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43 ! /B$ ,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+-«

576685 FG,HEI J+KIL¢¯? ´°§°4´¥¯ÂÌ(¯¥¡m¯B°4´1¯° Ë ¯K =¡§¥´9M2Ì(¯°#´+º-¥³¿?¯¢ ³:ºv¢ ´¢ ³´+ÉNB+°#´

Eº-¢V´¯?q°²¡ ¨ ¯? s¡s¢¿E´

Ω¿?¯¢ ³

E

9-"X

' ;>VMT%*M+Γ' ¤']++0Q .4Z

E .=<Y$+>?$++0G+- ω ∈ Ω | X(ω) ∈ Γ = X−1(Γ)

!' >?+-$:DL%(3.4RD 3QL%(Q "#.#"4

[X ∈ Γ]M

@' "4\-'Ω!¸ "=.=<Y++06Q .4

X(Ω) >K%(.4+' 3\-'¬DP+' D 3+*3

X!R.#' "%('X!"V¸ "ÁM

5766 *I<,+#-(- ; < ;<>=@?BA8<>C_-D >K%(3!",%(Q .4%(.4$U%("#3

X1+X2

$¸ "4DL%(3X1(i, j) = i ; X2(i, j) = j

3+D 3$+E+-.4A3$!' .#X%(% 3 I [$ODL%(3R.4&D 3+0]"4+3O+R.4: I $ ' /%(' !3¬>VMT%*M "#E$+3X%(-6!

S = X1 +X2 %("#!"

S(3, 4) = 7aGD Í $¸ "#O :.,%0G¹+0G:m%p>K%(3!",%(Q .4 %(.4$U%("#3&F>%(.4+' 3 %(

E«

!"ω = (x1, x2...., xi, ..., xp)

SDVK+3X%Xi(ω) = xi (i ∈ 1, 2, ..., p) M9-".=<Y"#02%(JK"#].=<Y$+D 3+' >?

ωI 0]0G' 6.#"463 $1 p0G&3 I '+"#.#.#"4DL%(3&' ®+\E'¹++' 3U

Xi(ω)3+D 3$+-+3X%Â.4

i − eme0

Ù

I +!:.#"4!G»s+Rj<T%]%(' I ' "#E$+3¹+DV' 3R.4:%(J?e¼¬M @A' 33+E!3+3 %(R.4¬>-"*'Z!'Ç!+/*"#!X%(.#.4R' DL%(3!!"#!"4' 3

E I !!"#!'$G 2Î;X'/++06Q .4E1

+E2

3+D 3$+EX%(-3DP I !"#>?+0G+-6.=<Y++0Q .42 DL%(3!!"4X%( I %(*"4%(6Ñ/+OÎ .4OÎ>%(3!",%(Q .4&%(.4$U%("#3O\-'¬.=<Y)%(0]"#+3X%G%>? 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.=<Y++06Q .4 0, 1, 2, ..., p +>?$+3!"¸L+-O.=< "4 +-!"#$Z1 + Z2 = p14D 7O'1Ç!+'Z*'Z.4 .,%10]0G6 $+.4$+0G+EO¸JK' 3X%(E&%(R.4:!"#3X%(J?

ω5GD(F,G) → ](F ∩G)

5766) <>C! I + < - C9v"4+-

E' ++0Q .4 +

F' Â'Á++0Q .4

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

1F : E → 0, 1 $¸ "4/DL%(3

1F (x) = 1!"x ∈ F

1F (x) = 0

!"x ∈ F c

9-' 3]' ¤!DL% I Z ;D 3QL%(Q "#.#"#$ .=< "#*" I %(+' 3S< ' ¤$+>?$++0G+EA

!] I ' ;>K%(3!",%(Q .4%(.4$U%("#38>K%H.4+' 3%( 0, 1 .4+' 3' !"#.#"4X%(!"4 DV+3!0G+Z ®3+0]D .,% I +3Z.4DV$+3X%(!"4QV-.4$+ Z' 3Â.4++06Q .4»§"#E+3X I !"4 3$+' "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 + 1A1C + 1B1C) + 1A1B1C

+.,%&Ç'!!"¸L¬.4RX%(!"4A ∩B = AB

+A ∪B = A+B si A ∩B = φ

M

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

X' ¬>PMT%*MLF>K%(.4+' 3%(

E+

Γ' ZX'++06Q .4

E 8%.=<Y$+JE%(.#"#$

1[X∈Γ] = 1Γ X »ÑKM ?¼

7 "#!"RV%(¬.=<Y)*+0]D .4 . ""aGD V $¸ "4X:.4p"#*" I %(+' 3 Y1, Y2, ..., Yp

DL%(3Yi = 1[Xi∈E1] =

1E1Xi ;

8%]%(.43.=<Y$+JE%(.#"#$Z1 =

∑i=pi=1 Yi

MÎ*M 9v"#

X' />VMT%*M3$+.#.4 ;»

a1, a2, ..., an¼R' $+-' 0G$+3X%(!"4®

X(Ω)È.4

aiE&*"4!!"# I

+Ai

.=<Y$+>?$++0G+E[X = ai]

M %]!' "#G»Ai

¼ I !!"#!':' /DL%(3!!"#!"4 Ω+R.=<YZDV+' O$ I 3!"#3

X =i=n∑

i=1

ai1Ai

»ÑKM -¼

%8 $+0G!!3X%(!"4¤ I ]ÎDP"#-]!"#0]0G$*",%(«·"#.R!'43G] 2>?$+3!"¸L+3G\-'.4s I !"4¸JK' 3X%(EÂ%('*)Î0G+06Q 3] ®»ÑKM ?¼/+1 8»ÑKM -¼6D 3+ +-].,%Z0G¹+0G>%(.4+' 31+' DV"#E

ωM '6"#->-"#1.4.4 I +' 3

"#)*DP$+3!"#0G+-$FRD 3+*3\E'+.4\-'A0]"#E' DV' 3A$ I 3!"#3.,% $+0G!!3X%(!"4 I 0]D .4W+ Á02%( "4W+3RF U<T%!' 3+3\-'j< "#.%G%!"#0]"#.4$¬.4 $¸ "#!"4&+RX%(!"4&*'Z $+Q ' O I I ' 36M

5766 :FC ?7 I 4< -IG< BA < 7AK +#K < ><>A8A8<bg%,d %§fPgP¡s

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

[X] =

∑

ω∈Ω

X(ω)

(ω)»ÑKMTÙ?¼

1 %(D DV+.4\-'+.4\E'D 3D 3!"4$+$ X+E!"4+.#.4O ¬.=<YDP$+3X%( I :3$' .#X%(EO"#0]0G$*"#+0G+-& 6X%1 $¸ "#!"4 «Ä Uh -c_g61?b 5(#Ag %sg'5&%1?_*dUb#·c 1 Í %

[1A] =A

\-'+.4\-':"#R.=<Y$+>?$++0G+EA

Ä ^ %sg-_ c+%d 1 9-"X

+Y

XE Î>VMT%*M 3O+!"α+β- Î06Q 3 3$+.4/ %

[αX + βY ] = α

[X] + β

[Y ]

Ä f 4%,d %'%d 1 9-"X

!/DVK!"#!"#>?Z· !DV$+3X%( I 2!6DPK!"#!"#>?+6j<Y!6E' .#.4Â\E'G"X

!/D 3\-'Æ 3+0G+E-' .#.4

Ä g·bq_ /%d 5·b 1?fqg Pb '%d 1 9-"φ : R 7→ R

!R' :s I !"4 I ->?)*2L%(.43 [φ(X)] ≥ φ(

[X])

Ä jb-` h/§b1 1O+' 3!6+ I 3]' s"4/F.=<Y) +0]D .4 aGD I 0]D G+E' .,%Â3+02%(3\-'m%("#GF.,%¸Z*'DL%(3X%(JK3X%(D [/D 3$ I $ +E%

[Z1] =

p∑

1

[Yi] =

p∑

1

[Xi ∈ E1] =

p∑

1

n1

n= p

n1

n

\-'L%(%DVK$]E1 = n1 , ]E2 = n2Ä jbEc+1 %1?b 1 9v"4+-A,B,C

!3"4 $+>?$++0G+E&S< ' !DL% I / ¬D 3QL%(Q "#.#"#$ Ω 0GE!3+3 .=<Y$+JE%(.#"#$

[A ∪B ∪ C] =

A+

B +

C − (

[A ∩B] +

[A ∩ C] +

[B ∩ C]) +

[A ∩B ∩ C]

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

5766 - I <KBA8< 7AK +# <P¡s

Xº-¢P´ Ì(¯¥¡m¯B°4´:¯° Ë ¯K =K¡¥´ MÌ¯°4´ºv¥X³:¿?¯¢ ³ E ¢qKºE³ ¯E ´°§°#´¥!¢ ³&°K¡£oK¢¿K¡§ ¯º-³X³¡³+ºvÌ(´¢q ¿K¡,³+ s¥¡KBº* s¡s¢?ª

¿E´X

°¯:q¥!B!¯B+¡°²¡§ Ë ΠX¿ Ë ¶¢¡=´:³º-¥:°À ´¢ ³´+ÉNB+°#´¶¢L¡ X(Ω)

*¯K¥

ΠX(k) =

[X = k] ∀k ∈ X(Ω)

´º¾ÂÌ,¯¯E¿É2´+ m =¯¢L Á°¯ÉÉ2´6°¡³U´¥!¢L ¿¡§ µ´³ Ë +º-¡m¿¡³ s¥¡JB+º Ë ´³1 +JE%(3 O\-'+.#.4EO.4R.4"4 >VMT%*M \-'¬' %>?O3+ I E!3$ÑKM /§f<% 5(#Ag %sg65%1?_*dUb#c /§f!%5Ab .1b-c(g·f!#//% Mq9v"#

A' $+>?$++0G+E: /D 3QL%(Q "#.#"#$

α .,%1>VMT%*M

1A

D 3+\-'2.4 +'*)>%(.4+' 31+ÑA 2X3!\-'2X%Z.4"Π

!.,%D 3QL%(Q "#.#"#$;!' 3 0, 1 >?$+3!"¸q%(EΠ(0) = 1 − α , et Π(1) = α %1D 3QL%(Q "#.#"#$

Π\E':.=<Y>v"4+E& : $¸ "#3O!' 3 0, 1 e<T%(D DP+.#.4]°#¡Á¿E´&´¥¢qºv°°²¡Á¿E´R*¯¥!¯É+ s¥´

αH %(.=<Y) +0]D .4 aGD -.4>K%(3!",%(Q .4O%(.4$U%("#3Yi

$¸ "4 F6.,%¸*'DL%(3X%(JK3X%(D [ . """ ER$\-' "4*"4H!3!"#Q '$« I [L% I ' S<Y+E!3/+.#.4 !' "#R' :.4"j -+3!' .#.#" /DL%(3X%(0GW+!3 n1

n

)

Î*M /§f<%ja&%§g·fq` %§_ /b M 1O!%(.=<Y)*+0]D .4 aGD + I %(. I ' .4.,%&.4" Π .,%&>VMT%*MZ1

"#E!3v*' "#O+$. "" 6' /S<T%(QP3 \E'

Z1D 3+:>K%(.4+' 3/%(:.=<Y++0Q .4 0, 1, 2, ..., p

!"k!:' +E!"4+3

≤ pV.=<Y$+>?$++0G+-

[Z1 = k]!:.=<Y+X+06Q .42 6!' "#/ ]X%("#.#.4

p E

k+3!0G6%(D DL%(3!!"4+ +-GF

E1 "#. I E!"4+- I p!

k!(p−k)!n1kn2

p−k $+.4$+0G+- I 6\E' "S'R0GW+F1.=<Y$+JE%(.#"#$

Π(k) =

[Z1 = k] =p!

k!(p− k)!(n1

n)k(

n2

n)p−k

Í %(D DP+.#.4+3X%¤°¡ B+¡¢qKÉ¡m¯°4´Z¿E´]*¯K¥¯É+ s¥´³(α, p) (α ∈]0, 1[

p ∈ N)

°#¯;P¥B¯B¡§° ¡s µ´Π

³º-¥0, 1, 2, ..., p ¿ Ë µ´¥É1¡¢ Ë ´*¯¥

Π(k) =p!

k!(p− k)!αk(1 − α)p−k

7 "#!"Z1

»§3!DjMZ2

¼!' "#O' ¬.4"jQ "#0]",%(.46 ¬DL%(3X%(0GW+!31» n1

n , p¼¬j»§3DjM» n2

n , p¼!¼:M

*M /§f<% :PijhbEc -fq` ?dUc;% #·b H $++3!0]"#&.,%2.4"· 6.,%]>VMT%*MX(F,G) = ](F ∩G)

6.=<Y)*+0]D .4 5 D «XD 3+Â>%(.4+' 3%(.=< "#E+3!>K%(.#.4

I = [(p+ q−n)+, p∧ q] !" k %(D DL%(3!!"4+EF I +"#E+3!>K%(.#.4E%

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

(

np

)(

pk

)(

n− pq − k

)

,

63!:\E'

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

(

pk

)(

n− pq − k

)

(

nq

)

»ÑKM ?¼

%]D 3QL%(Q "#.#"#$Π

DV3!$/DL%(3I $6DL%(3

Π(k) = H(n, p, q, k)! %(D DV+.4$/.4"·[EÅvDP+3!J?$0G$+!3!"4\-'

/DL%(3X%(0GW+!3n, p, q

MM jbEc+1 %1?b

»m%?¼ÂB¬$+3!"¸L+3]DL%(3]' I %(. I ' .D ' 3+0G+-%(.#J?$+Q 3!"4\E'\E' ∑

k H(n, p, q, k) = 1 »s DP' 3!3X%8' !"#.#"4X+3

.=< "4 +-!"#$(1 + x)p(1 + x)n−p = (1 + x)n +/$+JE%(.4+3:.4 I v 3 I "4+E6 xq ¸JK' 3X%(-6%(¬.46Î

0G+0Q 3¬¼&M»§Qq¼ H %(O.=<Y)*+0]D .4 5GD I %(. I ' .4+3 [F ⊂ G]

+[F ∩G = φ]

M» I ¼;7 '1Ç!+'*'.4Zj»s)*0]D .4 14D ¼¬ I %(. I ' .4+3 .,%D 3QL%(Q "#.#"#$6S<T%>?"#3 k E' 0G$+3KOJE%(JKL%(EM

% 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/«

c(fLhf! %d %§fqg%1P¡=´¢L

Xºv¢P´GÌ ¯;MZÌ(¯°#´+º-¥³¿?¯¢ ³

E´+ φº-¢P´Â¯Eq° ¡ ¨ ¯K s¡m¢ ¿E´

E¿E¯¢ ³

Rφ(X)

´³+ ¯°#¥³]º-¢P´GÌ¯¥¡m¯B°4´¯° Ë ¯K =¡§¥´¥ Ë ´+°°4´1¿?¢L °À ´³µ Ë ¥!¯¢ ¨ ´]´³ ¿?¢¢ Ë ´ ¯¥/°#¯D¥É1º-°4´

[φ(X)] =

∑

k∈X(Ω)

φ(k)ΠX (k).»ÑKM )K¼

Ë ÉG¢ ³ s¥!¯K s¡m¢«vO3$!' .#X%( $ I ' .4O"#0]0G$*",%(+0G+- .=<Y$+JE%(.#"#$]»mF:DV+'2D 3W$+>-"4 +-1E!".=<Y%:3$L$ I [ "F»ÑKM -¼¼

Φ(X) =∑

k∈X(Ω)

Φ(k)1X=k

+ :.,%.#"#$U%(3!"#$ :.=<Y!DV$+3X%( I H %(R.4 I %È X !+.#.4:3$+.#.4G 8%1+ZDL%(3!!" I ' .#"4+3

[X] =

∑

k∈X(Ω)

kΠX(k) ,

[Xr] =∑

k∈X(Ω)

krΠX(k) ,

[sX ] =∑

k∈X(Ω)

skΠX(k)»ÑKM ¼

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

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

L%

[X] =

k=p∑

k=0

k

(

pk

)

αk(1 − α)p−k.

I %(. I ' .*'ZX I ;0G+06Q 36' Z) +3 I " I I .,%"4\E'¬!' 3R.,%m3!06' .4/*'Q "# 0G»s $+>?+.4D DP+3(x+ a)p

$+3!"#>?+3]D ' "4 =%("#3a = 1 − x

%(&.=<Y$+JE%(.#"#$1Q +E'e¼ 8Q !"4+- [X] = pα

3+!3' >?]%("#!"S< ' %(' !3¬=%Z.,%>K%(.4+' 3!3' >?$/+G. " " DP' 3O.=<Y!DP$+3X%( I 6 Z1

»s%( I I %6 α = n1

n

¼

jbEc+1 %1?b ÑKM %(. I ' .4+3 %(.4¬0G¹+0G/D 3!"#6*.4¬0G0G+E&S<Y3*3Î1+.,%>K%(3!",%( I / Z1Î*M H %(O.=<Y)*+0]D .4 5GD *0GE!3+3&\E'

[ ](F ∩G)] = pqn

M ¢P¿¡ ¨ ¯K s¡sK¢ ³®« Ã .RU<T%(JK"#] I %(. I ' .4+3»=Á1»ÑKM ?¼!¼Â ∑

k k H(n, p, q, k) %(D 3W2%U>?"#3]0GE!3$.=< "4 +-!"#$

p(1 + x)p−1 =∑p−1

0 (k + 1)

(

pk + 1

)

xk q"4 +E!"¸L+3X%.4 +3!0G +xq−1 %( .=<Y$+JE%(.#"#$+-!3

DV.#Åv 0G(1 + x)p−1(1 + x)n−p = (1 + x)n−1

5766 - I - ? A < < CKBA8<>C$7AK +# <>C9v"4+-

X1 : Ω 7→ E1+X2 : Ω 7→ E2

+'*)Ø>%(3!",%(Q .4G%(.4$U%("#3 $¸ "4G!' 31.4Â0G¹+0GZ!DL% I ; D 3QL%(Q "#.#"#$

(Ω,

)M

@AKGPDP' 3¬' ω ∈ Ω , X(ω) = (X1(ω), X2(ω))

X

:%(.43¬' >K%(3!",%(Q .4G%(.4$U%("#3GFÂ>%(.4+' 3%(R.=<Y!DL% I /D 3-*' "# E = E1 ×E2

\-'¬.=<Y%(D DV+.#.4+3X%1.4 I ' D .4 (X1, X2)M@AK

X1(Ω) = F1 , X2(Ω) = F2 DV+' &%(D D .#"4\-'+3O.4& $¸ "#!"4&+O.4 3$!' .#X%(O*'DL%(3X%(JK3X%(D [D 3$ I $ +-

F.,%1>K%(3!",%(Q .46%(.4$U%("#3X

"#.+3$!' .#/\-']«Ä %1.4"

ΠX*' I ' D .4 (X1, X2)

!O' ¬D 3QL%(Q "#.#"#$6!' 3.=<Y+X+06Q .4:¸ "F1 × F2Ä 9v"#

f : E1 ×E2 7→ Rf(X1, X2)

!O%(.43R' :>K%(3!",%(Q .46%(.4$U%("#363$+.#.4/+

[f(X1, X2)] =

∑

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!"4f

Ú

Ä 9-"A1 ⊂ F1

[X1 ∈ A1] = ΠX1

A1 = ΠX(A1 × F2) =∑

x1∈A1;x2∈F2

ΠX(x1, x2)

Ä 9v"#φ1 : E1 7→ R

%

[φ1(X1)] =

∑

x1∈F1;x2∈F2

φ1(x1)ΠX(x1, x2)

.4"S I [L% I ' / R +'*)Â>%(3!",%(Q .4X1etX2

-O%(D DP+.4$¬°#¡³ ÉG¯¥m©¡¢P¯°#´³:*' I ' D .4 "#.V3!' .#: ÎDV"#EO\-' "SD 3$ I W +E&\E'j<Y.4 $*' "# %("4$+0G+- ¬.,%1.4"

ΠX DL%(3 )*+0]D .4G

∀x1 ∈ F1 , ΠX1(x1) =

∑

x2∈F2

ΠX(x1, x2)

¢Ì(´+¥X³U´É2´¢L ¢¢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¯°#´³M

@A' 3 U<Y+ I ->%("# I 3 Z3$L$ I [ "#3X%F.=<Y)*+3 I " I !' "#>%(-6«

jbEc+1 %1?b

Í ;DPKXΩ = 1, 2, 3, 4, 5, 62 , 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/!' 3Ω

*X%(*"4O\-' G< !R.,%]D 3QL%(Q "#.#"#$/' "m3!0G/!' 3R.,%]*",%(J?L%(.4 ΩÑe¼0]DL%(3+3O.4.4"4R02%(3!JK"#L%(.4 *' I ' D .4 (X1, X2)

+.4Z\-'Ω!0' " ' 2<

Î?¼0]DL%(3+3 [X2

1 +X22 ]

+ ′[X21 +X2

2 ] DP' >K%("#HµZD 3$+>?"#3 I /3$!' .#X%(OX%( I %(. I ' .

?¼!0]DL%(3+3 [(X1 +X2)

2]+ ′[(X1 +X2)

2]G<q+3! 0G- W+.41F]' )*DP$+3!"4+ I 6DL%(3m%("#+0G+-:"4*"4\-' " I 0]DV3!ÎG3$!' .#X%(&02%("4 È®.,%( I \-'j< ' ® $ ®D 3++!' "# I 0]0G

2ieme 3$!' .#X%(GP.,% I D "4]*'®D 3+0]"4+3 I <Y!/$+>v"4 +0]0G+E:' ])*DP$H3!"4+ I &!3W*" S$+3+E¬*'Ç+ ¬Î6 $\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

^_8gfLd %§fqg 5Abn−uplet (X1, X2, ..., Xn)

5·b0 _ c;%_ a&/§b _ /'E_*dUf!%§c(b !' J?$+$+3X%(.#"4X%(!"4;=% I "#.4 O.,%!"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 H3X%(. I I +3!L%("#.4 I ' 31 $+.4 I !"4D 3$"4 +E!"4+.#.4Z»§"#.Á3!X%("#\E'ÂÎ I %(*"4%(¼ DV+' UX%( I [L%( J?+3 S<YDL% I 6 :D 3QL%(Q "#.#"#$Kq3+>?+ "#3ÑUÙ&Ç!' 3O+8%(3!3!"4W+36+OU< "#-$+3+3¬%('ZD 3+0]"4+3O' 36qF]q I %(*"4%(:Y+R!' 3!' F6.4+' 3$+.4 I +' 3R\-' "Ps3!0G+-' DL%(3!!"#!"4

E1, E2, ...Eq &.,%/DVD ' .,%(!"4

E

DVK+3X%nj = cardEj (1 ≤ j ≤ q) (2 ≤ q ≤ n)

+ .=<Y®3X%(D DP+.#.4+3X%\-'n1 + n2 + ... + nq = n

M Í DVK:+!' "#

Zj = cardi |Xi ∈ Ej (1 ≤ j ≤ q)»Zj(ω)

3+D 3$+E2.4106Q 32S<Y$+.4 I +' 3*' I %(*"4%( j ¸JK' 3X%(E1!' 3:.,%;.#"4ω

p$+.4 I +' 3/3X%(0G+$

DL%(3!3/+\-'¹++' 3¼¬M O%('DL%%(J?/\E'Z1 + Z2 + ....Zq = p

+RDVKZ = (Z1, Z2, ..., Zq)

M

ÑU

I %(. I ' . ¬.4"4¬02%(3!JK"#L%(.4/ Z

!&=% I "#.41+¬X+3X%2.,%("4X$]%('8.4 I +' 32».,%.4" Zj

V!¬Q "#0]",%(.4] DL%(3X%(0GW+!31» nj

n , p¼!¼ "#E$+3X'R02%("#-+L%(E:F1.,%.4"

Z I <Y! F1*"#36%('*)\E'L%(-!"#$

ΠZ(k1, k2, ..., kq) =

[Z1 = k1;Z2 = k2; ....;Zq = kq]\-'L%(

k1 + k2 + ...kq = pM

'/%>?/ I F I %(. I ' .4+3:.4 I %(3*"#L%(. N 1.=<Y++0Q .4] :!' "#: 1X%("#.#.4p\E' "E

k1+3!0G:%(

E1k2

+3!0G%(E2

+ I M#M#MY@A' 3m%("#3 I +.,%· 3+02%(3\-'S<T%(QV3\-'].4206Q 3; 2!' "# E.4k1

q¥!´+É¡=´¥³ +3!0GXE%(E1

S.4k2

³ºv¡§Ì¯¢L §³%(E2

#M#M#M#Y+.4kq

¿E´¥¢¡m´+¥X³%(Eq

!$+JE%(.Fnk1

1 nk2

2 ....nkqqN

U<YQ !"4+-/+80' .#!"#D .#",%(E I +!\-'L%(E!"#$1DL%(3¬.406Q 3 P DL%(3!!"#!"4: 1, 2, ..., p+'R+X+06Q .4& I %(3*"#L%('*) k1, k2, ..., kq Í 3/ P !$+JE%(.·F

(

pk1

)(

p− k1

k2

)(

p− k1 − k2

k3

)

....

(

p− k1 − k2 − ...− kq−1

kq

)

=p!

k1!k2!....kq!.

Í Q "4+-O I ¸L%(.4+0G+E1«

ΠZ(k1, k2, ..., kq) =N

np=

p!

k1!k2!....kq !(n1

n)k1(

n2

n)k2 ....(

nq

n)kq1[k1+k2+....+kq=p]

+!:.4" ¬D 3QL%(Q "#.#"#$!' 3F = 0, 1, 2, ..., pq !O%(D DV+.4$:.4"j0' .#!"#0]",%(.4 ¬DL%(3X%(0GW+!3

» n1

n ,n2

n , ...,nq

n , p, q¼:M Í Z3+02%(3\E'+3X%G\E':!' D DP3!R.4/'++0Q .4/!!3!" I O 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¬'& +3DL%R.4 $+0G!3X%(!"4

g'5>EhÁb-g65·_g61?b5 (#·g·b &o_` %//b5 -g·b-` b-gPd $1 &m%(0]"#.#.4:¸ "4

(A1, A2, ..., An)S<Y$+>?$++0G+-&*"#:"# $+DP+%(-6!" ∀J ⊂ 1, 2, ..., n %

[⋂

j∈J

Aj] =∏

j∈J

Aj

g'5>EhÁb-g65·_g61?b5Ab 5Ab#' 1 /§_2 Ub $19v"4+- C + D +'*) I .,%S<Y$+>?$++0G+- *"#3X%]\E' C + D ER"# $+DV+%(E¬!"

∀C ∈ C ∀D ∈ D [C ∩D] =

CD.

6>VMT%*MY

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

ÑKÑ

g'5>EhÁb-g65·_g61?b5 (#·g·b &o_` %//b5·b!*_ c;%§_ a/§b G_ /'E_*dUf!%§c(b 9v"4+-»

Xi : Ω 7→ Ei¼' &=%(0]"#.#.4&¸ "4: ¬>%(3!",%(Q .4 %(.4$U%("#3 Z*"#\-' I +!&m%(0]"#.#.4:!R"# $+DV+*H

%(E!"∀Γ1 ⊂ E1,∀Γ2 ⊂ E2...,∀Γn ⊂ En,

[X1 ∈ Γ1;X2 ∈ Γ2; ...Xn ∈ Γn] =

[X1 ∈ Γ1]

[X2 ∈ Γ2]...

[Xn ∈ Γn]

@A' 3\E'j< "#.S+"#R%("#!"= "#.S!'43GR\E'&.4$+JE%(.#"#$D 3$ I $ +E "4+E>?$+3!"¸L$\-'L%(Â.4X'+X+06Q .4Γi-3*' "# F1 RDV"#EM

9-"fi : Ei 7→ Fi

XEn%(D D .#" I %(!"41+&!" (X1, X2, ...Xn) I !!"#!'6' /m%(0]"#.#.4/"# $+DP+%(-Â"#.+

! :0G¹+0G6 ¬.,%m%(0]"#.#.4(f1(X1), f2(X2), ..., fn(Xn))

g'5>EhÁb-g65·_g61?bbEdGh·c(f95&#6%d 5·b /f<% 219v"4+-

(F1,Π1)+

(F2,Π2)ÎZ!DL% I /D 3QL%(Q "#.#"4$ %(D DV+.#.4+3X%Öq¥!B!¯B+¡°²¡§ Ë q¥e¿º-¡s : Π1

+Π2

\-'.=<Y+3X%

Π1 × Π2q.,%ÂD 3QL%(Q "#.#"#$

Π!' 3

F1 × F2 ¸ "41DL%(3

Π(x1, x2) = Π1(x1)Π2(x2)M Í %

%(.43¬E!"Γ1etΓ2

$!"#JK+E3!DV I !"#>?+0G+EOÎ6'++06Q .4R F1etF2

Π(Γ1 ×Γ2) = Π1(Γ1)×Π2(Γ2)9-"RL02%("#-+L%(EÂ

X = (X1, X2)!&' I ' D .4] >K%(3!",%(Q .4/%(.4$U%("#3G

X1etX2-:"# $+DP+%(-/!"

++' .4+0G+- !"ΠX = ΠX1

× ΠX2

MB' I !X%(+Ò I e\-'%(.4 I %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+-E

' ++06Q .4&¸ "S+µ' .4"S &D 3QL%(Q "#.#"#$/!' 3

E .,%D 3DVK!"#!"4Z\E' "S!' "#R%DP' 3RQ*Ç!+.,%

I !!3!' I !"48S< ' /!' "#(X1, X2, ....., Xn)

¬>K%(3!",%(Q .4 %(.4$U%("#3 "# $+DP+%(- + $\E' "4*"4!!3!"#Q '$ .4"

µM 7 I +V+6*DVK

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 %§fqgÑKM= !R.=< ' "4\E'6D 3QL%(Q "#.#"#$6!' 3

ΩX'.,%\E'+.#.4:.4O>PMT%*M

X1, X2, ..., XnXE"# $+DV+%(E& ¬.4"

µÎ*M= !R.,%1D 3QL%(Q "#.#"#$/' "m3!0G/!' 3Ω"j+O+' .4+0G+E !"

µ!R.,%1.4"S' "m3!0G6!' 3

EM

jb-`h&/§bG1#·g·b_ #dUc(b_ hAh·c(f91+:·b 5Ab /§_ /§f!%a%sg·fq` %_ /§b9v"#e»

A1, A2, . . . , Ap¼G' ' "#8

p$+>?$++0G+-Z"# $+DV+%(E ®0G¹+0G®D 3QL%(Q "#.#"#$

α DVK

Z =∑i=p

i=1 1Ai

M '%(.#.4>?"#3\E'Z

' "#Â' ®.4"¬Q "#0]",%(.4 ®DL%(3X%(0GW+!3α, p

MN >?+3!!' .=< "# $+DV+%( I 6 Ai

%1+S+/ !"I!' ¬DL%(3!!"4 1, 2, ..., p I %(3*"#L%(. k

⋂

i∈I

Ai

⋂

i∈Ic

Aci = αk(1 − α)p−k ,

[Z = k] =

(

pk

)

αk(1 − α)p−k

ÑUÎ

Om%("#\-'Z%*0G+!&0G¹+0G&.4"P\-' .,%/>VMT%*M

Z1*';%(J?1»s)*+0]D .4 aGD ¼Ej<YDL%*Æ%('[L%X%(3SM H %(

I 0G- W+.4Kv+ÂS+U .,%6.4"q' "m3!0G&!' 3Ω = Ep +.4>K%(3!",%(Q .4%(.4$U%("#3

XiE.4%(D D .#" I %(!"4

I v3 $ S<T%(D 3W/.,%D 3DPK!"#!"4 D 3$ I $ +E I :>PMT%*MSXE6 I "# $+DV+%(E+/!' "#>?+-6' 1.4"' "m3!0G' 3

EM ¬$+>?$++0G+E

[Xi ∈ E1]-& I "# $+DV+%(E: D 3QL%(Q "#.#"#$ n1

n

I \-' "·DV+3!0G+S<T%(D D .#"4\-'+3O.,%m3!06' .4 I "Hµ X!'OF.,%>VMT%*M Z1 =

∑

i 1[Xi∈E1]M

Uh Ec(_g61?b5 (#·gÓh·c(f95#%d5Ab _ c;%§_ a/§b 2_ /-_*dUf!%§c(b %sg65 -hb-g65A_gPdUb $19v"4+-

X+Y

XE:Î2>K%(3!",%(Q .4:%(.4$U%("#3¬"# $+DP+%(-6F2>%(.4+' 3:%(E

f+g +'*)s I !"4

E' 0G$+3!"4\-'&!' 3E

%(.43

[f(X)g(Y )] =

[f(X)]

[g(Y )].NZDL%(3!!" I ' .#"4+3 !" X +

Y-3$+.#.46

[XY ] =

[X]

[Y ]M

7 D 3WO%>?"#3 3+02%(3\E'$/\-'j< "#.j!'43GO ¬0G-!3+3O.,%] +'*)*"4W+0G6%X+3!!"4 *DPKXZ = (X,Y )

%

[XY ] =

∑

x,y

xyΠZ(x, y) =∑

x,y

xyΠX(x)ΠY (y) = (∑

x

xΠx(x))(∑

y

yΠY )(y) =

[X]

[Y ]

Í Z+3X%]\-':!"S.=< ' / Î1>PMT%*M I +E!3$2 [XY ] = 0

:_ c+%§_g61?b 5(#Agb efq` ` b 5Ab "#_"#c"%sg'5>Ehbvg'5A_gPdUb 9v"#

(X1, X2, ..., Xn)' &=%(0]"#.#.4/

n>VMT%*M 3UM "# $+DP+%(- DPK

S = X1 +X2 + ...+XnM

Í %]%(.43V ar(S) = V ar(X1) + V ar(X2) + ...+ V ar(Xn).

Í X/3X%(0GW++S+&%("4$+0G+-:%(' I % È.4O>VMT%*MXi

XE I +E!3$ .,%2¸*' I %(. I ' .·>K%G+' "# "I 0]D :+-'*'DV"#ED 3$ I $ +-M < [-Å-DV![W/S< "# $+DV+%( I !R" I "j+-!"4+.#.42M

jbEc+1 %1?bG1N $*' "#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 +3DL%(3' 3X%(D DV+. R!"4 $+.4$+0G+EX%("#3M

3 <( 54: 5A< 0 : "%$ , . )+/) :9v"4+E

A+B

+'*)2$+>?$++0G+ES< ' Â!DL% I D 3QL%(Q "#.#"4$ (Ω,

)%U>? I B > 0

%:D 3QL%(Q "#.#"#$ I *"#!"4*H+.#.4

[A|B]

A\-'L%(

B»sZ*"#&%('!"GE+X% I [L%(E B EU¼+3X%Z*DL%(3O $¸ "#!"4

[A|B] =

ABB

Í %1$+>v"4 +0]0G+E .4$+JE%(.#"#$

(A+A′|B) =

(A|B) +

(A′|B) ;

[Ω|B] = 1 ;

[φ|B] = 0.

%m I !"4S<Y++0Q .4 B $¸ "4:DL%(3

B(A) = AB

B

! I ' :' >?+.#.4/D 3QL%(Q "#.#"#$!' 3 ΩM

+'*)D 3D 3!"4$+$ !' "#>%(-O- I *m3!0G F.=< "4 $:"#-!' "#!"#>?6\-':.=<YZ&=%("#OS< ' I *"#!"4 +0G+E]MÄ

B[B] = 1 »§.,%1D 3QL%(Q "#.#"#$\E'j< "#.SD .4+' >? X% I [L%(- \-'j< "#.SD .4+' 6!$+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+-"# $+DV+%(EÂ

B »§.,%®D 3QL%(Q "#.#"#$

A«R¡§°Ì¯;P°#´ºvÌ¡§¥!]0Gv*"¸L$ZDL%(3].,% I L%("4%( I Z

.=<Y$+>?$++0G+-«V°4´ µ´É L³´³ ¨ ºvÌ(´¥+ /*02%("4R¬.=<Y!RDL%DL%(3 I +.#.4/ ¬.=<Y$+>?$++0G+E&"# $+DV+%(E«À ¯¡ºv¢GB¥!´+°¯¢®³U´¥ÌU¡¼¬M

jbEc+1 %1?b $1ÑKM Í *"4!DVKÂ I %(3!G%("#" I !!"#!'$«.,%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<Y+.#.4 +3!3!"4W+3: K+>?'0GE!3:' : ¬=% I :«*+.#.4:!Q .,%( I [ \-'+.#.4:!.,%]D 3QL%(Q "#.#"#$6\-'¬.=<T%(' !3¬m% I 6"#RQ .,%( I [

Î*M H %(' ®DPD ' .,%(!"4 ' 0]"#.#.#"4 S< "#*"#>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<Y2DL%GX%(.4+0G+-2¸q%(Q .4 D .#'D 3$ I "4X$+0G+E«

Ñ+

Ä %(D D .#"4\E'$6F' /DP+3 /02%(.,% G*"#. :' Z3$' .#X%(DPK"#!"%>? I ' ¬D 3QL%(Q "#.#"#$6 0, 9Ä %(D D .#"4\E'$6F' "#*"#>-"4*'ZQ "4+DP3!X%(-6 "#.j!RDVK!"#!"%>? I .,%1D 3QL%(Q "#.#"#$ 0, 1Í 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 <T%( "#02%(+' 3S< ' /Ç+'G$+.4$+>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 <T%( "#02%(+' 36v\-' "PX%("#È !3' >? .=<T%(' 0GQ "#.4]*' >v3&%(.43' & Î6DP3!I ["4!"4]+m%("#+ØX3! 0GE!3+31' I [W+>v3 "#."#*"4\-';%(.43]%(' I %(*"4%(]\E'j< "#.DP+' 1+ I 30Gv*"¸L+3 I [")]U< "#.*.4R $!"#3¬M?+.#' "H I "q%H=H="#. "#E$+3¹+FO.4=%("#3 /»§3$+DV:«' " U< "#. $!"#JK.=<T%(' !3DV3!G*"#.%GÎm"4RD .#' I [L%( I O /JE%(JK+3.,%1>?"#!' 3e¼¬M

+6859685 I J+K II < A8A < - I < BA < 7AK +# <9v"#

X' G>%(3!",%(Q .4G%(.4$U%("#32FÂ>K%(.4+' 3/%(

E .,%8°K¡ ¨ ¢q¿¡s s¡sK¢¢P´°§°#´ ΠX/B

X

\E'L%(B

+3X%®DL%(3O $¸ "#!"4*.,%.4"

X\-'L%(

Ω!0' " :.,%D 3QL%(Q "#.#"#$

B %1 I

ΠX/B(x) =

B [X = 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!'nm"4GF' 8Ç!+' ;D "#.4Â'=% I ;Q ",%("4$ .4

k I ' D¤!13+D 3$+E$;DL%(3]' Â>VMT%*M

Xk

+3!' .#.#">?$+3!"¸q%(E [Xk = 1] = α

.,%;!' "#G 1>VMT%*M»Xk 1 ≤ k ≤ n

¼ !/!' D DVK$1"# $+DV+%(EKMA9-"p ≤ n

Sp =

∑pk=1Xk

3+D 3$+-;.40Q 3 ÂD "#.40QP$G%(' I ' 3] p D 3+0]"4+3 I ' D 'X%U>?O\-'¬.,%.4"

Sp!RQ "#0]",%(.4 :DL%(3X%(0GW+!3

(α, p)M

H 'R02%("#-+L%(E&' %(' !3/+-!"4+3q ≤ n

+ ) %(0]"#O.,%.4" Sp

\-'L%([Sn = q]

M Í 8%

[Sp = k | Sn = q] =

[Sp = k ;Sn = q]

[Sn = q]=

[Sp = k ; Sn − Sp = q − k]

[Sn = q]=

[Sp = k]

[Sn−p = q − k]

[Sn = q] %] +3! "4W+3$+JE%(.#"#$/D 3e>?+L%(E¬ I 6\-' Sn − Sp

!O"# $+DP+%(- Sp

+ %10G¹+0G6.4"\E'Sn−p

M OI %(. I ' .S=% I "#.4:0GE!3%(.43\-'

[Sp = k | Sn = q] =

(

pk

)(

n− pq − k

)

(

nq

) ((p+ q − n)+ ≤ k ≤ p ∧ q) »mÎ*M#Ñe¼

NZDL%(3!!" I ' .#"4+3*DP' 3 p = 1

[X1 = 1 | Sn = q] =

[S1 = 1 | Sn = q] =

q

n,

[X1 = 0 | Sn = q] = 1 − q

n

»mÎ*MTÎ?¼

H %(.4 I *"#!"4 +0G+E:DL%(3O.=<Y$+>?$++0G+-[Sn = q]

.,%].4"jQ "#0]",%(.4 Sp

! +>?+E'.,%].4"[-Å-DV+3!J?$(H0G$+!3!"4\-']3+ I E!3$;%('DL%(3X%(JK3X%(D [» . """ ¼ I +.#.4G

X1/\E'L%(-F;+.#.4]3!$G' ].4"Á 0+3!' .#.#"

02%("4/%>? I ' ®DL%(3X%(0GW+!3]*" S$+3+E]M Í 3+02%(3\-'+3X%;\E'.4¬.4"4 I *"#!"4 +.#.4:!3' >?$¬1 $+DP+ +-D .#'

α

ÑUÙ

+68596 :;< , 4=@ 0 et α1, α2, ..., αq

q reels > 0 DVK

H = k ∈ 0, 1, , ..., pq | k0 + k1 + k2 + ...+ kq = p Í Z*"# \E'j< ' :>VMT%*M

Z = (Z1, Z2, ...Zq)F¬>K%(.4+' 3%( 0, 1, 2, ..., pq !0' .#!"#0]",%(.4 ODL%(3X%(0G+!3

(α1, α2, ..., αq , p, q)» I

. "" ¼!"j+.#.4/%*0G+ODV' 3R.4"j.,%D 3QL%(Q "#.#"#$!' 3 0, 1, ..., pq *DV3!$:DL%(3H

$¸ "4:DL%(3

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

p!

k1!k2!...kq !αk1

1 αk2

2 ...αkqq ∀k ∈ H

9v"#Z' +.#.4>%(3!",%(Q .4R%(.4$U%("#3 0GE!3+3\-'.,%O.4" I *"#!"4 +.#.4 :» Z1, Z2, ..., Zq−1

¼·\-'L%([Zq = j]!R.,%1.4"S0' .#!"#0]",%(.4 /DL%(3X%(0GW+!3» n1

n−nq, n2

n−nq, ...,

nq−1

n−nq, p− j , q − 1

¼&M

+68596) :FC ?7 I 4< IK+KI I <>A A < -IG< K A < A +#K < #<>A A <9v"4+E

X' />K%(3!",%(Q .4%(.4$U%("#33$+.#.4+

B' $+>?$++0G+-: /D 3QL%(Q "#.#"#$-' .#.4 .=<q!DV$+3X%( I 6 X3+.,%(!"#>?+0G+EFG.,%2D 3QL%(Q "#.#"#$

B»§"#E!3v*' "#G%(' $+Q ' ¬ I I [L%(D "#!3e¼P+3X%2$

B[X]'

[X|B]++3X%G%(D DV+.4$/!DV$+3X%( I I *"#!"4 +.#.4 X \-'L%(B

M <Y$+JE%(.#"#$

[X|B] =

[X1B ]B

»mÎ*M ?¼

: $+0G-!3%("4$+0G+- !"S.=<Y%3++-'»ÑKM -¼ -'3+D 3+O.4RX%(!"4 8%1+V+

B[X] =

∑

i

ai

BAi =1B

∑

i

ai

[1Ai1B ] =

1B

[1B

∑

i

ai1Ai] =

1B

[X1B ] .

I %(. I ' .D 3X%(!"4\-' ;.=<Y!DV$+3X%( I I *"#!"4 +.#.4S< ' >K%(3!",%(Q .4Z%(.4$U%("#3ZU<YS I !'ZF8.=<T%("4 X%.4"I *"#!"4 +.#.4 %(D D .#"4\E'L%(-»ÑKM )K¼RF.,%1D 3QL%(Q "#.#"#$

BZQ !"4+- +S+

[X|B] =

∑

k∈X(Ω)

kΠX/B(k)»mÎ*M -¼

jb-`h&/§b $11 +>?+%('*) .4"4 I *"#!"4 +.#.413+ I E!3$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

[Sp | Sn = q] =

p q

n,

[X1 | Sn = q] =

q

n

»mÎ*MTÙ?¼

»§B"#3 F I /'Ç+R.=<Y) +3 I " I 6Î1*'ZDL%(3X%(JK3X%(D [ . "" ¼

+68596 :;< , 4<H %(R.=<Y) +0]D .4 _-D » Ç!+ /Î] $¼& $++3!0]"#+3 .,%1.4" I *"#!"4 +.#.4 X1

\E'L%([S = q]

(2 ≤ q ≤ 12 , S = X1 +X2) +Z $*' "#3\E'

[X1 |S = q] = q2

Ñ

+68596 0I J+KI I <>=@PMT%*M

X+

1BER"# $+DV+%(E DP' 3R\-'j< "#.j+Z"#%("#!"=

"#.V=%(' O+R"#.!'43G\-'[X = k]

"#R"# $+DV+%(E& B ∀k ∈ X(Ω) I /\-' "jU<Y$ I 3!"#O+ I 3

[X = k | B] =

[X = k] ∀k.

<Y$+JE%(.#"#$ΠX/B = ΠX

»mÎ*M ?¼! I ' I *"#!"4;$ I %("#3O+'432X%(E¬%!' 3X%(-Á.=< "# $+DV+%( I & X +

BM Í 2+3X%\E'»mÎ*M ?¼

+E!3X%4

[X|B] =

[X]»mÎ*M )K¼

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

X1, X2, ..., X8+].=<YÓDPK

S =∑8

1Xi

0GE!3+3&\E'

[S | X1 +X2 = k] = k + 21.»mÎ*M ¼

+68596 :;< , 4<9v"4+-

X1, X2, X3!3"4G>VMT%*M 3UM"# $+DV+%(E8

S = X1 + X2 + X3 ×%(D DV+.#.4

µ.,%8.4"O

X3

0GE!3+3&\E'¬.,%].4" I *"#!"4 +.#.4 S \-'L%([X1 = i;X2 = j]

! $6DL%(3

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

»mÎ*MTÚ?¼

+ $*' "#3:.=<Y$+JE%(.#"#$

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

[X3].»mÎ*M#ÑU?¼

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

+68596 :;< , 4<Í !' D DVK&.=<Y$+JE%(.#"#$»mÎ*M )K¼>?$+3¸L$:DP' 3R' />PMT%*M 3UM*D 3+L%(-&Î>K%(.4+' 3 0GE!3+3&\E'

X!O%(.43"# $+DV+*H

%(E B

M 3' >?+3&' I E!3Hµ) +0]D .40GE!3X%(-¬\-' I +.,%Gj<Y!OD .#' $ I X%("#3+0G+-¬.4 I %O\-'L%( XD 3+D .#' Î>%(.4+' 3M

3 43 <( 54: 5A< 0 : "%$ , . / $ , 5 $76 8 $=8 )$;:<5 ,-+6685 I ?' >?+E2$+$Z%(0G+$ÂF I !"4 $+3+32 ])*D 3!"4G ;.,%m3!0G

[X|Z = i]È

X!O' />VMT%*M3$+.#.4+

Z' />VMT%*ML\-'+. I \E'F]>K%(.4+' 3 %(O' ++0Q .4¸ "

FMqN

J?$+$+3X%(.j.4:3$' .#X%(O $+DP+ i I :\E' "S'&%(0GW+6F1 $¸ "#3O' :s I !"4 φ : F → R

DL%(3

[X|Z = i] = φ(i)

»mÎ*M#ÑKÑe¼

Ñ )

» Í ZDVK+3X%**DL%(3 I ->?+E!"4j φ(i) = 0!"

[Z = i] = 0¼M

Ã .j!O%(.43L%(!' 3+. ¬DVK+3U ¬02%( "4W+36D .#'R3X%(02%$K

[X|Z] = φ(Z)

»mÎ*M#ÑUÎ?¼

@' "4\-'Z.4 I 0G+06Q 38 »mÎ*M#ÑUÎ?¼G!G' >VMT%*M\-' ">K%(' φ(i)

O» I <Y!XF*"#3 [X|Z = i]

¼ E+\-'L%(Z = i

EM 1 +>?+' Â"#X%(E%('*)Â) +0]D .4*'ÂDL%(3X%(JK3X%(D [ " .]DP' 3RU< [L%(Q "#!'+3OF I +!&X%(!"4M $+JE%(.#"#$»mÎ*M#Ñe¼:j»mÎ*MTÎ?¼¬j»mÎ*MTÙ?¼¬»mÎ*M ¼ +>v"4+ +E¬%("#!"S3!DV I !"#>?+0G+E

[Sp = k|Sn] =

(

pk

)(

n− pSn − k

)

(

nSn

)

[X1 = 1|Sn] =

Sn

n;

[X1 = 0|Sn] = 1 − Sn

n

[Sp|Sn] = p

Sn

n;

[X1|Sn] =Sn

n[S|X1 +X2] = X1 +X2 + 21

Í %(' 3X%$\-'.,%>PMT%*M I *"#!"4 L%(E Z DP+' ¬D 3+*3]&>K%(.4+' 3¬%(¬' ®DL% I D 3v*' "# I <Y!&.4I %%(O.,%s3!0' .4»mÎ*M#ÑU?¼È.=<YZDVK+3X%

Z = (X1, X2) I :\-' " I *' "# F.=<Y$+JE%(.#"#$

[S|X1, X2] =

[S|Z] = X1 +X2 +

[X3]

+66 GIG< +# C+I G< :[X|Z]

c(fLhf! %d %§fqg

[X|Z]!R.=< ' "4\E'/>PMT%*M 3

Y>?$+3!"¸q%(E .4OÎ1D 3D 3!"4$+$ !' "#>%(-6«

ÑKMY

!' &m I !"4 $++3!0]"# "4! Z

Î*M [Y ψ(Z)] =

[Xψ(Z)] ∀ψ : E → R

Ë ÉG¢ ³ s¥!¯K s¡m¢ Ã .·! I .,%("#3 S<T%(D 3W&X%2 $¸ "#!"48\-'Y =

[X|Z]

>?$+3!"¸L6.,%]D 3D 3!"4$+$ÂÑKM H <T%(' !36DL%(3!U.4$+JE%(.#"#$

[φ(Z)ψ(Z)] =

∑

i

φ(i)ψ(i)

[Z = i] =∑

i

[X1[Z=i]]ψ(i) =

[X

∑

i

1[Z=i]ψ(i)] =

[Xψ(Z)]

0GE!3+-6\-'Y

>?$+3!"¸L]Î*M 1O$ I "#D 3v\-'+0G+EUS"#Y = h(Z)

' >VMT%*M 3UML>?$+3!"¸q%(-2ÑKMP+:Î*M I +!1 +3! "4W+3D 3D 3!"4$+$$ I 3!"#6DV' 3 ψ = 1j

I *' "#:F].=<Y$+JE%(.#"#$;»=Á!M»ÑKM ?¼!¼h(j)

[Z = j] =

[X1[Z=j]]

L"jQ "4+8\-'h = φ

S<YÈZ.4¬3$' .#X%(UM 02%("#E+L%(-&\-'+.4\-'RD 3D 3!"4$+$O!"#0]D .4 I +!/!DV$+3X%( I I *"#!"4 +.#.4KM

Ñ

+66) 9?7 C K+#I

ÑKM [aX + bX ′|Z] = a

[X|Z] + b

[X ′|Z] ∀a ∈ R ∀b ∈ R

Î*MX ≥ 0 =⇒

[X|Z] ≥ 0*M

[

[X|Z]] =

[X]M 9-"

X+Z

ER"# $+DV+%(E [X|Z]

Ù*M 9v"#

ψ : F → R' &m I !"4Z-' 0G$+3!"4\E'!' 3

F %

[Xψ(Z)|Z] = ψ(Z)

[X|Z].

»mÎ*M#Ñ?¼

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.,%]>%(3!",%(Q .46%(.4$U%("#3/3$+.#.4Y = H(., Z)

!"S.=<YZDVK/%(.43K(., i) =

[H(., i)|Z]

%

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

»mÎ*M#Ñ+-¼

)vM 9v"4+- ' :>VMT%]F>%(.4+' 3 %(E

"# $+DV+%(E Z

+h : E × F → R

DPKXv(i) =

[h(X, i)] (i ∈ F )

M Í %]%(.43

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

»mÎ*M#ÑUÙ?¼

$-` fqg6 dUc(_*d %fPg' ÑKM+OÎ*ML$+X%(EO$+>v"4 +EUDL%OF" 8%

[φ(Z)] =

∑

i∈Z(Ω)

φ(i)

[Z = i] =∑

i

[X1Z=i] =

[X] .

DV"#EMD 3U>v"4+E I \-'φ(i) =

[X] ∀i ∈ Z(Ω)

MR@A' 3Z0G-!3+38Ù*M#R 3+02%(3\-'+3X%¤\-'

[Xψ(Z)|Z = i] = ψ(i)φ(i) %1 I [Xψ(Z)|Z] = ψ(Z)φ(Z) = ψ(Z)

[X|Z]

M58E!3O+*¸.,%s3!0' .42»mÎ*M#Ñ+-¼M@AK DP' 3 I +.,% u(i, j) =

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

8%]%(.43

[Y |Z = i] =

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

S<YÈ [Y |Z] = u(Z,Z) = K(., Z)

M %m3!06' .42»mÎ*M#ÑUÙ?¼+3$!' .#6%("4$+0G+-« !"S.=<YZDVK

H(., i) = h(X, i),

.,%m I !"4 u >?$+3!"¸L6%(.43u(i, j) = v(i) ∀i ∀j +>?+3!!' ¬.=< [-Å-DV![W/S< "# $+DV+%( I 2M

Í %(' 3X%Z$G\-'].4:>VMT%*MU

\-' "ÁE :m I !"46 $++3!0]"# "4! Z Ç'+-6' 3.4GDL%(3!!" I ' .#"4+3%(R.4 I *"#!"4 +0G+- DL%(3

ZM N.#.4 E I %(3X% I $+3!"4$DL%(3.,%D 3D 3!"4$+$!' "#>K%(EG« \-'+.4\E'6"#

.=<Y$+D 3+' >?ω?.,% I L%("4X%( I Z(ω)

+-!3X%4 I +.#.4 U(ω)M Ã E!' "#!"#>?+0G+-UvDP+' 3+D 3$+E+3 I +.,%

I 0]0G:.4 I -!3X%("#3/ ¬.=< "# $+DP+%( I «*!"= +ZV+U*' ¬>VMT%*MV

!R"# $+DV+%(E Z*.,% I L%("4XX%( I

Z(ω)

j<Y!S<T%(' I ' I ' 3DP' 3D 3$*"#3/.4R>%(.4+' 3DPK"#Q .4 V (ω)M

ÑUÚ

+66 :;< , 4=@ 0 ∀ω +¬DPK |Ω| = n Í ®+3X% V .=<Y!DL% I >? I 3!"4+. &>PMT%*M 3¬!' 3 Ω

+ VZ.4

'+X+06Q .46 V s3!0G$6 s I !"4 $++3!0]"# "4!O Z DVK+3X%]+*¸ |Z(Ω)| = pM

ÑKM Í 06' "# V *'D 3v*' "# I %(.,%("#3 $¸ "DL%(3 < X,Y >=

[XY ] +3 .,%]*"#0G+"48 6.=<Y!DL% I

+' I .#"4*"4+®%("#!"j $¸ "A%("#"j\E' I +.#.4/*''!DL% I VZM

Î*M 9v"#X ∈ V 0GE!3+3 \-'

[X|Z]!R.,%D 3(Ç! I !"4+' I "4*"4+ / X !' 3 VZ

M*M Í 2 RZ

.4X'!DL% I R V s3!0G$ >PMT%*M 3UM?\-' "-Á m I !"4% 3GÁ Z

M '+.#.4 !X%*"#0G+"4 658E!3+3&\E' 3R.,%]D 3(Ç I !"4+' I "4*"4+ R[X|Z]

X

!' 3 RZ $/DL%(3R.=<Y$+JE%(.#"#$

Y =

[X] + r(Z − [Z])

\-'L%(Z%DVK$r = cov(X,Z)

var(Z)M58E!3+3&X%( I %(. I ' .4R.=< "#$+JE%(.#"#$ [(X − [X|Z])2] ≤

[(X −R[X|Z])2]M

Ù*M58E!3+3&\E' [X|Z] = R[X|Z]

!" |Z(Ω)| = 2

3 8 6,-! >: "%$ ,:54:5 <0! 'Â' D DPK+3%( I 8DL%(3X%(JK3X%(D [\-'.4®!' D DV3! ( !;$+JE%(./F

ΩM7>K%(E I 0]0G+ I +3

"#E+3!3J? 'O' 3R.,%1 $+DP+%( I + Z [X|Z]

M Ã E!3v*' "4OS<T%(QV3.,%!"4' "#>%(-G«

_ ced %d %§fqg×b-g Lbvg'5Ac -bhA_ c #Agb! _ c+%§_ a&/b_ /'E_*dUf!%§c(b "9v"#

Z' >VMT%*M >K%(.4+' 3:%(

E .,%2m%(0]"#.#.4¸ "41S<Y$+>?$++0G+E [Z = k] , k ∈ Z(Ω) I !!"#!'

' GDL%(3!!"#!"4¸ "42 Ω

%(D DV+.4$GDL%(3!!"#!"4Ø+ J?+*3$2DL%(3Z +.#.4G+3X%$ PZ

M 1 +>?+%(.431F;.,% $¸ "#!"4 ¬.=<Y+DV$+3X%( I I *"#!"4 +.#.4 %*\-'L%(%DVK$

Ak = [Z = k] , ck = [X;Ak]

Ak

[X|Z] =

∑

k∈Z(Ω)

ck1Ak

Í ®>?"#:!' 3 I +!6m3!06' .4]\-' [X|Z]1 $+DP+

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

+ J?+*32M 7 "#!"=

[X|Z] =

[X|Z + 1] =

[X|Z3]D ' "4\-'.4]DL%(3!!"#!"4Â+ J?+*3$2DL%(3

Z,Z + 1, Z3 E2"4 +E!"4\-' +¤3+>K%( I [K!" Z !F®>K%(.4+' 3%( −1 , +1 [X|Z]

+ [X|Z2]

-*" S$+3+EeM %/>?$+3!"#X%(Q .4&!"4;"#E$+3%(E¬! I I +.#.4& I *"#!"4 +0G+E¬DL%(3R' :DL%(3!!"#!"4 Z! I *' "# %('*) $¸ "#!"4&!' "#>%(-/«

/EaAcbb-g Lbvg'5Ac -bhA_ c #Agbh·_ cd %d %§fqg "

Ä " P ' DL%(3!!"#!"4 Ωq®%(D DV+.#.4+3X%8¯° © B¥´´¢v©E´¢q¿¥ Ë ´:*¯¥ P +&.=<Y8+3X% F(P)

»s'®D .#'"#0]D .4+0G+E F "%(' I ' I *s'!"4j<Y!&F I 3X%("#*3e¼.,% I .,%:S<Y$+>?$++0G+E m3!0G$6 O3$+' "4S<Y$+.4$+0G+-O P M %1 $/S< ' ¬DL%(3!!"#!"4$\-' "#>K%(.4+E/F I +.#.4: &.=<T%(.#J?W+Q 3/\-'j<Y+.#.4:+ J?+*3 D 3$sW+3$U%(0]0G"#3X%("4 +3]!' 36.41%(.#J?W+Q 3 I %(3+.#.4DPKXW +E1 6D 3D 3!"4$+$1 Â!X%(Q "#.#"#$

ÎK

3+.,%(!"#>?+0G+-;%('*)¤DV$+3X%(!"4GQV-.4$+ \E';j<Y-2DL%2.4]DL%(3!!"#!"4UMÁ7 "#!"=" F ].=<T%(.#J?W+Q 3+ J?+*3$6DL%(3 P

F1etF2 ∈ F =⇒ F1 ∪ F2 , F1 ∩ F2 , F c1 ∈ F

Í +3X%2$+JE%(.4+0G+E&\E'Ω et φ

%(D DL%(3!!"4+ +-¬F F $+.4$+0G+E& P +3E&%(D DP+.4$¯K =É2´³6 ¬.=<T%(.#J?W+Q 3 F MÄ 9v"4+- P + P ′ +'*)ØDL%(3!!"#!"4G

Ω ¤*"#3X%®\-' P 6P° º-³&¶Á¢P´\-' P ′ I Â\-';.=<Y +3X%

P P ′ L!"·' $+.4$+0G+-¬ P ! I E+-'®%( ' ®$+.4$+0G+E: P ′ >?$+3!"¸L1%("4$+0G+E:\-' I +!D 3D 3!"4$+$:$\E' "#>K%(.4+E .=< "# I .#'!"4 F ⊃ F ′ M Ã .S+!%("#!"j!"S+R+' .4+0G+-O!"V' O%(0G: F ′%( F M

+6)685 A < < ? /- I <47= A8A8< < CKBA8<>C27AK +#K <>C9v"#

Z' >PMT%*MVF2>%(.4+' 3:%(

EM <T%(.#J?W+Q 3 F(Z)

+ J?+*3$DL%(3Z

+3X%*LDL%(3¬ $¸ "#!"4jV.=<T%(.#J?W+Q 3+ J?+*3$/DL%(3O.,%DL%(3!!"#!"4 PZ

M Ã .j!m% I "#.46 ¬>?"#3O\E'

F(Z) = [Z ∈ Γ] ; Γ ⊂ E9-"Z

R' I ' D .4 (Z1, Z2)*' &%(0G

[Z1 = k1] F(Z1)

%(D DL%(3!!"4+E¬F F(Z)D ' "4\-'j<YZDP+' R.=<Y$ I 3!"#3

∑

k2[Z1 = k1;Z2 = k2]

"#.j+Z3$' .#:\E' F(Z1) ⊂ F(Z1, Z2)Í /J?$+$+3X%(.#"4%("4$+0G+E·DL%(3S3$ I ' 3!3+ I I +!!"#!'L%(!"4%(' I %j N >K%(3!",%(Q .4·%(.4$U%("#3«U"#Z1, Z2, . . . , ZN' /!' "#: />PMT%*M Fn

.=<T%(.#J?W+Q 3 F(Z1, Z2, . . . , Zn)M7 .43

F1 ⊂ F2 ⊂ . . . ⊂ FN

+6)6 CKBA8<>C$7AK +# <>C F =@<>C - BA8<>C9v"# P ' DL%(3!!"#!"4]+ J?+*3X%(E' R%(.#J?W+Q 3 F ' >PMT%*M

Y+3X% *"# F ¦oÉG´³ºv¥¯B+°#Ó!"-+.#.4! I !X%(E

!' 3 I [L%\E'6%(0G F »§"=MYKM!' 3 I [L%\-'/$+>?$++0G+-& P ¼M R3+02%(3\-'O' "#>%(-'OX+3E' !"#.4

Ä Y etY ′ - Î1>VMT%*M 3UM F H=0G' 3X%(Q .4O>?$+3!"¸q%(- [Y 1F ] =

[Y ′1F ] ∀F ∈ F L%(.43

Y = Y ′ MÄ 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 Es I !"4 $++3!0]"# "4!O Z E1M

cfqhf %d %§fqg@1 9v"#Y

' :>VMT%*M 3UM .4 +'*)D 3D 3!"4$+$O' "#>%(-O-O$\E' "#>K%(.4+EÑKMY

! F(Z)H=0G' 3X%(Q .42M

Î*M Ã .j)*"4!¬' ¬m 3!!"4ZE' 0G$+3!"4\-'φ!' 3

E+.#.4/\E'

Y = φ(Z)M

Ë ÉG¢ ³+ s¥¯? s¡s¢ Á9-"ÁÎ*MS!¬>?$+3!"¸L$KY

! I !X%(EG' 3:.4/%(0G/ F(Z)SS<YÈ¤ÑKM 3$ I "H

D 3v\-'+0G+EUvDPKXE = z1, z2, ..., zp

!"Y

! F(Z)H0G!' 3X%(Q .4Kv"#.q)*"4!K?DL%(3 $¸ "#!"4j

p I !X%(E c1, c2, ..., cp +.#.46\-'Y =

∑p1 ci1Z=zi

"#.+3$!' .#2\-'Y

>?$+3!"¸LÎ*M#%>? Iφ =

∑p1 ci1zi

»=Á!M4»ÑKM ?¼!¼

Î*Ñ

+6)6) 0I J+KI I <>=@PMT%*M 3UMP+ F .=<T%(.#J?W+Q 31+ J?+*3$DL%(3&' DL%(3!!"#!"4 P

ΩM Í %(D DV+.#.4+3X%

!DV$+3X%( I 2» I *"#!"4 +.#.4e¼O X \-'L%( F .,%;Ì¯¥¡s¯CB°#´¯° Ë ¯K =K¡¥´$ [X|F ] $¸ "4:DL%(3

[X|F ] =

∑

B∈P

[X|B]1B

Í 6+3X% \-' [X|F(Z)]!·$+JE%(.4FR.,%R>VMT%*M

[X|Z]"#-!3-*' "#+ "" .qMB" I "v02%("#E+L%(-.4jD 3D 3!"4$+$

.4RD .#'R"#0]DV3!X%(E& :.=<Y!DV$+3X%( I I *"#!"4 +.#.4KMÑKM ^ %sg-_ c+%dUbGbEd&hf! %d %'%,d 7 D DV+.4 V .=<Y!DL% I >? I 3!"4+.·»s *"#0G+!"4 |Ω| ¼· A>%(3!",%(Q .4Á%(.4$U%("#33$+.#.4

X 7→ [X|F ]

¸ "#/%(.43¬' G%(D D .#" I %(!"4.#"#$U%("#3]+&DVK!"#!"#>?1 V %(¬.41'¬!DL% I »s 6*"#0G+!"4 |P| ¼m3!0G$¬DL%(3.4R>VMT%*M 3UM F H=0G!' 3X%(Q .46M

Î*M g·bq_ /%d 5·b 1?fqg Pb '%d 9v"#u : R 7→ R

' 6%(D D .#" I %(!"4 I E>?) %1.=< "#$+JE%(.#"#$

[u(X)|F ] ≥ u(

[X|F ])»mÎ*M#Ñ?¼

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

[|X||F ] ≥ | [X|F ]| »mÎ*M#Ñ )K¼

[X2|F ] ≥ [X|F ]2.

»mÎ*M#Ñ ¼ Ë ÉG¢ ³+ s¥¯? s¡s¢9v"#

B ∈ P M %/>%(3!",%(Q .4&%(.4$U%("#3¸JK' 3X%(-%('D 3+0]"4+30G+0Q 3 »mÎ*M#Ñ?¼!$+JE%(.4' 3

BF

[u(X)|B]+.4X I 0G+0Q 3OF

u(

[X|B]) 1+3!0]"#.,%& $+0G!!3X%(!"42+G%(D D .#"4\-'L%(E

.=< "#$+JE%(.#"#$ I E>?)*"#$6F1.,%D 3QL%(Q "#.#"#$ B"#-!3-*' "#+ " .

*M Zg·b 1?_ c(_21KdEc;% U_*d %§fqg &µfPg'5A_`bvgPdU_ /§b 1 [X|F ]

!;.=< ' "4\E'>PMT%*M 3UM>?$+3!"¸q%(-.4ÎD 3D 3!"4$+$' "#>%(-6«%?¼Y

! F H=0G!' 3X%(Q .4 Qq¼ ∀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 3UMY

X%(!"4=%("4X%(E %('*);DV"#E %?¼+OQq¼!R$ I X%("#3+0G+- $+JE%(.46% [X|F ]N>?+3!!'Z /%?¼-"#.j)*"4!&' m%(0]"#.#.4: : I %(.,%("#3 (cA;A ∈ P)

+.#.4:\-'Y =

∑

A∈P cA1A .=<Y$+JE%(.#"#$

Qq¼& %(D D .4\-'$6F' Z$+>?$++0G+EB

P DP+3!0G+R I %(. I ' .4+3 cB %+ZS+:«

[Y 1B ] = cBB I

\-' "j+E!3X%("#cB =

[X|B]

I <Y!R.4:3$!' .#X%( I [+3 I [$MÂb-` _ c #·b 1 Í ZDV+' 3+0]D .,% I +3Qq¼DL%(3Qj<¼:« ∀Z F− 0G!' 3X%(Q .4K

[XZ] =

[Y Z]

»sØ' !"#.#"4;!"#0]D .4+0G+-G.42m%("#]\-'Z

' I 0Q "#L%("4 .#"#$U%("#3S< "#*" I %(+' 3GS<T%(0GG+.,%.#"#$U%(3!"# ¬.=<Y!DV$+3X%( I GM¼

M= fPg'5&%d %§fqgAgbv`b-gPd 0 4#'1 1Eb %'& $1 9v"4+E F + G +'*)%(.#J?W+Q 3 %U>? I F ⊃ G %]%(.43U

\-'+.4\-'"#.,%>VMT%*M 3UMX

[X|F|G] =

[X|G] ;

[

[X|F ]] =

[X]»mÎ*M#ÑUÚ?¼

Ã .'/!'43G3X%; 10G-!3+3/.,%ÂD 3+0]"4W+3G$+JE%(.#"#$ .4¬D 3+0]"4+3+/ I ®0G+06Q 3 I +.#.4H I " S\E''&%(D DP+.#.4+3O3!DV I !"#>?+0G+E

U et V$+X%(- G H=0G!' 3X%(Q .4U' O3+>-"4+- F0GE!3+3 .4$+JE%(.#"#$

[U1G] =

[V 1G] ∀G ∈ G.

ÎKÎ

5®%("4 [U1G] =

[

[X|F ]1G] =

[X1G].,% +'*)*"4W+0G®$+JE%(.#"#>?+L%(-Z*'Ó=%("#;\E'

G ∈ F M1 ++' I $\-'+ I D 3X%(!"4\-' I ®3$!' .#X%(«%>? I .4;X%(!"4 "" . " R %DL%(3) +0]D .4

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

[X|Z1, Z2]

Ù*M 7 %Y

b Ud F `b 4#c(_ a/§b [Y |F ] = Y

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

[XY |F ] = Y

[X|F ].

»mÎ*MTÎK?¼

Ë ÉG¢ ³+ s¥¯? s¡s¢ H $"#JKL%(EODL%(3UetV

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

[U1B ] =

[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$6 6.=<Y$+JE%(.#"#$»mÎ*MTÎK?¼/M O[EÅvDP![W&-O.4' "#>%(- «

X+Y

-Á +'*)G>PMT%*M-F¬>%(.4+' 3Á3!DV I !"#>?%( E +F

K'2!Á' RmE!"42-' 0G$+3!"4\E'' 3

E × F ' D DPK

Y F H=0G!' 3X%(Q .4 !"S.=<YDPKH(ω, y) =

[f(X, y)|F ](ω)

%«

[f(X,Y )|F ](ω) = H(ω, Y (ω))

»mÎ*MTÎ*Ñe¼

Ë ÉG¢ ³+ s¥¯? s¡s¢ 5-!3 ' &S<T%(QP3\E'6.,%]>VMT%*M*¸JK' 3X%(E/%('8 I Z0G+0Q 3 Â»mÎ*MTÎ*Ñe¼/»s\-''+3R' "#>%(-.=< 'X%(J?¬ D 3QL%(Q "#.#"4!/

H(., Y )¼ F H=0G!' 3X%(Q .4 I ["4"4DV' 3 I +.,%

' Ø%(0GB

F · $!"#JKDL%(3y+6DL%(3

c.46>K%(.4+' 3;» I !X%(E¼/D 3!"463!DP I !"#>?+0G+-1DL%(3

Y et

[f(X, y)|F ]!' 3

B Z>?"#U+Z3+JE%(3%(- X%1 $¸ "#!"4jq\E'

H(., Y )! $+JE%(.46F.,% I !X%(-

c!' 3

B +.#.4/ I F H=0G' 3X%(Q .4 +Z' !3GDV+' O$ I 3!"#3G«

[1BH(., Y )] =

[1BH(., y)] =

[1B

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

[1Bf(X, y)] =

[1Bf(X,Y )],

I 6\E' "S0G-!3/.=<Y$+JE%(.#"#$2»mÎ*MTÎ*Ñe¼/M*M 7 %

Xb Ud%§g65>Ehbvg'5A_gPdUb 5Ab F .,%>%(3!",%(Q .42%(.4$U%("#3

[X|F ]13$*' "#6FÂ' I !X%(- D .#'

D 3$ I "4X+0G+E«

[X|F ] =

[X]»mÎ*MTÎKÎ?¼

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

[f(X, y)]

%*+%(D D .#"4\-'L%(E1»mÎ*MTÎ*Ñe¼+»mÎ*MTÎKÎ?¼

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

»mÎ*MTÎ ?¼

+6)6 8C! I J+KI I <>A A <>CH %( I :DL%(3X%(JK3X%(D [

X $!"#JK+3X%]' :>%(3!",%(Q .4%(.4$U%("#36F1>K%(.4+' 3%(

EM

ÑKMK¢q¿¡s s¡s¢L¢P´É2´¢L ¯¥º-¢V´¯K° ©B¥!´ ¬9v"# F ' %(.#J?W+Q 3»s+ J?+*3$DL%(32' DL%(3!!"#!"4 P ¼ !"x ∈ E

ZDVK+3X%ΠX(x|F) =

[X = x|F ] =

[1X=x|F ]

»"4+ +3G\-'j< "#.RU<T%(JK"#]S<Y$+JE%(.#"#$2+E!3®Ì(¯¥¡m¯B°4´³¯° Ë ¯K =¡§¥´³ F H=0G' 3X%(Q .4¼ÂM ]>VMT%*MΠX(x|F)DVKW +-.4OÎ1D 3D 3!"4$+$ !' "#>K%(E3' .#X%(E"#0]0G$*",%(+0G+-¬ I 6\E'¬'O%U>?>v'+ " "

Î

»m%?¼ ∀x ∈ E ΠX(x|F)! F H=0G!' 3X%(Q .4+ ≥ 0»§Qq¼ ∀ω ∈ Ω

∑

x∈E ΠX(x|F) =

[1Ω|F ] = 1@A' 3¬Å;>?"#3 D .#' I .,%("#3UL&D 3e>-"4"#3+0G+-Q

.,%]s I !"4® ÎG>%(3!",%(Q .4& $¸ "4DL%(3Q(x, ω) =

ΠX(x|F)(ω) .4D 3D 3!"4$+$»m%?¼+»§Qq¼e<Y)vD 3!"#0G+- !' 3

Q /.,%m%' "#>%(-G«

»m%?¼ ∀x ∈ E ω 7→ Q(x, ω)! F H=0G!' 3X%(Q .4+ ≥ 0»§Qq¼ ∀ω ∈ Ω x 7→ Q(x, ω)!R' :D 3QL%(Q "#.#"#$6' 3

E+!m I !"4 Q q\-'6.=<Y®+3X%ΠX(.|F)

+3X%;%(D DP+.4$Â°#¡ ¨ ¢q¿¡s s¡sK¢¢P´°§°#´G¿E´X

+º*¯K¢q¿ F M Í X¬3X%(D DP+.#.4+3X%2\E'j< "#.jU<T%(JK"#j» I R»§Qq¼!¼¬ S< ' :m%(0]"#.#.4/ /.4"4 ¬D 3QL%(Q "#.#"#$6!' 3 E "# )*$/DL%(3

ΩÎ*MK¢q¿¡s s¡s¢L¢P´É2´¢L ·*¯K¥ºv¢P´GÌ¯¥¡s¯CB°#´¯° Ë ¯K =¡§¥´Â9v"#Z

' 1>%(3!",%(Q .4G%(.4$U%("#3GF>K%(.4+' 3/%(F

' D DPK+3X% I 0]0G8S< [L%(Q "#!' 8\-'Z(Ω) = F

M % °#¡ ¨ K¢q¿¡s s¡s¢L¢P´°§°#´8¿E´X

+º ¯¢q¿Z

+3X%DL%(3 $¸ "#!"4

ΠX(.|F(Z))MS+&Q*Ç!+& I I 0]0G]%('DV"#E]ÑKMqD 3$ I $ +E: E¬'&3+D 3+

.4X%(!"4U' ¬m I !"4 Q Î>%(3!",%(Q .4»)Vω¼ 02%("4 S<T%(D 3W»§Qq¼&+ I 0]D :+-'

"" ?"#.L)v"4q : E×F 7→ [0 1]

+.#.4O\E'Q(x, ω) = q(x,Z(ω))

qX I %(. I ' .,%(E!3WÁ!"#0]D .4+0G+-

DL%(3.,%Âm3!06' .4q(x, z) =

[X = x|Z = z]

% I .=<Y$+JE%(.#"# Π(X,Z)(x, z) = q(x, z)ΠZ(z)\-' "

DV+3!0G+1 20G-!3+3G G=% I $+.4$+0G+-X%("#3;' 3+.,%(!"4Ø\-'2>?']%>?+Ò;X%(1 ' Â $~Ç!F>-'%(>?!3RÇ+' X]«

[f(X,Z)] =

∑

z∈F

ΠZ(z)∑

x∈E

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

Í DP+' :3+!3' >?+3 I +!]s3!0' .4]DL%(3:' ]>?"41D .#'6%(Q!!3X%("#» I G\E'&Ç!>?' I +"#.#.4G =%("#3DV' 3Á>?'[L%(Q "#!'+3R%('202%( "4+0G+ER !DV$+3X%( I I *"#!"4 +.#.4¼ I <YÁ' I X$\E'+ I &!"#0]D .4 /.=<Y$+JE%(.#"#$

[f(X,Z)] =

[

[f(X,Z)|Z] ]+ G»mÎ*MTÎ*Ñe¼:M

*M¡³ ¨ ¢q¿¡s s¡sK¢¢P´°§°#´³6´ Á¡§¢q¿ Ë ´¢q¿?¯¢ ¨ ´Â« Í %1>v'Z+ »mÎ*MTÎKÎ?¼\E'¬.=< "# $+DV+%( I 6S< ' I ' D .42»X,Z

¼ G>PMT%*M 3UMj+-!3X%("#L%("#.=<Y$+JE%(.#"#$

[X|Z] =

[X] .,%;3$ I "#D 3v\E'G/JK3K!"4W+3+0G+-6"#)% I »§!3' >?+3

' I -!3Hµ)*+0]D .4e¼ "#.A)*"4!1+®3+>%( I [1' I *"#!"4$ I XX%("#31+¬'432X%(E]S< "# $+DV+%( I ++3!0GO ¬.4" I *"#!"4 +.#.4 %.4:3$!' .#X%(O' "#>%(-6«

X´ Z

³¢L ¡§¢q¿ Ë ´¢q¿?¯K¢L µ´³:³¡´ ³U´º-°4´É2´¢L Á³¡ΠX(.|Z) = ΠX+!&$+JE%(.#"#$&!"#JK "¸LO\-'.,%:m%(0]"#.#.4 D 3QL%(Q "#.#"#$"# )*$ODL%(3

Ω¸JK' 3X%(ER%('2D 3+0]"4+30G+06Q 3¬!

+;3$U%(.#"#$¬3$*' "#/F.,%+' .4&.4"V02%(3!JK"#L%(.4: X

M %1 $+0G!3X%(!"4Z I ¬3$' .#X%(U !3WR!"#0]D .4K !.,%("4X$6%('.4 I +' 36M

+6)6 :;< , 4<ÑKM= fPg'5&%d %§fqgAgbv`b-gPd05 (#·g·b/§f!%` #6/d %sg·fq` %§_ /§b " 9v"4+E

p+q +'*);+-!"4+3

> 0 et α1, α2, ..., αq

q reels > 0 ZDVK

H = k ∈ 0, 1, , ..., pq | k0 + k1 + k2 + ...+ kq = p Í Z*"#O\-'j< ' :>PMT%*M

Z = (Z1, Z2, ...Zq)F>K%(.4+' 3j%( 0, 1, 2, ..., pq S06' .#!"#0]",%(.4 DL%(3X%(0G+!3

(α1, α2, ..., αq , p, q)» I . "" ¼!"j+.#.4/%*0G+ DP' 3R.4".,%D 3QL%(Q "#.#"#$6!' 3 0, 1, ..., pq *DP3!$/DL%(3H

$¸ "4:DL%(3

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

p!

k1!k2!...kq!αk1

1 αk2

2 ...αkqq ∀k ∈ H

9v"#Z

' /+.#.4>%(3!",%(Q .4%(.4$U%("#3 0GE!3+3:\E'/.,%].4" I *"#!"4 +.#.4 Â»Z1, Z2, ..., Zq−1

¼R\-'L%(Zq

!R.,%.4"j06' .#!"#0]",%(.4 :DL%(3X%(0GW+!3» n1

n−nq, n2

n−nq, ...,

nq−1

n−nq, p− Zq , q − 1

¼ +3X%G\-'.46!' D DP3! I +!:.4" I *"#!"4 +.#.4O%(.4$U%("#32M

Î(

Î*M 7S_g6 :d %dUcb 9v"#(X1, X2, ..., Xn)

' !' "#O n>K%(3!",%(Q .4%(.4$U%("#33$+.#.4U?"# $+DV+%(Ee-\-' "H

*"4!3!"#Q '$ ZDVKSn = X1 +X2 + ...Xn

»m%?¼Z9v"#

f : R 7→ R 0GE!3+3 \-'

[X1f(Sn)] =

[X2f(Sn)] = ... =

[Xnf(Sn)]»§Qq¼;N $*' "#36\-'

[X1|Sn] = Sn

n

+6)6 ,+#>C = 8< IIG<>C -IG<= < A +#K < :·_ %§g·b 5·b _ c+3Lf 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<Y3*3 k 02%(3?e>-"4+ Â·»§"=MY

pij ≥ 0 ∀i, j + ∑

j pij = 1 ∀i) !"

fR' ¬s I "4E' 0G$+3!"4\-'6!' 3E

DPK+3X%Pf(i) =

∑

j pijf(j) !' "#(X0, X1, X2, · · · , XN )

>VMT%*MvF6>%(.4+' 3%(E

+3X%%(D DP+.4$ ¨Ê ¯K¡¢P´6¿E´®¯¥Ì¿E´¬É]¯? s¥¡ ¨ ´/¿-´ s¥!¯¢ ³¡s s¡m¢

P!"j\-'+.4\-'/"#R.,%m I !"4ZE' 0G$+3!"4\-'

f!' 3

E

[f(Xn+1)|Fn] = Pf(Xn) ∀n ∈ 0, 1, 2, · · · , (N − 1)

1.4 I +' 3:>?$+3!"¸L+3X%SFÂ!"#!3]S<Y) +3 I " I P' I $\E'+ I ] I +!]$+JE%(.#"#$]\-' ".#' "3X%(D DP+.#.4+3X%DP+' :¹+!3\-'+.4\E'X' >?+ "#36«

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

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

_*dUc+%1?b 5·bdUc_g6 %d %§fqg 5(#Agb` _ c41;:·b_ /'E_*dUf<%c(b9v"4+E

µ' .4"E D 3QL%(Q "#.#"#$!' 3

Rd Á!' D DP3!j¸ " F +

((Yn) ; n ≥ 1)' Á!' "# >VMT%*M"# $+DV+%(E

$\-' "4*"4!!3!"#Q '$O :.4"µF1>K%(.4+' 3%(

Rd M Í ZDVK

X0 = 0 , Xn =

i=n∑

i=1

Yi ∀n ≥ 1

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

(2.23)

[f(Xn+1)|Fn] =

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

∑

k∈F

f(Xn + k)µ(k) = Pf(Xn)

\-'L%(8%1DPKX$Pf(i) =

∑

k∈F f(i+ k)µ(k) =∑

j∈F+i f(j)µ(j − i) + $*' "#O"#0]0G$H

*",%(+0G+- .4¬3$' .#X%( %( I $2M

ÎKÙ

+

9v"#(Ω,

)' ;!DL% I & OD 3QL%(Q "#.#"#$G»¸ "¼ Â' D DPK+3X%%(' I I [L%(D "#!3¬\-'O.4&!' D DV3! !

$+JE%(.·%ΩN

+3X%1' 0Q 36+E!"4+3&!!3!" I +0G+E DPK"#!"·¸ ) $:%(R' :.,%]' "#]M

(5 84:>, $;:5A<!%! . 54:-!*$($ " :-)! >:*"0, )+/ 5 ! 5 698 !)+6859685 FG,HEI J+KIBC

Ä Í ;%(D DP+.#.4+3X%O¶° s¥¯? s¡s¢!' 3Ω' O' "# I 3"4X%(- S<T%(.#J?W+Q 3 (F0,F1,F2, ...FN )

' 3ΩMO!DL% I

D 3QL%(Q "#.#"4X$:06' "AS< ' ¬¸.#!3X%(!"4+3X%]%(D DV+.4$´³µ*¯ ¨ ´¶Á°² s¥ Ë

Ä Í Z*"#3X%G\E'j< ' /' "#(X0, X1, X2, ..., XN )

¬>PMT%*M !¯?¿?¯P Ë ´F I +!¬¸.#!3X%(!"4!"∀i ∈ 0, 1, 2, ..., N, Xi

FiH=0G' 3X%(Q .4GM

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 ->?HL%(-O\-' F−1 = F0

¼M

Ä 9v"#/»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 +!' "#R.,%¬¸.#!3X%(!"4 Fn = F(X0, X1, X2, ..., Xn)

»§'%>?>-'G+ "" .\E'j< "#.PU<T%(JK"4X%("#Q "4+ÂS< ' ¸.#!3X%(!"4q¼M %/!' "#?»

Xn¼!%%(D $&% I +!R¸.#!3X%(!"4ÂX%(*"4\-'.,%6!' "#»

Xn−1¼R4»m%>? I .,% I ->?+E!"4 X−1 =

X0¼&YRD 3$+>-"4!"#Q .4KM Í ZDP+' I !3!' "#3F.=<T%("4 / G»

Xn¼S<T%(' !3O!' "# %%(D $

"#-$+3X%(-]«L!' D DVKULDP' 3&!"#0]D .#"¸L+3UV\E'6.4O>VMT%*MXn

"4+-¬FG>%(.4+' 3&%(&.4/0G¹+0G!DL% I E

+O 'O' /!' "#2»φn : En+1 7→ R

¼S<T%(D D .#" I %(!"4O-' 0G$+3!"4\E' DV' 3' n*.,%1>VMT%*M 3UM

Yn = φn(X0, X1, X2, ...., Xn)! Fn

H=0G!' 3X%(Q .4q»s+.#.4 ! I X%(E&!' 3 I [L%\E'¬%(0G Fn¼RE

X3!:\E':.,%1' "#(Yn)

!O%%(D $ %(R.4 I %'.4>VMT%*MXn

XE3$+.#.4*.4!' "#Sn = X0 +X1 +X2 + .... +Xn et Πn = X0X1X2...Xn

XE %("#!"j !' "#O%%(D $M

Î

43 $ ,:5 $=8 !9v"#»

Ω, (Fn),¼' Z!DL% I :¸.#!3$GM

)+6685 FG,HEI J+KIBC9v"#»

Mn ; 0 ≤ n ≤ N¼¬' Â!' "#Â%%(D $;F.,%;¸.#!3X%(!"4

(Fn) '*"#31\E' I +!' "#!'

(Fn)H=02%(3!!"# JE%(.4Â»s'ZD .#'RQ 3!"4W+>?+0G+- ' :02%(3!!"# JE%(.4e¼q!"j+.#.4¬>?$+3!"¸L/.4O$+JE%(.#"#$

[Mn+1|Fn] = Mn ∀n ∈ 0, 1, 2, ..., N − 1. »*M#Ñe¼

'*"#3O\-' I <Y!R' 1³º-¥ÉG¯¥ s¡§¢-©?¯K°#´Â»§3!DjM*' 1³º-³ÉG¯¥+ s¡¢-©E¯°#´(¼!"j+.#.4/>?$+3!"¸L:.4R"#$+JE%(.#"#$

[Mn+1|Fn] ≤Mn»§3!DjM

[Mn+1|Fn] ≥Mn¼

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

)+66 <>=% <>C ?9?,+#>C9v"#

(Mn)' /02%(3!!"# JE%(.4

ÑKM 9-"n ≤ n+ p ≤ N ,

[Mn+p|Fn] = Mn

+DL%(3!!" I ' .#"4+3 Mn =

[MN |Fn]Î*M 9-"

φ : R 7→ R! I E>?) R»

φ(Mn)¼!1' '02%(3!!"# JE%(.4 I <Y!G+ DL%(3!!" I ' .#"4+3G.4 I %1DV' 3

|Mn| et M2n*M I 0Q "#L%("4.#"#$U%("#3/S< ' :m%(0]"#.#.4¬¸ "46 :02%(3!!"# JE%(.4O!R' /02%(3!!"# JE%(.42M

M [Mn] =

[M0] ∀n

3+!"4+*3X%G\-' [Mn]

/ $+DP+ZDL% n :D 3+0]"4+3ODV"#E3$' .#¬"#0]0G$*",%(+0G+E: G»mÎ*M#ÑUÚ?¼ %1+S+

[Mn+2|Fn] =

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

[Mn+1|Fn] = Mn,

S<YÈ.4:3$!' .#X%( %( I $:DL%(3O3$ I ' 3!3+ I KM / I Z!R' I $\E'+ I 6 :.=< "#$+JE%(.#"#$/ I ->?)v"#$Â»mÎ*M#Ñ?¼ %1+ZS+

[φ(Mn+1)|Fn] ≥ φ(

[Mn+1|Fn]) = φ(Mn),

.4 !3"4!"4W+0G¬!$+>-"4 +- "L>?' %>?+Ò¬DL%Á0G-!3+3R.4O\-'L%(!3!"4W+0G1 I <Y!\E'j< ' ÂDV"#E"#0]DP3!X%(-U»s%( ' 2»mÎ*M#ÑUÚ?¼!¼>?' %]$ I [L%(D DV$ "#.!RD .#'O\E'¬+0]D S< Å3+0G$*"4+3M

)+66) <>=% < C < ;<>=@?BA8<>CÑKM 7Sfq``b 5·b "#_"cC"=1?b-gPdUcEb %sg65 -hb-g65·_gPdUb 1 9v"#»

X0, X1, X2, ..., XN¼/' ' "# >VMT%*M 3UM

I +-!3$R"# $+DV+%(E DVK Sn = X0 +X1 +X2 + ...+Xn ; Fn = F(X0, X1, X2, ..., Xn) Ç!

*"4O\E'(Sn)

!R' :02%(3!!"# JE%(.4 +V+/D ' "4\E'Xn+1

!R"# $+DV+%(E Fn

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

[Xn+1|Fn] = Sn +

[Xn+1] = Sn

Î )

9-' D DVK /D .#'.4>PMT%*MXn

$\-' "4*"4!!3!"#Q '$ +.#.4&E&%(.43O' O.,%10G¹+0G/>K%(3!",%( I 6\-'¬'%(D DV+.#.4+3

σ2 %( I +!:"#!'L%(!"4

S2n − (n+ 1) σ2 R%('X!"V' ¬02%(3!!"# JE%(.4 I 0]0G:.4¬D 3' >?

.4/DP+!"# I %(. I ' .' "#>%(-6«

[S2

n+1|Fn] =

[(Sn +Xn+1)2|Fn] = S2

n + 2Sn

[Xn+1] +

[X2n+1] = S2

n + σ2

S<YÈ"#.j3$!' .#/%("4$+0G+- \-'

[S2

n+1 − (n+ 2)σ2|Fn] = S2n − (n+ 1)σ2

Î*M _ cd %§g L_ /§b 8hAc(f95#%d 1 9v"#Â»T0, T1, T2, ..., TN

¼1' G!' "# G>VMT%*MSDPK"#!"#>?2S"# $+DV+%(ES<YDP$+3X%( I 1

.,%8!' "#Mn = T0T1T2...Tn

!]%(.431' Â02%(3!!"# JE%(.4DL%(313X%(D DP3!GF.,%¸.#!3X%(!"4L%(!' 3+.#.4+ J?+*3$/DL%(3.4>PMT%*M

TnM NV+6

[Mn+1|Fn] = T0T1...Tn

[Tn+1] = Mn

Ã . Å%¬S<T%(' !3Á) +0]D .4 02%(3!!"# JE%(.4"#E$+3%(E%X I "4$F¬ !' "#"# $+DV+%(EUE02%("4 I j<Y!DL%:.,F;.4GD .#':"#0]DP3!X%(- .4G' II W/ I +!]![$3!"4G6*Æ %('=%("#6\E'j<Y+.#.4GDP+3!0G+/S<T%(.#.4+3!3W6%(' +.,F .=< "# $+DP+%( I «q' I +3!X%("#0Q 3] !3X%(!s3!02%(!"4/"#0]D .4: $+!3!' "4+E/+S+ I +.#.4 I "' :+D 3$+3!>K%(E.,%]D 3D 3!"4$+$6 ¬02%(3!!"# JE%(.4 '&%(.#.4$+!'*"4+3 +'*)S<Y+E!3+.#.46M

,-$ 0! < , )! (0 $ ,:5 $=8 !9-' 3.=<YDL% I ¬¸.#!3$ (Ω,Fn,

)Z: :DV' 3

0 ≤ n ≤ N«

Ä ' 6!' "#(Mn)

%%(D $F» Fn¼

Ä ' 6!' "#¬D 3$+>-"4"#Q .4(Un)Ã E!3v*' "42S<T%(' !3DL%(3!.,%8X%(!"4×!' "#>K%(E «!"

(Xn ; 0 ≤ n ≤ N)!]' Z!' "# >VMT%*M 3UM

$!"#JK+3X%1DL%(3(∆Xn ; 1 ≤ n ≤ N)

.,%]!' "#: O% II 3"4X+0G+E& 2»Xn

¼ $¸ "46DL%(3

∆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 = X0 +

k=n∑

k=1

∆Xk.»*MTÎ?¼

Í Z+3X%]\-'j< ' /!' "#/%%(D $(Mn)

!R' :02%(3!!"# JE%(.4!"S+O+' .4+0G+E&!"

[∆Mn|Fn−1] = 0 ∀n ≥ 1

»*M ?¼

Î

)+6)685 FG,HEI J+KI

Í %(D DV+.#.4+3X%1!3X%(m3!0G$/ (Mn)

DL%(3(Un)

+O.=<Y+3X%(U ∗M)n

.,%1!' "#: $¸ "46DL%(3

(U ∗M)n = U0M0 +

k=n∑

k=1

Uk∆Mk»*M -¼

Ã .S3$!' .#/ I /\-' "S>-"4+-OS<Y+!3/*"#O\-'¬.,%1!' "# (U ∗M)n! I %(3X% I +3!"4X$/DL%(3R.4R3+.,%(!"4«

(U ∗M)0 = U0M0 ∆(U ∗M)n = Un∆Mn ∀n ≥ 1»*MTÙ?¼

ZgbEb3 Xb-g65 fqgPdUb _ c+/§fÃ 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$!' .#X%(O%(R' Z+->-"#3 +0G+- 3X%(D DV+.,%(E I +.#' "S< ' :X%(.#.4 Ç+'*)®**"4D .#' R\E'&>?'0]"4+Ò:' "4\-'+0G+E/ÑÏ!' 3D "#.4 >?'JE%(JK+Ò:' s"4>?!3&0]"4:+ I % G!' II W2j+:.,%;DV+3 +Ò2+ I %/S<Y$ I [ I M Í !' D DVK+3X%\-'1>?'6*"4DPK+Ò2F>?!32%(3!3!"#>?$G%(/.4:Ç+'S< ' /0]0G6S<T%(3!J?+E

M0M

' "#!O0G -"# J2+R3QV/ ¬QL%(.Á*.4/+0]DS< "#E!3v*' "#3Ä .,%.4"·!Å-0G$+!3!"4\-':!' 3.=<Y+X+06Q .4 −1,+1 \-':.=<Y+3X%

Π»Π(−1) = Π(1) = 1/2

¼Ä ' 6!' "#: :>VMT%*M*"# $+DP+%(- $\-' "4*"4!!3!"#Q '$O /.4"

Π\-'¬'+3»

Xn ; 1 ≤ n ≤ N¼

Xn3+D 3$X+E+3X%].4¬JE%("#D 3 I ' 3$:DL%(3R.4¬*H="4+0G I ' DX%(*"4O\E'

Mn = M0 +X1 +X2 + ...+Xn+3X%.,%

\-'L%(E!"#$6S<T%(3!J?+- ->?'O*"4!DVK+3+Ò6F I +"#X%(E6M@' "4\E'¬.4O>PMT%*M XnE I +E!3$U (Mn)

!R' 02%(3!!"# JE%(.4;4» I <Y!&FGDP+'8 I [K&D 3WO' I %ODL%(3!!" I ' .#"4+3¬ /.=<Y) +0]D .4 """ Ñe¼:MqN3$U%(.#"#$ÂL%( ' I %!"#Z*.43W+JK.4O*'1Ç!+'ZER=%("# :+.#.4:m%Z\-':.,%.4" R>VMT%*M

XnE' /DP$+3X%( I ¬.4$+J?W+3+0G+-

$+JE%(!"#>?2 +\-':.=<Y$+>?.#' !"48 ¬>?!3¬s3!!' 6!R' /!' 3!02%(3!!"# JE%(.4M9-"j>?' ¹+' ZDV+']Ç!'+' 3LÎ I [K>?'ODL%(3X%("#!3E&"#!' D DV3!X%(Q .4« /DL%DV' >?"#3OJE%(JK+3 S<T%(3!J?+-!"q.4 I ' D;!=% I &S< ' ODL%(3!:?.#"#0]"#+3>?K0]"4%2ÑÏ\E'L%(>?'Á>?'+-+Ò&+Â>?+"#&S<T%(' !3 DL%(3! DV+' Á!' D D 3!"#0G+3 I "# I ->?$+ "4+E+G"#-!3-*' "4X%(-' R' "#6»

Un¼ 06Q 33$+.4Á !!"#$ F ¸JK' 3+3Á>?K

0]"41»§' 60]"4X6$+JE%(!"#>?1!"#JK "¸q%(-&\-'6>?'¬%>?+Ò1DL%(3!"4$!' 3=% I e¼6MP0]0G+E/ $++3!0]"#+3+Ò1>?'&>?!30]"4 /9-"V>?'¹+ I 0]0G&' .4 0G Kv>?'+3+Ò¬"# '+ I $¬DL%(3.=<Y$+>?.#' !"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%(-n+3X%m I !"4 Q+3!>K%(!"4O\-'¬>?' %(' 3+Ò:=%("#Ç'\-'j<TF I +R"#!X%(EM

NØ+3!0G102%(![$+02%(!"4\-'Z I +.,%X2!3X%*' "#]DL%(3.4m%("#1\E'Â>?!3Â0]"4Un

1%(.4$U%("#3+1\E'.,%!' "#

(Un)!6%%(D $MS58%(.#[+' 3+'+0G+-S.46*"#3 I +' 36 I %!"#2E:DL%/*"4DPK$F%*0G+!!3

j< "#0]DV3!&\-'+.#.4&!' "#&%%(D $1«-"#.P'43G3X%("#¬-!"P+.P$+X%("#.4 I % v O0]"4X+3 Un = 106(1Xn=1 − 1Xn=−1)DV' 3+0]DP I [+3' ]0]"#.#.#"4 §3X%( I =% I +3!X%("# F I [L%\E' I ' D ]>?'"#0]DPKX+3X%: I $ I "4 +3 >?!3 0]"4X

Un%('*H="4+0G I ' D+Â*"4DPKX%(-' "4\E'+0G+- "#*m3!02%(!"4s' 3! "4DL%(3.4ÁÇ+'Ç~'X\E'j<TF

.=< "#!X%(-n − 1

» I +!;3!!3!" I !"4¤!]DP$+3$F.,%3' .4+!ZF.=<T%("4 *'=%(0G+'*) E3!"4+ Â>K%D .#' EU¼ %!!3+"#-O I »

Un¼F1¹+!3:X+' .4+0G+E&%%(D $/02%("4q¥ Ë ÌU¡³¡KB°4´2M

7ÅE%(-O0]"4$: I +!:m% I j*>?!3¬m3!!' 6%('Z+0]Dn +>?+E'2L!"S.=<Y I E>v"4+E&\E' U0 = 1

M0 + U1X1 + U2X2 + ...+ UnXn = UOM0 + U1∆M1 + U2∆M2 + ...+ Un∆Mn.

ÎKÚ

B'O%U>?+Ò63 I E'.,%!3X%(m3!0G$/ :.,%02%(3!!"# JE%(.4(Mn)

DL%(3O.4:D 3 I X!'D 3+>-"4"#Q .4 (Un)M

Ã .)v"4!] RÇ'+' 3 I E>K%("# I '6 ].=<Y)*"4!+ I G G!3X%($+JK"4:JE%(JKL%(E\E'G.=<YDP+' I !3!' "#32%(D 3W%>?"#3 DL%$/ E' "#¬F1+3 +O$+!'*"4+3& $+3!"4 S< ' JK3X%(0Q 3 I ' D ' /+.#.4/!3X%($+JK"46!%(D DV+.4$:%(.4 .,%( JE%(J? I ' 3X%(-R' E02%(3!!"# JE%(.4 E]*\E'j< "#.P=%(' ROJE%(3 +3 I *m*3/%>? I .,%!"4Z 02%(3!!"# JE%(.4 02%(![$+02%(!" I "4+]M 6![$3W+0G\-' "!' "#&!O $DP$+3X%(-«"#.0GE!3\E'j<T%(' I ' !!3X%($+JK"4I I +3!L%(E>?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

: Efqc v`b9-' 3A' !DL% I Á¸.#!3$¬» Ω , (Fn) ,

¼j"4+-»Mn

¼V' 02%(3!!"# JE%(.4¬»§3DjMe' !' 3A'' X'H=02%(3!!"# JE%(.4e¼+Â»

Un¼:' !' "#2D 3$+>-"4"#Q .4»§3!DjM' !' "#2D 3$+>-"4"#Q .4GDVK!"#!"#>?e¼

(U ∗M)n!%(.43' 202%(3!!"# JE%(.4

»§3!DjM ' :' 3O'' /'H=02%(3!!"# JE%(.4e¼M Ë ÉG¢³ s¥!¯K s¡sK¢ 3! ' %(' I % O02%(3!!"# JE%(.4&+ DPKX

M ′n = (U ∗M)n

®%∆M ′

n = Un∆MnS<YÈ +' !"#.#"4X%(-O.,%1D 3$+>v"4!"#Q "#.#"#$/

(Un)

[∆M ′

n|Fn−1] = Un

[∆Mn|Fn−1] = 0

)+6)6 - AK ?9? K + < <#? >C < I +# +#I? C KBA8<9v"4+-2»

0 = M0,M1, ...,MN¼R' 02%(3!!"# JE%(.4» Fn

¼ X%G¸.#!3X%(!"4L%(!' 3+.#.4 *"#3X%2\-'(Mn)

DVKHW .,%Âq¥!q¥¡ Ë Ë ¿E´¥´µq¥ Ë ³U´¢L =¯K s¡m¢®q¥ Ë ÌU¡³¡KB°4´!"=A\-'+.4\E'"#.,%» Fn

¼µH=02%(3!!"# JE%(.4(Zn)

j"#.)v"4!2' I !X%(- c +' D 3 I X!'RD 3$+>v"4!"#Q .4 Un

+.4\-'(Zn) = c+ (U ∗M)n ∀n M

'&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 DjM¼M9v"#

(ε1, ε2, ..., εN )' !' "# >PMT%*M 3UMÁ"# $+DP+%(-$\-' "4*"4!!3!"#Q '$® I +E!3$ DVK ε0 = 0

+%(D DV+.4

(Fn).,%¸.#!3X%(!"4L%(!' 3+.#.46+ J?+*3$6DL%(3 I +!/!' "# I !"4 $+3O.,%102%(3!!"# JE%(.4

Mn = (ε0 +ε1 + ...+εn)ME@·' 3$+>-"#+3.4Á!3!"#>-",%(.#"#$ K'Á' D DPK+3

ε1Â $+J?$+$+3$6M ¸.#!3X%(!"4

L%(!' 3+.#.4G+ J?+*3$2DL%(3].4]!' "#(Mn)

+(εn)

XE]"4 +E!"4\-'U I 0]0G DV+' 1.4;>?$+3!"¸L+32F8!"#!3S<Y) +3 I " I KM

c(fLhf! %d %§fqg 1(Mn)

DPKXW .,%D 3D 3!"4$+$/ ¬3+D 3$X+EX%(!"4ZD 3$+>v"4!"#Q .4:!"S+R+' .4+0G+E !"ε1

D 3+>K%(.4+' 3%(' Z++0Q .4/3$*' "# F2ÎDP"#-6M

Ë ÉG¢³ s¥!¯K s¡sK¢×«9v"4+Ea et b > 0

!' D DVK¬' 6S<T%(QV3\-'ε1

D 3+ 2¬>%(.4+' 3%( −a, b +R0GE!3&\-'

(Mn)DVKW /%(.43O.,%DjM 3UM DjM DVK

p =

[ε1 = b] , q = 1 − p =

[ε1 = −a].@' "4\E'

ε1! I +-!3$K 0 = pb− (1 − p)a

S<YÈp = a

a+b , q = ba+b , p

q = ab

M"4 $+3Á02%("#-+L%(E' R>VMT%*M 3UM

Fn FnH=0G!' 3X%(Q .4>?+3!"¸q%(-

[Fn|Fn−1] = 0 '%(.#.4

0G-!3+3\-'j< "#.¬)*"4!γn Fn−1

H=0G!' 3X%(Q .4®+.#.4®\E'Fn = γnεn

S<T%(D 3W.,%D 3DPK"#!"4

K

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

' 3R

n+1 » I <Y!]FZ*"#3' 2s I !"4Ø3$+.#.4Â n + 1

>%(3!",%(Q .4]3$+.#.4¼+.#.4\E'Fn = fn(ε0, ε1, ε2, ..., εn)

MÁ0]0Gεn

!1"# $+DV+%(E Fn−1

.=<Y!DV$+3X%( I I *"#!"4 +.#.4 [Fn|Fn−1] I %(. I ' .4F].=<T%("4 /.,%m3!06' .4;»mÎ*MTÎ ?¼R

X3!:\E'&.=<YZ%.=<Y$+JE%(.#"#$pfn(ε0, ε1, ..., εn−1, b) + qfn(ε0, ε1, ...εn−1,−a) = 0

S<YÈ;.=<Y!"#3fn(ε0, ..., εn−1,−a) = −a

bfn(ε0, ..., εn−1, b)M

Í DP+' O%(.43 $ I 3!"#3

Fn = fn(ε0, ε1, ..., εn−1, b)1εn=b + fn(ε0, ε1, ..., εn−1,−a)1εn=−a

=fn(ε0, ε1, ..., εn−1, b)

b(b1εn=b − a1ε=−a)

=fn(ε0, ε1, ..., εn−1, b)

bεn = γnεn

\-'L%(®%]DVK$γn = fn(ε0,ε1,...,εn−1,b)

b

I +.,%2 $+0GE!3!3%X+3!!"4MV!"4 $+3O02%("#*H+L%(-' 02%(3!!"# JE%(.4

(Nn) %(D D .#"4\E'L%(- I \E' "6D 3$ I W F

Fn = ∆NnR"#.>-"4+-

Nn =N0 +

∑n1 γkεk

M Ã .Á3!2!"#0]D .4+0G+-GF3+02%(3\-'+3\-'N0

\-' " F0H=0G!' 3X%(Q .48!'

I X%(E/3$+.#.4KM@%&F.,%13$ I "#D 3v\E' DVK [ε21] =

[ε2k] = σ2 ML0]0G6.,%1!' "#

(ε2k −σ2; k = 1, 2, ..., N)Ám3!0G$ >VMT%*ME"# $+DV+%(E+ I +E!3$: ∑n

1 (ε2k −σ2)!'

02%(3!!"# JE%(.46E' .#.4F.=<Y3!"#JK"# DL%(3[-Å-DV![W] "#.j)*"4!¬' /!' "#(γn)

D 3$+>v"4!"#Q .4:+.#.4/\-'

k=n∑

k=1

(ε2k − σ2) =k=n∑

k=1

γkεk ∀n ∈ 1, 2, ..., N

Ã .+3$!' .#Â\E' ∀n ≥ 1 , ε2n − σ2 = γnεn D ' "4\E'

σ2 > 0

I +.,%Z+-!3X%4S<T%(QP3εn(ω) 6= 0 ∀ω

S<T%(' !3:DL%(3!*D ' "4\E' ε2n−σ2

εn

! Fn−1H=0G!' 3X%(Q .4G%

ε2n − σ2

εn=

[ε2n − σ2

εn|Fn−1] =

[ε2n − σ2

εn] =

[ε21 − σ2

ε1]

7 D DV+.,%(EC

.,% I !X%(E¸JK' 3X%(EG%(' +3! "4+30G+0Q 3Â I +! I [L%("#;S<Y$+JE%(.#"#$Â·>?"#\-'

ε2n − Cεn − σ2 = 0 ∀n ≥ 1M Ã .·+3$!' .#6\-'

εnD 3+8>K%(.4+' 3 %( .=<Y++0Q .4FÎ

DV"#E I !!"#!'$:DL%(3.43X% I "#O*'!3!"# 0Gx2 −Cx− σ2 M

"! ( 8$ , ,>:0]0G+ DL%(3"#E!3v*' "#3/ +'*)Z $¸ "#!"4"#0]DV3!X%(E«

$g%,d %§fPg' 219v"#

(Xn ; n ∈ 0, 1, 2, ..., N) ' !' "#O R>PMT%*ME+ T ' >VMT%*M?%¬>%(.4+' 3Á%( 0, 1, 2, ..., N ?2+3X%XT

.,%;Ì¯K¥¡m¯B°4´1¯° Ë ¯K =¡§¥!´ $¸ "46DL%(3XT (ω) = XT (ω)(ω)

*Ñ

'+ I 3

XT =n=N∑

n=0

Xn1T=n

+(XT

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

XTn = Xn1n<T +XT 1n≥T

'+ I 3XT

n = Xn∧T

Í *"#3X%1\-':.,%1!' "#XT !R.,%]!' "#

X¯¥¥ + Ë ´]¯ T M

@%(3!0]"S.4!3X%($+JK"4R\E' "VU<Y+V3+EF/!3Ç!'+' 3*';DL%(3X%(JK3X%(D [ " " . " -"#.S+Â!' 1v%(D DV+.4$¬!!3X%($+JK"4S<T%(3!3¹+ \-' ">K%1'O"#E$+3+3&02%("#E+L%(- +.#.4 I !"4!F]0]"4+36ÑÏÇ+< '\-'j<TF1' I +3!X%("#+0]D

T»m%(.4$U%H

"#3e¼D ' "4eq% II WO X%(J?%(D 3WO' ZJE%("#8'QV' V$6 $+D 3!"#>?%(D 3W ' /DV+3!KqFG\E' "#!+3 .4RÇ!+' Mq=%("4X%(EUv.=<Y$+>?.#' !"4

(Mn) &.,%:m3!!' :*'Ç'+' 3R!3+0]D .,% I $¬DL%(3.,%' "# (MT

n ) I +!¬ +3! "4W+3K I <Y!

I :\-':'O%(.#.4>?"#3O02%("#E+L%(EUqDV+' ¹+!3 I !"4 $+3$ I 0]0G/' :!3X%(m3!0G$/ (Mn)

M

c(fLhf! %d %§fqg%19v"4+E

((Mn) ; 0 ≤ n ≤ N)' /!' "#: :>VMT%*M +

T' ¬>VMT%GF>%(.4+' 3O%( 0, 1, ..., N ZDVK

Un = 1n≤T .,%!3X%(m3!0G$/

(Mn)DL%(3.,%1!' "#

(Un)!

(MTn ) Ë ÉG¢ ³ s¥!¯K s¡m¢ Ã .j!'43GO ¬>?$+3!"¸L+3U I :\E' "j!=% I "#.46%('Z>v'Z $¸ "#!"4UL\E'

∆(MT )n = UnMn ∀n ≥ 1

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

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%(' !3/DL%(3! \E'

T ≤ n ∈ Fn ∀n ⇔ T = n ∈ Fn ∀n »*M ?¼

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

$g%,d %§fPg 1L¢¤¯? ´°§°4´¥¯® µ´+ÉOq³¿À ¯K¥¥ + :¿E´2°¯¶° s¥!¯K s¡sK¢

(Fn)ºv¢P´2Ì¯¥¡m¯B°4´;¯° Ë ¯K =¡§¥!´$MÌ¯K°#´ºv¥³Â¿?¯¢ ³ 0, 1, 2, ...., N³+¯? 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/«

KÎ

)+6G685 < + = < ,+

9v"4+-(Ω, (Fn),

)' !DL% I ¸.#!3$O+ T ' 2+0]DS<T%(3!3+ "

(Mn)!'

(Fn)H=02%(3!!"# JE%(.4

»§3DjM ' /!' 3'' /X'H=02%(3!!"# JE%(.4e¼"#.+Z :0G¹+0G/ (MT

n )M

B" I "S<T%(' !3s3!0GR' !"#.4 *'![$3W+0G6S<T%(3!3¹+ %( I /\-' "j!' "#U (Mn) !"#J?+3X%1' :02%(3!!"# JE%(.4K

(Un)' /!' 3!02%(3!!"# JE%(.4Kq+

T' Z+0]DS<T%(3!3¹+UM

ÑKMMT∧n =

[MT |Fn] M0 =

[MT |F0]

Î*MUT∧n ≥

[UT |Fn]

[UT∧n|F0] ≥

[UT |F0]»*M )K¼

58E!3¬EDL%(3) +0]D .4E.4Á"#$+JE%(.#"#$ .,%/ +'*)*"4W+0GO.#"#JK D ' "4\-'(UT

n )!' !' 3!02%(3H

!"# JE%(.4UT

n ≥ [UT

N |Fn] I \E' "q0GE!3 .,%:D 3+0]"4W+3 "#$+JE%(.#"#$KP»s+.,%/ I 0GUÅ?+ L%(-'

I *"#!"4 +0G+-:!' D D .4$+0G+EX%("#33+.,%(!"#>?+0G+E¬% F0¼¬M

b-`h "5 #b-gPdUcEbG19v"#

(Xn)' !' "#%%(D $]FG' 6¸.#!3X%(!"4

(Fn) >PMT%*MVF]>K%(.4+' 3¬%(

E+¬"#

Γ' ®'&++06Q .4

E DPK

TΓ = infn ≤ N |Xn ∈ Γ»m%>? I .,% I E>?+-!"4 inf φ = N

¼ c(fLhf! %d %§fqg%1

TΓ!&%(.43R' Z+0]DS<T%(3!3¹+&\E'j<Y%(D DV+.#.4G µ´É L³¿À ´¢L s¥ Ë ´1¿E´6°#¯]³º-¡s µ´ (Xn)

¿?¯¢³/°!À ´¢ ³U´É B°#´Γ

Ë ÉG¢³ s¥!¯K s¡sK¢ T = 0 = X0 ∈ Γ ∈ F0 , T = N = X0 /∈ Γ;X1 /∈ Γ; ...;XN−1 /∈ Γ ∈ FN+RDP' 3

0 < n < N, T = n = X0 /∈ Γ; ...;Xn−1 /∈ Γ;Xn ∈ Γ ∈ FnÃ . Å1%¬QVU%(' I ' DS<T%(' !3)*+0]D .4 R+0]DÁS<T%(3!3¹+UE.4D .#'Á"#0]D .4ÁS<Y+E!3 +'*)1$+X%(-Á.4Á+0]DÁS<T%(3!3¹+I !X%(-UM Í DV' 3!3X%]S<T%(' !3/DL%(3!O>?$+3!"¸L+3 \-'¬.,%1QP3!/"#*s$+3!"4+' 3K .,%]QP3!/' DP$+3!"4+' 3/+O.,%10]0G/ Î+0]DS<T%(3!3¹+O- +0]DS<T%(3!3¹+ %("#!"=

T ∧ 1+T + 1

E +0]DRS<T%(3!3¹+ 3$L$ I [ "#3X%1' 0]"#E' %('1m%("#\-'

T−1j<Y!DL%$ I X%("#3+0G+EÁ' ]+0]DS<T%(3!3¹+O«' ¬Ç!'+' 3DP+' $ I "4 +3Á Re<T%(3!3¹++3

:Ç!'+3%(D 3W.4]D 3+0]"4+3 I ' D\-' "Á.#' "%Âm%("#DP+3*32D .#' ÑUKKÏÔ02%("4"#..#' "602%(.#[+' 3+'+0G+-"#0]DVK!"#Q .4: :.4&=%("#36F.=< "#!X%(-O"#0]0G$*",%(+0G+E&D 3$ I $ +EUM

)+6G6 I < ,8? - < - + = < ,+ c(fLhf! %d %§fqg%1

9v"#((Xn) ; 1 ≤ n ≤ N)

' !' "#>? I 3!"4+.#.4®%%(D $+.#.4\E'DP' 3G' !' "#ZD 3$+>v"4!"#Q .4>? I 3!"4+.#.4 (Un)

%("# [(U ∗X)N ] =

[< U0, X0 >]

(Xn)

!O%(.43O' ¬02%(3!!"# JE%(.4KM

Ë ÉG¢ ³+ s¥¯? s¡s¢ Í 3X%(0GW+/%("4$+0G+EO%(' I %' "4*"#0G+!"4 +.=ML9-"T' Â+0]DRS<T%(3!3¹+U*.,%

' "#Un = 1n≤T

!RD 3$+>-"4"#Q .4K+(U ∗X)N = XT

%] I

[XT ] =

[X0] ∀T9-"A

!R' $+>?$++0G+- Fn Z>?"#R=% I "#.4+0G+E&\E'¬.,%>K%(3!",%(Q .46%(.4$U%("#3

T = n1A + (n+ 1)1Ac

!O' Z+0]D S<T%(3!3¹+ULDP' 3 .4\E'+.XT = Xn1A +Xn+11Ac

ZDV+' I $ I 3!"#3

[X0] =

[Xn1A +Xn+1(1 − 1A)] =

[(Xn −Xn+1)1A] +

[X0] .

Í % I [Xn1A] =

[Xn+11A] ∀A ∈ Fn .,% I %(3X% I $+3!"4X%(!"4 » "" ¼:+E!3X%4%(.43

.=<Y$+JE%(.#"#$Xn =

[Xn+1|Fn]

I /\-' "S+3!0]"#:.,%] $+0G!!3X%(!"4jM

",<76 8 ( $, ,>:?< " :5 $=8)+6685 : I <>A9?G? < A A -IG< C- J+ < -?+##<9v"#

(Zn 0 ≤ n ≤ N)' 1!' "#1 >PMT%*M 3UMS%%(D $]F' 6¸.#!3X%(!"4

(Fn)M H $¸ "4X&DL%(3¬3$ I ' 3!3+ I

I +%(-/' /!' "# (Un 0 ≤ n ≤ N)DL%(3.4$+JE%(.#"#$

UN = ZN Un = sup(Zn,

[Un+1|Fn]) (0 ≤ n < N).»*M ¼

Ã . I .,%("#3:\-' (Un)!¬' 1!' "#]%%(D $ S<T%(' !31DL%(3!UV.4¬"#$+JE%(.#"#$

Un ≥ Zn+Un ≥

[Un+1|Fn]3$!' .#X%(- (24)

*0GE!3+-&\-'(Un)

!R' 6!' 3!02%(3!!"# JE%(.4/02%Ç!3X%(E(Zn)

MÑKM= cfqhf %d %§fqg 1

(Un)?\-' "L!%(D DV+.4$´¢LÌ(´°#? ´6¿E´ L¢P´+°°V

(Zn)?!Á.,%:D .#'DV+!"#O' 3!02%(3!!"# JE%(.4

02%Ç!3X%(E(Zn) Ë ÉG¢ ³+ s¥¯? s¡s¢ V+.#.4¬ 0GW+ DL%(33$ I ' 3!3+ I / I +%(E "# (Vn)

' &' 3!02%(3!!"# JE%(.402%Ç!3X%(E

(Zn) %

VN ≥ ZN = UN S<T%(' !3/DL%(3!.=< [EÅvDP![WX/ :3$ I ' 3!3+ I 6\-'¬.=<Y

$ I 3!"#3X% Vn+1 ≥ Un+1+E!3X%("#6.,%]!' "#:S< "#$+JE%(.#"#$

Vn ≥ [Vn+1|Fn] ≥

[Un+1|Fn] ≥ Un

S<YÈ.4:3$!' .#X%(6MÎ*M= cfqhf %d %§fqg 1 @AK

T0 = infn ≥ 0 | Un = ZnT0

!6' +0]D6S<T%(3!3¹++(UT0

n )/'

02%(3!!"# JE%(.4M Ë ÉG¢³ s¥!¯K s¡sK¢

T0!1' ¤+0]DGS<T%(3!3¹+ I %(3 I <Y!].4;+0]DGS<Y+E!3$S< ' !' "#

%%(D $»s\-'6' .,%("4:%('.4 I +' 3&.4"#8S<Y)vD .#" I "#+3¼P%(O' ®'&++06Q .41 R

M H <T%(' !3 DL%(3!&(UT0

n )!Á.,%!' "#O!3X%(s3!0G$& .,%6!' "#

(Un)DL%(3.,%6' "#D 3$+>v"4!"#Q .4

(φn) $¸ "4GDL%(3

φn = 1n≤T0M Í Ø% I ∆UT0

n = φn∆UnD ' "4

[∆UT0n |Fn−1] =

[1n≤T0∆Un|Fn−1] 02%("4!' 3.=<Y$+>?$++0G+E n ≤ T0 = n − 1 < T0

Un−1

!!3!" I +0G+E6!' DV$+3!"4+' 36F

Zn−1DL%(3¬ $¸ "#!"4

T0V ]3!1\E'

Un−1 =

[Un|Fn−1]

8%1 I

[∆UT0

n |Fn−1] =

[1n≤T0(Un − [Un|Fn−1])|Fn−1] = 1n≤T0

[(Un −

[Un|Fn−1])|Fn−1]

= 0

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

(

)+66 <>=@? C ,+(9?+8= -;1 +D 3+.4RX%(!"4 *'ZDL%(3X%(JK3X%(D [/D 3$ I $ +-MÑKM $g%d %§fqg 1 Í *"#¬\-'j< ' +0]D&S<T%(3!3¹+

T! D !"#02%(.!"

[ZT |F0] ≥

[ZS |F0]\E'+.4\-'6"# .4

+0]D S<T%(3!3¹+SM

:D .#'' >?+-*.=<T%(.#J?W+Q 3 F0R!3!"#>-",%(.4G "SQ "4+\E'¬.=< "#$+JE%(.#"#$6D 3$ I $ +E¬3$*' "#&F

[ZT ] ≥

[ZS ] \-'L%(.,%1!' "#2»

Zn¼3+D 3$+-/.=<Y$+>?.#' !"48 :.,%s3!!' 6S< ' ]Ç'+' 3*.,%

"#JK "¸ I %(!"4 :.=<YD !"#02%(.#"#$!R' F=%("# I .,%("#3GMÎ*M= cfqhf %d %§fqg@1 O;+0]DOS<T%(3!3¹+

T!OD !"#02%(.!"j+X+' .4+0G+E e< "#.S>?$+3!"¸L/.4OÎ I *"#!"4«

a) ZT = UT b) (UTn ) est une martingale

Ë ÉG¢ ³+ s¥¯? s¡s¢ «·9v"#T

' +0]DS<T%(3!3¹+>?+3!"¸q%(-a) et b)

DP+' 6$ I 3!"#3j\-'+.4\E'X"#.4¬+0]DOS<T%(3!3¹+

Sj» I O» 2.7 ¼!¼

[ZT |F0] =

[UT |F0] = U0 ≥

[US |F0] ≥

[ZS |F0] ,

I 6\E' "j0G-!36\-'T!OD !"#02%(.ÁM>1O+02%(3\-' %('ZDL%%(J?:\E'

T0!D !"#02%(.AS<T%(D 3W

.,%]D 3DPK!"#!"48Î1*'ZDL%(3X%(JK3X%(D [/D 3$ I $ +- %1 I [ZT |F0] =

[ZT0|F0].1 $ I "#D 3-\-'+0G+EY!"

T' +0]DOS<T%(3!3¹+ D !"#02%(. ZDV+' $ I 3!"#3

[UT |F0] ≥

[ZT |F0] =

[ZT0

|F0] = U0 ≥ [UT |F0]

Í 8%1 I [ZT ] =

[UT ]+ZT ≤ UT

I /\E' "j+-!3X%("# ZT = UTM

@%&%('ZDV"#E(b)

«%

[UT |F0] =

[ZT |F0] = U0 ≥

[UT∧n|F0] ≥

[UT |F0] =

[UT |Fn|F0] .

Ã .+3$!' .#2\E'].4/>PMT%*M 3UMUT∧n

+ [UT |Fn]

E/0G¹+0G!DP$+3X%( I I 0]0GG+.#.46-I 0]DL%(3X%(Q .4 +.#.4OXEO$+JE%(.4/ I :\E' "S+3!0]"#6.,%1 $+0G!!3X%(!"4M

*M= fqcf<//_ %§c(b 1T0

D !"#02%(.·+R' +0]D S<T%(3!3¹+OD !"#02%(.! ≥ T0M

'%>?+S+· $~ÇFR$\-'T0

$+X%("#D !"#02%(. .4Á3!D 3U>v"4+E I \E' e ÁDL%(3AX%O $¸ "#!"4T0

!R.,%1D .#'DP+!"#:>VMT%*M*>?$+3!"¸q%(E1»a¼¬M

M Âb-` _ c #·b 17H $!"#JKDL%(3 Tn.,%=%(0]"#.#.4/ R+0]DOS<T%(3!3¹+ ≥ n

+RDPK

Dn = infp ≥ n |Xp = Up;8%]%(.43

[ZDn |Fn] ≥

[ZT |Fn] ∀T ∈ Tn. 'R.,%("4XR.4:.4 I +' 33$L$ I [ "#3&F' / $+0G!!3X%(!"4M

)+66) 0 =@?BAK = <I+ C9v"4+E

(Zn)' :!' "#:%%(D $/+

(Un)+E>?+.4D DV: 69-+.#. Tn

+Dn

%(' 3-.,%!"#JK "¸ I %(!"4 $%(R.,%13+02%(3\-' I "Hµ '/M B" I "·Î]%(' !3m% I %(. I ' .4+3 .=<Y+E>?+.4D DV6 69-+.#. (Zn)

M

KÙ

c(fLhf! %d %§fqg

Í %.4$+JE%(.#"#$ !' "#>K%(E6«

(a) Un =∨

T∈Tn

[ZT |Fn] ; (b) Un =

[UDn |Fn] =

[ZDn |Fn]

Ë ÉG¢³ s¥!¯K s¡sK¢ 7OD DV+.4Xn

.,%>PMT%*Mv¸JK' 3X%(-&%('X I 0G+06Q 3 ¬.=<$+JE%(.#"#$2»m%?¼MÄ 9v"4+E

T1 et T2 ∈ Tn 0G-!3\-'j< "#.)v"41' !3"4!"4W+0G]+0]D/S<T%(3!3¹+

T ∈ Tn+.

\E'

[ZT |Fn] =

[ZT1|Fn] ∨ [ZT2

|Fn]@·KXDP' 3 I +.,% A = [ZT1

|Fn] ≥ [ZT2

|Fn] et T = T11A+T21Ac

'.,%("4X%('.4 I +' 3.4/"#Z /0GE!3+3 \-'

T>?$+3!"¸L:.,%1D 3D 3!"4$+$ I [+3 I [$ÂM

Ä 0]0G6.=<Y+X+06Q .4 Tn!O¸ "L®+® $*' "#:%("4X$+0G+E¬DL%(3 3$ I ' 3!3+ I 1\-'j< "#.A)*"4!6'

UMT%*MRn ∈ Tn

>?$+3!"¸q%(EXn =

[ZRn |Fn]

MÄ @' "4\-'.,% I EX%(- n ∈ Tn

Xn ≥ Zn

M H <T%(' !3DL%(3!6% [Xn+1|Fn] =

[ZRn+1

|Fn] ≤Xn

I %(3 Rn+1 ∈ TnM(Xn)

!%("#!"q' O!' 3!02%(3!!"# JE%(.4&02%Ç!3X%(E(Zn)

+ I m%("#≥ (Un)

MÄ @·' 3% I [+>?+3 0GE!3+3\E'

(Xn) = (Un);\E'Á\-'+.4\E'"#

T ∈ Tn

[ZT |Fn] ≤

[UT |Fn] ≤ Un ' DL%%(J?ÂFZ.,%ZQV3!' DP$+3!"4+' 3Â\E'L%(

T $ I 3!"# TnD 3' >?6%(.43R.=< "#$+JE%(.#"#$

Xn ≤ UnM

Ä @A%XZ%('*)Ô+JE%(.#"#$»§Qq¼ .,% 3+02%(3\-'®=%("#%(' DL%(3X%(JK3X%(D [D 3$ I $ +EZ0G-!3\-'Xn =

[ZDn |Fn]

:D .#'/ ¬DL%(3O.,%] $¸ "#!"4 Dn

ZDn = XDn

I /\E' "S+3!0]"#.,%1 $+0G!!3X%(!"4M

)+66 : I <>A9?G? < C < IG<>A8A <,+ I <>C G< 9v"#

((Xn) ; 0 ≤ n ≤ N)' I [L%("# 58%(3?e>F1>K%(.4+' 3 %(O' +DL% I 6S<Y$+X%(O¸ " E + %*0G+!X%(-

PDV' 302%(!3!" I !3X%(!"#!"4» I . "" ¼RM 9v"# f : N×R 7→ R

' m I !"4ÂE' 0G$+3!"4\-'&' 3E

$+DP+%(-*'Â+0]D '%(.#.4'"#-$+3+3F/.,%!' "#:%%(D $&3$+.#.4]»

Zn¼Á $¸ "4&DL%(3

Zn = f(n,Xn)+F

+E>?+.4D DV1 G9-+.#.(Un)

M '&%(.#.4&>?"#3:\-' I +.#.4 I "A¬%('"·' 6s I !"4*' I ' D .4 (n,Xn) D .#'

D 3$ I "4$+0G+-«

c(fLhf! %d %§fqgH $¸ "4RDL%(3O3$ I ' 3!3+ I 6 I +%(-/' ¬m I !"4 u : N ×E 7→ R

F.=<T%("4 6 O$+JE%(.#"#$

u(N, i) = f(N, i) ∀i ∈ E

u(n, i) = f(n, i) ∨ Pu(n+ 1, .)(i) (=∑

j∈E

piju(n+ 1, j)) ∀i ∈ E ∀n < N

(Un)!O%(.43O$+JE%(.4F

(u(n,Xn))M

Ë ÉG¢ ³+ s¥¯? s¡s¢«EN.#.4 XR0GW+ODL%(33$ I ' 3!3+ I ¬ I +%(E !"L'!' D DVK>v3X%("4.=<Y$+JE%(.#"#$

Un+1 = u(n+ 1, Xn+1) ,

8%(' 3X% +Z>?+3!!' ¬.,%1D 3D 3!"4$+$6 /58%(3?e> [Un+1|Fn] = Pu(n+ 1, .)(Xn)

LS<YÈ

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

)&>< " <!54:5A< ( < <=6 ( 8. !>.0, $ ,:5 $=8 )+6685 >=@<9v"#»

Un¼' /!' 3!02%(3!!"# JE%(.4 "#.)*"4!¬' :02%(3!!"# JE%(.4»

Mn¼+O' :!' "# I 3"4X%(-:D 3$+>v"4!"#Q .4

»!»An) , n ∈ 0, 1, ..., N ¼!¼+.#.4O\E' A0 = 0

+

(Un) = (Mn) − (An)»*MTÚ?¼

¬+.#.4/ $ I 0]DPK"#!"4»%(D DV+.4$2¿ Ë!¨ KÉO (³¡s s¡sK¢ ¿E´ UB¼R!R' "4\-']M Ë ÉG¢ ³+ s¥¯? s¡s¢ @A' 32$+X%(Q .#"#3G.=< ' " I "#$ 0GE!32\-';.,%® $ I 0]DVK!"#!"4 (2.9)

$++3!0]"#I 0]D .4W++0G+-»

An¼]Á»s+DL%(3 I $\-'+E»

Mn¼6¼ F;.=<Y3!"#JK"#8j' /=% I "#.4«

A0 = 0+

M0 = U0(!"EQ "4+1\E'j< "#.*!'43G $++3!0]"#+3.4% II 3"4+0G+- &»

An¼ 3+ I *"#!"4 L%(E

.4 Î10G+06Q 3 O$+JE%(.#"#$∆Un = ∆Mn − ∆An

DL%(3 Fn−1 Q !"4+E DP' 3

n ≥ 1

∆An = − [∆Un|Fn−1]»*M#ÑU?¼

S<YÈ .=< ' " I "#$ .=<Y)*"4!+ I D 3' >?+ $¸ "4X%(-(An)

DL%(3.,% m3!06' .4(2.10)

.,%!' "#%("#" I !!3!' "#!02%( "m!+0G+E¬D 3$+>v"4!"#Q .4 :D .#'

∆An! ≥ 0

D ' "4\E'(Un)

!' ' 3!02%(3!!"# JE%(.4

(An)1 I ' !' "# I 3"4%(E®M Ã .'13!F0GE!3+32\-'2.,%8!' "#

(Mn) $¸ "46DL%(3

Mn = Un +AnR' :02%(3!!"# JE%(.4 $ I 3!"#>?ODP' 3 I +.,%

∆Mn = ∆Un − [∆Un|Fn−1]

+ I *"#!"4 .4Î0G+06Q 3«v"#.S>-"4+- [∆Mn|Fn−1] = 0

&M MYÏM H M

)+66 I < - ; >=@< + <>=@? C ,+!9?+# = A1O+D 3+.4X%(!"4 " +S I !'.,%/ $ I 0]DVK!"#!"4 H -Q8

Un = Mn −An O.=<Y+E>?+.4D DV

9-+.#. (Zn)

DVK+!' "#

T1 = infn | An+1>01AN >0 +N1AN =0

Í Z>?$+3!"¸L6%("4X$+0G+E*JK3 I F.,%1D 3$+>-"4"#Q "#.#"#$/ G»An

¼& \-'T1

!R' +0]DOS<T%(3!3¹+ :D .#'6

c(fLhf! %d %§fqg%1

T1!D !"#02%(. "#.jj<Y)*"4!¬DL%OS<T%(' !3/+0]DOS<T%(3!3¹+OD !"#02%(. ≥ T1

M Ë ÉG¢ ³+ s¥¯? s¡s¢¤« ' %(.#.4 ' !"#.#"4+3&.,% I %(3X% I $+3!"4X%(!"4 "" MTÎG O+0]D&S<T%(3!3¹+&D !"#02%('*)MÑKM

(UT1n )

´³ ºv¢P´:ÉG¯¥+ s¡§¢-©?¯°4´«*DL%(3O $¸ "#!"4 T1

%.4R"#0]D .#" I %(!"4

n ≤ T1 =⇒ An = 0 =⇒ Un = Mn .

Ã .+3$!' .#\-'Un∧T1

= Mn∧T1∀n I \-' "·U<Y$ I 3!"#¬+ I 3 (UT1

n ) = (MT1n )

PS<YÈ.43$' .#X%(6M

)

Î*MUT1

= ZT1

« Í %UT1

=∑N

0 Un1T1=n =∑N−1

0 Un1T1=n + ZN1T1=NMV@.,%

'DP' 3n ≤ N − 1

!' 3].=<Y$+>?$++0G+- T1 = n !' 3G.4\-'+.An = 0

»s+2 IUn = Mn

¼&An+1 > 0

+OÈ.=<YZDV+' O I $ I 3!"#3

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

5®%("4 I 0]0GUn −An+1 < Un

%1$ I X%("#3+0G+-Un = Zn

"#.+3$!' .#6\E'

UT1=

N∑

0

Zn1T1=n = ZT1.

*M Ã .S'3/F0GE!3+3O.4 I %(3X% I W+3/02%)*"#02%(. T1

«L9-"S' +0]DOS<T%(3!3¹++.\-'

S ≥ T1 et

[S > T1] > 0 %(' 3X%

[US ] =

[MS −AS] <

[MS ] =

[M0] =

[U0]

" Q "4+G\E'(US

n )DP+' DL%¹+!3' 02%(3!!"# JE%(.4

Sj<Y! I DL%ÁD !"#02%(.?&M MYÏM H M


5 &:5A< $=5 , Ä '%(D DP+.#.4+3¬ÉG¯¥ ¨Ê*Ë ¶¢q¯¢ ¨ ¡m´+¥:' Â!DL% I ¸.#!3$O¸ " (Ω, (Fn),

; n = 0, 1, 2, · · · , N)

%(!"4m%("H%(E %('*);[EÅvDP![WX!' "#>K%(E6«

F0 = Ω, φ ; FN = F = P(Ω) ;

(ω) > 0 ∀ω ∈ Ω

N+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:$

0, 1, 2, · · · , d .=<T% I !" 0Ç'+3X%6' 3.4 DL%(3!!" I ' .#"4+3 "#.L3+D 3$X+E+3X%1+;V+' ®¯ ¨ s¡ DO³+¯¢³ ¥¡,³ +º ´G»§DL%(3

) +0]D .4Â' 8.#"#>-3+& I %("4X6S<Y$+DL%(3!JK1'' QP8*'!3$X3¬%GX%('*)Z¸ )*e¼ .4d% I !"m ' "#>%(-U

\-'j<YDV+' O&¸JK' 3+3 I 0]0G/ &% I !"4U+3- %(D DV+.4$6¯ ¨ s¡ DX³:¥¡,³ +º Ë ³MÄ ·´³q¥¡¾Âº ¨ º-¥³¿E´³¯ ¨ s¡ DX³F.=< "#!X%(- n +3-3+D 3$+E$ DL%(3

d+ 1>K%(3!",%(Q .4 %(.4$U%("#33$+.#.4

DVK!"#!"#>?/»§.4 I ' 3RS< ' /% I !"4Zj<Y·Ç%(02%("4$+JE%(!"Á¼S0

n, S1n, · · · , Sd

n

%%(D $&F;» Fn¼ .4&>? I +' 3

%(.4$U%("#3F1>%(.4+' 3O%(R

d+1+

%("#!"j $¸ "A+3X%$Sn

MÄ ·´/ =¯Kº¾;¿À ¡¢q Ë ¥ X+3X%' 3$+.

r > 0J?' >?+3!L%(- .4¬D 3!") ¬.=<T% I !"%(R3!"4\-'« Z' D DPK+3X%1%(

' /.,%1!' "#S0

0 = 1 ZDVK+3X%1 I S0

n = (1 + r)n MÄ ¨ sº ¯° ¡³+¯? s¡s¢¤« %2>%(.4+' 3& /.,%20G L%("4 $+D 3$ I "4%(' I ' 3 *'8+0]D DP' 3 DP' >?"#3 I 0]DL%(3+3.4&D 3!");S<*% I !"mQ+3!>?$F R%(R*" V$+3+-/*;' !"#.#"4¬' >?+ER.,%>K%(.4+' 3\E'j<T%U>%("#.,%0G L%("4F.=< "#!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+

rL"#-!3-*' "#3X%2' ¨ e´ ¨ ¡=´¢L ¿À ¯ ¨ sº ¯°²¡,³¯K s¡m¢

βn = 1S0

n= 1

(1+r)n

ML9-"Pn

R.4¬D 3!")S< ' 8% I !"Q+3!>?$F1.=< "#!X%(E n X;q¥¡¾Â¯ ¨ sº ¯°²¡,³ Ë X+3X%*DL%(3 $¸ "#!"4×

Pn = βnPn»§.4]D 3!")% I !'L%(.#"4$Z I %(. I ' .4$%(:' ]0G L%("4]D .#'¬m3!ZS!:"#*m$+3!"4+' 3

%('ÂD 3!") I ' 3X%(-¼&MEB" I "V' :%(' !3 m% & 3+D 3$+E+3.4 I v 3 I "4+ES<T% I !'L%(.#"4X%(!"4«βn

!.,%\-'L%(E!"#$2S<T%(3!J?+E6\-'j< "#.Am%(' :"#->?!!"#3%.=< "#!X%(E%(/.=<T% I !"X%(¬3!"4X\E'DV' 3/*"4!DPKX+3¬ ÑÏ %.=< "#X%(E

n Z+3X%]\-'

S0n = 1 ∀n M

Ä P s¥¯K Ë ©¡m´³¿-´Z©E´³+ s¡sK¢ ´+ & ¥ µ´ D´ºv¡§°°4´³×« Ã 02%(JK"#' "#->?!!"4+' 3DVK$%(-Z' DP3!m+' "#.#.4 F.=< "#X%(E

n I 0]DV3!X%(E]' Â\E'L%(-!"#$φk

n

.=<T% I !" k %('>v' "#*m3!02%(!"4] E1"#.*"4DPK;FI +R"#X%(E6»s$+!' 6 ¬.=<Y+>?.#' !"4 I ' 3 ¼&*"#.>% $ I "4 +3 ¬0Gv*"¸L+3OX% I 0]DVK!"#!"4 M 5®%("4R"#.'&=%(' :D 3$ I "4+3 I ]\-'.=<Y+E+)% I +0G+E/DL%(3:.,F*M9-' D DVK:\-'

Sn3+D 3$+-].4 I ' 3

¬% I !"m&F2.,% I .(!' 31 .,%G$U%( I QP' 3"4W+3 .,%G%(nM % I 0]DVK!"#!"4®*'DV3!m+' "#.#.4

φnF

KÚ

I +6"#!X%(E1%;$+$2 $ I "4 $2%(.43\E' I I ' 3/j<Y+X%("#6DL% I E'×«j.,%+' .4]"#*m3!02%(!"4*"4!DV "#Q .4$+X%("#¬.=< [ "4!3!"4\-'1 I ' 3ÁÇ'\-'j<T%G.,%Â%(

n − 1M Í 8X63+!3' >?] I %(&.,%0G¹+0G"#!'L%(!"4

\-'!3]>v"4+"#.Á%(0]"3+ I E!3$]L%(JK'W+3» I " " .*¼ %(:' I %!"#ZF;\-' ".=<Y %>K%("#:"#0]DPKX$]\E'.,%' "#&s3!0G$:DL%(3 0]"4!' II X!"#>?O"#RD 3$+>v"4!"#Q .4GML+.,% I *' "# F1.,%1 $¸ "#!"4' "#>%(-G« 'O%(D DV+.#.4+3:³+ s¥¯K Ë ©¡m´1¿-´ ©-´³ s¡m¢®'+ I 3&*¥+ µ´'DU´ºv¡°§°#´' :' "#&q¥ Ë ÌU¡³¡KB°#´(φn , n ∈ 1, ..., N ¼ O>K%(3!",%(Q .4%(.4$U%("#3OF6>%(.4+' 3R%( Rd+1 M Í I 0]D .4$++3X%1.,% $¸ "#!"4Z φF.,%]%(

0+DPKX%(-

φ0 = φ1Í 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<T% I !"m¼DV+3!0G+!+E¬S<T%>?"#3¬ O +! .#"#QV+.#.4$O+!"#!3¬%('Z.#"4+'8 /.=<Y¹+!36+E' 0G$+3X%("#3 I 0]0G/.4EU?Ç.=< "#02%(JK"#KL.4R> !3UM

Ä ¯K°#´ºv¥¿À º-¢*K¥ µ´'DU´ºv¡°§°4´Ó« H $+X%(EZDL%(3<,>

.48D 3-*' "# I %(.,%("#3®''+.¬' 3R

d+1 .,%>%(.4+' 3F.=< "#X%(E

nS< ' ZDV3!s+' "#.#.4% I "4$/%.,%]!!3X%($+JK"4 (φn)

+3X%**DL%(3O $¸ "#!"4j

Vn(φ) =< φn, Sn >=

k=d∑

k=0

φknS

kn

»§M#Ñe¼

+!1 $¸ "#!"4 XDL%X I 0]0G+-X%("#3 ®+3X%Â' m"4:\E'(Vn(φ)) I !"#!'1' ' "#

%%(D $6 :>PMT%*M 3UMSM %;Ì¯°4´ºv¥¯ ¨ sº ¯°²¡,³ Ë ´ I :DV3!s+' "#.#.46+3X%1 I

Vn(φ) = βnVn(φ) = βn < φn, Sn >=< φn, Sn > .

43 , $;: ) 5 !*$. :< 0$ 0&')!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»

φn¼>?$+3!"¸q%(E

< φn, Sn >=< φn+1, Sn > ∀n < N»§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%(n+R.,%

%(n+1

» I <Y!F/*"#33+0]D .,% I %(E φnDL%(3

φn+1¼-+=%("4X%(EÁ+Â3!\-' I R3$U%(0G$+L%(J?+0G+- I +3!>?R.,%

>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<T%(3!J?+EO§3X%("4eM

6685 9?7 C K+#I %]!3X%($+JK"4

(φn)!O%(' (¸L%( I $!"j+O+' .4+0G+- !"= \-'+.4\E':X"#

n ≥ 1

Vn(φ) =< φ0, S0 > +

k=n∑

k=1

< φk,∆Sk > .»§M ?¼

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

Vn(φ) =< φ0, S0 > +k=n∑

k=1

< φk,∆Sk > .»§M -¼

Ë ÉG¢ ³+ s¥¯? s¡s¢ V@A' 3n ≥ 1

v%< φn,∆Sn >=< φn, Sn > − < φn, Sn−1 >

¬3!&\E'»§MTÎ?¼R!$\E' "#>K%(.4+E%('*);$+JE%(.#"#$

< φn,∆Sn >= ∆Vn(φ) I <Y!&F1*"#36F»§M ?¼¬M

E

Âb-`_ c #·b/1 Í ;DV+' R +3R' Om3!0G&D .#' I +$/% (4.3)+

(4.4) "#.V'43GDV' 3 I +.,%S<Y$++*3

.,% !"4] ' "#Á!3X%(m3!0G$%(' I %>? I 3!"4+.j«e3+>?+F .,% $¸ "#!"4 "" . " +]!' D DPK%(E02%("#E+L%(-\-'

(Mn)+

(Un)EOF]>%(.4+' 3%(O' Z!DL% I /N' I .#"4*"4+ .=<Y$+JE%(.#"#$

(U ∗M)n = U0M0 +

k=n∑

k=1

< Uk,∆Mk >

$¸ "#' /' "#]¥ Ë ´°°4´ I %(3X% I $+3!"4$/DL%(3O.4R3+.,%(!"4

(U ∗M)0 =< U0,M0 > ∆(U ∗M)n =< Un,∆Mn >

7·>? I:I X%(!"46 (4.3) + (4.4)e<Y$ I 3!"#>?+E3!DV I !"#>?+0G+E

Vn(φ) = (φ ∗ S)n Vn(φ) = (φ ∗ S)n»§MTÙ?¼

%D 3DPK"#!"4Ø' "#>%(-2>K%Z'0GE!3+3G\-'j< "#.!'43G 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

66 9?7 C K+#I

9v"4+EV0

' >VMT%*M 3UM F0H=0G!' 3X%(Q .4G»§"=MYKM*' I !X%(Ee¼+ ((φ1

n, φ2n, ..., φ

dn) ;n ∈ 0, 1, ..., N) ' ¬!' "#D 3$+>v"4!"#Q .42FÂ>K%(.4+' 3/%(

Rd "#.)*"4!]!!3X%($+JK"42%(' (¸L%( I $ φ = (φ0

n, φ1n, φ

2n, ..., φ

dn)

' "4\-']+.#.4G\-'V0(φ) = V0

M Ë ÉG¢ ³+ s¥¯? s¡s¢Ô«j9-"

u = (u0, u1, ..., ud) et v = (v0, v1, ..., vd) ∈ Rd+1 V':"#-!3-*' "#36.,%

X%(!"4

[u, v] =

k=d∑

k=1

ukvk =< u, v > −u0v0.»§M ?¼

9-"φ!' 6!3X%($+JK"46 /J?!"4X%(!"4=%("4X%(- %('*) I *"#!"4 6.=<Y$+ I $K.4O3+.,%(!"41»§MTÎ?¼

S<Y$¸ "4XX%(E .=<T%(' (¸L%( I +0G+E:E $\E' "#>K%(.4+E6%('*)$+JE%(.#"#$

< ∆φn, Sn−1 >= 0 ∀n ≥ 1

\-' "jU<Y$ I 3!"#>?+- + I 3∆φ0

n = −[∆φn, Sn−1]»§M )K¼

Í 8%(' 3X%]+Z' !3V0 = V0(φ) = φ0

0 + [φ0, S0]»§M ¼

'+ I 3Kφ0

0 = V0 − [φ0, S0]»§MTÚ?¼

$+JE%(.#"#$1»§M )K¼R+1»§MTÚ?¼0GE!3+-:\E':.,%G!' "#(φ0

n)! I 0]D .4W++0G+E/ $++3!0]"#$DL%(3O.4

' "#(φk

n , k ≥ 1) I ]\-' "0GE!3].=< ' " I "#$2

φMV@A' 3/0GE!3+3)v"4+ I KV3+>?+

%('*);$+JE%(.#"#$/»§M )K¼+:»§MTÚ?¼\-' "V'DV+3!0G+!+E $¸ "#3R' ¬!' "#(φ0

n)+' &!3X%($+JK"4&

J?!"4φ = (φ0

n, φ1n, ..., φ

dn)M Ã . I .,%("#36\E' φ !/%(' (¸L%( I $ "#.3!GFÂ0GE!3+3\E'

φ0nRD 3$+>-"4!"#Q .4K I /\E' "S3$' .#¬m% I "#.4+0G+- G»§M )K¼M

Ñ

fLc(f!//§_ %§cbG1 <Y++0Q .4 V VN (φ) = VN (φ)

φDL%(3 I ' 3X%(ER.=<Y++0Q .4: !!3X%($+JK"4R%(' (¸L%( I $!' Â'

!DL% I :>? I 3!"4+.· :.=<Y!DL% I :>? I 3!"4+. E R>PMT%*M 3' 3ΩM

Í 8%1+V+

VN (φ) = c+

n=N∑

n=1

< ψn,∆Sn >; c ∈ R , ψ ∈ Pd

\-'L%( Pd $!"#JK2.=<Y!DL% I >? I 3!"4+. !' "# G>VMT%*MjD 3$+>v"4!"#Q .4 (ψ1n), (ψ2

n), ...., (ψdn) F

>K%(.4+' 3O%(R

d

66) >+# + <>C EJ+# 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

V0(φ) = 0 , VN (φ) ≥ 0 ,

[VN (φ) > 0] > 0»§M#ÑU?¼

2+.#.4!!3X%($+JK"4ÂDV+3!0G+] I F' "#E>?!!"4X+' 3]X%(.4+0G+EG $+0' " %('+0]DG8 j<T%>?"#32F I ' D!Æ 36%(' I ' +!2F;.=< "#!X%(-

N' +JE%(JKL%(E ].=<T%(3!J?+E1%>? I ' GD 3QL%(Q "#.#"#$2!!" I +0G+-6DVK!"#!"#>?KM

H %(' 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 E+ $+.#"#¬S< "# "#!"4$ EU¼M <T%)*"40G] 6>v",%(Q "#.#"#$1\E'':%(.#.4¬02%("#E+L%(-"#E!3v*' "#3O I S< ' RJK3X%( R"#0]DV3!X%( I KM-&<Y!F I +! I *"#!"42\E'!3O!DL% I RD 3QL%(Q "#.#"4$R0' "S< ' /!' "#6 ¬>PMT%*M*DVK!"#!"#>?DV' 3!3X%1+3!>v"#3 :0Gv W+.46F' Z02%(3 I [$/¸L%( I "4+3UM

$ ,-&)! / 5 $7698 ! 6)685 FG,HEI J+KI

Í *"#3X%1\-'j< ' Z02%(3 I [$/¸L%( I "4+3 !:Ìe¡s¯B+°#Ú< "#.Sj<Y)*"4!:DL%%( I :02%(3 I [$ :!!3X%($+JK"4/S<T%(3!Q "#!3X%(J?2M

< "4 $\E'j< ' 02%(3 I [$K»s8j<YK6*"#36' I "4$+$e¼ m I !"4 3 I % I +0G+E/U< "#.jj<Y)*"4!DL% !!3X%($+JK"4S<T%(3!Q "#!3X%(J?OÇ':' 3.4¬s%(0G+-X%(.=M Ã .SÅZ%1DV' 3 I +.,%] 3X%("4X I [ "4\-'« !"j.4 I ' 3 S< ' % I !"4%6' &$+>?.#' !"4\-' "q3+ÂDVK!"#Q .4O' &!3X%($+JK"4&S<T%(3!Q "#!3X%(J?K +.#.4 >K%:=%("#3O.=<YQ*Ç!+RS<T% I [L%(02%"sU I ¬\E' "%(' 3X%:DV' 3S+ O=%("#30GE+3 I ' 3=%("4X%(E%("#!"qe<Y$+>%(' "#3.4Á!DV"#3 JE%("#X%(3!"4\-'O\-'j<Y+.#.4D 3 I ' 3X%("#UM Í 8DV+' & I %*0G+!!3 E+\E'j<Y+®3$+JK"#0G!X%(!"4 L%("#3 E;q' 02%(3 I [$]! >v",%(Q .4ÂM ¬\E'I &3+' 3F/.,%>v",%(Q "#.#"#&S<T%(' X%(-RD .#'3X%(D "4 &\-' .,%DP$ I ' .,%(!"4!D .#'"#0]DP3!X%(- I +.#.4 I "=v%(';.#"4+'S<T%(D DL%(3X%("#!3 I 0]0G2' 2[EÅ**3202%(.=%("4X%(-K»sX'/.,%+3!3+' 3UA.,%!DP$ I ' .,%(!"4!' 36.4JK3X%("# I *' "4X%("#.4FE%4S%(0G+' 3 EF2.,%.,%(E+3!e¼S +>v"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<Y%(D DV+.#.46X' >?+EO' JE+$ I 0]"4 I %!"##E I :j<Y!DL%.4:m%("#O JE+U<Y+ 3!" I [ "#3¬+ 3!02%(E E\-' "j"#*"#JKL%("#OL%(JK'W+36' ZD 3$!"4 +E¬ :.,%13$+D ' Q .#"4\-'2 I <Y! I +.#' " GU<Y+ 3!" I [ "#3X%(/3!"4\E'K I G\E'1'%(D DP+.46" I "' %(3!Q "#!3X%(J?M H %(6.,%;' >?+.#.4G0G3X%(.4KS¸.#.42 1.,%' >?+.#.4¬$ I 0]"4K*.,%/>?+3!!';.,%6D .#' I $+.4$+Q 3$¬DL%(3.40G$*",%j<Y!RS<T%("#.#.4+' 3D .#'.=<T%(3 +' 3R%('Â!3X%>K%("#.V02%("4.4¬J?Æ O*'Z3!"4X\E']M

EÎ

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

V0(φ) ≤ 0 ; VN (φ) ≥ 0 ;

[VN (φ) > 0] > 0.»§M#ÑKÑe¼

Î*M Ã .S)v"#&' +E!"4+3p ∈ 1, 2, . . . , N +' ;>? I +' 3O%(.4$U%("#3 Fp−1 −mesurable η = (η1, η2, . . . , ηd)F]>%(.4+' 3%(

Rd +.4\-'

[η,∆Sp ≥ 0] ; [

[η,∆Sp > 0]

> 0.»§M#ÑUÎ?¼

*M Ã .j)*"4!¬' /!!3X%($+JK"4/S<T%(3!Q "#!3X%(J?KMA»s7O' !3+0G+-&*"#R.4/02%(3 I [$:j<Y!ODL%R>-",%(Q .4KM¼M Ã .j)*"4!¬' /!!3X%($+JK"4/S<T%(3!Q "#!3X%(J?

φ>?$+3!"¸q%(E

Vn(φ) ≥ 0 ∀n ≤ N. Ë ÉG¢ ³+ s¥¯? s¡s¢ S<T%(QP3Z\-'4. =⇒ 3. =⇒ 1.

ZD 3 I W /+!' "# I 0]0G6!' "#/«»m%?¼

1. =⇒ 2.«L9v"#

φ = (φ0, ψ)' DP3!m+' "#.#.46%(' (¸L%( I $/>?$+3!"¸q%(-1»§M#ÑKÑe¼ DPK

u(n) =

[Vn(φ) ≥ 0] ; v(n) =

[Vn(φ) > 0] ; p = infn | u(n) = 1 et v(n) > 0Ã .L3$' .# [EÅvDV![W=%("#!' 3

φ\-'

1 ≤ p ≤ N S<T%(' !3 DL%(3!U ∀n = Vp(φ) − Vp−1(φ) ≥ Vp(φ)

I 6\E' "j+-!3X%4»§M#ÑUÎ?¼"#"=M 9-' D DVK 02%("#E+L%(-

u(p − 1) < 1 'DVK+3&%(.43

η = ψp1Vp−1(φ)<0M Í ZDV+'

%(.43 $ I 3!"#3

[η,∆Sp] = 1Vp−1<0 < φp,∆Sp >= 1Vp−1<0[Vp(φ)− Vp−1(φ)] ≥ 1Vp−1<0(

− Vp−1(φ))

I 6\E' "j+-!3X%46$+JE%(.4+0G+E1»§M#ÑUÎ?¼»§Qq¼

2. =⇒ 4.«9v"#

η' ¤>? I +' 3Â%(.4$U%("#3>?$+3!"¸q%(E»§M#ÑUÎ?¼ $¸ "42.,%!' "#D 3$+>v"4!"#Q .4Z

>? I +' 3&%(.4$U%("#3 (ψn)F>%(.4+' 3%(

Rd DL%(3

ψp = η et ψn = 0 pour n 6= p

+2%(D DV+.4(φn)

.4;DV3!m+' "#.#.4%(' (¸L%( I $ ;>K%(.4+' 3]"# "#!",%(.4E' .#.4% I "4$ZF(ψn)

DL%(3].,%D 3DVK!"#!"4 "" M Í %

Vm(φ) =∑

1≤n≤m

< φn,∆Sn >=∑

1≤n≤m

[ψn,∆Sn],

"Q "4+\-'Vm(φ)

OE' .·DP' 3m K%(' [η,∆Sp]

DP' 3m ≥ p

M 63$' .#X%( + $ I ' .4"#0]0G$*",%(+0G+-UM

! "# %$&(')"&+*-,".0/%)21"34.!$$.-56 /87

6)6 I < < C I - + >=@< G-",%(Q "#.#"#$S< ' 02%(3 I [$+/.,%![$3!"4 /02%(3!!"# JE%(.4U'%(' 3QPX"#S< ' Ø3$!' .#X%(]S<T%(L%(.#Å*K I -'¤'.420 8 Ê*Ë ¥ É2´¿-´/¯ Ê ¢¦ ¯¢q¯ ¨+Ê M ''QV3!+3/F+ $+0G-!3+3/' >?+3!"4 %4j%("#Q .#"4\-' "+3X%Â!'432X%(-1DP' 3¬.,%Â!' "# %( I DL%(3X%(JK3X%(D [K»E,< ., . >

¼ $"#JK+3X%1' !DL% I /+' I .#"4*"4+\E'+. I \-'GM

: Efqc v`b9v"4+E

V' X'/!DL% I ]>? I 3!"4+.+ K ' GDL%(3!!"4 I E>?) I 0]DL% I

E]3+ I -!3X%(EDL% V "#.

)*"4!/%(.43' >? I +' 3 a E

>?$+3!"¸q%(E

< a, x >> 0 ∀x ∈ K < a, y >= 0 ∀y ∈ V

Ë ÉG¢ ³+ s¥¯? s¡s¢« A' S<T%(QV31\E'V('!DL% I S< ' ]!DL% I Á>? I 3!"4+. *"#0G+!"4

¸ "4K(!Sm+3!0G$OMe@·KC = K+V = x+y | x ∈ K ; y ∈ V

'js+3·.,%R $+0G!!3X%(!"4+8Î1$+X%(DPUMÑKMC

!s+3!0G$K I E>?) /+ I E!"4+- DL% 0 'R'QV3!+3OF10G-!3+3O.4¬D 3+0]"4+3

DV"#EU.42%(' !32$+X%(-G=% I "#.4 "# zn = xn + yn (xn ∈ K ; yn ∈ V )' Z!' "#

I ->?+3!J?+E; 1DV"#E6 C "#.ÁU<T%(JK"#/ ]0G-!3+3\-'].,%Â.#"#0]"#

z ].,%!' "#

(zn)!

%(C D ' "4\-'

K! I 0]DL% I Uj"#.)v"4!1' ]X':!' "#Z»

xnk

¼O (xn)

\-' " I E>?+3!J?>?+3O' Z>? I +' 3 x

Kynk

= znk− xnk

I E>?+3!J?1%(.43>?+3 z − x.4\E'+.·%(D DL%(3!!"4+-

%VD ' "4\-'

V!jm+3!0G$ "#.-3!\E'j<TF $ I 3!"#3 z = x+(z−x) DV' 3·D 3' >?+3\-' z ∈ C

M

Î*M H <T%(D 3W1' Ø3$!' .#X%( I .,%"4\E';!' 3.4]!DL% I ]+' I .#"4 +UA"#.R)v"4Â%(C

' ØDV"#Ea' "4\-' 63!0G0]"# "#02%(.4KM9v"#

x' 8>? I +' 3&\-'+. I \E'1 C +

θ ∈ ]0, 1] D ' "4\-'

C I ->?)*K a+ θ(x− a) ∈ C

v :3!¬\E' ‖a‖2 ≤ ‖a+ θ(x− a)‖2 $+>?+.4D DL%(EO.4

X I 0G+06Q 3 Q !"4+E .4R"#$+JE%(.#"#$

2 < a, x− a > +θ‖x− a‖2 ≥ 0 ∀θ ∈ ]0 1] .

Ïq%("4X%(E +*3θ>?+3O1+ $ I 3"4%(EU "#.j>v"4+E < a, x >≥ ‖a‖2 > 0 ∀x ∈ C

MÃ .S'3/F0GE!3+3 \-':.,%m3!0G:.#"#U%("#3

< a, . >!RE' .#.4!' 3

V

X"4+Ex ∈ K , y ∈ V , λ ∈ R

x+ λy

! %(.43%(C /3!:\-'

< a, x + λy >≥ ‖a‖2 .,%1m I !"4®% 3G

λ 7→< a, x > +λ < a, y > $¸ "4!' 3

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

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%(-

Ω I 0]0G!' D DV3!'R.,%\-'+.#.4:.4 I ' 3&% I !'L%(.#"4$O O% I !"m ((Sk

j ) 0 ≤ j ≤ N)XE 02%(3!!"# JE%(.4eM

K

Ë ÉG¢³ s¥!¯K s¡sK¢«%?¼9-' D DPKX\-'j< "#.q)*"4!R' OD 3QL%(Q "#.#"#$ !' D DP3!

Ω'Á.,%\E'+.#.4 .4 I ' 3

% I !'L%(.#"4X$/ 6% I !"m (Skj 0 ≤ j ≤ N)

E/ ¬02%(3!!"# JE%(.4UMA9v"#φ = (φk

j )'

!3X%($+JK"48%(' (¸L%( I $8+.#.4\-'V0(φ) = 0

D ' "4\-'φ!;%(' (¸L%( I $K %*

\-'+.4\-'/"#n ≥ 1

Vn(φ) =

i=n∑

i=1

< φi , ∆Si >=

k=d∑

k=0

(φk ∗ Sk)n.

9v' q(Vn(φ))

!R' ¬02%(3!!"# JE%(.46E' .#.4F.=<Y3!"#JK"# +.#.4:>?$+3!"¸L/ I

[VN (φ)] = 0 "#.Á!6%(.43 I .,%("#36\-' VN (φ)

]DV+' :DL%/¹+!3]' G>PMT%*M ≥ 0

-' .#.4KMQq¼ 1 $ I "#D 3-\-'+0G+EU!' D DPKX¬.402%(3 I [$]>v",%(Q .4 '6%(.#.46%(D D .#"4\-'+3:.41![$(H3W+0G6 -*&%([ *H R%(L% I [%(R.,%]"#!'L%(!"4!' "#>K%(EG«Ä E

+3X%1.=<Y!DL% I 6+' I .#"4*"4+ >VMT%*M 3UM !' 3Ω0' "·*'D 3v*' "#O I %(.,%("#3

< X,Y >=∑

ω∈Ω

X(ω)Y (ω) .

Ä $!"#JK+3X%G.4'&++06Q .4 E

m3!0G$/DL%(3&.4O>VMT%*M 3UMVN (φ)

\E'L%(φ $ I 3!"#

.=<Y++06Q .4 70 ¬!X%($+JK"4:%(' (¸L%( I $¬+.#.4¬\E' V0(φ) = 0

M Ã .A!Om% I "#.4 >?"#3O\E' !R' ZX'!DL% I :>? I 3!"4+. E M

Ä Í Z+3X% .4/X'++06Q .46 E

m3!0G$/ R>VMT%*M 3UMX

DVK!"#!"#>?R>?$+3!"¸q%(-∑

ω∈ΩX(ω) = 1 '.,%("4X%('Â.4 I +' 3.4 "#Â O>?$+3!"¸L+3R\-' ! I E>?)

+ I 0]DL% I UM Í Z3+02%(3\-'+3X%]\-'j< ' :>PMT%*MX

>?$+3!"¸L[X > 0] 6= φ

D ' "4\-'j< "#. j<Y)*"4!DL%Â !!3X%($+JK"4®S<T%(3!Q "#!3X%(J?K + -*"4sÇ"#-UM

7 D D .#"4\-'¬%(.43¬.4![$3W+0G1 *¬%([ *H R%(L% I [ F + «"#.)v"4/' >PMT%*M 3UMα+.#.46\E'

∑

ω∈Ω

α(ω)X(ω) > 0 ∀X ∈ ;∑

ω∈Ω

α(ω)VN (φ)(ω) = 0 ∀φ ∈ 70

»§M#Ñ?¼

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

‖ α ‖=∑

ω∈Ω

α(ω) ;

=α

‖ α ‖ .

!1' ;D 3QL%(Q "#.#"#$ I [L%(3!J?U%(E2'G.41DP"#-G Ω %>? IZI +!;X%(!"4¤.,%

+'*)*"4W+0G6$+JE%(.#"#$6 G»§M#Ñ?¼RU<Y$ I 3!"#

[VN (φ)] = 0 ∀φ ∈ 7

0 .»§M#Ñ+-¼

EÙ

':%(.#.4&0GE!3+3/\-'(Sn)

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

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

F>K%(.4+' 3Z%(R

d +Z $!"#JKZDL%(3(φn)

R.,%Ø!3X%($+JK"4 70%X I "4$F

ψ+

%(D D .#" I %(!"48*'Z![$3W+0G "" »m%>? I V0 = 0)M

(σn).,%1' "#:>? I 3!"4+.#.4K

F>K%(.4+' 3¬%(R

d :% I !"m&3!"4\E'$:% I !'L%(.#"4$:+ [ , ].4D 3-*' "#: I %(.,%("#3'!'+.

%(R

d M2 I %(. I ' .!"#0]D .4 I *' "#¬F.=<Y$+JE%(.#"#$

(ψ ∗ σ)N = [ψ0 , σ0] + VN (φ)

@3+2.=<Y!DV$+3X%( I Z ÂÎ80G+06Q 3Â'N "#.>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.=<T% I !"&%(63!"4\-';% I !'L%(.#"4$ (S0n)

!' 02%(3!!"# JE%(.46!3!"#>-",%(.4KM

" :5 <0! . ,< "J) !1&%(D DP+.4·S<T%(QP36Q 3!"4W+>?+0G+EA.4j0G$ I %( "4!0G·QP' 3"4+3»s $~ÇF $ I 3!"#%(·.=< "#E!3v*' I !"4q¼·F.=<Y3!"#JK"#

¬ÎGD !"4&+' 3DP$+ &.4 D .#'O"#0]DV3!X%(E1«.4P s¡m¢ ³¿·À ¯ ¨+Ê ¯K »s+Zs3X%( JK.,%("4 ¨ ¯°§°s¼O+O.4P s¡sK¢ ³¿E´6Ì(´¢q µ´»s'qºv ¼M

GDPKX+' 3:S< ' I %(.#.Á+' 3DV$+ q¥¡¾8¿À ´!¾E´¥ ¨ ¡ ¨ ´ K > 0+/ ¿E¯K µ´2¿·À ´!¾-´¥ ¨ ¡ ¨ ´ N $+0]"46DL%(3:'

QL%(\-'O!' 3.=<T% I !"·F:3!"4\-'¬³+KºE³¦ ¯ ¨ ´¢L l !3' >?O"#E>?!!"V*'*3"#S<T% I [++3F¬.,%/%( N .=<T% I !" l %('2D 3!")KM N) %(0]"# /\-'+.#.4&m%Z"#.SDV+' 0G L%UÅ?+3 I /*3"# F.,%1%(6S<Y)*+3 I " I KMÄ 9-"

SlN > K

Ã .A% I [W+1.=<T% I !" l FG.,%QL%(\-'%('®D 3!")K

»s8*"#/%(.43¬\-'j< "#.R´!¾-´¥ ¨ ´]³¢V s¡s¢j¼D ' "4e<Y+0]D 3 63+>?+*3 I /0G¹+0G]% I !"' 3O.402%(3 I [$]%('D 3!")

SlN

.,%G>K%(.4+' 3¬ 6.=<YD !"4FG.,%%(N

!R.4¬QV$+$¸ I 6\-'¬.#' "SD 3 I ' 3 I +!6DP$+3X%(!"4jPF1X%>?"#3 SlN −K

MÄ 9-"=(%(' I -!3X%("#3K Sl

N < KU.4Á*3"#·D 3 I ' 3$ÁDL%(3.,%O $++E!"4]*' I %(.#.-j<T%RD .#'·%(' I ' "#E$+3¹+AD ' "4\-'j<Y

DV+' D 3 I ' 3+3' 3A.402%(3 I [$.=<T% I !"VX'H#Ç!% I +-ÁF ' ]D 3!") < KM H %( I I %e.,% >K%(.4+' 3 .=<YD !"4

S<T% I [L%(&F.,%]%( N !-' .#.4KMNZ3$' 0G$K .,%>%(.4+' 3 'Z.4¬D 3!");*' I %(.#. M]³¯¿E¯K µ´¿À ´¾E´¥ ¨ ¡ ¨ ´ N +3X%

(SlN −K)+

M

/P s¡m¢Z¿EÓÌ(´¢L µ´ +' 3DV$+ »s'qº* +' 3DP$+q¼ D 3!")1S<Y) +3 I " I K + %(RS<Y) +3 I " I N !' 3.=<T% I !"AX'ÁH#Ç% I +E l DP+3!0G+F; $++E+' 3O Ì(´¢q¿K¥!´

l.,%%(

N%(';D 3!")

KM-@%(3R' Â3X%("4 +0G+E

+Z' RDV"#EO"#0]"#.,%("#3%('D 3$ I $ +EUP I I .#' \-'¬.4¬D 3!")Z I :D ' OF1.,%1%( N !(K − Sl

N )+M

@%(3 %(Q ' &.,%( JE%(J?/;"4 +E!"¸L/' >?+-' :D !"4Z+' 3DV$+ 6FX%>%(.4+' 3OF.,%1%(:S<Y) +3 I " I I :\E' ")*D .#"4\E':.,%1 $¸ "#!"48!' "#>K%(EG«

6G685 FG,HEI J+KI

'%(D DV+.#.4+3&V s¡s¢´ºv¥! Ë ´¢L¢P´6¿·À Ë!¨+Ê*Ë ¯¢ ¨ ´;»s'+ I 3O ¿?¯K µ´¬¿À ´!¾E´+¥ ¨ ¡ ¨ ´(¼ N ' O>%(3!",%(Q .4&%(.4$U%("#3h ≥ 0

.,%]>%(3!",%(Q .46%(.4$U%("#3h := βNh

+3X%]%(D DV+.4$2P s¡m¢¯ ¨ sº ¯°²¡,³ Ë ´ %X I "4$/F h M

jb-`h&/§b 9v"#

(Un).,%1!' "#6 I ' 3OS< ' % I !"3!"4\-'$ u

ÑKM /P s¡m¢¿À ¯ ¨Ê ¯K h »s' ¨ ¯°§°s¼+' 3DV$+ q¥¡¾1¿À Ë ¾E´+¥ ¨ ¡ ¨ ´ K +Á /¿?¯? µ´ ¿·À ´!¾-´¥ ¨ ¡ ¨ ´ N !' 3.=<T% I !"³º-³¦ ¯ ¨ ´+¢L u +3X%1 $¸ "4/DL%(3

h = (UN −K)+M

Î*M Í S<Y$¸ "#O :.,%0G¹+0G¬=%Z' 2V s¡s¢¿E´6Ì(´¢q µ´h»s';qº* !¼+' 3DV$+ /DL%(3

h = (K − UN )+M

*M Í 1DV+' ·3+0]D .,% I +3.,%O>PMT%*M UNDL%(3.,%O>%(.4+' 3A0GUÅ?+

UI = 1|I|

∑

n∈I Un Á.=<T% I !" (Un)

!' 3A' "#-+3H>K%(.#.4¬ O+0]D

I ⊂ 0, 1, 2, . . . , N D 3$ $++3!0]"#$ .4D !"4 I !!3!' "#R I +!O=%;D 3+ +E.4&0 S<4P s¡m¢ ³/¯(³¡s¯K s¡+º´³M*@%(3R)*+0]D .4¬' ÂD ' /»s+' 3DP$+q¼%!",%(!"4\-'

h OD 3!");S<Y)*+3 I " I K !' 3

.=< "#-+3!>%(.#.4I+3X%1 $¸ "DL%(3

h = (K − UI)+ M

M 9v"4+-a et b

+'*);3$+.4> 0

%>? I a < U0 ?!"4+' 3Á!3' >K%(E!3D3!"4X\E'$ .=<T% I [L%( &.=<T% I !" u D 3$mW+3+3X% I +.#' "V h M Ã .PX+3X%%("#!" I +3!X%("# &.#"#0]"#+3RDP+3!R%('; "#>?U%('a\E' "#!F]% II +D +3 :.#"#0]"#+3OXRJE%("#O%('Z "#>?U%('

bM

6G6 <>C ? ;@<#; A - I + A K+# 9< ! "#$% -IG<G9? +I < -9?7#< I I < IBC!-I= BA <

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.=<Y+V3+].,%® +02%( KM Í DV+' G I .4 I !"4 $+3+3 I 0]0GZ % I !"s2F83!"4X\E'!' D D .4$+0G+-X%("#3*']02%(3 I [$¸L%( I "4+3UM '%(.#.4 I !"4 $+3+3' ]D 3!") I 0]0GR3X%("4 L%(Q .4OU< "#. j< "#E!3v*' "#DL%S<YD DV3!!' "#$6S<T%(3!Q "#!3X%(J?K I /\E' "S' I *' "# F.,%] $¸ "#!"48!' "#>%(-G«

$g%,d %§fPg9v"#

h' D !"4+' 3DV$+ ]%(&' 02%(3 I [$¸L%( I "4+3&>v",%(Q .4 U< "#.)v"4!' !' "#%%(D $ (Hn) :>VMT%*M ≥ 0+.#.4/\-'

Ä HN = hÄ .4¬02%(3 I [$:¸L%( I "4+31» Ω, (Sn), (Hn),

¼X"#+ I 3:>-",%(Q .4K*"#3X%1\-'

H0!R' Âq¥¡#¾;´¾ ¨ °²º*¯¢q °À ¯¥ B+¡§ s¥!¯U©E´ ¬.=<YD !"4 h M

Hn+3X%]%(D DP+.4$/DjMYKMT%*MLF.=< "#!X%(E

n ¬.=<YD !"4

hM

7·>%(- &*"4 I ' +3.=<Y)v"4+ I +.=< ' " I "#$¬S< ' O+.#.4&!' "#K-'%(.#.4>?"#3Uv!' 3Á.=<Y) +0]D .4&"#0]DP3!X%(-\-' "j!' "#U\-' I +!/)v"#J?+ I 6 ¬>-",%(Q "#.#"#$6"#E!3v*' "#O I E!3X%("#- s3!' 3R.4:D 3!"); I %(.#.+O RD ' UM

6G6) *I<>A +#I <?K+ 7A8A? - + IGC(-I= A <

9v"4+E(Ω, (Sn),

)' Ó02%(3 I [$Z¸L%( I "4+3>-",%(Q .4+

l ∈ 1, 2, . . . , N ' ×% I !"/3!"4\-'$KM "#!!"#!' !"4¸L%( I "4W+36$+0G+ !' 3R.=<T% I !"'H#Ç% I +E l ' I %(.#.+' 3DV$+ C = (Sl

N −K)+%("#!"j\-'j< ' ZD ' +' 3DV$+

P = (K − SlN )+

60G¹+0G/D 3!")ZS<Y)*+3 I " I K > 0MV9v"#

(Cn)»§3!DjM

(Pn)¼R' /!' "# /DjMYKMT%*ML /.=<YD !"4

S<T% I [L%( C »§3!DjM :.=<YD !"4 :>?+EP¼

)

c(fLhf! %d %§fqg%1

Í %.4$+JE%(.#"#$

Cn − Pn = Sln − K

(1 + r)N−n∀n ≤ N

»§M#ÑUÙ?¼

Ë ÉG¢ ³+ s¥¯? s¡s¢« !3Á02%(3 I [$ I 0]DP3!02%("#E+L%(-.4d+3

% I !"s 0, 1, . . . , d, c, p %*0G+HX%(- DV' 3 I ' 3¬F].=< "#X%(E n Sn = Sn, Cn, Pn

+ O!' D DVK$/>v",%(Q .4KM '&%(.#.4 0GE!3+3\-'KS!".=<Y$+JE%(.#"#$Z»§M#ÑUÙ?¼ j<Y!:DL%:X%(!"4!m%("#KV"#.)*"4!' ]!!3X%($+JK"4]S<T%(3!Q "#!3X%(J? DV' 3:DL%!3D%(.4' 3*"#3 I U-'Á'QV3!+3%(' I % n = 0

+.,%("4X%(-%('Â.4 I +' 3U-\-'Ç!OX%("4s3!",%(ZS<Y) +3 I " I U .4/X"#ZS<Y+ $*' "#3:' / $+0G!!3X%(!"4J?$+$+3X%(.4KMÑKM 9-' D DVK

C0 − P0 > Sl0 − K

(1+r)N

M9v"#]»

(ψkn) ; 0 ≤ n ≤ N ; k ∈ 1, 2, . . . d, c, p ¼.,%G' "#/>? I 3!"4+.#.4D 3$+>-"4!"#Q .4F]>K%(.4+' 3

%(R

d+2 $¸ "4:DL%(3

ψln = 1 ; ψc

n = −1 ; ψpn = 1 ∀n ≥ 0 ; ψk

n = 0 ∀n ≥ 0 ∀k /∈ l, c, pH <T%(D 3W .,%1D 3DVK!"#!"4 "" "#.)*"4!¬' 6X%($+JK"46%(' (¸L%( I $ φ I m I "4%(-¬%U>? I ψ' 3R.4 % I !"mR3!"4\-'$R++.#.4/\-' V0(φ) = 0

MÍ 8%1 I

Vn(φ) = Sln − Cn + Pn + φ0

nS0n ∀n ≥ 0

»§M#Ñ?¼+ +DL%(3!!" I ' .#"4+3U

Sl0 − C0 + P0 + φ0

0 = 0»§M#Ñ )K¼

@' "4\E'φ!O%(' (¸L%( I $KP%]S<T%(' !3:DL%(3!

Vn(φ) =< φn+1, Sn >= Sln − Cn + Pn + φ0

n+1S0n

»§M#Ñ ¼ % I 0]DL%(3X%("4 »§M#Ñ?¼+:»§M#Ñ ¼0G-!3 \-'

φ0n

$+DP+DL% n "#.L3$' .#&%(.43

»§M#Ñ )G¼\E'

φ0n = φ0

0 = C0 − P0 − Sl0

»§M#ÑUÚ?¼

Vn(φ) = (C0 − P0 − Sl0)(1 + r)n + Sl

n − Cn + Pn»§MTÎK?¼

VN (φ) = (C0 − P0 − Sl0)(1 + r)N +K

»§MTÎ*Ñe¼ %]!!3X%($+JK"4

φ I !!"#!'/ I ' %(3!Q "#!3X%(J? I /\-' "S"#E+3*"#¬%('02%(3 I [$6S<Y¹+!3/>-",%(Q .4KMH %(' 2X%(.#.42 G02%(3 I [$8S.,%Z02%( "4W+32D 3X%(!"4\-'2 20G+!!32+v+' >v3 I +!2!!3X%($+JK"4X+3X%("#O.,%G' "#>%(-«LFG.=< "#!X%(E

0>?'¬% I [++Ò' % I !"4®+O' D ' +O>?'O>?+ +Ò'

I %(.#. I +.,%6>?'ÁD 3 I ' 3' &\E'L%(-!"#$»m%(.#J?$+Q 3!"4\-'e¼S<T%(3!J?+E.#"4\-' "4 O$+JE%(.4¬F C0 −P0 −Sl0\-'&>?'D .,% I +Ò6%('X%('*)

r"V+.#.4¬!

> 0 +R\-' >?'R+0]D 3!' -+Ò%(';0G¹+0G¬X%('*)Â!"S+.#.4

< 0

>?'/1' I [+ÒFÂ3!"4+Ç'\-'j<TFÂ.,%%(N

>?!32\E'L%(-!"#$2 ]-' 0G$+3X%("#32!%(.43

(C0 − P0 − Sl0)(1 + r)N

+'*) I %RDV+' >?+E 02%("#E+L%(-&¬D 3$X+E+36«»m%?¼

SlN > K

.=<T% I [++' 3O*' I %(.#.S.=<Y) +3 I K >?'.#' "S.#"#>v3+Ò:.=<T% I !"4\-' >?'DPKX$ +Ò:%('D 3!")

Kj+/>?!32D ' /]>K%(' /3!"4+jM %>%(.4+' 3 ]>?!3GDV3!s+' "#.#.42!6 I Q "4+

I +.#.46\E'¬'&%>? I %(. I ' .4$KM

»§Qq¼Sl

N < K .=<T% I [++' 3&*' I %(.#.·%DP+3*'X%(3!J?+-U>?'R>?+ +Ò6>?!3/% I !"4%('D 3!")

K I ¬\E' >?'%(' 3!"4&.,%DPKX!"4;S< ' ÂD ' .,%>K%(.4+' 3R >?!3&DV3!s+' "#.#.4¬!

+ I 36.,%0G¹+0GGMB'&%>?+Ò6 I 3$+'!"F]Q + "#3' Es3:.#' I [ EM

Î*M 9-"%(' I E!3X%("#3 C0−P0 < Sl0− K

(1+r)N

.,%&!!3X%($+JK"4 I !"4!X%(-ÁF¬% I [++3.4 I %(.#.=KFO>?+*3.4D ' +A+0]D 3!' -+3.=<T% I !"4 I *' "#Á%('0G¹+0G3$!' .#X%(R»§' G%(3!Q "#!3X%(J?e¼ .4Á.4 I +' 3DV' 3!3X%%U>? 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/!'432%(E:DV' 3:"#02%(JK"#+3:' ]!!3X%($+JK"41D 3$+>v"H"#Q .4DV+3!0G+!X%(ES<Y$+X%(Q .#"#3:.=<Y$+JE%(.#"#$

(4.15)j\-' "=L>?'¬.=<T%U>?+Ò]$KS!&' $+JE%(.#"#$]+-!3

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

^_ &o_ a·c;%1?_*d %§fqg 5(#Ag¤hfqcedUbC&µb#6%//§b U_g6 Gc+% #·b / #_ %5·b 5#_21Kd %'& ~dUc Dc+% # %]3+.,%(!"4 :DL%(3!"#$:DV+' OU<Y$ I 3!"#3

Sln − Cn + Pn =

K

(1 + r)N(1 + r)n »§MTÎKÎ?¼

I :\-' " I *' "# %('Z3$!' .#X%(O' DP+'' 3!D 3+L%(E ' "#>%(-/« DP3!m+' "#.#.4 I !!"#!'$:DL%(3

Ä .=<T% I [L%( ¬.=<T% I !" l +S< ' ZD ' O!' 3 I + % I !" :D 3!")S<Y)*+3 I " I KÄ .,%>?+-6S< ' I %(.#.j /0G¹+0G6% I !"'H#Ç% I +E %('0G¹+0G:D 3!")ZS<Y) +3 I " I

%(' 3X%:.4O0G¹+0G I 0]DV3!+0G+ER\-'R.=<T% I !"·X%(Á3!"4\-'Q +E'+2D .,%U%(-.,%60]0GK = K

(1+r)N

F:.,% I %("4S<Y$+DL%(3!JKKM

" :5 <0! ! 5 . 8 $7698 ! <. , )+"98A5 &'$76 8 ! ($!*. $ , & ) / 5 $7698 H %(R' I :DL%(3X%(JK3X%(D [KL'!' D DVK $:' 02%(3 I [$:¸L%( I "4+3O>-",%(Q .4KMÍ *"#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 $ φ +.#.4/\-'

h = VN (φ) =< φN , SN >@' "4\-'

φ!O%(' (¸L%( I $Kq.=<Y$+JE%(.#"#$:D 3$ I $ +EDP+' + I 36U<Y$ I 3!"#3

h = V0(φ) +k=N∑

k=1

< φk,∆Sk > ou h = V0(φ) +N

∑

k=1

< φk,∆Sk >»§MTÎ ?¼

$g%,d %§fPg' 'O%(D DV+.#.4+3 P .=<Y++06Q .4 RD 3QL%(Q "#.#"#$ %*0G+!X%(-

ΩDV' 3!' D DV3!'R.4\-'+.#.4

(Sn)!

' 102%(3!!"# JE%(.41>? I 3!"4+.#.4 '/%U>?:>-'+®M *M ;\-'KP%(¬' 02%(3 I [$1>-",%(Q .4K P j<Y!¬DL%¬>v"4 KM %!!3X%($+JK"4

φ! %(D DV+.=<Y1³+ s¥¯K Ë ©¡m´Â£~º*¥+ µ´'DU´ºv¡°§°#´~ªÂ¿E´ ¨ ºvÌ(´¥+ sº-¥´G¿-´/°À P s¡m¢

h

EÚ

c(fLhf! %d %§fqg9v"4+-

h' /D !"4+' 3DP$+ K

φ' DP3!m+' "#.#.4/!"#0' .,%(E

h+= ∈ P

%.=<Y$+JE%(.#"#$

Vn(φ) =

[h|Fn] ∀n ≥ 0»§MTÎ(-¼

Ë ÉG¢³ s¥!¯K s¡sK¢×«9v'P(Vn(φ))

!¬.,%Â!3X%m3!0G$G (Sn)

DL%(3(φn)

I <Y!/ I ' 02%(3!!"# JE%(.4 :>K%(3!",%(Q .4:+3!0]"#L%(.4

hM

Âb-`_ c #·b 19-"¬.4802%(3 I [$Â>v",%(Q .4K' !!3X%($+JK"4%(' (¸L%( I $!"#0' .,%(E

hÂ$ I XX%("#3+0G+E%*0]"4X!"#Q .4K

I 0]0GZ.4ZD 3' >?Z.4DV+!"#23X%("4 +0G+-Â' "#>%(-«"#φ%(' (¸L%( I $+.#.4Z\-'

VN (φ) = h ≥ 0 !"

∈ P 'R>?+O />?"#3O\E'Vn(φ) = βN

[h|Fn] ≥ 0

M Ã .j+3$!' .#6\E'Vn(φ) ≥ 0 ∀n M

^bh·c;% 5(#AgbZfqhd %§fPg¤cEh&/%1?_ a&/§b "N .=<T%(Q+ I 2S< [-Å-DV![WG!' D D .4$+0G+-X%("#3K»§>?"#3.41DL%(3X%(JK3X%(D [' "#>%(-¼ j<Y/DL%:' "4\-'KjDL%

D .#'&\E'.,%G!!3X%($+JK"4φ I ' >?+3!!' 31 h M< )%(0G+8 ;»§MTÎ(-¼ I *' "#¬$U%(0]0G"#:F2' 63+02%(3\-'

"#0]DV3!X%(E/«.4D 3+0]"4+30G+0Q 3O I +!$+JE%(.#"#$R $+DV+1DL% 7 ∈ P X%(*"4Á\E'.4R I 10G+0Q 3O $+DV+GDL% .,%6!!3X%($+JK"4O I ' >?+3!!' 3 φ M %:>K%(.4+' 3 I 0]0' ¬%('*)Î:0G+06Q 3 »§MTÎ(-¼O $+DV+ I \-': h +O

n I 6\E' "*Ç'!!"¸L¬.4RX%(!"4&!' "#>K%(E/«

πn(h) = Vn(φ) =1

(1 + r)N−n

[h|Fn] ; π0(h) = V0(φ) =

1

(1 + r)N

[h]

»§MTÎKÙ?¼

%Ì¯¥¡s¯B+°#´1¯° Ë ¯K =K¡¥´ FnH=0G!' 3X%(Q .4

πn(h)U<T%(D DP+.#.4+3X%q¥¡#¾;¿E´6°!À P s¡m¢

hMG°#¯;¿?¯K µ´

nM

% ¨ ¢³ =¯¢L µ´®¥ Ë ´+°°4´*³¡s s¡Ì(´π0(h)

:»s\E'.=<YÔ+3X%D .#'Â"#0]D .4+0G+Eπ(h)

¼U<T%(D DV+.#.4+3X%P¥¡#¾ ¿E´°À V s¡s¢

hMv&<Y!RF I D 3!")2\E'j< ' Â02%(![$+02%(!" I "4+Z+0]D .4UÅ?$&DL%(3' OQL%(\-' I +"#.#.4+3X%6 >?+*3 .=<YD !"4

hF.,%1%(

0 +Z>?" I "S' :D 3+0]"4W+363X%("4 «

<Y$+0G+!+' 3 .=<YD !"4h*"4DPKX%(-6%('®+0]D

0 .,%Â0]0G

V0(φ)!/%' 3$DV' >?"#3¬=%("#36=% I ]F

¬Q .#"#JE%(!"4U» I <Y!/FÂ*"#31 $+QP' 3X+3&.,%Â0]0Gh\-'L%(8.=<YD !"4+3X%2) +3 I $1DL%(3:% I [++' 3F2.,%

%(6S<Y) +3 I " I e¼ "#..#' "!'43G DP' 3 I +.,%GS<"#->?!!"#3 I +¬%(3!J?+E¬!' 3.4602%(3 I [$+!' "#>K%(EO.,%2!!3X%($+JK"4φM Ã .

! I L%(!' 3+.·\E'j<K"#.S>?+ I +!/D !"4%('ZD 3!")π0(h)

DV' 3O%(DL%("4+3OXO%( J?"4UM %&D 3DVK!"#!"42!' "#>K%(EK'Á+1s' 3! "#' R I KKm%("4X%(-"#E+3!>?+ "#3.,%&L%(!' 3D 3(s S< ' ]02%(3 I [$¸L%( I "4+3»m%(QX+ I /S<T%(3!Q "#!3X%(J?e¼M

c(fLhf! %d %§fqg9v"#

h' D !"4+' 3DV$+ 3$+D .#" I %(Q .4 π(h)

!.=< ' "4\-'RD 3!")G) I .#'L%(EÁ.=<T%(3!Q "#!3X%(J? I +!OD !"4jM Ë ÉG¢ ³ s¥!¯K s¡m¢Ø«

ÑKM58E!3S<T%(QV32\-'π(h)

!Á' 2DjMYKMT%*ME .=<YD !"4h 3+>?+%('*)]X%(!"4 RM MTÎ/+ÁDPK

Hn = Vn(φ)MP9v"#¬S<T%(' !36DL%(3! ∈ P

I 0]0G(Sn)

+(Hn)

- ?H=02%(3!!"# JE%(.4Uq"#.·+8! ¬0G¹+0G6

(Sn, Hn))M Ã .S+Z3$' .#:\E'¬.4¬02%(3 I [$:¸L%( I "4+3O$++*' (

Ω, (Sn) := (Sn,Hn), ) !

>v",%(Q .4KM

ÙK

Î*M@%;F.=< ' " I "#$ $!"#JKDL%(3(hn)

' !' "#8%%(D $ >PMT%*M 3UM ≥ 0>?$+3!"¸q%(E

hN = h+

DVKπ = h0

M '%(.#.4Á0G-!3+3R\E'K-!"π 6= π(h)

v2DV+' I !!3!' "#3 ' OX%($+JK"4 S<T%(3!Q "#!3X%(J?%(.4&02%(3 I [$&¸L%( I "4+3$++*' (

Ω, Sn := (Sn, hn), ) MV» Í I "4 W+3 02%("#-+L%(E&\E' .=<YD !"4j

"#-!3-*' "#%(.4602%(3 I [$K I !!"#!'/' ®% I !"F]3!"4\E'6!' D D .4$+0G+EX%("#3$d+ 1

-O.4 I ' 3¬F.,%G%(

n

hn¼M

9-' D DVK DL%(3 ) +0]D .4π > π(h)

DPK&%(.43ψn = (φ1

n, φ2n, . . . , φ

dn,−1)

+ $!"#JK DL%(3(φn).4&DV3!s+' "#.#.4/%(' (¸L%( I $ I m I "4%(- %>? I ψ ' 3.4% I !"mRF3!"4\E'¬*';02%(3 I [$:$++*'Z+>?$+3!"¸q%(-

V0(φ) = 0»=ÁMMTÎ*MTÎ?¼M Í %

Vn(φ) = φ0nS

0n + φ1

nS1n + . . .+ φd

nSdn − hn

»§MTÎ ?¼

= (φ0n − φ0

n)S0n + Vn(φ) − hn

»§MTÎ )K¼

»§MTÎ )K¼R$ I 3!"#6+ n = 0 I *' "#&F.=<Y$+JE%(.#"#$

φ00 − φ0

0 = π − π(h)»§MTÎ ¼

0]0G6.4!!3X%($+JK"4φ+φ- %(' (¸L%( I $UA»§MTÎ ?¼U<Y$ I 3!"#O+ I 3

Vn(φ) = φ0n+1S

0n + φ1

n+1S1n + . . .+ φd

n+1Sdn − hn

»§MTÎKÚ?¼

= (φ0n+1 − φ0

n+1)S0n + Vn(φ) − hn

»§M K?¼

0]DL%(3X%(-1»§MTÎ )K¼R+6»§M K?¼LZQ !"4+E .4R3+.,%(!"4

φ0n+1 − φ0

n = φ0n+1 − φ0

n

0]D 6+E' »§MTÎ ¼%1 I

φ0n = π − π(h) + φ0

n

Vn(φ) = (π − π(h))(1 + r)n + Vn(φ) − hn

VN (φ) = (π − π(h))(1 + r)N > 0

%]!!3X%($+JK"4φ I !!"#!'/ I ' %(3!Q "#!3X%(J?6\-' "SDP+' O¹+!3¬0]"4 +D 3X%(!"4\E'6 :.,%m%!' "#>%(-G«

.4Â!DV$ I ' .,%(+' 3G%(!!"#3$DL%(3.,%!' 3$+>%(.#'L%(!"4 .=<YD !"4h.=<Y+0]D 3!' -ZFZ.,%%(

0.,%>?+"#0]0G$H

*",%(+0G+-!' 3¬.402%(3 I [$%('8D 3!")πP+:D 3$+.4W+>?]' 3 I +!]X0]0G1.,%\-'L%(E!"#$

π(h)\E' ".#' "DP+3!0G+

S<T% I [++3/.4DP3!m+' "#.#.41 I ' >?+3!!' 3 φ .=<YD !"4h .,%Â0]0G3!X%(EK

π − π(h)S!¬D .,% I $

FÂ.,% I %("4X]S<Y$+DL%(3!JKKMj7OD 3W/%U>?"#3/.,%("4$]%('.4 I +' 3:.4]"#S<K"#02%(JK"#+36' ]!!3X%($+JK"4]S<T%(3!Q "#!3X%(J?%(O.4 I %È π < π(h)

q I I .#' \-'j<Y%$ I !"#3+0G+Eπ = π(h)

M

6685 <>C ? ; - I < 9? +I< ; A- I + A K+ < IBC A8< 7C I 7A@.,%'Á%(' ]02%(3 I [$>v",%(Q .4R+ '' D !"4+' 3DP$+

hG$ I X%("#3+0G+-

!"#0' .,%(Q .4KM %1D 3DVK!"#!"4' "#>%(-:>%1+DL%(3!!" I ' .#"4+3 0GE!3+3 .=<Y)v"4!+ I 6 :DjMYKMT%*M DV' 3 I +!/D !"4M " # ""

Ù*Ñ

c(fLhf! %d %§fqg%19v"#

Π(h).4/X'++06Q .46

R+I -!"#!'$/DL%(3.4RD 3!")) I .#'L%(EO.=<T%(3!Q "#!3X%(J? ¬.=<YD !"4 h 8%

Π(h) = [h] ; ∈ P »§M *Ñe¼

Ë ÉG¢ ³ s¥!¯K s¡m¢ÑKM 9v"#

π ∈ Π(h) "#.)*"4!G%(.43:' G!' "#%%(D $

(hn)>?$+3!"¸q%(-

hN = h et h0 = π+.#.4G\-'1.4

02%(3 I [$ (

Ω, (Sn, hn), ) "#6>-",%(Q .4KM Ã .Á)*"4!] I »=Á!MVM *M ?¼¬' GD 3QL%(Q "#.#"#$ ¯ I [L%(3!J?U%(E1'

.4¬DV"#E6 Ω

'¬.,%\-'+.#.4(Sn)

+(hn)

-/ :02%(3!!"# JE%(.4 "#.+®3$!' .#]+DL%(3!!" I ' .#"4+36\-'¯ ∈ P M H :D .#'U

π = h0 = ¯ [h] I :\-' "S0GE!3\E'

π%(D DL%(3!!"4+-&%('Z0G+0Q 36 :*3"#/ 2»§M *Ñe¼M

Î*M Ã E>?+3X+0G+EUV"4+E ∈ P +π =

[h]

DPKXhn =

[h|Fn] = βnhn

MV0]0G(Sn) et (hn)XEZ ?H=02%(3!!"# JE%(.4UR.4®02%(3 I [$ $++*' (

Ω, (Sn, hn), ) Â>v",%(Q .4 S<T%(' !3®DL%(3!UR >?"#

"#0]0G$*",%(+0G+-:\E'h0 = π

+hN = h

"#.j+3$!' .#/\-'π ∈ Π(h)

M

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"#EUMÄ H %(1.4 I % I E!3X%("#3K Π(h)

!1' Ø"#E+3!>K%(.#.4' >?+3!G RM»=Á!MÏ.#.#0G+3G+29 I [ "4 «9- I [L%!!" I

ÏA"#L%( I e¼

$ , & )! &>< "98 >: ! 6685 FG,HEI J+KI

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.=<T%)v"40G; Â>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<Y)v"#J?+3\E'' OD !"42D ' "4¹+!3 !"#06' .4$ODL%(3' !!3X%($+JK"4 I E>?+L%(Q .4&!Q "4+G3$U%(.#"4! G+3X%:' R0G¹+0G\-'.=<YG +02%( DL%ÁDP' 3.=< "#X%(ES<Y)*[ "#QV+3 I +!!!3X%($+JK"4«KX%>?"#3Á\-'j<Y+.#.4)*"4!!'432X%(EDV' 3Q + "#3 3$!' .#X%(UM H <T%(' !3:DL%(3!U .4/0G- W+.4/ .,% I %(89 I [.4&\E' "j!R.4/D .#' I ' 3X%(0]0G+E&' !"#.#"4$m' 3! "#Á' ) +0]D .4"#0]DV3!X%(E R02%(3 I [$ I 0]D .4+ "#.L+Á 0G¹+0G *'20Gv W+.4O*"4 I 3+Á\-''%(.#.4$+!'*"4+3 %(' I [L%(D "#!36!' "#>K%(EUM

66 +# 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

%?¼9-' D DPKX.4R02%(3 I [$O>v",%(Q .4O+ I 0]D .4++Á"# h ' R>VMT%*M 3UM ≥ 0 "#.q)v"4!' !!3X%($+JK"4

%*0]"4X!"#Q .4φ+.#.4\E'

h = VN (φ) I 0]0G

φ!·%(' (¸L%( I $¬UDP+' A$ I 3!"#3 » I !MU.,%3+02%(3\-'

"" .*¼

βNh = VN (φ) = V0(φ) +

N∑

1

< φj ,∆Sj > .

9-"C1+

2E· +'*)&D 3QL%(Q "#.#"#$A'S.4\-'+.#.4

(Sn)!S' 02%(3!!"# JE%(.4Ke%(' 3X%

βN

1[h] =βN

2[h]

I :\-' "j+E!3X%("#$+>-"4 +0]0G+- 1 =

2Qq¼ Ã E>?+3X+0G+EO!' D DVKA.402%(3 I [$R>v",%(Q .4+A I 0]D .4+ "#. )*"#R%(.43' >VMT%*M h ≥ 0®!"#0' .,%(Q .4 3+D 3+L%(E:.4 X%(!"4:*' I 3.#.,%("#3 "" I +.,%!"#JK "¸L1\E' V !&' 8'

DL% I >? I 3!"4+.!!3!" I ¬ E \-'6'&0' "#3:02%("#+L%(E/*'8D 3-*' "#: I %(.,%("#3 < X,Y >= [XY ]

S<T%(D 3W I ¬\-' .=<YX%("#R R!DL% I R+' I .#"4*"4+e*"#.V)*"4!¬%( E ' >? I +' 3 X -' . »§"=MYKMV' 1>VMT%*M 3UMSE' .#.4e¼¬3!![J?L%(.F V I +.,%;+E!3X%("#+DL%(3!!" I ' .#"4+3UjD ' "4\-' X !3!![J?L%(.4%('*) I !X%(EU [X] = 0

M H $¸ "4&%(.43=∗DL%(3R.=<Y$+JE%(.#"#$

∗ =

k +X

k

= (1 +

X

k)

Èk' 3$+.

> 0%X+Ò:JK3X%(DP' 3 \E'

k +X > 0 "#.j!&%(.43m% I "#.46 :>?"#3\E'

∗

' 6D 3QL%(Q "#.#"#$6*" S$+3+E :D .#'U\-'+.4\E'6"#.,%]!' "#/D 3$+>-"4!"#Q .4ψF1>K%(.4+' 3O%(

Rd

∗[

N∑

1

< ψn,∆Sn >] =

[X

N∑

1

< ψn,∆Sn >] = 0.

Ã .S3$!' .#6%(.43O*'Z.4+0]0G " " \E'G»Sn

¼!R' :02%(3!!"# JE%(.4'=∗M

, 5 >: &><.0/+, :-. , ( 8. %< " :5A<. ,< "J) ($!. $ ,-&) & < " 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/.=<Y!D 3!"#!Á.,%6>%(.4+' 30GUÅ?+ ¬ O.=<YD !"4F6X%%(OS<Y) +3 I " I vF6%>?"#3

[h] I <Y' Q .#"4+3\-'Á.,% >?+- .=<YD !"41D 3 I ' 3F&1$+0G+!+' 3R W·.=< "#X%(E

0e' 0]0GS<T%(3!J?+E

\-'j< "#.DP+' /m%("#31s3!' I !"¸L+3F%;JK' "4 "#.DV+' USDL%(36) +0]D .4KjD .,% I +3 I +%(3!J?+E]FÂ.,% I %("42S<Y$+DL%(3!JK2+U%( I +!:[-Å-DV![WKX¬DP+3!0G+!!3 ¬>?+*3/0G"# I [+3Uq%('D 3!")

βN

[h] =

[h] =1

(1 + r)N

[h]

»§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

hU< "#. I X+E&F]"#E>?!!"#3¬%(O ¬% I !"s&F13!"4\-' "#.·*"4DPK%("#!"S< ' 6!3X%($+JK"4 φ »m%(D DV+.4$

³ s¥!¯K Ë ©K¡m´¿E´ ¨ º-Ì´¥ sºv¥´ 8.=<YD !"4q¼Â.#' "¬DP+3!0G+!X%(- I !!"#!'+3' DV3!s+' "#.#.4 -.,%>%(.4+' 3F.=< "#!X%(-

N!R)% I +0G+E

hM Ã .S!R I L%(!' 3+. ¬ $¸ "#3.4&D 3!") V0

¬.=<YD !"4hF.=< "#!X%(-

0+

D 3!")Vn

F1.=< "#!X%(EnDL%(3

V0 = V0(φ) ; Vn = Vn(φ).

Ù

Í G+3X%6\E'V0

!' I X%(E6q» I <Y!.4RD 3!") I +"#.#.4$ F:.=<Y$+0G+!+' 3 R.=<YD !"4q¼X%(*"4\-'Vn

!' >K%(3!",%(Q .4O%(.4$U%("#3 Fn

H=0G!' 3X%(Q .4 I <Y!F:*"#3' m I !"4 .=<Y$+>?.#' !"4Â I ' 3Q+3!>?$jÇ~'X\E'j<TF.,%1%(

nj»s%(.4 I %R'!'+.4 I /X+3X%' ¬s I !"4*' +3! "4+3 I ' 3 Sn

¼¬ I :\-' "SDL%(3X%("#O3X%("4 L%(Q .42MH %(.,%G3$U%(.#"#$KL!3D !"4

h+3X%10]"4!' 3.4602%(3 I [$' I 0]0G/' % I !"4jL+& +>-"4+*3X%%("#!"

' % I !"AF/3!"4X\E'&!' D D .4$+0G+EX%("#3 +.#.4¬%(' 3X%' D 3!")Wn

F/.=< "#!X%(-n»*" S$+3+ER

Vn¼Á $++3!0]"#$ DL%(3.,%

.4" :.=<Y+V3/+O /.,%] +02%( MWn

!O.,%]0]0G/S<T%(3!J?+-&\-'j< "#.j>?'=%('*3X%G $+QV' 3+3OU< "#.>?'D 3+.,%=%(EX%("4!"46S<T% I [++3 I +!6D !"4F1.,%1%(

n \-':DP+' DP++3e< "#.j)v"4!/' Z$ I %(3!"#0]DP3!X%(- +-!3

Vn+Wn

3"4 I [KXO*" S$+3+EO' "#>%(-.4R+0]DV$+3X%(0G+E«Ä \-'R.=<Y;%¬DP+3*'ÂG+0]D+G.#"4X%(- I ÁÁ\-' "L)vDVK+-Á' ![$3!"4OX%(.4+0G+ER$+!3X%( J?W+3¬%('*)3$U%(.#"#$ *'02%(3 I [$M

Ä \-'&.,%![$3!"4¬D 3QL%(Q "#.#"4!/!R>K%(.#"4 ¬02%("4R\-'¬.4R!X%(!"4!!" I "4+-R02%(.Vm%("#R.4+' 3R!3X%U>%("#.«*.,%.4"S D 3QL%(Q "#.#"#$ 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.,%]%(

n' ZD 3!")

Wn!3WR"#*m$+3!"4+' 3&F

Vn >?'"#->?!!"4+Ò%(.43>?!3:m3!!'

+ I +.#.46 :' :>?!3¬=%(0]"#.#.4/%(R.=<YD !"4 h M"4+¤+E+*'j".4]DV$+3X%(+' 3]' !"#.#"4+- I 10G- W+.4 I <YDL%(3 I \-'Â.41D 3!")\-'j< "#.41DP+3!0G+!+-2 I %(. I ' .4+31;E1DL%!3D$+.4"#JK$] 1D 3!")3$+.4 I X%($]!' 3.402%(3 I [$¸L%( I "4+3 02%("4"#.!1 02%(' >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$+3X%(+' 3"4+E I ->%("# I ' &X%DV+3!!"#+ I ]«v"P.,%X%(.#"#$:S<Y+-!3/+'*)Â-RDP+3'L% $R\E'j< "#.S)*"4!&' EoÇ'!&D 3!")9E&DP' 3' ¬D !"4j I <Y!$+>v"4 +0]0G+E6F I 6D 3!").,F\-'U<YV I !'+3-:.4&!3X%(X% I !"4UMS5®%("4 3+>?+/%('*)02%(![$+02%(!"4\E' %.4¬3$!' .#X%(O' "#>%(-6«

: Efqc v`b

V0 =1

(1 + r)N

[h] ; Vn =

1

(1 + r)N−n

[h|Fn]

»§M ?¼

Í DV' 3!3X% I !X%(+3O\-'¬.,%m% I ' .#$: $6%('>?+ +' 3 ¬.=<YD !"4 : I ' >-3!"#3ODL%(3 % I !"sOF3!"4X\E'!3X%*' "#/+®.,%( JE%(J?102%(![$+02%(!"4\E'1DL%(3¬.,%Â!' Q!!"#!' !"4 9F0 » I 0]DL%(3+3 (4.13)

+(4.14)

¼]M D 3!")2S< ' OD !"4; I %(. I ' .,%(EF¬.=<T%("4 & ?DP+' ' Q .#"4+3.,%/D 3QL%(Q "#.#"#$ E+0G¹+0G %( I +3!X%("#I %/*:DL%.,% I L%("#!36%>? I D 3$ I "4!"4 M 'O%(.#.4O+>?"#3O' )*+0]D .4%(' I [L%(D "#!36' "#>%(-UM

Ë ÉG¢ ³+ s¥¯? s¡s¢Ø«@' "4X\E'(Vn(φ))

!R' :02%(3!!"# JE%(.4'=8 %

V0(φ) =

[VN (φ)] = βN

[h] , βnVn(φ) = βN

[h|Fn]

Rm3!06' .4(3.12)

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

Âb-`_ c #·b m3!06' .4D 3$ I $ +E DP+3!0G+!+-O : +3R' : $+0G!!3X%(!"4D .#'3X%(D "4 »§02%("4\E' "Vj<Y!R>K%(.,%(Q .4\-':%(' 02%(3 I [$ I 0]D .4+¼R ¬.,%13+.,%(!"4 :DL%(3!"#$ I %(.#.H=D ' (36)

%1+V+

Cn − Pn =1

(1 + r)N−n

[SN −K|Fn] = (1 + r)n [SN |Fn] − K

(1 + r)N−n= Sn − K

(1 + r)N−n

Ù(

+

!>& ,5 " : 5A< (0. < ( 8 9v"4+-

a+b +'*)3$+.4>?$+3!"¸q%(- −1 < a K%(E ]0Gv W+.4G02%(![$+02%(!"4\-'ÂF.,%!"#!'L%(!"4 QV' 3!"4W+3!"#0]D .#"4!G!' "#>K%(E«S"#.Áj<Y)*"4!G\-'j< ' % I !" F;3!"4X\E'((Sn) ; 0 ≤ n ≤ N)

+ I 0]0G2S< [L%(Q "#!' S' % I !"R3!"4\-'$S0

n = (1 + r)n M K:[EÅvDV![W!' 3R.4 I 0]DV3!+0G+E& ¬.=<T% I !"4 S +3E I +.#.4 I "«v+-!3:.4R"#!X%(- n +

n+ 1*.4 I ' 3 Sn

¬.=<T% I !"3!"4\-'$:!R06' .#!"#D .#"4$ &=%%(.4$U%("#3/"#DL%(3

1 + a "#DL%(3

1 + b %(' !3+0G+-&*"#/

Sn+1 = Tn+1Sn

È(Tn)

!' O' "# >VMT%/F¬>K%(.4+' 3%(Á.=<Y++06Q .4¬F/ +'*)]DV"#E (1+a), (1+ b) M-+!O[EÅvDP![W!!3WR3!!3!" I !"#>?]«*"#0]DVK+3\-'j< ' I ' 3 ¬QP' 3X&X"# I [L%\-'Ç' 3/*'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' 33X%("4«

Ä "#.jR02%(![$+02%(!"4\E'+0G+-&!3WO!"#0]D .4/+RDV+3!0G+O I %(. I ' .4O)*D .#" I "#6MÄ I 0]0G/'.4¬>?+3!3 I "Hµ X'/ I <Y!R' 02%(3 I [$ I 0]D .4+MÄ "#.jR.=<T%(L%(.4JK'6*"4 I 3+O*'0G- W+.4/F1+0]D I E!"#E'® .,% I Z%(9 I [.46M

N3+>%( I [2j< "#0]DPK%(' I ' I E!3X%("#- "!' 3.4O.4"4O >VMT%*M Tn»s\-' "jDV+' >?+E¬ $+DP+*3

n¼¬

"j!' 3.4+' 3"#E+3X% I !"41»s+ZDL%(3!!" I ' .#"4+3 Z:' D DPK¬DL%O.4+' 3R"# $+DP+%( I e¼6M

+6859685 <>C ?4< G<>C #? < - <>CÍ ÂDPK

Ω = 1 + a, 1 + bN ' &$+D 3+' >?ω+3X% I ' &!' "# t1, t2, ..., tN

N3$+.4>%(.,%(-

1+a'1 + b

M s I !"4 I v3 $ TiA $¸ "41DL%(3

Ti(ω) = tiA+3E1.4

N>K%(3!",%(Q .4G%(.4$U%("#3

I +E!3X%(.4& I :0Gv W+.4]M Í +3X%(Fn)

.,%¸.#!3X%(!"4L%(!' 3+.#.4/+ J?+*3$6DL%(3.,%1!' "#(Tn)

I 0]D .4$+$F.=<Y3!"#JK"#/DL%(3 F0 = Ω, φ M9v"#

s0 ∈ R $¸ "#' /' "#¬3$+.#.4

((Sn) ; 0 ≤ n ≤ N)+DPKX%(-

S0 = s0 , S1 = s0T1, ....., Sn = Sn−1Tn = s0T1T2.....Tn

ÙKÙ

'.,%("4X+3R%('.4 I +' 3R.4&"#; 0G-!3+3\E'(Fn)

!R%('!"P.,%:¸.#!3X%(!"4L%(!' 3+.#.4¬+ J?+*3$¬DL%(3.,%!' "#

(Sn)M 1 %(D DV+.4\-':'*"4!DVKS< ' / +'*)*"4W+0G/!' "#

S0 $++3!0]"# "4!GL $¸ "4:DL%(3

S0n = (1 + r)n =

1

βn

Í Z0' "#ΩS< ' :D 3QL%(Q "#.#"#$ ' 0]"46F' /X+' .4¬3!!3!" I !"4« I [L%(3!J?+3O'R.4RDV"#E Ω

M

43 5 $7695A8A54: ) >: & < "98 )>: .( ( & $ , &0)9v"#! ' ®D 3QL%(Q "#.#"#$\E'+. I \-'!' 3

Ω )%(0]"#F \-'+.#.4 I *"4*"#!"4.,% !' "#

(Sn)!'

02%(3!!"# JE%(.46X'= M Í %

[Sn+1|Fn] = βn+1

[Sn+1|Fn] = βn+1Sn

[Tn+1|Fn] = βSn

[Tn+1|Fn] =

1

1 + rSn

[Tn+1|Fn],

3!\-'j< ' I *"#!"4®$ I X%("#31+&'432X%(EDV' 3&\-'j< "#.A+®"#:%("#!"!« [Tn+1|Fn] = 1 + r

Ã .+Z3$!' .#G»§D 3+*36.=<Y!DV$+3X%( I / OÎ10G+0Q 3¼¬\-'

[Tn] = 1 + rM @' "4X\E'

1 + a < (1 + a)

[Tn = 1 + a] + (1 + b)

[Tn = 1 + b] = 1 + r < 1 + b,

%$ I %("#3+0G+Ea < r v",%(Q .4KZDV+' %(D D .#"4\-'+3 I /\-' "j>v"4+EOS<Y¹+!3/*"#F.,%1D 3QL%(Q "#.#"#$ %("#!"«

¢V´ ¨ K¢q¿¡s s¡s¢8¢ Ë!¨ ´³X³¯¡¥Ó*Kº-¥ º ´°4´6É]¯K¥ ¨Ê Ë ³¡s ÌU¡m¯B°4´]´³ r ∈]a, b[ ¢qKºE³:³ºU? (³´¥!¢ ³ ¨ ´+ m µ´

¨ ¢q¿K¡§ s¡m¢8Ì Ë ¥¡ ¶ Ë ´1¿?¯¢ ³/ =º* µ´6°¯2³ºv¡s µ´ Ã .j<Y$+X%("#]DL%1$ I %("#3Â Â3 I ' 3!"#32%('*)3$!' .#X%(]*'ØDL%(3X%(JK3X%(D [ "" DV' 31Q + "#3 I Â3$!' .#X%(«!".=<Y®!' D DPKXK DL%(3&) +0]D .4K

r > bq"#.·=% I "#.46 /JE%(JK+3¬ .=<T%(3!J?+E:F I ' D®!Æ 3ULX%(¬%(D DV3!O"# "#!",%(.+

X%(O3!"4\E'2«qF1.=< "#!X%(-0q+0]D 3!' E' % I !"48\E'6.=<YZ>?+"#0]0G$*",%(+0G+E¬DP' 3 D .,% I +3O.=<T%(3!J?+-

Q +E'ÔF8.,% I %("4S<Y$+DL%(3!JK» I <Y! I Z\-'j<YÓ%(D DV+.#.4' E>?+E8F® $ I ' >?+3! E «Ó>?+ ' % I !"4\-'j<YÂ DVKW DL%¼vD ' "4UvF6.=< "#!X%(-

NvÂ3X% I [W+¬.=<T% I !"4!' 3.4&02%(3 I [$¬DP' 33+06QV' 3+3X% +!KM

' .,%("4X+3¬.46.4 I +' 3/$ I 3!"#3.,%!X%($+JK"4φ I 3!3!DV%(E]+¬>?$+3!"¸L+3/\-'6.=<Y!¬+®D 3$+ I 1S< '

%(3!Q "#!3X%(J? \-'&=%("#3/!"r < a

@A' 3&%(.#.4+3D .#'R.4"#j'O%(' 3QV"#Z*'Z.4+0]0G/' "#>%(-6«

+6685 <>= =@<

9v"#T

' ]>%(3!",%(Q .4Â%(.4$U%("#323$+.#.4ÂF;>K%(.4+' 3%(6.=<Y+06Q .4;F +'*)DV"#E α, β (α < β)+"#

G ' '¬%(.#J?W+Q 3 F 8!' D DPKX6\-'j< "#.A)*"4!/' I !X%(- m +.#.4\-' [T |G] = m

T

!¬%(.43"# $+DV+%(E6 G +%.4"! $:DL%(3.4O$+JE%(.#"#$

[T = α] =

β −m

β − α;

[T = β] =

m− α

β − α

Ù

Ë ÉG¢ ³+ s¥¯? s¡s¢ Í Z0G-!36S<T%(QV3 I 0]0G:D 3$ I $ +0]0G+E¬\-'α < m < β

M@AK

A =

[T = α|G] , B =

[T = β|G] + I *"#!"4 L%(-ODL%(3R3X%(D DV3!OF G .4R$+JE%(.#"#$

1[T=α] + 1[T=β] = 1 , T = α1[T=α] + β1[T=β]

Q !"4+EA+B = 1 ,

[T |G] = αA+ βB.

Ã .j+;3$!' .#/\-'&.4R>%(3!",%(Q .4 %(.4$U%("#3A

+B

EX.#' !"4*'Z!Å*!W+0G: :Î1$\E'L%(!"4OFÎ]"# I E'6«

A+B = 1 , αA+ βB = m , -O.=< ' "4\-'/.#' !"4

A =β −m

β − α, B =

m− α

β − α.

Ã .V!%(.43"#0]DV3!X%(ER &+3R\E' .4>%(3!",%(Q .4%(.4$U%("#3A+B

XER I !X%(E I +.,%"#JK "¸L+8V+¬\-'6.,%2.4" I *"#!"4 +.#.4] T \-'L%( G $+DP+DL%¬

ωM %G>VMT%*M

T!

I "# $+DP+%(- ¬.=<T%(.#J?W+Q 3 G .=<T%X+3!!"4 I I +3!L%(E&X%.4"!OFDV+'D 3WO$+>v"4 +E2M

fLdU_*d %fPg' H %(R.,%]' "#: I I [L%(D "#!3% D +3X%].4RX%(!"4 !' "#>K%(E6«Ä Í DVK+3X%

p = b−rb−a , q = r−a

b−aÄ µ $!"#JK+3X%/.,%6.4"V OD 3QL%(Q "#.#"#$&DV3!$ODL%(3 1+ a, 1+ b $¸ "4 DL%(3

µ(1+ a) = p , µ(1+ b) = qÄ Í +3X%G+*¸.,%GD 3QL%(Q "#.#"#$

µN = µ× µ× ... × µ 3X%(D DP+.41» I M . " ¼\-' ;!.=< ' "4\-'

D 3QL%(Q "#.#"#$Â!' 3Ω

'/.,%\-'+.#.4G.4/s I !"4 I -3 $ T1, T2, ...., TNI !!"#!'+E' ]=%(0]"#.#.4

¬>VMT%*M "# $+DV+%(E&$\E' "4*"4!3!"#Q '$O :.4"µM

c(fLhf! %d %§fqg%1;!O.=< ' "4\-'/D 3QL%(Q "#.#"#$1'O.,%\E'+.#.4.,%]!' "#

((Sn) ; 0 ≤ n ≤ N)' /02%(3!!"# JE%(.4KM 602%(3 I [$

U)V 1 K+1' Q "#+"#! I >v",%(Q .4/+ I 0]D .4+UM Ë ÉG¢ ³+ s¥¯? s¡s¢ ÑKMRB¬$+3!"¸L ' S<T%(QV3\-'K'q

(Sn)!' 602%(3!!"# JE%(.4 I 0]0G

Tn+1!O"# $H

DV+%(E Fn 8%

[Tn+1|Fn] =

[Tn+1] =

1

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

I 6\E' "=*'O.=<T%>?>v'%('Z $+Q ' I /DL%(3X%(JK3X%(D [KL+E!3X%("#6.4¬3$!' .#X%(UMÎ*M 9v"# ' %(' !3D 3QL%(Q "#.#"#$!' 3

Ω"#*' "4X%(E.,%80G¹+0GD 3D 3!"4$+$!' 3].,%®' "#

(Sn)

%(' 3X% %(.43[Tn+1|Fn] = 1 + r

"#.&3$' .#%(.43*' .4+0]0G "" .¤\E'Tn+1

"# $+DV+%(E8 Fn

» I <Y!F®*"#3Z (T1, T2, ..., Tn)

¼1+%8DV' 3G.4"µM Í Ó+¤ $*' "#

%("4X$+0G+E1\E'].46>PMT%*M I -3 $ Tn-U' Gj"# $+DV+%(E1+6 2.4"

µ %

I q I MY\qM !MYSM

Ù )

+66 ; -IG<9? +I <- 9?7>< IIG< IGC A < = < ; */ C C < +!* - IGC + < I ![$3W+0G\-' "O!' "#Â!G' I X$\E'+ I "#0]0G$*",%(8 Z.=<Y$+!' Z=%("#%(' I [L%(D "#!38D 3$ I $ +E;+ $+JE%(.#"#$» ¼ : Efqc v`b 9v"#

h' /D !"4+' 3DV$+ + "#Â.,%1D 3QL%(Q "#.#"#$!' 3

Ω $¸ "4+ " " .4/D 3!")

Vn I +!GD !"4FÂ.,%%(n!6 $GDL%(3/.,%s3!0' .4

Vn = 1(1+r)N−n

[h|Fn]

+ DL%(3!!" I ' .#"4+3XD 3!")ZF.,%G%(

0! 1

(1+r)N

[h]

MU%(' I ' D½S<Y+E!3 >?'!3' >?+3E I +!®m3!06' .4!3D ![$3!"4\-' I DV+' *"#JKS<T%(!+E!"4 .4ÎI $\-'+ I O!' "#>K%(EODP+' >?+-DV+' O¹+!3¬.4=%("#3 I [L%( J?+3&S<T%>v"46«

Ä 9v"4+-71+;

2Î6D 3QL%(Q "#.#"#$' 3

Ω>?$+3!"¸q%(E3!DV I !"#>?+0G+E7

1[Tn = 1+ b] = 0, 99 ∀n »§02%(3 I [$!3W¬[L%('X!"4+3¼&+

2[Tn = 1 + a] = 0, 99 ∀n » v3X% I [DV+3!02%(+E¼ I 0]0G &.,%Â0G¹+0G]%(.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.=<Y)vD .#" I %(!"4!' "#>%(-G« %(R.=< [-Å-DV![W[L%('X!"4W+3®·DL%(3G) +0]D .4A.4Â>?+ +' 3GS< ' D !"4 A*"4DPKX%(-GS<T%(3!J?+E]s3X%("4]F8.=< "#!X%(E

0%.,%

DVK!"#Q "#.#"#$.4=%("#36s3!' I !"¸L+3/+ % I [+X%(-6 /% I !"m¬3!"4\E'$ "#.DV+' : I 6DV+3!0G+!!3] 1>?+*3XZD !"4F1' D 3!")D .#'RQL%\-' I +.#' "·%('\-'+.ZDV' 3!3X%("#OU<T%(!+*3GM

Ä %1D 3QL%(Q "#.#"#$ Â!RD .#'O"#0]D .4:\E':.,%D 3QL%(Q "#.#"#$ J?' >?+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 '$Á0G+' D 3QL%(Q "#.#"4! %(.=<Y$+X%(+' D [3!"4\-'\-'R'Á%(.#.4Á$+D 3' >?+3F&.,%¬.4 I !' 3*'DL%(3X%(JK3X%(D [!' "#>K%(EUM

$=8 &'. 8 ! " 8 5 & 5 : ! '%(.#.402%("#-+L%(E I !"4 $+3+3' I %(.#.q+' D ' Á!' 3.=<T% I !"S3!"4\-'$ OD 3!")]S<Y) +3 I " I K -Á.4D 3!")F.,%]%(

n+3-O$

Cn+Pn

^bv` ` b9v"#

T1, T2, ..., Tl' !' "#

l>%(3!",%(Q .4¬%(.4$U%("#3&"# $+DP+%(-:$\E' "4*"4!3!"#Q '$& .4" I 0]0' µ

+"#ψ' ¬s I !"4E' 0G$+3!"4\-'6!Åv0G$+!3!"4\E'6!' 3

Rl %Z !"S.=<YZ

s(j).=<Y$+.4$+0G+E

(s, s, ..., s)

Rj

[ψ(T1, T2, ...., Tl)] =

j=l∑

j=0

(

lj

)

pj(1 − p)l−jψ((1 + a)(j), (1 + b)(l−j)) (39)

Ë ÉG¢ ³+ s¥¯? s¡s¢ «S@AKJ(ω) = ]i |Ti(ω) = 1 + a

.,% $+0G!!3X%(!"4!/!3W6"#0]D .4G!".=<Y>?+' Q "4+Z3$L$ I [ "#3¬%('*)Î]DP"#-!' "#>K%(E6«Ä .,%.4"

J!Q "#0]",%(.4 DV+' +ØV+$ I 3!"#3 J =

∑li=1 Ui

\E'L%(%DVK$Ui =

1Ti=1+aM Ã .3%(.43¬FG3+02%(3\-'+3¬\E'

(Ui ; 0 ≤ i ≤ l) I !!"#!'' ' "#/"# $+DV+*H%(E6 ¬>VMT%*M -+3!' .#.#"j+.#.4 \E' [Ui = 1] = p ,

[Ui = 0] = 1 − p

Ä I 0]D O+-'Â O.,%/!Åv0G$+!3!"4 ' D DPK$ !' 3ψ?.,%/>PMT%

ψ(T1, T2, ....Tl)!' Rm I !"4Â

J

D .#'RD 3$ I "4$+0G+Eψ(T1, T2, ..., Tl) = ψ((1 + a)(J), (1 + b)(l−J))

Ù

Í 8%1 I

[ψ(T1, T2, ...., Tl)] =

j=l∑

j=0

ψ((1 + a)(j), (1 + b)(l−j))

[J = j]

=

j=l∑

j=0

(

lj

)

pj(1 − p)l−jψ((1 + a)(j), (1 + b)(l−j))

+6)685 >=@<

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

Cn =1

(1 + r)N−n

j=N−n∑

j=0

(

N − nj

)

pj(1 − p)N−n−j[Sn(1 + a)j(1 + b)N−n−j −K]+ »mÙ*M#Ñe¼

Ë ÉG¢ ³+ s¥¯? s¡s¢Ø«

(1 + r)N−nCn =

[(SN −K)+|Fn] =

[(SnTn+1....TN −K)+|Fn]

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

1

(1 + r)N−n

[(sTn+1Tn+2....TN −K)+]

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

!3!"4\-'ψs(tn+1, tn+2, ..., tN ) = (stn+1tn+2....tN −K)+

I *' "#¬F.=<Y$+JE%(.#"#$

c(n, s) =1

(1 + r)N−n

j=N−n∑

j=0

(

N − nj

)

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

Í +Z $*' "# "#0]0G$*",%(+0G+E&.,%s3!0' .4(4.1)

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

+6)6 0- < #+- < - 7A8A1 %(D DV+.4¬\E'.=<Y$+0G+!+' 3/ !3 I %(.#.Aj<T%D '®.4>?+*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 [$1! I 0]D .4+¼P' !!3X%($+JK"4 I ' >?+3!!' 31 I +!D !"4Mj5®%("46!"Á.4G02%(![$+02%(!" I "4+ØGX%(!"4=%("#%("4$+0G+-S< ' ![$3W+0G2S<Y)*"4!+ I .,%;QL%(\-'2\-' ".=<Y+0]D .4"4)*"#J?+3X%O\-'j< "#.*.#' "Em' 3! "4)vD .#" I "#+0G+-.4Á0GeÅ?+] I ' >v3!"#3 I <Y! I \E'Á'%(.#.4Am%("#302%("#E+L%(-ML9v"#

(Φn) = (φ0n, φn)

' /!X%($+JK"4:D 3$+>v"4!"#Q .4/!"#0' .,%(E .=<YD !"4S<T% I [L%( %

Cn = c(n, Sn) = Vn(Φ) = φ0n(1 + r)n + φnSn,

ÙKÚ

I 1\E'.=<Y$ I 3!"#3X% c(n, TnSn−1) = φ0n(1 + r)n + φnTnSn−1

3X%(D DV+.4¬'&\-'Tn

D 3+®\-'6.4/Î>K%(.4+' 3

1 + a+

1 + b %1 I

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

c(n, (1 + b)Sn−1)1[Tn=1+b] = [φ0n(1 + r)n + (1 + b)φnSn−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!F.=<T%(.#J?W+Q 3 Fn−1

I 0]D :+-' I /\-'Ä φn

! Fn−1H=0G' 3X%(Q .4

Ä TnR"# $+DP+%(- Fn−1Q !"4+-

c(n, (1 + a)Sn−1)p = [φ0n(1 + r)n + (1 + a)φnSn−1]p

c(n, (1 + b)Sn−1)(1 − p) = [φ0n(1 + r)n + (1 + b)φnSn−1](1 − p)

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

φ0n =

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

(b− a)(1 + r)n

»mÙ*MTÎ?¼

φn =c(n, (1 + b)Sn−1) − c(n, (1 + a)Sn−1)

(b− a)Sn−1

»mÙ*M ?¼

Í +3X%;\-'.=<YQ !"4+-:Q "4+ Î;!' "#¬D 3$+>v"4!"#Q .4 "#.Å%ÂD .#']«SF.=< "#!X%(EnP"#.!'43G/ I L%("#!3

.4 I ' 3] Â.=< "#!X%(E]"#0]0G$*",%(+0G+-2D 3$ I $ +-GDP' 313$3!JE%( "4+3GXDV3!m+' "#.#.48M '1.,%("41+) +3 I " I :.,% I ' >?+3!!' 36S< ' D !"4 ¬>?+EM

K


H %(R' I I [L%(D "#!32*''D .,% I +3O%(O' 02%(3 I [$ I 0]D .4+M1 %(D DV+.4 ' \-'/.4 $++E+' 3:S< ' I %(.#.%(0G$+3!" I %("#®%G.,%]DVK!"#Q "#.#"#$S<Y)*+3 I +3¬D !"4F]j< "#0]DV3!\-'+.·0G0G+- I 0]D 3!"4:+E!3 0

»ÁX%%(1S<Y$+0]"4!"4 ¼O+N

»sX%2%(1S<Y) +3 I " I e¼ "#.& I L%(!' 3+. 3+D 3$+-+3 I I %(.#.SDL%(3.,%G!' "#: :>VMT%*M

(Sn −K)+M@.#'J?$+$+3X%(.4+0G+E1

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

$8 . , ( . 0 < " :5A< $ )+, 5 & $=5 0 's+3%(D DV+.j%('*)Â0G¹+0G"4 $\-'&DP' 3.4RD !"4+' 3DV$+ /«v.,%>K%(.4+' 3

UnS< ' :D !"4

(Zk)F.,%%(n+3X%.,%X0]0GS<T%(3!J?+-G E1.42>?+ +' 31 "#]*"4!DVK+3F I +!;%(2DV' 3¹+!3Â+0G!' 3

S< [3+3] I -!3X%(]FZ' %(DVK!$+3!"4+' 3FnU< "#.Á3 I ' 3!]F' !!3X%($+JK"4ÂD !"#02%(.4 3X%("4X

DL%(3]3$ I ' 3!3+ I I +%(EZ+G!' D DVK I -'F8.,%®%(n.,%®0]0GS<T%(3!J?+E

Un$ I X%("#3F.,%

I ' >?+3!!' 36 /.=<YD !"4+E!3n+N

Un−1

"# %(.43 X%(!"4=%("#3¬.4OÎ I *"#!"4 ' "#>%(-6«Ä DV+3!0G+!!3 I ' >v3!"#3' $+>?+-!'+.v) +3 I " I .=<YD !"4]FR.,%%( n−1

I \E' "-+E!3X%("# Un−1 ≥ Zn−1M

Ä ¹+!3&%!' 3$ O*"4!DVK+3.4O.4+ +02%("#; O.,%:0]0GUn

I \E' "qDP+' ¹+!3OQ +E';+"#06' .,%(E.,%/>PMT%*MUn

F.,%]%(n Z "# I %>?"#3 Un−1 ≥ V (φ)n−1

Èφ!R' ZDV3!s+' "#.#.4:>?$+3!"¸q%(-

Vn(φ) = UnM @.,%O'X'= I 0]0G

(Vn(φ))R' :02%(3!!"# JE%(.42

βn−1Vn−1(φ) = βn

[Vn(φ)|Fn−1]LS<YÈ.=<YZ!"#3

Vn−1(φ) = 11+r

[Un|Fn−1]+.,%1' I *' "#¬F.,%] $¸ "#!"4!' "#>K%(E2«

+6859685 9?7 C K+#I <,+ HEIK+KIÑKM 9v"#

(Zk)' ;D !"4Ó%(0G$+3!" I %("# $¸ "4GDL%(33$ I ' 3!3+ I I +%(E;.,%®!' "#

(Un)DL%(3].4

$+JE%(.#"#$

UN = ZN , Un−1 = Zn−1 ∨1

1 + r

[Un|Fn−1]

Í *"#3X%]\E'Un

!R.,%]>%(.4+' 3 :.=<YD !"4(Zk)

F.,%]%(nÎ*M

(Un)!/L'7.=<Y+E>?+.4D DV6 69-+.#. :.,%1' "#

(Zn)

*Ñ

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

[Un|Fn−1]

M

jbEc+1 %1?b

Í ZDVK

Vn =∨

0≤p≤N−n

1

(1 + r)p

[Zn+p|Fn]

ÑKM58E!3+3&\E'Vn ≤ Un ∀n »sZDP' 3!3X% I 0]DL%(3+3 Vn

%>? I .,%>PMT%*M Xn ¬.4) +3 I " I "" . ¼

Î*M1 $L$ I [ "#3%('*):3X%("4DV' 3j.4\-'+.#.402%(.#JK3$Á.4A%(D DL%(3+ I .,%0]0GS<T%(3!J?+EV0

!+6D 3X%(!"4\-'"#'432X%(E6DP' 3 I ' >v3!"#3.=<YD !"48%(0G$+3!" I %("#F' "#!X%(-6M

43 < "%$ ,-$5 ! < (0! <" :5 <0! $ )+, 5 & $=5 0!*>: .0,< "J) 0! ZD !"4×%(0G$+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 <Y! I :\E' I *¸3!0G/.,%D 3DPK!"#!"4 I "Hµ '%(R.,%\-'+.#.4:' % D +3O.4RX%(!"4&!' "#>%(-/«

Ä (Zn 0 ≤ n ≤ N) $!"#JK+3X%1' 6D !"4%(0G$+3!" I %("#

Ä Cn+3X%.4:D 3!") I +!/D !"4F.,%G%(

nÄ cn

$!"#JK+3X%].4¬D 3!")ZF1.,%1%(n ¬.=<YD !"4+' 3DV$+

ZN

+6685 9?7 C K+#IÑKMCn ≥ cn ∀nÎ*M 9-"

(cn) ≥ (Zn)%(.43

(Cn) = (cn) Ë ÉG¢ ³+ s¥¯? s¡s¢ ÑKM 0]0G

(Cn)!' :' 3!02%(3!!"# JE%(.46+

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

Cn ≥ [CN |Fn] =

[cN |Fn] = cnÎ*M

(cn)!' G»s!' 3¼o02%(3!!"# JE%(.4:02%Ç!3X%(E

(Zn) .4 I %(3X% I W+3¬0]"# "#02%(.S< ' :+->?+.4D DP/

9-+.#.·+EX%("#%(.43Cn ≤ cn

+66 0 A A7= IGC < +( 7A8A < - ?7#< IGC %1D 3DVK!"#!"4ZD 3$ I $ +-%1DV' 3 I $\-'+ I :.4:3$!' .#X%( !' 3!D 3+L%(E&!' "#>%(-6«

c(fLhf! %d %§fqg9v"4+E

(Sn)' ®% I !"3!"4\-'$6+ K ' 3$+. ≥ 0

.4OD 3!")*' I %(.#.A%(0G$+3!" I %("# ((Sn −K)+ 0 ≤ n ≤ N)+

*' I %(.#.j+' 3DV$+ (SN −K)+EO$+JE%('*)M

Ë ÉG¢ ³+ s¥¯? s¡s¢ v@·K%(EZn = (Sn−K)+

(+%(D DP+.,%(-cn

.4D 3!")FO.,% %(n*' I %(.#.*+' 3DP$+

(SN −K)+*"#.!'43G /0GE!3+3 \-'

cn ≥ Zn ∀n M Í 3/cn =

[(SN − βNK)+|Fn] ≥

[SN − βNK|Fn] = Sn − βNKM

5®%("4 I 0]0Gcn ≥ 0

%G%('"cn ≥ (Sn − βNK)+ ≥ (Sn −K)+

S<YÈZ.4¬3$' .#X%(6M :3$' X%( %(L%(.4JK':j<Y!DL%>-3X%("SDV' 3.4D !"4 ¬>?+E2M

KÎ

, 5 >:*&><.0/+, : .0, ( . "%.:*$ )+,5 &'$=5 (0$!%8 < ( 8 ( +6)685 9?7 C K+#I9v"#

(n, x) 7→ u(n, x) (n ∈ 0, 1, 2, ..., N ; x ∈ R+¼ .,%;' "#] s I !"4 ¸ "4]DL%(3:3$ I ' 3!3+ I

I +%(-/DL%(3.4$+JE%(.#"#$

u(N,x) = (K − x)+ ; u(n− 1, x) = (K − x)+ ∨ pu(n, (1 + a)x) + (1 − p)u(n, (1 + b)x)

1 + r.

D 3!")Un

*'ÂD ' %(0G$+3!" I %("# OD 3!")S<Y) +3 I " I K !' 3.=<T% I !"3!"4\-'$ S !%(.43Un = u(n, Sn)

F:.,%%(n

Ë ÉG¢ ³+ s¥¯? s¡s¢ Í %SS<T%(D 3W " " Un−1 = Zn−1 ∨ 1

1+r

[Un|Fn−1]

3X%("4 /DL%(33$ I ' 3!3+ I I +%(E +O!' D DVK\-'

UnD ' "4:U<Y$ I 3!"#3 u(n, Sn)

.=<Y$+JE%(.#"#$/D 3$ I $ +E +>v"4+E¬%(.43

Un−1 = (K − Sn−1)+ ∨ 1

1 + r

[u(n, Sn−1Tn)|Fn−1]

Ã .S:3!¬D .#' \E'j<TF]%(D D .#"4\-'+3.,%m3!06' .4»Ñ?¼DP' 3Q + "#3 .4¬3$!' .#X%(6M

+6)6 :;< , 4<Í ZDVK+3X%

u = u(o, .)ÑKM58E!3+3DL%(33$ I ' 3!3+ I ¬\E'Y\-'+.4\E'&"# n

u(n, .)!QP3!&!' DV$+3!"4+' 3&S< ' &!' "#& m I !"4

% 3G +S<Y$*' "#3/\-'u!O% 3G/DL%(30G3 I U%('*) I E!"#E'Â $ I 3"4X%(-6+ I ->?)*M

58E!3+3&+Z' !3/\-'u!

> 0!' 3O.4:+JK0G+E

[0, K(1+a)N [

+-' .#.4/!' 3 I 0]D .4$+0G+-X%("#3MÎ*M Í DPK

A = x|u(x) = (K − x)+ 0G-!3+3 \-'j< "#.)v"4¬' 3$+.

x∗ ≥ 0+.·\E'

A = [0, x∗]*M58E!3+3&\E'/DP' 3R.4/ $++-+' 3 S< ' D ' %(0G$+3!" I %("#!' 3R.=<T% I !" S -O.4 I ' 3&F.,%]%( 0

!s0

"#.·)v"#¬' X+' "#.

σ ≥ 0+.\E'

Ä !"s0 ≤ σ

"#.jR"#*" V$+3+- S<Y) +3 I +3O' :>?+*3/XZD !"4F1.=< "#!X%(-0

Ä !"s0 > σ

"#.RD .#'R3$+06' $+3X%(+' 3¬ :>?+*3:.=<YD !"4D .#'!\E'6 ¬.=<Y)*+3 I +36M

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

Un = Vn(φ) = φ0nS

0n + φnSn.

3X%("4 +0G+-Á+1'DP"#-%(L%(.4JK'F """ I *' "#Á%(.43FO.,%Om3!06' .4R L%(-.,% \-'L%(E!"#$RS<T% I !"3!"4\-'$/F] $++ "#3 F.=< "#!X%(-

nDV' 3 I ' >-3!"#3!36D !"4 :>?+E2«

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

(b− a)Sn−1

+ +

$. ( K5 :-)+, >:?5 !,:-$ : $)!9v"#

s : R+ 7→ R

∗+

' s I !"4!!3!" I +0G+-/DPK"#!"#>?+: $+3!"#>K%(Q .4]!' D DPKX$63+D 3$+-+3:.=<Y$+>?.#' !"4 I ' 3OS< ' % I !"X%(3!"4\E'G»s $++3!0]"# "4e¼%(' I ' 3*'+0]D6» I E!"#-'q¼/ML9-" t < u

*.,%]\-'L%(E!"#$

rtu =1

s(t)

s(u) − s(t)

u− t

!.4/X%('*)S< "#E$+3¹+¬+E!3/.4O"#!X%(Et+uM /X%('*)ZS< "#-$+3¹+ "#!X%(-X%($

r(t)F1.=< "#!X%(E

t+3X% DL%(3

$¸ "#!"4

rt = limu↓t

rtu =s′(t)s(t)

» )vM#Ñe¼

% I L%("4X%( I : .,%:m I !"4 t 7→ rt+ &.,%/>K%(.4+' 3F6.=<Y3!"#JK"#

s0 &!3&% I !"·%(3!"4X\E' DP+3!0G+

3 I !!"#!'+3 .,%s I !"4 s I +.#.4H I "·!+ZS+.=< ' "4\-'/.#' !"4 ¬.=<Y$\-'L%(!"4*" S$+3+E!"4+.#.4

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

Í %1 Is(t) = s0 exp(

∫ t

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

∫ u

tr(v)dv)

» )vMTÎ?¼

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

s(t) = s0ert , s(u) = s(t)er(u−t) » )vM ?¼

43 ! . 5 : (0 ", < &'!>! .0! ( < < !>! >: . 695 0! : 5 H %(:' I 1DL%(3X%(JK3X%([V'/!' D DVK+3/ $ I X%(E!!3!" I +0G+EDPK!"#!"#>?

t, R, σMS@·' 3

I [L%\-'/+-!"4+3 N > 0'O%(.#.4 I !!3!' "#3/' Z02%(3 I [$/¸L%( I "4+3O &M 16M 1M ¬.,%=%Z!' "#>%(-G«

Ä R>%(.4+' 3 OÎ]% I !"m+3E Q+3!>?$ %('*) N + 1"#X%(E

0, t/N, 2t/N, · · · , Nt/N = tÄ :X%('*);S< "#-$+3¹+

rN+E!3/.4OÎ]%(

nt/N+

(n+ 1)t/N+3X%1$+JE%(.·F rT

N

M

(

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%(TN

1 , TN2 , · · · , TN

N )' !' "#

N>PMT%*ML"# $+DP+%(-/$\-' "4*"4!!3!"#Q '$/FG>K%(.4+' 3&%(

1 + aN , 1 + bN Z!' D DVK+3X%+Z' !3

N [TN1 ] = 1 + rN

I :\-' " '.=<T%>?>v' $++3!0]"#.,%.4"· R>PMT%*M

TNn

MÄ I ' 3: .=<T% I !"3!"4\E'$Â $

SNn

& $1DL%(3¬.,%2s3!0' .4SN

n = TN1 TN

2 · · · TNn

»s!' D DVKSN

0 = 1¼

DL% ·!' QV*"#>v"4!"4\E'Á'%U>? I !!3!' "#!' 3 [0 t]+>?+3

0\-'L%(

N → ∞ M Ã .EDL%(3X%("#AL%(!' 3+. :DV++3O\-'¬!36!' "#: /D 3 I 'O*"4 II 3+O>% I ->?+3!J?+31»s+Z' +\-':'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

tM@' "4X\E'

limN→∞

(1 + rN )N = limN→∞

(1 +Rt

N)N

= eRt ,

R%(D DL%(3X%("# I 0]0G/.4¬X%('*)ZS< "#-$+3¹+O"#!X%(-X%($6*'D 3 I !'R.#"#0]"#GM

@A' 3&%>?"#3O' :"#E+3!D 3$+X%(!"4%(L%(.4JK'6 σ*'O%(' 3OQPX"#Z*'Z.4+0]0G/!' "#>K%(E

6685 <>= =@<@A' 3Á'

N ≥ 1?2X ' ' "#

(XNn ; 1 ≤ n ≤ N)

N

>%(3!",%(Q .4%(.4$U%("#3"# $+DP+%(-+$\-' "4*"4!!3!"#Q '$F>%(.4+' 3A%(·.=<Y++06Q .4FOÎDV"#E − σ√

N,+ σ√

N !' D DVK+' !3Á\-'.=<Y!DV$+3X%( I

µN

XN1

!R+.#.4/\-'lim

N→∞NµN = µ

» )vM -¼ %' "#

(YN ) $¸ "4GDL%(3

YN = XN1 + XN

2 + · · · + XNN

I E>?+3!J?;%(.436+ .4">?+3/' ]JE%('X!"4+ G 0GUÅ?+

µ+O ¬>%(3!",%( I σ2 M

$-`fPg' UdUc_*d %§fqg@1 9-"φN

!R.,%m I !"4 I %(3X% I $+3!"4!!"4\-'/ YN 8%

φN (λ) =

[eiλYN ] = (

[eiλXN1 ])N

!"#.#"4X%(-&%(.43R.4:![$3W+0G6 /@A%(' . $+>vÅ3+.,%(!"ÁF.,% I ->?+3!J?+ I +.4"Á >?"#O\E'j< "#.S'=%(' 0G-!3+3 \-'

limN→∞

φN (λ) = exp(iλµ− λ2σ2/2)» )vMTÙ?¼

%s3!0' .46 %Åv.43&F.=<Y3*36ÎG%U>? I 3!/ %(JK3X%( J?/DV+3!0G+OS<Y$ I 3!"#3

eiλx = 1 + iλx− (λx)2/2 +R(x)%U>? I |R(x)| = 1

6 |λ|3|x|3 %1 I

eiλXN1 = 1 + iλXN

1 − 1

2λ2(XN

1 )2 +R(XN1 )

» )vM ?¼

%>? I |R(XN1 )| ≤ |λσ|3

6N√

N

M@3+L%(E¬%(.43R.=<Y!DV$+3X%( I / Î0G+0Q 3 G»*M ?¼: Q !"4+E

[eiλXN

1 ] = 1 + iλµN − λ2σ2

2N+

[R(XN1 )]

» )vM )K¼5®%("4 I 0]0G

KÙ

Ä | [R(XN1 )]| ≤

[|R(XN1 )|] ≤ |λ|3σ3

6N√

N= o(1/N)

Ä µN = µN + o(1/N)

DP+' $ I 3!"#3

[eiλXN

1 ] = 1 + iλµ

N− λ2σ2

2N+ o(1/N)

» )vM ¼1 +>?+ %(.43OF].=<Y$+!' /

φN« %

φN (λ) = exp[N log(1 + iλµ

N− λ2σ2

2N+ o(1/N))]

= exp[iλµ− λ2σ2

2+No(1/N) + ε(1/N)]

»ε $!"#JKL%(-1' Gm I !"4Ø+%(E1>?+3

0F.=<Y3!"#JK"#8A+.42!Åv06QV.4

log.,% $++3!0]"#L%(!"4

D 3!"# I "#DL%(.4]*'8.4JE%(3!"#![ 0G I 0]D .4)*e¼M Ã .A3!6D .#'¬\-'LF]=%("#3+*3 N >?+3&.=< "#*¸ "DP' 3Q + "#31»*MTÙ?¼

66 9?7 C K+#I@AK

YN = log SNN

.,%®' "# »YN

¼ I E>?+3!J?+¤.4"R>?+3]' JE%('"4+ Z 0GUÅ?+ −σ2/2+G

>K%(3!",%( I σ2 \E' "*3X%(D DV+.4.42%DP' 3O +"#$: :D 3QL%(Q "#.#"#$/.,%m I !"4 ν $¸ "46' 3RDL%(3

ν(y) =1

σ√

2πexp(−(y + σ2

2 )2

2σ2)

» )vMTÚ?¼

$-`fPg' UdUc_*d %§fqg 1 Í ;%log SN

N = XN1 +XN

2 + · · ·XNN

\-'L%(2%:DVK$XN

n = log T Nn

1+rNB¬$+3!"¸LO\-'¬.4!' "#(XN

n ; 1 ≤ n ≤ N)X%(!"4!s- %('*)Â[-Å-DV![W*'.4+0]0G "" .8«

Ä XNn

D 3+ZR>K%(.4+' 3%(.=<Y+X+06Q .4F]Î]DP"#-

log 1 + aN

1 + rN, log

1 + bN1 + rN

= − σ√N

, +σ√N

Ä .4R[EÅvDV![WS< "# $+DV+%( I 6- I .,%("#3+0G+E&>?$+3!"¸L$Ä .,%>?$+3!"¸ I %(!"4 2»*M -¼$ I !"#¬.4 I %(. I ' .' "#>%(-6«

9v"4+-f+u : R+ 7→ R

$¸ "4DL%(3

f(x) = logx

1 + rN; u(x) =

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

bN − aN

0]0Gu!$+JE%(.4OF

f%('*)GÎ&DP"#-

1+aN+

1+bN

N [f(TN1 )] =

N [u(TN

1 )] S<T%(' !3

DL%(3!*D ' "4\E'/'Z!O% 3GG

N [u(TN

1 )] = u(

N [TN1 ]) = u(1 + rN )

Í 8%1 I %(D 3W %U>?"#3O$:\-'f(1 + bN ) = −f(1 + aN ) = σ/

√N

µN =

N [XN1 ] =

N [f(TN

1 )] = u(1 + rN ) =σ(2rN − (aN + bN ))

(bN − aN )√N

1&%(D DP+.4 %(.43O\-'

1 + aN = (1 + rN ) exp(− σ√N

) = (1 + rN )[1 − σ√N

+σ2

2N+ o(1/N)]

1 + bN = (1 + rN ) exp(σ√N

) = (1 + rN )[1 +σ√N

+σ2

2N+ o(1/N)]

Ã .j+Z3$!' .#6\E'

bN − aN =2σ√N

+ o(1/N) ; 2rN − (aN + bN ) = −σ2

N(1 + rN ) + o(1/N)

Í 8%1 Ilim

N→∞NµN = lim

N→∞−σ2(1 + rN ) +No(1/N)

2(1 + rN ) + o(1/N)= −σ2/2

I 6\E' " I 0]D 6+E'*'Z.4+0]0G "" . $+0G-!3/.,%D 3DVK!"#!"4M

! < , . 8 !*(0 8 $& $( & <=8 ! '&%(.#.4¬Q + "#3¬.4OD 3!")8 &D !"4¬S<T% I [L%(¬+¬ >?+E EFG+0]D I E!"#-' E1+DL%XX%(E:FG.,%2.#"#0]"#%(G.4]s3!0' .4 I 3!3!DV%(EÂ*'¤0Gv W+.4Z " !" (" M@A' 3G' ;m"4Z'2'G"#E$+3X+3S<T%(QV3×»§"#.ÁÅ %DP' 3 I +.,% 3X%("4 I [ "4\-']\-'G'1 $ I ' >-3!"#3D .#'QL%¼GA%('*) D !"41 >?+EM@.,%/X' .,%D 3QL%(Q "#.#"#$N ∗

N

$ I 3!"#]%(¬.4DL%(3X%(JK3X%([1D 3$ I $ +E .4D 3!") PN0

F.=< "#!X%(E0*'®D '

:D 3!")S<Y)*+3 I " I K ! $/DL%(3.4$+JE%(.#"#$

PN0 =

1

(1 + rN )N ∗

N [(K − SNN )+] =

∗N [(

K

(1 + rN )N− SN

N )+] = ∗

N [(K

(1 + rN )N− eYN )+]

Ã .·U<T%(JK"# 02%("#E+L%(Eµ$+!'*"4+3 .,%2.#"#0]"# I +!\-'L%(E!"#$1\-'L%( N → +∞ MV9v"#ψ.,%]m I !"48 $¸ "4

DL%(3ψ(y) = (Ke−Rt − ey)+

8%

|PN0 − ∗

N [ψ(YN )]| = | ∗N [(

K

(1 + rN )N− eYN )+ − (Ke−Rt − eYN )+]|

≤ ∗N [|( K

(1 + rN )N− eYN )+ − (Ke−Rt − eYN )+|]

≤ K| 1

(1 + rN )N− e−Rt| → 0

» %] +3! "4W+36.#"#JK:D 3U>?+L%(- :.=< "#$+JE%(.#"#$+E!3:0Q 3O3$+.46« |a+ − b+| ≤ |a− b| ¼ "#.j+3$!' .#:\E'

limN→∞

PN0 = lim

N→∞ ∗

N [ψ(YN )]

5®%("4 vS<T%(D 3W.,%6 $¸ "#!"4;0G¹+0G¬ O.,% I E>?+3!J?+ I :+Â.4"-%G-D ' "4\vψ! I E!"#E':+QV3!$P» I

\-' "S/+3X%("#RDL%.4 I %DP' 3.4 I %(. I ' .A%(L%(.4JK'/3+.,%(!"Á%(' I %(.#.¼¬

limN→∞

PN0 =

∫

R

ψ(y)ν(y)dy =1√2π

∫

R

[Ke−Rt − exp(−σ2

2+ σy)]+e−y2/2dy

» )vM#ÑU?¼

)

6)685 I + - +K I -IG< 7A <- /IK+K7A < '%U>?AÇ~'\-'j<TF:D 3$X+E!' D DPKX$ \-'.,%6>%(.4+' 3"# "#!",%(.4

SN0

+.=<T% I !"3!"4\-'$ $+X%("#$+JE%(.4¬F 1 3+D 3+

02%("#E+L%(-.4 I %(. I ' .4%(' $+Q ' *'DL%(3X%(JK3X%(D [ " +!' D DVKX%(-\-' SN0

!' I !X%(- $++3!0]"# "4!S0

»§/ $+DV+%(E DL% N¼&M

7 D DP+.,%(-S′ + P ′ .4O>%(.4+' 3 3!DV I !"#>?O 6.=<T% I !"X'Ç% I +E +&*'ZD ' %( I +!/' >?+.#.4!"#!'L%(!"4

%S′N

N = S0SNN

63!:\E'

P ′N0 =

1

(1 + rN )N ∗

N [(K − S′NN )+] =

S0

(1 + rN )N ∗

N [(K

S0− SN

N )+]

%1.#"#0]"#P

I +!:)vD 3"4\E'L%(N

+>?+3R.=< "#*¸ "·+3X%.4¬D 3!");*'D ' %(R.4:0Gv W+.4/F+0]DI E!"#-' "#.je<Y$ I 3!"#6 +Z>?+3!!' G»*M#ÑU?¼

P =S0√2π

∫

R

[K

S0e−Rt − exp(−σ

2

2+ σy)]+e−

y2

2 dy =1√2π

∫

R

[Ke−Rt − S0exp(−σ2

2+ σy)]+e−

y2

2 dy

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

F (x) =1√2π

∫ x

−∞e−

−y2

2 dy , d1 =1

σ(σ2

2+ Log

S0

K+Rt) , d2 = d1 − σ.

» )vM#ÑKÑe¼

Í Z>?"#=% I "#.4+0G+E&\E'P = Ke−RtF (−d2) − S0F (−d1).

» )vM#ÑUÎ?¼

Í D 3 I W : ¬.,%60G¹+0G:m%;DV' 3 $¸ "#3 EF+0]D I E!"#-' E:.,%>%(.4+' 3*' I %(.#.j &D 3!")S<Y)*+3 I " I K + 6%(S<Y)*+3 I " I t « C +3X%Z DL%(3& $¸ "#!"4 .,%].#"#0]"#6 &\-'L%(E!"#$

C ′N0

\E'L%(N → ∞ M %1m3!06' .4

:DL%(3!"#$/$ I 3!"#/DP' 3R.4RD 3 I !'O " !" (" I *' "# %('*)$+JE%(.#"#$

C ′N0 − P ′N

0 = S0 −K

(1 + rN )N

H <YÈ+DL%X%(-FO.,%O.#"#0]"#&C−P = S0−Ke−Rt

]%O I C = S0[1−F (−d1)]−Ke−Rt[1−F (−d2)]

I :\-' "jDP+' + I 36U<Y$ I 3!"#3C = S0F (d1) −Ke−RtF (d2)

» )vM#Ñ?¼ $+JE%(.#"#$»*M#ÑKÑe¼R+6»*M#Ñ?¼ I !"#!'+EO.40Es3!0' .4 .0/§_2143 _g65 7 1;:f</§b 2" E

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cabanal/Annales/Azema/Finances/Polly/... · !3 o d !"4 f¬.,% i 3!qp +"#.#.4 + f x%( +3u e + 2je%(3 %( - .4 ' 3!"#3 k e\-' >? ' % >? +Ò&dp +3 *' .,%¬ x%(.#"# $& >? !3