3! "4+30G+0Q 3Â I +! I [L%("#;S?"# C \-' 2 2 = 0 ∀n ≥ 1 M Ã .·+3$!' .#6\-' D 3+8>K%(.4+' 3 %( .=

" ! ( 8$ , ,

>:

0]0G+ DL%(3"#E!3v*' "#3/ +'*)Z ${¸ "#!"4"#0]DV3!X%(E«

$g%,d %§fPg'21 9v"#

' !' "#O R>PMT%*ME+ ' >VMT%*M?%¬>%(.4+' 3Á%( ?2+3X% T {0, 1, 2, ..., N } .,%;Ì¯K¥¡m¯B{°4´1¯° Ë ¯K =¡§¥!´ ${¸ "46DL%(3 XT (Xn ; n ∈ {0, 1, 2, ..., N })

XT (ω) = XT (ω) (ω) *Ñ

' + I 3 XT =

+

n=N X n=0

(XnT )

Xn 1{T =n}

.,%³ºv¡§ µ´¿E´6Ì¯¥{¡m¯B{°4´³¯° Ë ¯K =¡§¥!´{³ ${¸ "4:DL%(3

XnT = Xn 1{n
' + I 3

XnT = Xn∧T

Í *"#3X%1\-':.,%1!' "# X T !R.,%]!' "# X ¯¥{¥ + Ë ´]¯ T M @%(3!0]"S.4!3X%($+JK"4R\E' "VUK%1'O"#E$+3+3&02%("#E+L%(- +.#.4 I !"4!F]0]"4+36ÑÏÇ+< '\-'j?%(D 3W ' /DV+3!KqFG\E' "#!+3 .4RÇ!+' Mq +!¬ +3! "4W+3K I ?.#' !"4 &.,%:m3!!' :*'Ç'+' 3R!3+0]D .,% I $¬DL%(3.,%' "# (Mn ) (MnT ) I M I :\-':'O%(.#.4>?"#3O02%("#E+L%(EUqDV+' ¹+!3 I !"4 $+3$ I 0]0G/' :!3X%(m3!0G$/ (Mn )

(c fLhf! %d %§fqg%1 9v"4+E ' /!' "#: :>VMT%*M + ' ¬>VMT%GF>%(.4+' 3O%( {0, 1, ..., N } ZDVK ((Mn ) ; 0 ≤ n .,≤%N !3X%()m3!0G$/ DL%(T3.,%1!' "# ! T Un = 1{n≤T } n) Ã .j!' GO ¬>?$+3!"¸L+3U I :\E' (M "jn!) =% I "#.46%('Z>v'Z(U ${¸ "#(M !"4nU)L\E' Ë ÉG¢ ³{ s¥!¯K s¡m¢

43

∆(M T )n = Un Mn ∀n ≥ 1

:D 3$+>-"4!"#Q .4KM %;3$+DVG!¬m' 3! "4 '6%(.#.4/02%("#-+L%(E'6 +02%( +3F\-'+.#.4 I *"#!"4 (Un ) DL%(3.4$\-' "#>K%(.4+ I O!' "#>K%(E/« ∀n

{n ≤ T } ∈ Fn−1 ⇔ {n > T } ∈ Fn−1 ⇔ {T ≤ n} ∈ Fn .

Í Z3+02%(3\-'+3X%]S
{T ≤ n} ∈ Fn

∀n

I :\-' "j' I *' "#&%.,%1 ${¸ "#!"48!' "#>K%(EG«

$g%,d %§fPg

⇔

{T = n} ∈ Fn

∀n

»*M ?¼

1

¢¤¯? ´{°§°4´{¥¯® µ´+ÉOq³¿À ¯K¥¥ + :¿E´2°¯¶° s¥!¯K s¡sK¢ ºv¢P´2Ì¯¥{¡m¯B{°4´;¯° Ë ¯K =¡§¥!´$MÌ¯K°#´{ºv¥³Â¿?¯¢ ³ (Fn ) {0, 1, 2, ...., N } ³+¯? s¡,³'D+¯¡³+¯K¢L M2°!À º-¢V´¿-´³ ¨ ¢q¿K¡§ s¡m¢ ³ Ë +ºv¡§Ì¯°4´{¢L µ´³»*M ?¼ /\-'¬'R>?+O />?"#3R3+"#0]0G$*",%(O.4:3$' .#X%(O!' "#>K%(E/« L

KÎ

+6 G685 )

<

+

= <

,+

9v"4+- ' !DL% I ¸.#!3$O+ ' 2+0]DS
B" I "S
Î*M

UT ∧n ≥

[MT |Fn ]

M0 =

[UT |Fn ]

[MT |F0 ]

[UT ∧n |F0 ] ≥

[UT |F0 ]

»*M )K¼

5 E!3¬EDL%(3{) +0]D .4E.4Á"#$+JE%(.#"#$ .,%/ +'*)*"4W+0GO.#"#JK D ' "4\-' 8 !' !' 3!02%(3H (UnT ) !"# JE%(.4 T T \E' "q0GE!3 .,%:D 3+0]"4W+3 "#$+JE%(.#"#$KP»s+.,%/ I 0GUÅ?+ L%(-' ] I ¼¬M I *"#!"4 Un+0G≥+-:[U !' ND |F D .4$+n0G +EX%("#33+.,%(!"#>?+0G+E¬% F0

-b `h"5 #b-gPdUcEbG1 9v"# ' !' "#%%(D $]FG' 6¸.#!3X%(!"4 >PMT%*MVF]>K%(.4+' 3¬%( +¬"# ' ®'&++06Q .4 (X ) (F ) E Γ n n DPK E

TΓ = inf{n ≤ N |Xn ∈ Γ}

»m%>? I .,% I E>?+-!"4 ¼ inf φ = N c(fLhf! %d %§fqg%1 !&%(.43R' Z+0]DS
{T = N } = {X0 ∈ / Γ; X1 ∈ / Γ; ...; XN −1 ∈ / Γ} ∈ FN {T = n} = {X0 ∈ / Γ; ...; Xn−1 ∈ / Γ; Xn ∈ Γ} ∈ Fn

Å %¬QVU%(' I ' DS?$+3!"¸L+3 \-'¬.,%1QP3!/"#*s$+3!"4+' 3K .,%]QP3!/' DP$+3!"4+' 3/+O.,%10]0G/ Î+0]DS
+6 G6 )

I

<

c(fLhf! %d %§fqg%1

8?

,

-

<

-

+

= <

,+

9v"# ' !' "#>? I 3!"4+.#.4®%%(D $ +.#.4\E' DP' 3G' !' "#ZD 3$+>v"4!"#Q .4 >? I 3!"4+.#.4((X n ) ;1%("# ≤ n ≤ N ) !O%(.43O' ¬02%(3!!"# JE%(.4KM (Un ) [(U ∗ X)N ] = [< U0 , X0 >] (Xn )

Ë ÉG¢ ³+ s¥¯? s¡s¢ ' #"

Í 3X%(0GW+/%("4$+0G+EO%(' I %' 4" *"#0G+!"4 +.=ML9-" T ' Â+0]DRS-"4"#Q .4K+

Un = 1{n≤T }

(U ∗ X)N = XT

[XT ] =

9-"

[X0 ]

∀T

!R' $+>?$++0G+- Z>?"#R=% I "#.4+0G+E&\E'¬.,%>K%(3!",%(Q .46%(.4$U%("#3 A n ZDV+' !OF' Z +0]D S
Í % I .=

[X0 ] =

[Xn 1A + Xn+1 (1 − 1A )] =

[(Xn − Xn+1 )1A ] + [X0 ] .

%(3X% $+3!"4X%(!"4 » "" ¼:+E!3X%4%(.43

.,%

I I [Xn 1A ] = [Xn+1 1A ] ∀A ∈ Fn I /\-' "S+3!0]"#:.,%] $+0G!!3X%(!"4jM Xn = [Xn+1 |Fn ]

" , ( $, , " $ +6 685 : <>A 9?G? < < >< A A G< C < -? #<

)

<76

I

>:?<

8

I

-I

:5

=8

- J+

+#

9v"# ' 1!' "#1 >PMT%*M 3UMS%%(D $]F' 6¸.#!3X%(!"4 M (Zn 0 ≤ n ≤ N ) (Fn ) DL%(3.4$+JE%(.#"#$ I +(% -/' /!' "# (Un 0 ≤ n ≤ N )

UN = Z N

Un = sup(Zn , [Un+1 |Fn ])

H

${¸ "4X&DL%(3¬3$ I ' 3!3+ I »*M ¼

(0 ≤ n < N ).

!¬' 1!' "#]%%(D $ S

[Vn+1 |Fn ] ≥

[Un+1 |Fn ] ≥ Un

S v"4!"#Q .4 R (Un ) ${¸ "4GDL%(3 (Un ) M Í Ø% I D ' "4 (φn ) φn = 1{n≤T0 } ∆UnT0 = φn ∆Un [∆UnT0 |Fn−1 ] = 02%("4!' 3.=?$++0G+E ! ![1 3!" I {n≤T +0G0+}E∆U 6!' nDV|F$+3!n−1 "4+' ]36F DL%(3¬ ${¸ "#!"4 {n ≤V T]0 }= 3!1{n \E'− 1 < T0 } Un−1 Zn−1 T0 Un−1 = [Un |Fn−1 ] 8%1 I

H

[∆UnT0 |Fn−1 ] =

= 0

[1{n≤T0 } (Un − [Un |Fn−1 ])|Fn−1 ] = 1{n≤T0 } [(Un − [Un |Fn−1 ])|Fn−1 ]

I 6\E' "S0G-!3/.,%D 3DVK!"#!"4 M (

+6 6 <>=@? C 9? 8= ; 1 +D 3+.4RX%(!"4 *'ZDL%(3X%(JK3X%(D [/D 3$ I $ +-M )

,+(

+

-

ÑKM $ \E'+.4\-'6"# .4 g%d %§fqg 1 Í *"#¬\-'j< ' +0]D&S-",%(.4G "SQ "4+ \E'¬.=< "#$+JE%(.#"#$6D 3$ I $ +E¬3$*' "#&F :D .#'' >?+-*.=?.#' !"48 :.,%s3!!' 6S< ' ]Ç'+' 3*.,% [ZS ] "#JK[Z T"¸ ]I ≥ %(!"4 :.=?$+3!"¸L/.4OÎ I *"#!"4« T

a)

Z T = UT

(UnT ) est une martingale

b)

Ë ÉG¢ ³+ s¥¯? s¡s¢ «·9v"# T ' +0]DS?+3!"¸q%(- a) et b) DP+' 6$ I 3!"#3j\-'+.4\E' X"#.4¬+0]DOS

[ZT |F0 ] =

[UT |F0 ] = U0 ≥

>1

[US |F0 ] ≥

[ZS |F0 ] ,

I 6\E' "j0G-!36\-' !OD !"#02%(.ÁM O+02%(3\-' %('ZDL%%(J?:\E' T0 !D !"#02%(.AS
$ I "#D 3-\-'+0G+EY!" ' +0]DOS
1

[UT |F0 ] ≥

[ZT |F0 ] =

[ZT0 |F0 ] = U0 ≥

[UT |F0 ]

Í 8%1 I T ] = [UT ] + ZT ≤ UT I /\E' "j+-!3X%("# ZT = UT M @%&%('ZDV[Z "#E « % (b)

[UT |F0 ] =

[ZT |F0 ] = U0 ≥

[UT ∧n |F0 ] ≥

[UT |F0 ] =

[UT |Fn |F0 ] .

+ E/0G¹+0G!DP$+3X%( I I 0]0GG+.#.46- Ã .+3$!' .#2\E'].4/>PMT%*M 3UM UT ∧n [UT |Fn ] I 0]DL%(3X%(Q .4 +.#.4OXEO$+JE%(.4/ I :\E' "S+3!0]"#6.,%1 $+0G!!3X%(!"4M *M= fqcfv"4+E I \E' e ÁDL%(3AX%O ${¸ "#!"4 '%>?+S+· $~ÇFR$\-' T0 !R.,%1D .#'DP+!"#:>VMT%*M*>?$+3!"¸q%(E1» ¼¬M T0 a M Âb-` _ c #·b 1 $!"#JKDL%(3 .,%=%(0]"#.#.4/ R+0]DOS
7H

Tn

Dn = inf {p ≥ n |Xp = Up };

8%]%(.43

≥n

[ZDn |Fn ] ≥

[ZT |Fn ] ∀T ∈ Tn .

'R.,%("4XR.4:.4 I +' 33$L$ I [ "#3&F' / $+0G!!3X%(!"4M

+6 6 =@?BA = < C )

)

0

K

I +

9v"4+E ' :!' "#:%%(D $/+ +E>?+.4D DV: 69-+.#. + %(' 3 -.,%!"#JK " ¸ I %(!"4 $ %(R.,%1(Z3+n02) %(3\-' I "Hµ '/M B" I (U "·Î]n%()' !3m% I %(. I ' .4+3 .=T?+.4DD nDV6 6 9-+.#. M (Zn )

KÙ

(c fLhf! %d %§fqg Í %.4$+JE%(.#"#$ !' "#>K%(E6«

(a)

_

Un =

[ZT |Fn ]

;

(b)

Un =

[UDn |Fn ] =

[ZDn |Fn ]

É ¢³{ s¥!¯K s¡sK¢ 7OD DV+.4 .,%>PMT%*Mv¸JK' 3X%(-&%(' X I 0G+06Q 3 ¬.=<$+JE%(.#"#$2»m%?¼{M ËG Xn Ä 9v"4+E T1 et T2 ∈ Tn 0G-!3\-'j< "#.{)v"41' !3"4!"4W+0G]+0]D/S

Ä Ä

+6 6 : <>A 9?G? < C < G<>A8A <

[ZT1 |Fn ] ∨ [ZT2 |Fn ]

H

Ä

'.,%("4X @·KXDP' 3 I +.,% A = { [ZT1 |Fn ] ≥ [ZT2 |Fn ]} et T = T1 1A +T2 1Ac %('.4 I +' 3.4/"#Z /0GE!3+3 \-' >?$+3!"¸L:.,%1D 3D 3!"4$+$ I [+3 I [$ÂM 0]0G6.=?$+3!"¸q%(E M @' "4R\-'n∈.,% TI nEX%(- Xn = [ZRn |FnM ] ?+3 0GE!3+3\E' (Xn ) = (Un ) T ∈ Tn ' DL%%(J?ÂFZ.,%ZQV3!' DP$+3!"4+' 3Â\E'L%( $ I 3!"# T Tn D 3[Z ' T>?|F 6%(n.4]≤ 3R.=< "#[U $+TJE%(|F.#"#n$ ] ≤ Un M @A%XZ%('*)Ô+JE%(.#"#$»§QqX¼ n ≤ .,% U3n+02%(3\-'®=%("#%(' DL%(3X%(JK3X%(D [D 3$ I $ +EZ0G-!3\-' :D .#'/ ¬DL%(3O.,%] ${¸ "#!"4 /\E' "S+3!0]"# Xn = [ZDn |Fn ] D n Z Dn = X Dn I .,%1 $+0G!!3X%(!"4M

Ä

)

[ZT |Fn ] =

I

I

,+

I

<>C G <

9v"# ' I [L%("# 58%(3 ?e> F1>K%(.4+' 3 %(O' +DL% I 6S?+.4D DV1 G9-+.#. M '&%(.#.4&>?"#3:\-' I +.#.4 I "A¬%('"·' 6s I !"4*' I ' D .4 D .#' (Un ) (n, Xn ) D 3$ I "4$+0G+-«

H

c(fLhf! %d %§fqg ${¸ "4RDL%(3O3$ I ' 3!3+ I 6 I +%(-/' ¬m I !"4 F.=
∀i ∈ E

u(n, i) = f (n, i) ∨ P u(n + 1, .)(i) (Un )

(=

X

j∈E

pij u(n + 1, j)) ∀i ∈ E ∀n < N

!O%(.43O$+JE%(.4F M (u(n, Xn )) I +%(E !"L'!' D DVK>v3X%("4.=
8%(' 3X% +Z>?+3!!' ¬.,%1D 3D 3!"4$+$6 /58%(3 ?e>

><

+6 685

)

<

[Un+1 |Fn ] = P u(n + 1, .)(Xn )

LS
Un = f (n, Xn ) ∨ P u(n + 1, .)(Xn ) = u(n, Xn )

)& " ! >=@<

54:5A<

(

<

<=6

( 8. !>.0, $ ,

:5

$

=8

9v"#» ¼' /!' 3!02%(3!!"# EJ %(.4 "#.{)*"4!¬' :02%(3!!"# JE%(.4» ¼+O' :!' "# I 3"4X%(-:D 3$+>v"4!"#Q .4 Un Mn »!» !¼ ¼+.#.4O\E' + An ) , n ∈ {0, 1, ..., N }

A0 = 0

»*MTÚ?¼

(Un ) = (Mn ) − (An )

¬+.#.4/

$ I 0]DPK"#!"4»%(D DV+.4$2¿ Ë!¨ KÉO (³¡s s¡sK¢ ¿E´ UB¼R!R' "4\-']M @A' 32$+X%(Q .#"#3G.=< ' " I "#$ 0GE!32\-';.,%® $ I 0]DVK!"#!"4 $++3!0]"# Ë ÉG¢ ³+ s¥¯? s¡s¢ I 0]D .4W++0G+- » ¼]Á»s+DL%(3 I $\-'+E» ¼6¼ F;.=
43

∆Un = ∆Mn − ∆An

Fn−1

n≥1

∆An = − [∆Un |Fn−1 ]

»*M#ÑU?¼

.,%!' "# DL%(3.,% m3!06' .4 S?+ ${¸ "4X%(- (An ) %("#" I !!3!' "#!02%( "m!+0G+E¬D 3$+>v"4!"#Q .4 :D .#' ! D ' "4\E(2.10) ' !' ∆A ≥ 0 (U ) n n ' 3!02%(3!!"# JE%(.4 1 I ' !' "# I 3"4%(E®M Ã .'13!F0GE!3+32\-'2.,%8!' "# ${¸ "46DL%(3(An ) R' :02%(3!!"# JE%(.4 $ I 3!"#>?ODP' 3 I +.,% (Mn ) Mn = Un + An + *"#!"4 .4Î0G+06Q 3«v"#.S>-"4+- ∆Mn = ∆Un − [∆Un |Fn−1 ] I [∆Mn |Fn−1 ] = 0 &M MYÏM M

+6 6

H

)

I

< ; > =@< <>=@? C -

1O+D 3+.4X%(!"4 9-+.#.

(Zn )

+

9? = A

,+!

+#

" +S I !'.,%/ $ I 0]DVK!"#!"4 DVK+!' "#

H

-Q8

Un = M n − A n

O.=?+.4D DV

T1 = inf{n | An+1>0 }1{AN >0} + N 1{AN =0}

» ¼& \-' !R' +0]DOS?$+3!"¸L6%("4X$+0G+E*JK3 I F.,%1D 3$+>-"4"#Q "#.#"#$/ G An T1 c(fLhf! %d %§fqg%1 !D !"#02%(. "#.jj
(Un )

T1

n ≤ T1 =⇒ An = 0 =⇒ Un = Mn .

\-' "·U
Î*M

P P −1 « Í % MV@.,% U T1 = N Un 1{T1 =n} = N 0 'DP' 3 !' 3].=?$++0G+- 0 Un 1{T1!=n} ' 3G.4+\-'Z+N. 1{T1 =N } »s+2 I An = 0 1 = n} ¼& n ≤ N −+1OÈ.=
U T1 = Z T1 Un = M n

An+1 > 0

Un = Zn ∨ [Un+1 |Fn ] = Zn ∨ (Mn − An+1 ) = Zn ∨ (Un − An+1 ) .

"#.+3$!' .#6\E' 5®%("4 I 0]0G %1$ I X%("#3+0G+- Un − An+1 < Un Un = Z n U T1 = *M

N X

Zn 1{T1 =n} = ZT1 .

0

«L9-" ' +0]DOS T1 ] > 0 [US ] = [MS − AS ] < [MS ] = [M0 ] = [U0 ] " Q "4+G\E' S DP+' DL%¹+!3' 02%(3!!"# JE%(.4 j

S

H

Ä

&

5 :5A<

$ ,

=5

'%(D DP+.#.4+3¬ÉG¯¥ ¨{Ê*Ë ¶¢q¯¢ ¨ ¡m´+¥:' Â!DL% I ¸.#!3$O¸ " (Ω, (Fn ), %(E %('*);[EÅvDP![WX!' "#>K%(E6«

F0 = {Ω, φ}

;

FN = F = P(Ω)

;

; n = 0, 1, 2, · · · , N )

(ω) > 0 ∀ω ∈ Ω

%(!"4m%("H

+3X%]%(D DV+.4$]°!À Ê ¥{¡ U¢¿ºÉG¯¥ ¨{Ê*Ë I :+3X%1.4¬D .#'O' >?+E .,%¿?¯K µ´¿À ´!¾E´{¥ ¨ ¡ ¨ ´ D !"46M Ä ¨ s¡ DX³¶¢q¯K¢ ¨ ¡=´{¥X³2« Í Z' D DPK+3X%G\E'j< "#.j{)*"4!:!' 3 I ¬02%(3 I [$ d + 1 ¯ ¨ s¡ DX³¶¢q¯K¢ ¨ ¡=´{¥X³:$ .=-3+& 0 I %("4X6S%(-U d \-'jK%(3!",%(Q .4 %(.4$U%("#33$+.#.4 .4&>? I +' 3 DVK!"#!"#>?/»§.4 I ' 3RS< ' /% I !"4Zj%(.4+' 3O%( d+1 %("#!"j ${¸ "A+3X%$ SnM , Sn , · · · , Sn R+ Sn Ä ·´/ =¯Kº¾;¿À ¡¢q Ë ¥ X+3X%' 3$+. r > 0 J?' >?+3!L%(- .4¬D 3!") ¬.= ( % 4 . + ' & 3 / , . 2 % G 0 L ( % 4 " + $ D 3 $ 4 " ( % ' I ' 3 *'8+0]D DP' 3 DP' >?"#3 I 0]DL%(3+3 I Ä ¨ .4&D 3!");S<*% I !"mQ+3!>?$F R%(R*" V$+3+-/*;' !"#.#"4¬' >?+ER.,%>K%(.4+' 3\E'j%("#.,%0G L%("4 F.=< "#!X%(E ' ×D 3!")¤%("#" I %(. I ' .4$8+3X%%(D DV+.4$;q¥{¡¾Ø¯ ¨ sº*¯°²¡³ Ë MN×!' D DVKX%(EÂ\E' .4ZX%('*)¤ $+D 3$ I ",%(!"480G$+X%("#3! $+JE%(.%(' X%('*)ZS< "#E$+3¹+ L "#-!3-*' "#3X%2' ¨ e´ ¨ ¡=´{¢L ¿À ¯ ¨ sº ¯°²¡,³¯K s¡m¢ ML9-" R.4¬D 3!")S< ' 8% I !"Qr+3!>?$F1.=< "#!X%(E X;q¥{¡¾Â¯ ¨ sº ¯°²¡,³ Ë X+3X%*DL%(3 1 1 βn = S 0 = (1+r)n Pn n ${¸ "#!"4n× ˜ »§.4]D 3!")% I !'L%(.#"4$Z I %(. I ' .4$%(:' ]0G L%("4]D .#'¬m3!ZS!:"#*m$+3!"4+' 3 %('ÂD 3!") I ' 3XP%(n-= ¼&MEβBnP" I n"V' :%(' !3 m% & 3+D 3$+E+3.4 I v I "4+ES?!!"#3%.=< "#!X%(E%(/.=?!!"4+' 3DVK$%(-Z' DP3!{m+' "#.#.4 F .=< "#X%(E I 0]DV3!X%(E]' Â\E'L%(-!"#$ k .=v' "#*m3!02%(!"4] E1"#.*"4DPK;F n φ k I +R"#X%(E6»s$+!' 6 ¬.=?.#' !"4 I n' 3 ¼&*"#.>% $ I "4 +3 ¬0Gv*"¸L+3OX% I 0]DVK!"#!"4 M 5®%("4R"#. '&=%(' :D 3$ I "4+3 I ]\-'.=
3

KÚ

I + 6"#!X%(E1%;$+$2 $ I "4 $2%(.43\E' I I ' 3/j?] I %(&.,%0G¹+0G"#!'L%(!"4 \-'!3]>v"4+"#.Á%(0]"3+ I E!3$]L%(JK'W+3 » I n" −" .*1¼ %(:' I %!"#ZF;\-' ".=K%("#:"#0]DPKX$]\E'.,% ' "#&s3!0G$:DL%(3 0]"4!' II X!"#>?O"#RD 3$+>v"4!"#Q .4GML+.,% I *' "# F1.,%1 ${¸ "#!"4' "#>%(-G« 'O%(D DV+.#.4+3:³+ s¥¯K Ë ©¡m´1¿-´ ©-´³{ s¡m¢®' + I 3&*¥+ µ´'DU´{ºv¡°§°#´' :' "#&q¥ Ë ÌU¡³¡KB{°#´ ¼ O>K%(3!",%(Q .4%(.4$U%("#3OF6>%(.4+' 3R%( d+1 M Í I 0]D .4$++3X%1.,% ${¸ "#!"4Z (φn , n ∈ {1, ..., N } R F.,%]%( + DPKX%(- φ 0 φ 0 = φ1 Í Â3+02%(3\-'+3X%\-'P» I -!3X%("#3+0G+E %('*) I ' 3R % I !"m¼{ I >VMT%*M 3UMEDV+' >?+E+!3 $+JE%(!"#>? I +.,% !"4+-GF I ;\-'2D .#'!"4+' 3 I [ "4\E']QV' 3!"4W+3»§>?+EGF8 $ I ' >?+3!UAD 3!"41+ØDP+"4S?"#3¬ O +! .#"#QV+.#.4$O+ !"#!3¬%('Z.#"4+'8 /.= !3UM Ä ¯K°#´{ºv¥¿À º-¢*K¥{ µ'´ DU´{ºv¡°§°4´Ó« $+X%(EZDL%(3 <, > .48D 3-*' "# I %(.,%("#3®''+.¬' 3 Rd+1 .,%>%(.4+' 3 F +3X%**DL%(3O ${¸ "#!"4j .=< "#X%(E S< ' ZDV3!{s+' "#.#.4% I "4$/%.,%]!!3X%($+JK"4 n (φn )

H

Vn (φ) =< φn , Sn >=

k=d X

φkn Snk

»§M#Ñe¼

k=0

+!1 ${¸ "#!"4 XDL%X I 0]0G+-X%("#3 ®+3X%Â' {m"4:\E' !"#!'1' ' "# (Vn (φ)) I %%(D $6 :>PMT%*M 3UMSM %;Ì¯°4´{ºv¥¯ ¨ sº ¯°²¡,³ Ë ´ I :DV3!{s+' "#.#.46+3X%1 I V˜n (φ) = βn Vn (φ) = βn < φn , Sn >=< φn , S˜n > .

, $ ) !*$. 0$ 0&')!

43

;:

5

:<

Í %(D DV+.#.4+3X%Â³{ s¥!¯K Ë ©K¡m´1¯º* =¶¢P¯¢ ¨XË ´' 6!' "#¬D 3$+>-"4"#Q .4:>? I 3!"4+.#.4» φn ¼>?$+3!"¸q%(E

»§MTÎ?¼ +.,% I 3!3!DV %(' I 0]DV3!+0G+E&S< ' Z"#E>?!"4+' 33$3!JE%( "4X%(-&XDP3!{m+' "#.#.4/+E!36.,%1%( +R.,% %( » ?R.,% n+1 I φn φn+1 >K%(.4+' 3 X%(.41*'8DP3!{m+' "#.#.4KM Ã .·e< "#E+3*"#: I ' D 3$+.4W+>?+0G+-/ !!"#$FX% I 0]02%(!"4 %("#!"\E' ' %(D DV3!OS=< φn+1 , Sn > ∀n < N

6 685

9? C

%]!3X%($+JK"4

7

(φn )

K+# I

!O%(' (¸L%( I $!"j+O+' .4+0G+- !"= \-'+.4\E':X"# n≥1 Vn (φ) =< φ0 , S0 > +

k=n X

< φk , ∆Sk > .

k=n X

< φk , ∆S˜k > .

E

k=1

Í Z+3X%]\-' I +!/$+JE%(.#"#$/O$\E' "#>K%(.4+EF1.=

V˜n (φ) =< φ0 , S0 > +

E

Ë ÉG¢ ³+ s¥¯? s¡s¢ V@A' 3 n ≥ 1 v% < φn , ∆Sn >=< φn , Sn > − < φn , Sn−1 > ¬3!&\E' »§MTÎ?¼R!$\E' "#>K%(.4+E%('*);$+JE%(.#"#$ I
< φn , ∆Sn >= ∆Vn (φ)

E

»§M ?¼

»§M -¼

43

"#.V' GDV' 3 I +.,%S? I 3!"4+.j«e3+>?+F .,% ${¸ "#!"4 "" . " +]!' D DPK%(E02%("#E+L%(- + EOF]>%(.4+' 3%(O' Z!DL% I /N' I .#"4*"4+ .=
(Mn )

(Un )

(U ∗ M )n = U0 M0 +

${¸ "#' /' "#]¥ Ë {´ °°4´ I %(3X% I $+3!"4$/DL%(3O.4R3+.,%(!"4 (U ∗ M )0 =< U0 , M0 >

k=n X

< Uk , ∆Mk >

k=1

∆(U ∗ M )n =< Un , ∆Mn >

7·>? I:I X%(!"46 + e?+E3!DV I !"#>?+0G+E (4.3) (4.4) I Vn (φ) = (φ ∗ S)n

43

»§MTÙ?¼

˜n V˜n (φ) = (φ ∗ S)

% D 3DPK"#!"4Ø' "#>%(-2>K%Z'0GE!3+3G\-'j< "#.!' G I L%("#!3;.,%J?!!"4 1% I !"s]FZ3!"4\E'ÂS< ' "#E>?!"4+' 3Á+.,%&>K%(.4+' 3"# "#!",%(.4 R]DV3!{m+' "#.#.4RDP' 3+G $*' "#3R.,%&X%(.#"#$O RX %¬!3X%($+JK"4 J?!!"4jM

6 6

9? C

7

K+# I

9v"4+E ' >VMT%*M 3UM H=0G!' 3X%(Q .4G»§"=MYKM*' I !X%(Ee¼+ ' ¬!' "# 1 2 d D 3 $+>v"4!"#Q V.402FÂ>K%(.4+' 3/F %(0 d "#.{)*"4!]!!3X%($+JK"42%(' (¸L%( ((φ $I n , φn , ..., φ0 n ) 1 ; n2 ∈ {0, d1, ..., ' N"4\-}) ']+.#.4G\-' R φ = (φn , φn , φn , ..., φn ) M V0 (φ) = V0 V':"#-!3-*' "#36.,% Ë ÉG¢ ³+ s¥¯? s¡s¢Ô«j9-" u = (u0 , u1 , ..., ud ) et v = (v 0 , v 1 , ..., v d ) ∈ Rd+1 X%(!"4 [u, v] =

k=d X k=1

uk v k =< u, v > −u0 v 0 .

»§M ?¼

-9 " !' 6!3X%($+JK"46 /J?!"4X%(!"4=%("4X%(- %('*) I *"#!"4 6.=K%(.4+E6%('*)$+JE%(.#"#$ \-' "jU?+- + I 3 Í 8%(' 3X%]+Z' !3 '+ I 3K

< ∆φn , S˜n−1 >= 0

∀n ≥ 1

∆φ0n = −[∆φn , S˜n−1 ] V0 = V0 (φ) = φ00 + [φ0 , S0 ]

»§M )K¼ »§M ¼

§» MTÚ?¼ $+JE%(.#"#$1»§M )K¼R+1»§MTÚ?¼0GE!3+-:\E':.,%G!' "# 0 ! I 0]D .4W++0G+E/ $++3!0]"#$DL%(3O.4 ' "# k ]\-' "0GE!3].=< ' " I "#$2 (φnMV)@A' 3/0GE!3+3{)v"4+ I KV3+>?+ (φn , k ≥ 1) I %('*);$+JE%(.#"#$/»§M )K¼+:»§MTÚ?¼\-' "V'DV+3!0G+!+E φ ${¸ "#3R' ¬!' "# 0 +' &!3X%($+JK"4& J?!"4 M Ã . I .,%("#36\E' !/%(' (¸L%( I $ "#.(φ 3!n)GFÂ0GE!3+3\E' 0 φ = (φ0n , φ1n , ..., φdn ) φ φn RD 3$+>-"4!"#Q .4K I /\E' "S3$' .#¬m% I "#.4+0G+- G»§M )K¼{M φ00 = V0 − [φ0 , S0 ]

Ñ

Lf c(f!//§_ %§cbG1 DL%(3 I ' 3X%(ER.=? I V3!"4+ .·{V :N.=(φ)} ?V NI (φ)} 3!"4+. φ R>PMT%*M 3' 3 M E Ω Í 8%1+ V+

V˜N (φ) = {c +

n=N X n=1

< ψn , ∆S˜n >; c ∈ R , ψ ∈ P d }

-\ 'L%( d $!"#JK2.=? I 3!"4+. !' "# G>VMT%*MjD 3$+>v"4!"#Q .4 F P {(ψn1 ), (ψn2 ), ...., (ψnd )} >K%(.4+' 3O%( d R

6 6

)

>+# +

<>C

E J+#

9<

Í ®*"#3X%2\-'j< ' !3X%($+JK"4 φ %(' (¸L%( I $]! ' G³+ s¥¯? Ë ©¡=´G¿À ¯¥B{¡§ s¥!¯U©E´G' D .#'&Q 3!"4W+>?+0G+E¬' ¯¥B{¡§ s¥!¯U©E´ ! "j+.#.4:>?$+3!"¸L/.4 I *"#!"4O!' "#>K%(E

»§M#ÑU?¼

[VN (φ) > 0] > 0

I F' "#E>?!!"4X+' 3]X%(.4+0G+EG $+0' " %('+0]DG8 j?"#32F I ' D !Æ 6 3 %(' I ' +!2F;.=< "#!X%(- ' +JE%(JKL%(E ].=? I ' GD 3QL%(Q "#.#"#$2!!" I +0G+-6DVK!"#!"#>?KM N %(' Â02%(3 I [$&¸L%( I "4+3R3$+.= I +! DVK!"#Q "#.#"#$&!+;D 3!"# I "#DP&"#E+3*"#¬DL%(3.4R%(' 3!"#$R &3$+JK' .,%(!"4 »s+.#.4¬3$+D 3!"#0G+E/+8DL%(3!!" I ' .#"4+3:.4 + $+.#"#¬S< "# "#!"4$ U¼{M v",%(Q "#.#"#$1\E'':%(.#.4¬02%("#E+L%(- "#E!3v*' "#3O I S< ' RJK3X%( R"#0]DV3!X%( I KM-&PMT%*M*DVK!"#!"#>?DV' 3!3X%1+3!>v"#3 :0Gv W+.46F' Z02%(3 I [$/¸L%( I "4+3UM

H

2+.#.4!!3X%($+JK"4ÂDV+3!0G+]

V0 (φ) = 0 , VN (φ) ≥ 0 ,

E

6 685 )

E

$ ,-& )! / $ ! 5 7698

FG,HEI J+ K I

Í *"#3X%1\-'j< ' Z02%(3 I [$/¸L%( I "4+3 !:Ìe¡s¯B+°#´U< "#.Sj
3

< "4 $\E'j< ' 02%(3 I [$K»s8j?.#' !"4\-' "q3+ÂDVK!"#Q .4O' &!3X%($+JK"4&SK%:=%("#3O.=%(' "#3.4Á!DV"#3 JE%("#X%(3!"4\-'O\-'jv",%(Q .4ÂM ¬\E' I &3+' 3F/.,%>v",%(Q "#.#"#&Sv"4+E¬' ®02%(.·$ I XX%("#3DP+3!0G+!X%(-:.,%G3$+JK' .,%(!"4*'®02%(3 I [$KMS7 .#.4 ' ZDV+'ZD .#'.4"#« I /\-' "!R"#E+3*"#¬%( I 6\E'j?+EO' +$ I 0]"4 I %!"# I :j?+.#.4G0G3X%(.4KS¸.#.42 1.,% ' >?+.#.4¬$ I 0]"4K*.,%/>?+3!!';.,%6D .#' I $+.4$+Q 3$¬DL%(3.40G$*",%jK%("#.V02%("4 .4¬J?Æ O*'Z3!"4X\E']M

E

FE

JE

E

E

E

H

EÎ

JE

#E

jbEc+1 %1?bG1 ]%+3!!"4' "#>%(-O-O$\E' #" >K%(.4+E ÑKM Ã .j{)*"4!¬' /!!3X%($+JK"46%(' (¸L%( I $ +.#.4/\E'

φ

V0 (φ) ≤ 0

;

VN (φ) ≥ 0

;

[VN (φ) > 0] > 0.

»§M#ÑKÑe¼

+' ;>? I +' 3O%(.4$U%("#3 Î*M Ã .S{)v"#&' +E!"4+3 p ∈ {1, 2, . . . , N } Fp−1 − mesurable η = (η 1 , η 2 , . . . , η d ) F]>%(.4+' 3%( d + .4\-' R

[η, ∆S˜p ≥ 0]

*M

;

»§M#ÑUÎ?¼

[η, ∆S˜p > 0 > 0.

Ã .j{)*"4!¬' /!!3X%($+JK"4/S-",%(Q .4KM¼ M Ã .j{)*"4!¬' /!!3X%($+JK"4/S?$+3!"¸q%(E (φ) ≥ 0 ∀n ≤ N. ZD 3 I W /+!' "# I 0]0G6!' #" /« S?$+3!"¸q%(-1»§M#ÑKÑe¼ DPK 0 1. =⇒ 2.

u(n) =

φ = (φ , ψ)

[Vn (φ) ≥ 0] ; v(n) =

[Vn (φ) > 0] ; p = inf{n | u(n) = 1 et v(n) > 0}

S
η = ψp

φ

[η, ∆S˜p ] =< φp , ∆S˜p >= V˜p (φ) − V˜p−1 (φ) ≥ V˜p (φ)

I 6\E' "j+-!3X%4»§M#ÑUÎ?¼ 'DVK+3&%(.43 "#"=M 9-' D DVK 02%("#E+L%(- M ZDV+' u(p − 1) < 1 η = ψp 1{Vp−1 (φ)<0} Í %(.43 $ I 3!"#3 [η, ∆S˜p ] = 1{Vp−1 <0} < φp , ∆S˜p >= 1{Vp−1 <0} [V˜p (φ) − V˜p−1 (φ)] ≥ 1{Vp−1 <0} − V˜p−1 (φ)

I 6\E' "j+-!3X%46$+JE%(.4+0G+E1»§M#ÑUÎ?¼ »§Qq¼ «9v"# ' ¤>? I +' 3Â%(.4$U%("#3 >?$+3!"¸q%(E»§M#ÑUÎ?¼ ${¸ "42.,%!' "#D 3$+>v"4!"#Q .4Z 2. =⇒ 4. >? I +' 3&%(.4$U%("#3η F>%(.4+' 3%( d DL%(3 (ψn )

R

ψp = η

et ψn = 0 pour n 6= p

+2%(D DV+.4 4. ;DV3!{m+' "#.#.4 %(' (¸L%( I $ ;>K%(.4+' 3]"# "#!",%(.4E' .#.4% I "4$ZF DL%(3].,% (ψn ) D 3 DVK!"#!"4 (φ" n" ) M Í % V˜m (φ) =

X

< φn , ∆S˜n >=

1≤n≤m

X

[ψn , ∆S˜n ],

1≤n≤m

"Q "4+\-' ˜ OE' .·DP' 3 +O>K%(' DP' 3 M 63$' .#X%( + $ I ' .4 Vm (φ) m

6 6

< < C

-

>=@< G <

@A' 3%(QV3 +36.=-",%(Q "#.#"#$S< ' 2 0 %(3 I [$+/.,%![$3!"4 /02%(3!!"# JE%(.4U' %(' 3QPX"#S< ' Ø3$!' .#X%(]S?+3!"4 %4j%("#Q .#"4\-' "+3X%Â!' 2X %(-1DP' 3¬.,%Â!' "# %( I DL%(3X%(JK3X%(D [K » ¼ $"#JK+3X%1' !DL% I /+' I .#"4*"4+\E'+. I \-'GM E, < ., . > )

I

I

+

I

I

43

Efqc v`b :

v9 "4+E ' X '/!DL% I ] >? I 3!"4+.+ ' GDL%(3!!"4 I E>?{) I 0]DL% I ]3+ I -!3X%(EDL% "#. V K E V {)*"4!/%(.43' ?> I +' 3 ?> $+3!"¸q%(E a

E

< a, x >> 0 ∀x ∈ K

< a, y >= 0 ∀y ∈ V

É ¢ ³+ s¥¯? s¡s¢« A' S? I 3!"4+. *"#0G+!"4 ËG V 'js+3·.,%R $+0G!!3X%(!"4 ¸ "4K(!Sm+3!0G$OMe@·K C = K+V = {x+y | x ∈ K ; y ∈ V } +8Î1$+X%(DPUM ÑKM !s+3!0G$K I E>?{) /+ I E!"4+- DL% 'R'QV3!+3OF10G-!3+3O.4¬D 3+0]"4+3 C 0 DV"#EU.42%(' !32$+X%(-G=% I "#.4 "# ' Z!' "# zn = xn + yn (xn ∈ K ; yn ∈ V ) ]0G-!3+3\-'].,%Â.#"#0]"# ].,%!' "# ! I ->?+3!J?+E; 1DV"#E6 "#.ÁU?)+3!J? >?+3O' CZ>? I +' 3 K E>?+3!J?1%(.43>?+3x nk (x.4n\E)'+.·%(D DL%(3!!"4+- I \E'jx?+3\-' M % D ' "4\-' !xjm+3!0GK$ y"#.-nk =3z!nk− V V z = x+(z−x) z∈C

Î*M

H

? I +' 3&\-'+. I \E'1 + D ' "4\-'a x C θ ∈ ]0, 1] I ->?{)*K v :3!¬\E' $+>?+.4D DL%(EO.4 C θ(x − a) ∈ C kak2 ≤ ka + θ(x − a)k2 X I 0G+06Q 3a + Q !"4+E .4R"#$+JE%(.#"#$ 2 < a, x − a > +θkx − ak2 ≥ 0 ∀θ ∈ ]0 1] .

Ïq%("4X%(E +*3 >?+3O1+ $ I 3"4%(EU "#.j>v"4+E M θ < a, x >≥ kak2 > 0 ∀x ∈C !RE' .#.4!' 3 Ã .S'3/F0GE!3+3 \-':.,%m3!0G:.#"#U%("#3 X"4+E ! %(.43<a, %(.> /3!:\-'V x ∈ K , y ∈ V , λ ∈ R x + λy C .,%1m I !"4®% G ${¸ "4!' 3 < a, x + λy >≥ kak2 λ 7→< a, x > +λ < a, y > R DV+' ¹+!36' "s3!0G$+0G+-¬0]"#3$\-'6!"·+.#.46! I !X%(E 8%2 I < a, y >= 0 I \-' "S+3!0]"#/.,%1 $+0G!3X%(!"4jM

3

6 6 )

)

C

C

K

A <>C < = ,+

#+ I

7A < C

'R'RD 3DVKO%( I /DL%(3X%(JK3X%(D [6 ¬0GE!3+3 .4:3!' .#X%(O' "#>%(-6«

Efqc v`b 1 Ó02%(3 I [$¸L%( I "4+3;!2>v",%(Q .48!" +Â+' .4+0G+EU< "#.&{)v"4! ' D 3QL%(Q "#.#"#$/¤%*0G+!X%(- I 0]0G !' D DV3!'R.,%\-'+.#.4:.4 I ' 3&% I !'L%(.#"4$O O% I !"m ˜k XE 02%(3!!"# JE%(.4eM Ω :

K

((Sj ) 0 ≤ j ≤ N )

Ë ÉG¢³{ s¥!¯K s¡sK¢« %?¼9-' D DPKX\-'j< "#.q{)*"4!R' OD 3QL%(Q "#.#"#$ !' D DP3! 'Á.,%\E'+.#.4 .4 I ' 3 % I !'L%(.#"4X$/ 6% I !"m ˜k E/ ¬ 02%(3!!"# JEΩ%(.4UMA9v"# k ) ' (Sj 0 ≤ j ≤ N ) φ = (φ D ' "4\-' !3X%($+JK"48%(' (¸L%( I $8+.#.4 \-' !;%(' (¸L%( I $Kj %* V0 (φ) = 0 φ \-'+.4\-'/"#

n≥1

i=n X

V˜n (φ) =

< φi , ∆S˜i >=

i=1

k=d X k=0

(φk ∗ S˜k )n .

q (V˜ (φ)) !R' ¬02%(3!!"# JE%(.46E' .#.4F.=
+.#.4:>?$+3!"¸L/ I "#.Á!6%(.43 I ,. %("#36\-' ˜ ]DV+' :DL%/¹+!3]' G>PMT%*M ≥0 VN (φ) N (φ)] = 0 -' .#.4KM Qq¼ $ I "#D 3-\-'+0G+EU!' D DPKX¬.402%(3 I [$]>v",%(Q .4 '6%(.#.46%(D D .#"4\-'+3:.41![$(H 3W+0G6 *&%([ *H R%(L% I [%(R.,%]"#!'L%(!"4 !' "#>K%(EG« +3X%1.=VMT%*M 3UM !' 3 0' "·*'D 3v*' "#O I %(.,%("#3 Ä 9v'

[V˜

n

1

E

< X, Y >=

Ω X

X(ω)Y (ω) .

ω∈Ω

$!"#JK+3X%G.4'&++06Q .4 m3!0G$/DL%(3&.4O>VMT%*M 3UM ˜ \E'L%( $ I 3!"# E VN (φ) .=?"#3O\E' 0!R' ZX'!DL% I :>? I 3!"4+. M E m3!0G$/ R>VMT%*M 3UM DVK!"#!"#>?R>?$+3!"¸q%(- % .4/X'++06Q .46 Ä Í Z+3X E P '.,%("4X%('Â.4 I +' 3.4 "#Â O>?X $+3!"¸L+3R\-' ! I E>?{) + ω∈Ω I 0]X(ω) DL% I UM Í=Z1 3+02%(3\-'+3X%]\-'j< ' :>PMT%*M >?$+3!"¸L X D ' "4\-'j< "#. j 0] 6= φ *"4sÇ"#-UM 7 D D .#"4\-'¬%(.43¬.4![$3W+0G1 *¬%([ *H R%(L% I [ F + «"#.{)v"4/' >PMT%*M 3UM α +.#.46\E' Ä

X

ω∈Ω

;

α(ω)X(ω) > 0 ∀X ∈

X

α(ω)V˜N (φ)(ω) = 0

ω∈Ω

∀φ ∈

7

0

»§M#Ñ?¼

» I !"4 $+3+3R.4>VMT%*M Í ;+3X%1\-'&.,%/D 3+0]"4W+3¬"#$+JE%(.#"#$:+-!3X%("# X \-' "j-.4R"#*" I %(+' 3& RDP"#-¼{M@AK %(α(ω) .43 > 0 ∀ω k α k=

!1' ;D 3QL%(Q "#.#"#$

X

α(ω)

;

=

ω∈Ω

α . kαk

I [L%(3!J?U%(E2'G.41DP"#-G Ω %>? IZI +!;X%(!"4¤.,% +'*)*"4W+0G6$+JE%(.#"#$6 G»§M#Ñ?¼RU

[V˜

N (φ)]

=0

EÙ

∀φ ∈

7

0

.

»§M#Ñ+-¼

':%(.#.4&0GE!3+3/\-' ˜ !&' 02%(3!!"# JE%(.4]' Z+' !"#.#"4X%(-¬.,% I %(3X% I H $+3!"4%(!"4 " " F I +V+(USLn ) 'R' /' "#:>? I 3!"4+.#.4/D 3$+>v"4!"#Q .4

(ψn ) = (ψnk ) , k = 1, 2, . . . , d , n = 0, 1, 2, . . . , N

F >K%(.4+' 3Z%( d +Z $!"#JKZDL%(3 R.,%Ø!3X%($+JK"4 7 %X I "4$F + R (φn ) ψ 0 %(D D .#" I %(!"48*'Z![$3W+0G " " »m%>? I M .,%1' "#:>? I 3!"4+.#.4K V = 0) (σn ) F>K%(.4+' 3¬%( d :% I !"m&3!"4\E'$:% I 0!'L%(.#"4$:+ .4D 3-*' "#: I %(.,%("#3'!'+. [ , ] . I ' .!"#0]D .4 I *' "#¬F.=
@3+2.=v"4+EU I 0]D +E'Ó »§M#Ñ+-¼{ [(ψ ∗ σ)N ] = [ψ0 , σ0] I /\-' "S0GE!3KA»=ÁM " " " ¼{ \-' (σn) !R' EH 02%(3!!"# JE%(.4M Ã .Á3!F3+02%(3\-'+3G\-'G.=-",%(.4KM

"

1

0! . , "J) !

:5 <

<

&%(D DP+.4·S?+0G+EA.4j0G$ I %( "4!0G·QP' 3"4+3»s $~ÇF $ I 3!"#%(·.=< "#E!3v*' I !"4q¼·F.=?O"#E>?!!"VK *'>*30"#S K Ã .A% I [W+1.=?+*3 I /0G¹+0G]% I !"' 3O.402%(3 I [$]%('D 3!") l .,%G>K%(.4+' 3¬ 6.=?"#3 l SN M N SN − K Ä 9-"=(%(' I -!3X%("#3K SNl < K U.4Á*3"#·D 3 I ' 3$ÁDL%(3.,%O $++E!"4]*' I %(.#.-jK%(.4+' 3 .=%(.4+' 3 'ZN.4¬D 3!");*' I %(.# . M]³¯¿E¯K µ´¿À ´¾E´{¥ ¨ ¡ ¨ ´ +M

H

N

/P s¡m¢Z¿E´OÌ(´{¢L µ´ +'

(SN − K)

3 DV$+ »s'qº* +' 3DP$+q¼ D 3!")1S?+-' :D !"4Z+' 3DV$+ 6FX%>%(.4+' 3OF.,%1%(:SK%(EG«

6 G685

FG,HEI J+ K I

'%(D DV+.#.4+3&V s¡s¢ ´{ºv¥! Ë ´{¢L¢P´6¿·À Ë!¨+Ê*Ë ¯¢ ¨ ; ´ »s'+ I 3O ¿?¯K µ´¬¿À ´!¾E´+¥ ¨ ¡ ¨ ´(¼ ' O>%(3!",%(Q .4&%(.4$U%("#3 .,%]>%(3!",%(Q .46%(.4$U%("#3 ˜ +3X%]%(D DV+ .4$2P s¡m¢¯ ¨ sº ¯°²¡,³ Ë ´ %X I 4" $/N F M h≥0 h := βN h h

jb-`h&/§b 9v"# .,%1!' "#6 I ' 3OS< ' % I !"3!"4\-'$ (Un ) u ÑKM /P s¡m¢¿À ¯ ¨{Ê ¯K »s' ¨ ¯°§°s¼+' 3DV$+ q¥{¡¾1¿À Ë ¾E´+¥ ¨ ¡ ¨ ´ +Á /¿?¯? µ´ ¿·À ´!¾-´{¥ ¨ ¡ ¨ ´ !' 3.=PMT%*M DL%(3.,%O>%(.4+' 3A0GUÅ?+ Á.=K%(.#.4¬ O+0]D .4&0 S<4P s¡m¢ I³/⊂¯(³{{0, ¡s¯K s¡ 1,+º2,´³{.M*.@. %(, N 3R{})*+0]D .4¬' ÂD ' /»s+' 3DP$+q¼%!",%(!"4\-' OD 3!");S%(.#.4 +3X%1 ${¸ "DL%(3 M I h = (K − UI )+ .=? I !O$+0]"4/DL%(3 = UN ∨ a ∧ b I +3!X%("#aet"#!b!"#!' !"4Á¸L> %( I0"4W+3a'<.4UO0< 0 b Rq°¯ ¨ ´{É2´{¢L ©E¯¥¯K¢L s¡ h' 2 "#E>?!"4+' 3Á!3' >K%(E!3D 3!"4X\E'$ .=?U%(' \E' "#!F]% II +D +3 :.#"#u0]"#+3OXRJE%("#O%('Z h"#>?U%(' M a b

6 G6

>< C ? @; <#; A A 9? #< < BC =

7

I I

-

I

I

+

9< ! "#$% G< 9? BA <

K+#

!-I

-I

G

+ I

<

-

NÓD 3X%(!"4\E'K.42D !"42+' 3DV$+ 2-G0]"4G!' 32.4;02%(3 I [$ +GX;$+J? I "4+-ÂF8' G"#!X%(-ÂF ' D 3!") \E' "$\-' "#.#"#Q 3.=%(-G«

$g%,d %§fPg v9 "# ' D !"4+' 3DV$+ ]%(&' 02%(3 I [$¸L%( I "4+3&>v",%(Q .4 U< "#.{)v"4!' !' "#%%(D $ (Hn ) :>VMT%*M h +.#.4/\-' ≥0 Ä HN = h Ä .4¬02%(3 I [$:¸L%( I "4+31» Ω, (Sn ), (Hn ), ¼X"#+ I 3:>-",%(Q .4K !R' Âq¥{¡#¾;´¾ ¨ °²º*¯¢q °À ¯¥ B+¡§ s¥!¯U©E´ ¬.=%(- &*"4 I ' +3.=?"#3Uv!' 3Á.=-",%(Q "#.#"#$6"#E!3v*' "#O I E!3X%("#- s3!' 3R.4:D 3!"); I %(.#.+O RD ' UM

6 G6 I<>A

)

*

+# I

<?

K+

7A8A ?

-

+

GC =

I

(-I

A<

9v"4+E ' Ó02%(3 I [$Z¸L%( I "4+3>-",%(Q .4+ ' ×% I !"/3!"4\-'$KM "#!!"#!' !"4 ), ) ¸L%( I "4W+(Ω, 36$+(S 0Gn+ !' 3R=. 0 MV9v"# (C ) »§3!DjM (P ) ¼R' /!' "# /DjMYKMT%*ML /.=?+E ¼ C P

)

(c fLhf! %d %§fqg%1 Í %.4$+JE%(.#"#$ Cn − Pn = Snl −

K (1 + r)

N −n

»§M#ÑUÙ?¼

∀n ≤ N

% !"s %*0G+H É ¢ ³+ s¥¯? s¡s¢« !3Á02%(3 I [$ I 0]DP3!02%("#E+L%(-.4 ËG d+3 I {0, 1, . . . , d, c, p} X%(- DV' 3 I ' 3¬F].=< "#X%(E +

O ! ' D V D K / $ v > , " ( % Q 4 . K M ' & ( % # . 4 .

G 0 E!3+3 n S¯n = Sn , Cn , Pn \-'KS!".= S0l − K N (1+r) 9v"#]» k ¼.,%G' "#/>? I 3!"4+.#.4D 3$+>-"4!"#Q .4F]>K%(.4+' 3 0 ≤ n ≤ N ; k ∈ {1, 2, . . . d, c, p} n ) ; ${ %( (ψd+2 ¸ "4:DL%(3

R

H

ψnl = 1 ; ψnc = −1 ; ψnp = 1 ∀n ≥ 0 ; ψnk = 0 ∀n ≥ 0 ∀k ∈ / {l, c, p} ? φ 3R.4 % I !"mR3!"4\-'$R++.#.4/\-' M V0 (φ) = 0

' Í 8%1 I

+ +DL%(3!!" I ' .#"4+3U @' "4\E'

Vn (φ) = Snl − Cn + Pn + φ0n Sn0

∀n ≥ 0

S0l − C0 + P0 + φ00 = 0

φ

!O%(' (¸L%( I $ KP %]S
Vn (φ) =< φn+1 , S¯n >= Snl − Cn + Pn + φ0n+1 Sn0

I ψ

»§M#Ñ?¼ »§M#Ñ )K¼ »§M#Ñ ¼

¼ 0G-!3 \-' 0 $+DP+DL% "#.L3$' .#&%(.43 % I 0]DL%(3X%("4 »§M#Ñ?¼+:»§M#Ñ φn n »§M#Ñ )G¼\E' φ0n = φ00 = C0 − P0 − S0l

Vn (φ) = (C0 − P0 − S0l )(1 + r)n + Snl − Cn + Pn

VN (φ) = (C0 − P0 − S0l )(1 + r)N + K

»§M#ÑUÚ?¼ »§MTÎK?¼ »§MTÎ*Ñe¼

%]!!3X%($+JK"4 I !!"#!'/ I ' %(3!Q "#!3X%(J? I /\-' "S"#E+3*"#¬%('02%(3 I [$6S-",%(Q .4KM φ %(' 2X%(.#.42 G02%(3 I [$8S.,%Z02%( "4W+32D 3X%(!"4\-'2 20G+!!32+v+' >v3 I +!2!!3X%($+JK"4 X+3X%("#O.,%G' "#>%(-«LFG.=< "#!X%(E >?'¬% I [++Ò' % I !"4®+O' D ' +O>?'O>?+ +Ò' 0 I %(.#. I +.,%6>?'ÁD 3 I ' 3' &\E'L%(-!"#$»m%(.#J?$+Q 3!"4\-'e¼S?'D .,% I +Ò6%('X%('*) "V+.#.4¬! +R\-' >?'R+0]D 3!' -+Ò%(';0G¹+0G¬X%('*)Â!"S+.#.40 r > 0 >?'/1' I [+ÒFÂ3!"4+Ç'\-'j?!32\E'L%(-!"#$2 ]-' 0G$+3X%("#32! N %(.43< 0 + * ' ) % R V D + ' ? > + E

2 0 ( % "#E+L%(-&¬D 3$X+E+36« I (C0 − P0 − S0l )(1 + r)N .=?'.#' "S.#"#>v3+Ò:.=?'DPKX$ +Ò:%(' »m%?¼ l SN > K D 3!") j+/>?!32D ' /]>K%(' /3!"4+jM %>%(.4+' 3 ]>?!3GDV3!{s+' "#.#.42!6 I Q "4+ I +.#.46K \E'¬'&%>? I %(. I ' .4$KM

H

.=?'R>?+ +Ò6>?!3/% I !"4%('D 3!") 3!"4&.,%DPKX!"4;S< ' ÂD ' .,%>K%(.4+' 3R >?!3&DV3!{s+' "#.#.4¬! + I 36.,%0G¹+0GGM B'&%>?+Ò6 I 3$+'!"F]Q + "#3' s3:.#' I [ M Î*M 9-"%(' I E!3X%("#3 .,%&!!3X%($+JK"4 I !"4!X%(-ÁF¬% I [++3.4 I %(.#.=KFO>?+*3 C0 −P0 < S0l − K N (1+r) .4D ' +A+0]D 3!' -+3.=? I D 3(¸O$ I 3!"#3 I 0]D .4W++0G+E&.4&s3!02%(.#"4!0G/02%(![$+02%(!"4\E' I 3!3!DP%(-UM *MRB'6*"4!DPKX+Ò02%("#-+L%(E :$+.4$+0G+E/!' 2%(E:DV' 3:"#02%(JK"#+3:' ]!!3X%($+JK"41D 3$+>v"H j\-' "=L>?'¬.=?+Ò]$KS!&' $+JE%(.#"#$]+-!3 "#Q .4DV+3!0G+!X%(ESK%(3!",%(Q .4 %(.4$U%("#3eM »§Qq¼

I l ?'%(' K

E

E

43

^_ o& _ · a c;%1?_*d %§fqg 5(#Ag¤hfqcedUbC&µb#6%//§b U_g6Gc+% · # b DL%(3!"#$:DV+' OU
#_ %5·b

D

5#_21Kd %'&

~dUc c+% #

»§MTÎKÎ?¼

K (1 + r)n (1 + r)N

Snl − Cn + Pn =

\-' " I *' "# %('Z3$!' .#X%(O' DP+' ' 3!D 3+L%(E ' "#>%(-/« I: DP3!{m+' "#.#.4 I !!"#!'$:DL%(3 Ä .=?+-6S< ' I %(.#.j /0G¹+0G6% I !"'H#Ç% I +E %('0G¹+0G:D 3!")ZS
H

F:.,% I (% "4

0! ! . $ ! . , )+" &'$ ! ( $ !*. $ , & ) / $

"

K (1+r)N

:5 <

5

8 7698

<

98A5

76

8

5 7698

%(R' I :DL%(3X%(JK3X%(D [KL'!' D VD K $:' 02%(3 I [$:¸L%( I "4+3O>-",%(Q .4KM Í *"#3X%\-'j< ' 6D !"4®+' 3DP$+ h !/³{¡É1ºv°¯B+°#´;»s®*"#&%('"Á¥ Ë q°²¡ ¨ ¯CB{°#´(¼ U< "#.·{)*"4!6' 6!3X%($+JK"41%('*H (¸L%( I $ +.#.4/\-' φ

@' "4\-'

h = VN (φ) =< φN , SN >

!O%(' (¸L%( I $ Kq.=
k=N X

˜ = V0 (φ) + < φk , ∆Sk > ou h

k=1

N X

< φk , ∆S˜k >

»§MTÎ ? ¼

k=1

$g%,d %§fPg' 'O%(D DV+.#.4+3 .=? I 3!P"4+.#.4 '/%U>?:>-'+®M *M ;\-'KP%(¬' Ω02%(3 I [$1>-",%(Q .4K jv(S"4 nK)M % P !!3X%($+JK"4 ! %(D DV+.=
φ

h

EÚ

c(fLhf! %d %§fqg 9v"4+- ' /D !"4+' 3DP$+ K

h

φ

' DP3!{m+' "#.#.4/!"#0' .,%(E

V˜n (φ) =

[h|F ˜ ] n

h

+=

∈P

%.=
∀n ≥ 0

K%(3!",%(Q .4:+3!0]"#L%((.4 Vn˜(φ)) M

»§MTÎ(-¼

h

Âb-`_

c · # b 1

9-"¬.4802%(3 I [$Â>v",%(Q .4K' !!3X%($+JK"4%(' (¸L%( I $ !"#0' .,%(E Â$ I XX%("#3+0G+E%*0]"4X!"#Q .4K !" I 0]0GZ.4ZD 3' >?Z.4DV+!"#23X%("4 +0G+-Â' "#>%(-«"# (% ' (¸L%( I h$+.#.4Z\-' φ VN (φ) = h ≥ 0 'R>?+O />?"#3O\E' ˜ M Ã .j+3$!' .#6\E' M Vn (φ) = βN

∈P

[h|Fn ] ≥ 0

Vn (φ) ≥ 0 ∀n

^bh·c;% 5(#AgbZfqhd %§fPg¤cEh&/%1?_ a&/§b " N .=?"#3.41DL%(3X%(JK3X%(D [' "#>%(-¼{ j?+3!!' 31 M< {)%(0G+8 ;»§MTÎ(-¼ I *' "#¬$U%(0]0G"#:F2' 63+02%(3\-' "#0]DV3!X%(E/«.4D 3+0]"4+φ30G+0Q 3O I +!$+hJE%(.#"#$R $+DV+1DL% 7 X%(*"4Á\E'.4R I 10G+0Q 3 P )Î: 0G+06Q 3 »§MTÎ(-¼O $+DV+ O $+DV+GDL% .,%6!!3X%($+JK"4O I ' >?+3!!' 3 M %:>K%(.4+' 3 I 0]0' ¬%('*∈ I \-': +O I 6\E' "*Ç'!!"¸L¬.4RXφ%(!"4&!' "#>K%(E/« h n πn (h) = Vn (φ) =

1 (1 + r)N −n

[h|F ] ; n

π0 (h) = V0 (φ) =

1 (1 + r)N

[h]

»§MTÎKÙ?¼

H=0G!' 3X%(Q .4 U?+*3 .=?" I "S' :D 3+0]"4W+363X%("4 « h *"4DPKX%(-6%('®+0]D .,%Â0]0G !/%' 3$DV' >?"#3¬=%("#36=% I ]F ?!!"#3 I h+¬%(3!J?+E¬!' 3.4602%(3 I [$+ !' "#>K%(EO.,%2!!3X%($+JK"4 M Ã . φ ! I L%(!' 3+.·\E'j?+ I +!/D !"4%('ZD 3!") DV' 3O%(DL%("4+3OXO%( J?"4UM m%("4X%(-"#E+3!>?+ "#3.,%&L%(!' 3D 3(s S< ' ]02%(3 I [$ %&D 3DVK!"#!"42!' "#>K%(EK'Á+1s' 3! "#' R I πK0K(h) ¸L%( I "4+3»m%(QX+ I /S
43

c(fLhf! %d %§fqg 9v"# ' D !"4+' 3DV$+ 3$+D .#" I %(Q .4 !.=< ' "4\-'RD 3!")G{) I .#'L%(EÁ.=?+%('*)]X%(!"4 RM MTÎ/+ÁDPK π(h) 0]0G ˜ h + ˜ - ?H=02%(3!!"# JE%(.4Uq"#.·+8! MP9v"#¬Sv",%(Q .4KM ÙK

Î*M@%;F.=< ' " I "#$ $!"#JKDL%(3 ' !' "#8%%(D $ >PMT%*M 3UM >?$+3!"¸q%(E + DVK M '%(.#.4Á0G-!3+3R(h \E'nK)-!" v2DV+' I !!3!' "#3≥ ' 0OX%($+JK"4 hSN I mπ(h) "I 4%(- %>? I ¯ ' 3ψ.4n=% I (φ !"mnR,Fφ3!n"4,.\E.'.¬, φ*n';, 02 %(3 I [$:$++*'Z+>?$+3!"¸q%(n-) ψ ¯ = 0 »=ÁMMTÎ*MTÎ?¼{M Í % V0 (φ) »§MTÎ ?¼ »§MTÎ )K¼

¯ = φ¯0 S 0 + φ1 S 1 + . . . + φd S d − hn Vn (φ) n n n n n n 0 0 0 ¯ = (φ − φ )S + Vn (φ) − hn n

»§MTÎ )K¼R$ I 3!"#6+ *' "#&F.=
n

n

»§MTÎ ¼

φ¯00 − φ00 = π − π(h)

0]0G6.4!!3X%($+JK"4 ¯ + - %(' (¸L%( I $UA»§MTÎ ?¼U
¯ = φ¯0 S 0 + φ1 S 1 + . . . + φd S d − hn Vn (φ) n+1 n n+1 n n+1 n = (φ¯0 − φ0 )S 0 + Vn (φ) − hn n+1

n+1

n

0]DL%(3X%(-1»§MTÎ K) ¼R+6»§M K?¼{LZQ !"4+E .4R3+.,%(!"4

φ¯0n+1 − φ¯0n = φ0n+1 − φ0n

0]D 6+E' »§MTÎ {¼ %1 I

φ¯0n = π − π(h) + φ0n ¯ = (π − π(h))(1 + r)n + Vn (φ) − hn Vn (φ) ¯ = (π − π(h))(1 + r)N > 0 VN (φ)

6 685

%]!!3X%($+JK"4 ¯ I !!"#!'/ I ' %(3!Q "#!3X%(J?6\-' "SDP+' O¹+!3¬0]"4 +D 3X%(!"4\E'6 :.,%m%!' "#>%(-G« .4Â!DV$ I ' .,%(+' φ3G%(!!"#3$DL%(3.,%!' 3$+>%(.#'L%(!"4 .=?+"#0]0G${H *",%(+0G+-!' 3¬.402%(3 I [$%('8D 3!") P+:D 3$+.4W+>?]' 3 I +!h]X0]0G1.,%\-'L%(E!"#$ 0 \E' ".#' "DP+3!0G+ S?+3!π!' 3 .=?"#3/.,%("4φ$]%('.4 I +' 3:.4] "#S
<>C ? ;

- I

< 9? < ; A

+ I

-

I

+

A

K+

< BC A8< 7C I

I

7A

@.,%'Á%(' ]02%(3 I [$>v",%(Q .4R+ '' D !"4+' 3DP$+ G$ I X%("#3+0G+- !"#0' ,. %(Q .4KM %1D 3DVK!"#!"4 ' "#>%(-:>%1+DL%(3!!" I ' #. "4+3 0GE!3+3 .=
Ù*Ñ

c(fLhf! %d %§fqg%1 .4/X'++06Q .46 9v"#

Π(h)

R+

I -!"#!'$/DL%(3.4RD 3!"){) I .#'L%(EO.=
Π(h) = {

[h] ;

∈P }

Ë ÉG¢ ³{ s¥!¯K s¡m¢ "#.{)*"4!G%(.43:' G!' "#%%(D $ ÑKM 9v"# >?$+3!"¸q%(- +.#.4G\-'1.4 π ∈ Π(h) (hn ) hN = h et h0 = π 02%(3 I [$ "#6>-",%(Q .4KM Ã .Á{)*"4!] I »=Á!MVM *M ?¼¬' GD 3QL%(Q "#.#"#$ ¯ I [L%(3!J?U%(E1' Ω, (S , h ), .4¬DV"#E6 n n'¬.,%\-'+.#.4 ˜ + ˜ -/ :02%(3!!"# JE%(.4 "#.+®3$!' .#]+DL%(3!!" I ' .#"4+36\-' Ω ( Sn ) ( hn ) ¯ ∈ P M :D .#'U π = h0 = ¯ [h] ˜ I :\-' "S0GE!3\E' π %(D DL%(3!!"4+-&%('Z0G+0Q 36 :*3"#/ 2»§M *Ñe¼{M + Î*M Ã E>?+3X+0G+EUV"4+E ˜ n = [h|F ˜ DPKX h ˜ n ] = βn hn MV0]0G (S˜n ) et (h ˜ ∈P π= [h] XEZ ?H=02%(3!!"# JE%(.4UR.4®02%(3 I [$ $++*' Â>v",%(Q .4 S?n"#) "#.j+ 3$Ω, "#0]0G$*",%(+0G+-:\E' + !' .#(S /n\-,'h n ), M

H

h0 = π

hN = h

π ∈ Π(h)

fq` h /'v`b-gPd Í ZDV+' O%(.#.4+3D .#'R.4"#%(.,%1 I 3!"#D !"48 Π(h) %(R' Z02%(3 I [$6>-",%(Q .4]« Ä Í 'Q "4+ h !R3$+D .#" I %(Q .4+U I 0]0G/'R>?+ ¬.4:>?"#3U Π(h) I 0]DP3!6' Z+' .SDV"#EUM Ä %(1.4 I % I E!3X%("#3K Π(h) !1' Ø"#E+3!>K%(.#.4' >?+3!G R M»=Á!MÏ.#.#0G+3G+29 I [ "4 «9- I [L%!!" I ÏA"#L%( I e¼

H

$ , & )! & " ! ><

6 685

98 >:

FG,HEI J+ K I

Í *"#\-'j< ' Z02%(3 I [$/¸L%( I "4+3O! I 0]D .4+ !"S' /D !"4 +' 3DV$+ 6!"#06' .,%(Q .4KM E!3X%("#3+0G+-ÂF .=v",%(Q "#.#"#$Â\-' "DL%(3X%4]"# [$+3+E2F .,% L%(!' 3;S< ' Ø02%(3 I [$K I +!Â[-Å-DV![W (% D DL%(3X%4 I 0]0G' I 0]0Gv*"#$02%(![$+02%(!"4\-' DV+' ]0G¹+0G +02%( +32!"R.4=%("#GS?+L%(Q .4&!Q "4+G3$U%(.#"4! G+3X%:' R0G¹+0G \-'.=?"#3Á\-'jK%(EUM

43

H

6 6

+# I

7A8<>C < =

,+

>C = ? A < C

,+

Efqc v`b 802%(3 I [$1>-",%(Q .41! I 0]D .4+/!"+:+' .4+0G+-/U< "#.{)v"4!' D 3QL%(Q "#.#"#$ºv¢¡+º ´2'&.,%\-'+.#.41.4¬D 3!") % I !'L%(.#"4$ O% I !"sOEO 02%(3!!"# JE%(.4 Ë ÉG¢ ³+ s¥¯? s¡s¢ :

ÙKÎ

"#.q{)v"4!' !!3X%($+JK"4 ?% ¼9-' D DPKX.4R02%(3 I [$O>v",%(Q .4O+ I 0]D .4++Á"# ' R>VMT%*M 3UM h ≥ 0 0]0G !·%(' (¸L%( I $¬UDP+' A$ I 3!"#3 » I !MU.,%3+02%(3\-' %*0]"4X!"#Q .4 +.#.4\E' φ h = VN (φ) I φ " " .*¼ βN h = V˜N (φ) = V0 (φ) +

9-"C +

N X

< φj , ∆S˜j > .

1

E· +'*)&D 3QL%(Q "#.#"#$A'S.4\-'+.#.4 ˜ !S' 02%(3!!"# JE%(.4Ke%(' 3X% (Sn ) βN 1 [h] = :\-' "j+E!3X%("#$+>-"4 +0]0G+- βN 2 [h] I = Qq¼ Ã E>?+3X+0G+EO!' D DVKA.402%(3 I [$R1>v",%(Q .42+A I 0]D .4+ "#. {)*"#R%(.43' >VMT%*M ®!"#0' .,%(Q .4 3+D 3+L%(E:.4 X%(!"4:*' I 3.#.,%("#3 " " I +.,%!"#JK "¸L1\E' !&' 8h≥ '0 DL% I >? I 3!"4+.!!3!" I ¬ \-'6'&0' "#3:02%("#+L%(E/*'8D 3-*' "#: I %(.,%("#3V S? I <+' X, 3 Y >= -' . [XY »§"=MYKMV] ' 1>VMT%*M 3UMSE' .#.4e¼¬3!![J?L%(.F I +.,%;+E!3X%("#+DL%(3!!" I ' E.#"4+3UjD ' "4\-' X ! X 3!![J?L%(.4%('*) I !X%(EU M ${V¸ "4&%(.43= DL%(3R.=
2

H

∗

=

k+X k

= (1 +

X ) k

"#.j!&%(.43m% I "#.46 : ' 3$+. %X+Ò:JK3X%(DP' 3 \E' >?"#3\E' >0 k + X > 0 ' 6D 3QL%(Q "#.#"#$6*" S$+3+E :D .#'U\-'+.4\E'6"#.,%]!' "#/D 3$+>-"4!"#Q .4 1 F >K%(.4+' 3 O∗%( ψ d È

k

R

∗[

N X

"

"

5

>:

><

n

n

>] = 0.

\E'G» ˜ ¼!R' :02%(3!!"# JE%(.4'= Sn

:

∗

M

. , "J) ( $ !. $ ,-& ) &

, & .0/+, -. , ( 8. " "

N 1

1

Ã .S3$!' .#6%(.43O*'Z.4+0]0G

[X X < ψ , ∆S˜

< ψn , ∆S˜n >] =

:5A<

%<

<

<

8 >:

']+ %(3!3!"#>?F8.,%\-'!!"4 I +-!3X%(.4«F\-'+.RD 3!") "#GØ>?+*3;' D !"4¤+' 3DV$+ % h D 3+0]"4W+3¬!!"#02%(!"4;\E' "P>-"4+-F/.=%(.4+' 30GUÅ?+ ¬ O.=?"#3 I ?+- .=?+*3/0G"# I [+3Uq%('D 3!")

βN [h] =

˜ = [h]

1 [h] N (1 + r)

»§M KÎ?¼

5 %("4Uj%(:' 02%(3 I [$ I 0]D .4+U"#.DV+' :m%("#3G0]"4+'*) «V"#.%»m%('0G"#/![$3!"4\-'+0G+E¼/.,%ÂDVK!"#Q "#.#"#$ ® 6"#06' .4+3 U< "#. I X+E&F]"#E>?!!"#3¬%(O ¬% I !"s&F13!"4\-' "#.·*"4DPK%("#!"S< ' 6!3X%($+JK"4 »m%(D DV+.4$ ³{ s¥!¯K Ë ©K¡m´¿E´ h ¨ º-Ì´{¥{ sºv¥´ 8.=%(.4+' 3 F .=< "#!X%(- !R{)% I +0G+E M Ã .S!R I L%(!' 3+. ¬ ${¸ "#3.4&D 3!") ¬.=
n

V0 = V0 (φ)

Ù

;

Vn = Vn (φ).

!' I X%(E6q» I K%(3!",%(Q .4O%(.4$UV%(0"#3 H=0G!' 3X%(Q .4 I ?.#' !"4Â I ' 3Q+3!>?$jÇ~'VXn\E'j-"4+*3X%%("#!" ' % I !"AF/3!"4X\E'&!' D D .4$+0G+EX%("#3 h+.#.4¬%(' 3X%' D 3!") F/.=< "#!X%(- »*" S$+3+ER ¼Á $++3!0]"#$ DL%(3.,% .4" :.=?n'=%('*3X%G $+QVV' n3+3OU< "#.>?'D 3+ n .,%=%(EX%("4!"46S%(-.4R+0]DV$+3X%(0G+E« Vn Wn Ä \-'R.=K%(.#"4 ¬02%("4R\-'¬.4R!X%(!"4!!" I "4+-R02%(.Vm%("#R.4+' 3R!3X%U>%("#.«*.,%.4"S D 3QL%(Q "#.#"#$ J?' >?+3!L%(E .4 I ' 3 &% I !"sR"# I 3!3 I 2M Ä \-'/.402%(3 I [$1%23! DL%(3 I \E'j< "#.·%(' 3X%("#¬%U>?"#3 3X%("4 I E!3' 3X%("4 +0G+E:02%(![$+02%(!"4\-' "S>?'OQ+3!>?+Ò1F1.,%]%( ' ZD 3!") !3WR"#*m$+3!"4+' 3&F >?'"#->?!!"4+Ò%(.43>?!3:m3!!' Vn + I +.#.46 :' :>?!3¬=%(0]n"#.#.4/%(R.=K%("4R!D 3!"#DP' 3D 3$++*3:\-' j< "#0]DV3!¬\E'+.#.4 m3!06' .4¬02%(JK"4\-' s' 3! "4&DL%(3' I %(3!02%( I "4+ 'Â' %!3.4JK'¬%(QV' !"#3X%("#%('0G¹+0G&3$!' .#X%(:-DP' 3!>v';\E' '.4DV$+3X%(+' 3"4+E I ->%("# I ' &X% DV+3!!"#+ I ]«v"P.,%X%(.#"#$:Sv"4 +0]0G+E6F I 6D 3!").,F\-'U?+/%('*)02%(![$+02%(!"4\E' % .4¬3$!' .#X%(O' "#>%(-6«

H

E

:

9E

Efqc v`b

V0 =

1 N (1 + r)

[h]

;

Vn =

1 N −n (1 + r)

[h|F ]

»§M ? ¼

n

Í DV' 3!3X% I !X%(+3O\-'¬.,%m% I ' .#$: $6%('>?+ +' 3 ¬.=-3!"#3ODL%(3 % I !"sOF3!"4X\E' !3X%*' "#/+®.,%( JE%(J?102%(![$+02%(!"4\E'1DL%(3¬.,%Â!' Q!!"#!' !"4 9 F0 » I 0]DL%(3+3 + ¼]M (4.13) D 3!")2S< ' OD !"4; I %(. I ' .,%(EF¬.=? I D 3$ I "4!"4 M 'O%(.#.4O+>?"#3O' {)*+0]D .4%(' I [L%(D "#!36' "#>%(-UM !R' :02%(3!!"# JE%(.4'=8 % Ë ÉG¢ ³+ s¥¯? s¡s¢Ø«@' "4X\E' (V˜n (φ)) Rm3!06' .4 Âb-`_

V0 (φ) = (3.12)

[V˜

N (φ)]

= βN

[h]

,

+Z3$' .#+EO"#0]0G$*",%(+0G+-M

βn Vn (φ) = βN

[h|F ] n

c · # b

m3!06' .4D 3$ I $ +E DP+3!0G+!+-O : +3R' : +$ 0G!!3X%(!"4 D #. '3X%(D "4 »§02%("4\E' "VjK%(.,%(Q .4 \-':%(' 02%(3 I [$ I 0]D .4+¼R ¬.,%13+.,%(!"4 :DL%(3!"#$ I %(.#.H=D ' %1+V+ (36) Cn − P n =

1 (1 + r)

N −n

[S

N

− K|Fn ] = (1 + r)n

Ù(

[S˜

N |Fn ]

−

K (1 + r)

N −n

= Sn −

K (1 + r)N −n

+

!>& , " 5

: 5A<

(0. ( <

8

'Á%(.#.4 I !!3!' "#3R' ]02%(3 I [$¸L%( I "4+3¸ "+3!>K%(E v9 "4+- + +'*)3$+.4>?$+3!"¸q%(- a b −1 < a K%(E«S"#.Áj
1+b

Sn+1 = Tn+1 Sn

È !' O' "# >VMT%/F¬>K%(.4+' 3%(Á.=?]«*"#0]DVK+3\-'j< ' I ' 3 ¬QP' 3X&X"# I [L%\-'Ç' 3/{(1 *' 06' !"#D .#"4$/DL%(3OÎ;v'Z*"#>v"4$&DL%(3 ]E%(D DL%(3X%("# I 0]0G[L%(' +0G+-"#3!3$U%(.#"4 +.#.4 I *' "#$U%(0]0G"4RF¬' G0Gv W+.4O\-' "'"#-$+3+3X%:DV' 3 3X%("4« Ä "#.jR02%(![$+02%(!"4\E'+0G+-&!3WO!"#0]D .4/+RDV+3!0G+O I %(. I ' .4O{)*D .#" I "#6M Ä I 0]0G/'.4¬>?+3!3 I "Hµ X'/ I %( I [2j< "#0]DPK%(' I ' I E!3X%("#- "!' 3.4O.4"4O >VMT%*M »s\-' "jDV+' >?+E¬ $+DP+*3 ¼¬ n "j!' 3.4+' 3"#E+3X% I !"41»s+ZDL%(3!!" I ' .#"4+3 Z:' D DPK¬DL%O.4+' 3R"# $+TDPn+%( I e¼6M

+6859685

<>C ? 4< G <>C #? < <>C

-

Í ÂDPK Ω = {1 + a, 1 + b}N ' &$+D 3+' >? ω +3X% I ' &!' "# {t1 , t2 , ..., tN } N 3$+.4>%(.,%(- 1 + a ' M s I !"4 I v3 $ A ${¸ "41DL%(3 A+3E1.4 >K%(3!",%(Q .4G%(.4$U%("#3 Ti I +E!13X%(+.4b& I :0Gv W+.4]M Í +3X% .,%¸.#!3X%(!"4 L%(!T' i3(ω) +.#.4/=+ J?ti+*3$6DL%(3.,%1N !' "# 0]D .4$+$ (Fn ) (Tn ) I F.=
((Sn ) ; 0 ≤ n ≤ N )

S0 = s0 , S1 = s0 T1 , ....., Sn = Sn−1 Tn = s0 T1 T2 .....Tn

ÙKÙ

'.,%("4X+3R%('.4 I +' 3R.4&"#; 0G-!3+3\E' !R%('!"P.,%:¸.#!3X%(!"4L%(!' 3+.#.4¬+ J?+*3$¬DL%(3.,% !' #" M %(D DV+.4\-':'*"4!DVKS< ' /(F +'*n)*) "4W+0G/!' "# 0 $++3!0]"# "4!GL ${¸ "4:DL%(3

1

(Sn )

S

1 βn

Sn0 = (1 + r)n =

Í Z0' #" Ω S< ' :D 3QL%(Q "#.#"#$

$

43

5 7695A8A54:

' 0]"46F' /X+' .4¬3!!3!" I !"4« I [L%(3!J?+3O'R.4RDV"#E M Ω

) & " ) . ( ( & $ , &0) >:

<

98 >:

v9 "#! ' ®D 3QL%(Q "#.#"#$\E'+. I \-'!' 3 {)%(0]"# F \-'+.#.4 I *"4*"#!"4 .,% !' "# ˜ !' Ω ( Sn ) 02%(3!!"# JE%(4. 6X'= M Í % [S˜n+1 |Fn ] = βn+1 [Sn+1 |Fn ] = βn+1 Sn [Tn+1 |Fn ] = β S˜n [Tn+1 |Fn ] =

3!\-'j< ' I *"#!"4®$ +Z3$!' .#G»§D 3+*36.=
43

I X%("#31+&' 2X%(EDV' 3&\-'j< "#.A+®"#:%("#!"!« M @' "4X\E' I / OÎ10G+0Q 3¼¬\-'

1 ˜ Sn [Tn+1 |Fn ], 1+r

[Tn ] = 1 + r

[Tn+1 |Fn ] = 1 + r

Ã.

1 + a < (1 + a) [Tn = 1 + a] + (1 + b) [Tn = 1 + b] = 1 + r < 1 + b,

%$ I %("#3+0G+E M 5®%("4U!"S.4:02%(3 I [$6!R>v",%(Q .4KZDV+' %(D D .#"4\-'+3 I /\-' "j>v"4+EOS
¢qKºE³:³{ºU? (³´{¥!¢ ³ ¨ ´+ m µ´ ¢V´ ¨ K¢q¿¡s s¡s¢8¢ Ë!¨ ´³X³¯¡¥´O*Kº-¥ º ´°4´6É]¯K¥ ¨{Ê Ë ³¡s ÌU¡m¯B{°4´]´³{ ¨ ¢q¿K¡§ s¡m¢8Ì Ë ¥{¡ ¶ Ë ´1¿?¯¢ ³/ =º* µ´6°¯2³ºvr¡s µ∈]a, ´ b[ Ã .j+0]b D 3!' E' % I !"48\E'6.=?+"#0]0G$*",%(+0G+E¬DP' 3 D .,% I +3O.=?+E8F® $ I ' >?+3! «Ó>?+ ' % I !"4 \-'j?$+3!"¸L+3/\-'6.=%(-6«

E

+6 685

E

<>= =@<

9v"# ' ]>%(3!",%(Q .4Â%(.4$U%("#323$+.#.4ÂF;>K%(.4+' 3%(6.=

[T = α] =

β−m β−α

Ù

;

[T = β] =

m−α β−α

Ë ÉG¢ ³+ s¥¯? s¡s¢ Í Z0G-!36S
1[T =α] + 1[T =β] = 1 , T = α1[T =α] + β1[T =β]

Q !"4+E

A+B =1

,

[T |G] = αA + βB.

Ã .j+;3$!' .#/\-'&.4R>%(3!",%(Q .4 %(.4$U%("#3 + EX.#' !"4*'Z!Å*!W+0G: :Î1$\E'L%(!"4OF A B Î]"# I E'6« -O.=< ' "4\-'/.#' !"4

A+B =1

A=

,

β−m β−α

αA + βB = m ,

,

B=

m−α . β −α

Ã .V!%(.43"#0]DV3!X%(ER &+3R\E' .4>%(3!",%(Q .4%(.4$U%("#3 + XER I !X%(E I +.,% \-'L%( A $+DP+BDL%¬ M %G>VMT%*M ! "#JK "¸L+8 V+¬\-'6.,%2.4" I *"#!"4 +.#.4] I "# $+DP+%(- ¬.=v"4 +ET2M G

fLdU_*d fPg H %(R.,%]' "#:

% '

I I [L%(D "#!3 % D +3X%].4RX%(!"4 !' "#>K%(E6«

Ä Í DVK+3X% p = b−r , q = r−a 3QL%(Q "#.#"#$&DVb−a 3!$ODL%(3 ${¸ "4 DL%(3 Ä µ $!"#JK+3X%/.,%6.4"V OD b−a {1 + a, 1 + b} µ(1 + a) = p , µ(1 + b) = q Ä Í +3X%G+*¸.,%GD 3QL%(Q "#.#"#$ µN = µ × µ × ... × µ 3X%(D DP+.41» I M . " ¼\-' ;!.=< ' "4\-' '/.,%\-'+.#.4G.4/s I !"4 I -3 $ !!"#!'+E' ]=%(0]"#.#.4 D 3QL%(Q "#.#"#$Â!' 3 T1 , T2 , ...., TN I ¬>VMT%*M "# $+DV+Ω%(E&$\E' "4*"4!3!"#Q '$O :.4" M µ

c(fLhf! %d %§fqg%1

;!O.=< ' "4\-'/D 3QL%(Q "#.#"#$1'O.,%\E'+.#.4.,%]!' "# ' /02%(3!!"# JE%(.4KM 602%(3 I [$ ˜ U)V K+ ' Q "#+"# ! I >v",%(Q .4/+ I 0]((D S.4+nU)M ; 0 ≤ n ≤ N ) Ë ÉG¢ ³+ s¥¯? s¡s¢ ÑKMRB¬$+3!"¸L ' S
1

1

[T

Fn

n+1 |Fn ]

=

[T

n+1 ]

=

1 [(1 + a)(b − r) + (1 + b)(r − a)] = 1 + r, b−a

I 6\E' "=*'O.=?>v' %('Z $+Q ' I /DL%(3X%(JK3X%(D [KL+E!3X%("#6.4¬3$!' .#X%(UM Î*M 9v"# ' %(' !3D 3QL%(Q "#.#"#$!' 3 "#*' "4X%(E.,%80G¹+0GD 3D 3!"4$+$ !' 3].,%®' "# ˜ Ω ( Sn ) %(' 3X% %(.43 "#.&3$' .#%(.43*' .4+0]0G "" .¤\E' [Tn+1 |Fn ] = 1 + r Tn+1 "# $+DV+%(E8 » I PFMTn%*M I -3 $ -U' Gj"# $+DV+%(E1+µ6 2.4" % Tn µ I q I MY\qM !MYSM

Ù )

+6 6

;

G< 9?

-I

< 9? >< G<

+ I

-

7

II

GC A < =

I

< ; C C <

*/

+!*

-

GC ?+3E I +!®m3!06' .4!3D ![$3!"4\-' I DV+' *"#JKSK%(EODP+' >?+-DV+' O¹+!3¬.4=%("#3 I [L%( J?+3&Sv"46« +; Î6D 3QL%(Q "#.#"#$' 3 >?$+3!"¸q%(E3!DV I !"#>?+0G+E7 »§02%(3 I [$ Ä 9v"4+-7 Ω = 0, 99 ∀n !3W¬[L%('X1!"4+3¼&+2 » v3X% I [DV+3!02%(+E1¼ [T nI =0]10G+ b] &.,%Â0G¹+0G]%( 2 [Tn = 1 + a] = 0, 99 ∀n .4¬Î I %q"#.·+®+3X% /0G¹+0G1*'D 3!") S< ' 1D !"4Ø« I -!3X%("#3+0G+E6F2' 6"#-!' "#!"4®!30]DP+'; .46D 3!")S< ' D !"4 $+DP+DL%& 6.,%2+%( I *'802%(3 I [$;M Í ®Q+3!>?.460G¹+0GD [$+0GW+ %(.4¬0G- W+.4: .,% I Z%( 9 I [.46M Í DV+' O%>K%( I +3.=%(-G« %(R.=< [-Å-DV![W [L%('X!"4W+3®·DL%(3G{) +0]D .4A.4Â>?+ +' 3GS< ' D !"4 A*"4DPKX%(-GS?0+*3 XZD !"4F1' D 3!")D .#'RQL%\-' I +.#' "·%('\-'+.ZDV' 3!3X%("#OU?+3!L%(-O.4:02%(3 I [$ .4&=%("#O\E'¬.4/D 3!") S< ' RD !"4 I %(. I ' .4' !' QP$+$*" I !"4 «(=%("#3R I %(. I ' .4%U>? I >VMT%*M"# $+DP+%(- $\-' "4*"4!!3!"#Q '$Á0G+' D 3QL%(Q "#.#"4! %(.=?+3F&.,%¬.4 I !' 3 *'DL%(3X%(JK3X%(D [!' "#>K%(EUM

$ &'. ! " & !

=8

8

8 5

5 :

'%(.#.402%("#-+L%(E I !"4 $+3+3' I %(.#.q+' D ' Á!' 3.=
Cn

Pn

^bv` ` b 9v"# ' !' "# >%(3!",%(Q .4¬%(.4$U%("#3&"# $+DP+%(-:$\E' "4*"4!3!"#Q '$& .4" I 0]0' ¬sT l I !"4 E' 0G$+3!"4\-l'6!Åv0G$+!3!"4\E'6!' 3 l %Z !"S.=
s

R

j=l X l pj (1 − p)l−j ψ((1 + a)(j) , (1 + b)(l−j) ) [ψ(T1 , T2 , ...., Tl )] = j j=0

(s, s, ..., s) (39)

.,% $+0G!!3X%(!"4!/!3W6"#0]D .4G!" «S@AK Ë ÉG¢ ³+ s¥¯? s¡s¢ .=?+' Q "4+Z3$L$ I [ "#3¬%('*J(ω) )Î]DP= "#-]{i !|T ' "#i>K(ω) %(E6=« 1 + a} Ä .,%.4" J !Q "#0]",%(.4 DV+' +ØV+$ I 3!"#3 J = Pli=1 Ui \E'L%(% DVK$ Ui = M Ã .3%(.43¬FG3+02%(3\-'+3¬\E' !!"#!'' ' "#/"# $+DV+*H 1{Ti =1+a} (Ui ; 0 ≤ i ≤ l) I %(E6 ¬>VMT%*M +3!' .#.#"j+.#.4 \E' [Ui = 1] = p , [Ui = 0] = 1 − p Ä I 0]D O+-'Â O.,%/!Åv0G$+!3!"4 ' D DPK$ !' 3 ψ ?.,%/>PMT% ψ(T1 , T2 , ....Tl ) !' Rm I !"4Â J D .#'RD 3$ I "4$+0G+E ψ(T1 , T2 , ..., Tl ) = ψ((1 + a)(J) , (1 + b)(l−J) )

Ù

R

Í 8%1 I

[ψ(T1 , T2 , ...., Tl )] =

j=l X

ψ((1 + a)(j) , (1 + b)(l−j) ) [J = j]

j=0

+6 685

)

j=l X l pj (1 − p)l−j ψ((1 + a)(j) , (1 + b)(l−j) ) = j j=0

>=@<

:D 3!")S< ' I %(.#.+' 3DV$+ ! $/DL%(3.,%s3!0' .4 Cn =

1 (1 + r)

N −n

Ë ÉG¢ ³+ s¥¯? s¡s¢Ø«

j=N X−n j=0

(1 + r)N −n Cn =

N −n j

[(S

N

@AK&%(.43 c(n, s) =

pj (1 − p)N −n−j [Sn (1 + a)j (1 + b)N −n−j − K]

− K)+ |Fn ] =

1 (1 + r)

N −n

[(sT

[(S T

n n+1 ....TN

n+1 Tn+2 ....TN

»mÙ*M#Ñe¼

+

− K)+ |Fn ]

− K)+ ]

3!\E'Â4» I &»Ñ?¼!¼6 M .4+0]0GD 3$ I $ +E6%(D D .#"4\-'$F2.,%]m I !"4®!Å-0G${H Cn = c(n, Sn ) !3!"4\-' ψs (tn+1 , tn+2 , ..., tN ) = (stn+1 tn+2 ....tN − K)+

I *' "#¬F.=
j=N X−n

1 (1 + r)

N −n

j=0

N −n j

+

pj (1 − p)N −n−j [s(1 + a)j (1 + b)N −n−j − K] .

M Ã .j!R"#E' !"#.4 ¬3 I 0]0G+ I +3 ' I %(. I ' .%(L%(.4JK':DP' 3 Í +Z $*' #" "#0]0G$*",%(+0G+E&.,%s3!0' .4 Q + "#3.4:D 3!" )*'ZD ' /« "#.jRD .#'O!"#0]D .4:(4.1) /X:+3!>-"#3 /.,%s3!0' .4/ :DL%(3!"#$GM

+6 6

)

0 -

<

#+-

<

-

7A8A

1 %(D DV+.4¬\E'.=
I %(.#.Aj?+*3GFG' ®D 3!") 3X%("4 L%(Q .4]\-'DL%(3 I \E'1'&.#' " %>?¬*"#¬\E'j< "#.A{)v"4X%("#A»§D ' "4\E'.4/02%(3 I [$1! I 0]D .4+¼{P' !!3X%($+JK"4 I ' >?+3!!' 31 I +!D !"4 Mj5®%("46!"Á.4G02%(![$+02%(!" I "4+ØGX%(!"4=%("#%("4$+0G+-S< ' ![$3W+0G2Sv3!"#3 I v"4!"#Q .4/!"#0' .,%(E .=
Cn = c(n, Sn ) = Vn (Φ) = φ0n (1 + r)n + φn Sn ,

ÙKÚ

I 1\E'.=K%(.4+' 3 + 1+a

1+b

{

c(n, (1 + a)Sn−1 )1[Tn =1+a] = [φ0n (1 + r)n + (1 + a)φn Sn−1 ]1[Tn =1+a] c(n, (1 + b)Sn−1 )1[Tn =1+b] = [φ0n (1 + r)n + (1 + b)φn Sn−1 ]1[Tn =1+b]

@.,%O'R'=2+RD 3+.4R!DP$+3X%( I I *"#!"4 +.#.4 Î1$+JE%(.#"#$RD 3 $ I $ +-ODL%(3R3X%(D DV3! 0]D :+-' I /\-' F.=
c(n, (1 + a)Sn−1 )p = [φ0n (1 + r)n + (1 + a)φn Sn−1 ]p c(n, (1 + b)Sn−1 )(1 − p) = [φ0n (1 + r)n + (1 + b)φn Sn−1 ](1 − p)

Ã .S3!/F13$'*3 I :Åv!W+0G¬.#"#$U%("#3F]Î"# I E' DP' 3 Q + "#3 φ0n = φn =

(1 + b)c(n, (1 + a)Sn−1 ) − (1 + a)c(n, (1 + b)Sn−1 ) (b − a)(1 + r)n c(n, (1 + b)Sn−1 ) − c(n, (1 + a)Sn−1 ) (b − a)Sn−1

»mÙ*MTÎ?¼ »mÙ*M ?¼

43

Í +3X%;\-'.=v"4!"#Q .4 "#.Å%ÂD .#']«SF.=< "#!X%(E P"#.!' G/ I L%("#!3 .4 I ' 3] Â.=< "#!X%(E]"#0]0G$*",%(+0G+-2D 3$ I $ +-GDP' 313$3!JE%( "4+3GXDV3!{m+n' "#.#.48M '1.,%("41+ {) +3 I " I :.,% I ' >?+3!!' 36S< ' D !"4 ¬>?+EM

K

H %(R' [L%(D "#!32*''D .,% +3O%(O' 02%(3 [$ 0]D .4+M 1 %(D DV+.4 'I \-I '/.4 $++E+' 3:S< ' I %(.#I .%(0G$+3!" I %("#®%G.,%]DVKI !"#Q "#I .#"#$S
\-'+.·0G0G+- 3+D 3$+-+3 I

$g%,d %§fPg

¼ + »sX%2%(1SVMT%*M M @ . # ' ? J + $ $+3X%(.4+0G+E1 (Sn − K)+

'O%(D DV+.#.4+3OD !"4%(0G$+3!" I %("#6' 6!' "#/%%(D $/ />%(3!",%(Q .4&%(.4$U%("#3DVK!"#!"#>?/M

$ . , ( . 0 "

8

<

:5A<

$ )+, & $ 0 5

=5

's+3%(D DV+.j%('*)Â0G¹+0G"4 $\-'&DP' 3.4RD !"4+' 3DV$+ /«v.,%>K%(.4+' 3 S< ' :D !"4 Un F .,% %( +3X% .,% X0]0GS?+ +' 31 "#]*"4!DVK+3F I +!;%(2DV' 3¹+!3Â+0G(Z !' k3) n S< [3+3] I -!3X%(]FZ' %(DVK!$+3!"4+' 3F U< "#.Á3 I ' 3!]F' !!3X%($+JK"4ÂD !"#02%(.4 3X%("4X DL%(3]3$ I ' 3!3+ I I +%(EZ+G!' D DVK I -' nF8.,%®%( .,%®0]0GS?+3!!' 36 /.=%(-6« Ä DV+3!0G+!!3 I ' >v3!"#3' $+>?+-!'+.v{) +3 I " I .=PMT%*M F.,%]%( Z "# I %>?"#3 È !R' ZDV3!{s+' "#.#.4:>?$+3!"¸q%(- Un n−1 M n@.,%O'X'= U n−1 I 0]≥0GV (φ) R' φ:02%(3!!"# JE%(.42 Vn (φ) = Un (V˜n (φ)) LSK%(E2«

+6859685

9? C

7

K+# I

<

,+

HEI K+ K I

ÑKM 9v"# ' ;D !"4Ó%(0G$+3!" I %("# ${¸ "4GDL%(33$ I ' 3!3+ I I +%(E;.,%®!' "# DL%(3].4 (Zk ) (Un ) $+JE%(.#"#$ UN = Z N

,

Un−1 = Zn−1 ∨

Í *"#3X%]\E' Un !R.,%]>%(.4+' 3 :.=?+.4D DV6 69-+.#. :.,%1' "# ˜ (Un )

( Zn )

*Ñ

1 1+r

[U |F n

n−1 ]

[U˜ |F

Í Z>?$+3!"¸L/+ZS+R=% I "#.4+0G+E&\E' U˜n−1 = Z˜n−1 ∨

n

jbEc+1 %1?b Í ZDVK

n−1 ]

M

_

Vn =

0≤p≤N −n

1 p (1 + r)

[Z

n+p |Fn ]

»sZDP' 3!3X% I 0]DL%(3+3 KÑ M58E!3+3&\E' %>? I .,%>PMT%*M ¬.4{) +3 I " I "" . ¼ Vn ≤ Un ∀n Vn Xn Î*M $L$ I [ "#3%('*):3X%("4DV' 3j.4\-'+.#.402%(.#JK3$Á.4A%(D DL%(3+ I .,%0]0GSv3!"#3.=
1

43

"%$ ,-$ !

43

<

(0! "

<

5

<

0! $ )+, & $ 0!* .0, "J) 0!

:5 <

5

=5

>:

<

ZD !"4×%(0G$+3!" I %("#8+P3ZFX×% I [++' 3D .#'2 DVK!"#Q "#.#"#$G\-'ZÓ[0G.4JK' +' 3DV$+ "#. DL%(3X%("#O I >v3X%("4+06Q .,%(Q .4\E':XÂD 3!")+3X%D .#'$+.4+>?$ I %(-/« Ä (Zn 0 ≤ n ≤ N ) $!"#JK+3X%1' 6D !"4%(0G$+3!" I %("# Ä Cn +3X%.4:D 3!") I +!/D !"4F.,%G%( n Ä cn $!"#JK+3X%].4¬D 3!")ZF1.,%1%( n ¬.=
+6 685

9? C

7

K+# I

KÑ M Cn ≥ c n ∀n Î*M 9-" %(.43 (cn ) ≥ (Zn ) (Cn ) = (cn ) Ë ÉG¢ ³+ s¥¯? s¡s¢ ÑKM 0]0G !' :' 3!02%(3!!"# JE%(.46+

Î*M

0

n

Cn ≥ !' (cn )

(cn )

' :02%(3!!"# JE%(.42 %

|F ] = c

N 4. I %(3X% I W+3¬0]"# "#02%(.S< ' :+->?+.4D DP/ G »s!' n3¼o02%(3!!"# EJ N%(.4:n02%Ç!3Xn%(E (Zn ) 9-+.#.·+EX%("#%(.43

+6 6 A A 7=

(C[C )|F ] = [c

Cn ≤ c n

GC < 7A8A %(-6« (c fLhf! %d %§fqg .4OD 3!") *' I (% .#.A%(0G$+3!" I %("# 9v"4+E ' ®% I !"3!"4\-'$6+ ' 3$+. + K ≥0 ((Sn − K)+ 0 ≤ n ≤ N ) *' I %(.#.j(S +' n3)DV$+ E O + $ E J ( % * ' ) M (SN − K)+ v@·K%(E Z = (S −K)+ (+%(D DP+.,%(- c .4D 3!")FO.,% %( n *' I %(.#.*+' 3DP$+ Ë ÉG¢ ³+ s¥¯? s¡s¢ * # " . !' G /0GnE!3+3 n\-' M 3/n (SN − K)+ cn ≥ Zn ∀n Í M c˜n = [(S˜N − βN K)+ |Fn ] ≥ [S˜N − βN K|Fn ] = S˜n − βN K 5®%("4 I 0]0G %G%('" S-3X%("SDV' 3.4D !"4 ¬>?+E2M

43

KÎ

, *& .0/+, .0, ( . "%. *$ )+, &'$ +6 685 9? C

5

)

>:

7

><

:

:

5

=5

(0$ ! ( (

%8

<

8

K+# I

9v"# ¼ .,%;' "#] s I !"4 {¸ "4]DL%(3:3$ I ' 3!3+ I 7→ u(n, x) (n ∈ {0, 1, 2, ..., N } ; x ∈ R + I +(n, (% x)-/ DL%(3.4$+JE%(.#"#$ u(N, x) = (K − x)+ ; u(n − 1, x) = (K − x)+ ∨

D 3!")

Un

pu(n, (1 + a)x) + (1 − p)u(n, (1 + b)x) . 1+r

*'ÂD ' %(0G$+3!" I %("# OD 3!")S
Ë ÉG¢ ³+ s¥¯? s¡s¢ Í %SSv"4+E¬%(.43

n

Un−1 = (K − Sn−1 )+ ∨

1 1+r

[u(n, S

n−1 Tn )|Fn−1 ]

Ã .S:3!¬D .#' \E'j
+6 6 : ;< 4<

)

,

Í ZDVK+3X% u = u(o, .) KÑ M58E!3+3DL%(33$ I ' 3!3+ I ¬\E'Y\-'+.4\E'&"# !QP3!&!' DV$+3!"4+' 3&S< ' &!' "#& m I !"4 % G + S?{)*M 58E!3+3&+Z' !3/\-' u! !' 3O.4:+JK0G+E +-' .#.4/!' 3 I 0]D .4$+0G+-X%("#3M K u >0 [0, (1+a) N [ 0G-!3+3 \-'j< "#.{)v"4¬' 3$+. ∗ Î*M Í DPK +.·\E' A = {x|u(x) = (K − x)+ } x ≥0 A = [0, x∗ ] *M58E!3+3&\E'/DP' 3R.4/ $++-+' 3 S< ' D ' %(0G$+3!" I %("#!' 3R.=?+*3/XZD !"4F1.=< "#!X%(- 0 Ä !" s0 > σ "#.RD .#'R3$+06' $+3X%(+' 3¬ :>?+*3:.=
3

+6 6

)

)

3

>+# +

< <

<

#+-

<

@' "4\-':.4¬02%(3 I [$6! I 0]D .4+*"#.j{)v"4!¬' 6!!3X%($+JK"4 D 3$+>-"4"#Q .4/+O%(' (¸L%( I $/>?$+3!"¸q%(- φ Un = Vn (φ) = φ0n Sn0 + φn Sn .

3X%("4 +0G+-Á+1'DP"#-%(L%(.4JK'F "" " I *' "#Á%(.43FO.,%Om3!06' .4R L%(-.,% \-'L%(E!"#$RS-3!"#3!36D ! "4 :>?+E2«

n

φn =

u(n , (1 + b)Sn−1 ) − u(n , (1 + a)Sn−1 ) (b − a)Sn−1

+

$. ( -)+,

K5 :

+

! -$ $ )!

>:?5

,:

:

' s I !"4!!3!" I +0G+-/DPK"#!"#>?+: $+3!"#>K%(Q .4]!' D DPKX$63+D 3$+-+3:.=?.#' !"4 9v"# s : R+ 7→ R∗+ *.,%]\-'L%(E!"#$ I ' 3OS< ' % I !"X%(3!"4\E'G»s $++3!0]"# "4e¼%(' I ' 3*' +0]D6» I E!"#-'q¼/ML9-" t
rtu =

1 s(u) − s(t) s(t) u − t

!.4/X%('*) S< "#E$+3¹+¬+E!3/.4O"#!X%(E + M /X%('*)ZS< "#-$+3¹+ "#!X%(-X%($ F1.=< "#!X%(E +3X% LD %(3 t u r(t) t ${¸ "#!"4 » )vM#Ñe¼ s0 (t) rt = lim rtu = u↓t

s(t)

.,%:m I !"4 + &.,%/>K%(.4+' 3F6.=
Í %1 I s(t) = s0 exp(

H

Z

ds(t) = s(t)r(t)dt s(0) = s0

t

r(v)dv) ; s(u) = s(t) exp( 0

Z

» )vMTÎ?¼

u

r(v)dv) t

%(R.4 I % È.4/X%('*)Â"#!X%(-X%($6!R' I !X%(E *.4$+JE%(.#"#$OD 3 $ I $ +E& +>-"4+ +- r

s(t) = s0 ert ,

43

H

s(u) = s(t)er(u−t)

! . (0 " , &'!>! .0! (

5 :

<

<

<

!>!

>:

» )vM ?¼

. 0! 695

:

5

(% :' I 1DL%(3X%(JK3X%([V'/!' D DVK+3/ $ I X%(E!!3!" I +0G+EDPK!"#!"#>? MS@·' 3 t, R, σ I [L%\-'/+-!"4+3 N > 0 'O%(.#.4 I !!3!' "#3/' Z02%(3 I [$/¸L%( I "4+3O &M 6M M ¬.,%=%Z!' "#>%(-G« Ä R>%(.4+' 3 OÎ]% I !"m+3E Q+3!>?$ %('*) N + 1 "#X%(E 0, t/N, 2t/N, · · · , N t/N = t M Ä :X%('*);S< "#-$+3¹+ rN +E!3/.4OÎ]%( nt/N + (n + 1)t/N +3X%1$+JE%(.·F rT N

1 1

(

Ä Í Z ${¸ #" aN + bN DL%(3.4O$+JE%(.#"#$ 1 + aN = (1 + rN )e−σ/

√

N

, 1 + bN = (1 + rN )eσ/

√

N

Ä Í 8+3X% (T1N , T2N , · · · , TNN ) ' !' "# N >PMT%*ML"# $+DP+%(-/$\-' "4*"4!!3!"#Q '$/FG>K%(.4+' 3&%( Z!' D DVK+3X%+Z' !3 I :\-' " '.=?>v' $++3!0]"# N {1 + aN , 1 + bN } N [T1 ] = 1 + rN .,%.4"· R>PMT%*M N M Tn Ä I ' 3: .=?+3 \-'L%( M Ã .EDL%(3X%("#AL%(!' 3+. DL%S 0·=!' 1QV*"#>v"4!"4\E'Á'%U>? I !!3!' "#!' 3 :DV++3O\-'¬!36!' "#: /D 3 I 'O*"4 II 3+O>% I [0 ->?+t]3!J?+31»s+Z' 0 +\-'N :→ '∞ R:D 3$ I "4+3 DL% " I "¼>?+3R' 02%(3 I [$:¸L%( I "4+3&F+0]D I E!"#-'8S< [3!"#ÒU M@' "4X\E' t

lim (1 + rN )N = lim (1 +

N →∞

N →∞

Rt N ) = eRt , N

(% D DL%(3X%("# I 0]0G/.4¬X%('*)ZS< "#-$+3¹+O"#!X%(-X%($6*'D 3 I !'R.#"#0]"#GM R @A' 3&%>?"#3O' :"#E+3!D 3$+X%(!"4%(L%(.4JK'6 *'O%(' 3 OQPX"#Z*'Z.4+0]0G/!' "#>K%(E

6 685

σ

<>= =@<

@A' Á 3 ' ?2X ' ' "# >%(3!",%(Q .4%(.4$U%("#3"# $+DP+%(-+ N ≥1 (XnN ; 1 ≤ n ≤ N ) N !' D DVK+' !3Á\-'.=%(.4+' 3A%(·.=
N →∞ N X2 + · · ·

${¸ "4GDL%(3 %' "# N N I E>?+3!J?;%(.436+ .4">?+3/' ]JE%('X!"4+ G + XN 0GUÅ?+ (YN+ O) ¬>%(3!",%( I Y2N M = X1 + µ σ !R.,%m I ! "4 I %(X3 % I $+3!"4!!"4\-'/ 8% $-`fPg'UdUc_*d %§fqg@1 9-" φN YN !"#.#"4X%(-&%(.43R.4:![$3W+0G6

=%(' 0G-!3+3 \-' %s3!0' .46

φN (λ) =

N

[eiλYN ] = ( [eiλX1 ])N

/@A%(' . $+>vÅ3+.,%(!"ÁF.,% I ->?+3!J?+ I +.4"Á >?"#O\E'j< "#.S' lim φN (λ) = exp(iλµ − λ2 σ 2 /2)

» )vMTÙ?¼

%Åv.43&F.=? I 3!/ %(JK3X%( J?/DV+3!0G+OS
%1 I %U>? I |R(x)| = 61 |λ|3 |x|3 %>? I |R(X1N )| ≤

1 N eiλX1 = 1 + iλX1N − λ2 (X1N )2 + R(X1N ) 2 3

|λσ| √ 6N N

5®%("4 I 0]0G

» )vM ?¼

M@3+L%(E¬%(.43R.=
[eiλX1 ] = 1 + iλµN −

KÙ

λ2 σ 2 + [R(X1N )] 2N

» )vM )K¼

Ä | [R(X1N )]| ≤ [|R(X1N )|] ≤ µ Ä DP+' µN$ = 3I !"#3N + o(1/N )

|λ|3 σ 3 √ 6N N

= o(1/N )

N

[eiλX1 ] = 1 + iλ

1 +>?+ %(.43OF].=
« %

φN

» )vM ¼

λ2 σ 2 µ − + o(1/N ) N 2N

φN (λ) = exp[N log(1 + iλ

λ2 σ 2 µ − + o(1/N ))] N 2N

λ2 σ 2 + N o(1/N ) + (1/N )] 2

= exp[iλµ −

» $!"#JKL%(-1' Gm I ! "4Ø+%(E1>?+3 F .= +3&.=< "#*¸ "DP' 3 N Q + "#31»*MTÙ?¼

6 6

9? C

7

K+# I

@AK ¼ I E>?+3!J?+¤.4"R>?+3]' JE%('"4+ Z 0GUÅ?+ +G 2 ˜N .,%®' "# » >K%(3!",%( I YN2 = \E' "log *3XS(% D NDV+.4.42%YDPN' 3O +"#$: :D 3QL%(Q "#.#"#$/.,%m I !"4 ${¸ "46' 3 LD −σ %(3 /2 σ

ν

R

» )vMTÚ?¼

2

(y + σ2 )2 1 ν(y) = √ exp(− ) 2σ 2 σ 2π

$-`fPg'UdUc_*d %§fqg

Í ;% log S˜N = X N + X N + · · · X N \-'L%(2%:DVK$ XnN = log TnN 1 N B¬$+3!"¸LO\-'¬.4!' "# X%(!"4!2s- %('*)ÂN[-Å-DV![W*' .4+0]0G " " .8« 1+rN N (Xn ; 1 ≤ n ≤ N ) N D 3+ZR>K%(.4+' 3%(.=

Xn

{log

1 + bN σ 1 + aN σ , log } = {− √ , + √ } 1 + rN 1 + rN N N

Ä .4R[EÅvDV![WS< "# $+DV+%( I 6- Ä .,%>?$+3!"¸ I %(!"4 2» *M -¼$ I !"#¬.4 I 9v"4+- + ${¸ "4DL%(3 f

I .,%("#3+0G+E&>?$+3!"¸L$ %(. I ' .' "#>%(-6«

u : R+ 7→ R

f (x) = log

x 1 + rN

;

u(x) =

(x − (1 + aN ))f (1 + bN ) + (1 + bN − x)f (1 + aN ) bN − a N

0]0G !$+JE%(.4OF %('*)GÎ&DP"#- + 1+aN 1+bN DL%(3!*D ' u"4 \E'/'Z!O% fGG

3

N N [u(T1 )]

Í 8%1 I %(D 3W %U>?"#3O$:\-' µN =

N N [X1 ]

=

= u(

N N [f (T1 )]

N N [T1 ])

= u(1 + rN ) √ f (1 + bN ) = −f (1 + aN ) = σ/ N

N N [f (T1 )]

= u(1 + rN ) =

=

N N [u(T1 )]

σ(2rN − (aN + bN )) √ (bN − aN ) N

S
1&%(D DP+.4 %(.43O\-' σ σ σ2 + o(1/N )] = (1 + rN ) exp(− √ ) = (1 + rN )[1 − √ + 2N N N σ σ2 σ = (1 + rN ) exp( √ ) = (1 + rN )[1 + √ + + o(1/N )] 2N N N

1 + aN 1 + bN

Ã .j+Z3$!' .#6\E' 2σ σ2 bN − aN = √ + o(1/N ) ; 2rN − (aN + bN ) = − (1 + rN ) + o(1/N ) N N

Í 8%1 I

−σ 2 (1 + rN ) + N o(1/N ) = −σ 2 /2 N →∞ 2(1 + rN ) + o(1/N )

lim N µN = lim

N →∞

" I 0]D 6+E' *'Z.4+0]0G I 6\E'

$+0G-!3/.,%D 3DVK!"#!"4M

" " .

! , . !*(0 $& $ ( & <

8

8

!

<=8

E

E

'&%(.#.4¬Q + "#3¬.4OD 3!")8 &D !"4¬S?+E FG+0]D I E!"#-' 1+DL%XX%(E:FG.,%2.#"#0]"# %(G.4]s3!0' .4 I 3!3!DV%(EÂ*'¤0Gv W+.4Z " !" (" M@A' 3G' ;m"4Z'2'G"#E$+3X+3 S-3!"#3D .#'QL%¼GA%('*) D !"41 >?+EM @.,%/X' .,%D 3QL%(Q "#.#"#$N ∗ $ I 3!"#]%(¬.4DL%(3X%(JK3X%([1D 3$ I $ +E .4D 3!") N F.=< "#!X%(E *'®D ' P0 0 ! N$/DL%(3.4$+JE%(.#"#$ :D 3!")S
K

P0N =

1

(1 + rN )

N

∗ N [(K

N + − SN ) ]=

∗ N [(

K (1 + rN )

N

N + − S˜N ) ]=

∗ N [(

K − e YN ) + ] (1 + rN )N

MV9v"# .,%]m I !"48 ${¸ "4 Ã .·U
−e )

|P0N −

∗ N [ψ(YN )]|

K − eYN )+ − (Ke−Rt − eYN )+ ]| (1 + rN )N ∗ K ≤ − eYN )+ − (Ke−Rt − eYN )+ |] N [|( (1 + rN )N 1 − e−Rt | → 0 ≤ K| (1 + rN )N

= |

∗ N [(

» %] +3! "4W+36.#"#JK:D 3U>?+L%(- :.=< "#$+JE%(.#"#$+E!3:0Q 3O3$+.46« lim P0N = lim

N →∞

N →∞

|a+ − b+ | ≤ |a − b|

¼ "#.j+ 3$!' .#:\E'

∗ N [ψ(YN )]

5®%("4 vS?+3!J?+ I :+Â.4"-%G-D ' "4\v ! I E!"#E':+QV3!$P» I ψ \-' "S/+3X%("#RDL%.4 I %DP' 3.4 I %(. I ' .A%(L%(.4JK'/3+.,%(!"Á%(' I %(.#.¼¬ lim

N →∞

P0N

1 = ψ(y)ν(y)dy = √ 2π R Z

Z

R

)

[Ke−Rt − exp(−

σ2 2 + σy)]+ e−y /2 dy 2

» )vM#ÑU?¼

6 685 )

I

+

-

+ K

I

G< 7A <

-I

-

7A ?AÇ~'\-'j%(.4+' 3"# "#!",%(.4 N +.=%(.4+' 3 3!DV I !"#>?O 6.=?+.#.4!"#!'L%(!"4 % 0 NS P N 63!:\E'

S

N

= S 0 SN

1

N

P 00 =

(1 + rN )

N

S0

N

∗ N [(K

− S 0 N )+ ] =

(1 + rN )

N

K ∗ N [( S0

N + − SN ) ]

% .#"#0]"# I +!:{)vD 3"4 \E'L%( +>?+3R.=< "#*¸ "·+3X%.4¬D 3!");*' D ' %(R.4:0Gv W+.4/F+0]D 1 I E!"#-' "#P .jeY< $ I 3!#" 6 +Z>?+3!!' G»*M#ÑU?¼ N S0 P =√ 2π

Z

y2 1 σ2 K [ e−Rt − exp(− + σy)]+ e− 2 dy = √ 2 2π R S0

Z

R

[Ke−Rt − S0 exp(−

Í {)*D 3!"#0G/S< [L%(Q "#!' I ¬3$' .#X%(O ¬=%.4$+J?W+3+0G+E¬*" V$+3+-2«*DVK 1 F (x) = √ 2π

Í Z>?"#=% I "#.4+0G+E&\E'

Z

x

e−

−y 2 2

dy , d1 =

−∞

y2 σ2 + σy)]+ e− 2 dy 2

S0 1 σ2 ( + Log + Rt) , d2 = d1 − σ. σ 2 K

» )vM#ÑKÑe¼

» )vM#ÑUÎ?¼ Í D 3 I W : ¬.,%60G¹+0G:m%;DV' 3 ${¸ "#3 F+0]D I E!"#-' :.,%>%(.4+' 3*' I %(.#.j &D 3!")S
E

E

P = Ke−Rt F (−d2 ) − S0 F (−d1 ).

H

N

N

C 0 0 − P 0 0 = S0 −

K (1 + rN )N

0E

E

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