Essais sur le risque de crédit des obligations : Analyse de la migration des notes et des effets de contagion

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1 Essais sur le risque de crédit des obligations : Analyse de la migration des notes et des effets de contagion Myriam Ben Ayed Ghamgui To cite this version: Myriam Ben Ayed Ghamgui. Essais sur le risque de crédit des obligations : Analyse de la migration des notes et des effets de contagion. Business administration. Université de Cergy Pontoise; Institut supérieur de gestion (Tunis), French. <NNT : 2013CERG0639>. <tel v1> HAL Id: tel Submitted on 8 Jan 2014 (v1), last revised 27 Jan 2014 (v3) HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 UNIVERSITE DE CERGY-PONTOISE E.D. ECONOMIE, MANAGEMENT, MATHEMATIQUES CERGY LABORATOIRE DE RECHERCHE THEMA INSTITUT SUPERIEUR DE GESTION DE TUNIS E.D. SCIENCES DE GESTION ESSAIS SUR LE RISQUE DE CREDIT DES OBLIGATIONS: ANALYSEDELAMIGRATIONDESNOTESETDESEFFETS DE CONTAGION THESE pour l obtention du titre de DOCTEUR EN SCIENCES de GESTION de l Université de Cergy-Pontoise et de l ISG de Tunis Présentée et soutenue publiquement le 22 Mars 2013 par Mme Myriam BEN AYED JURY Directeurs de thèse: Monsieur Adel KARAA Professeur à l ISG de Tunis Monsieur Jean-Luc PRIGENT Professeur à l Université de Cergy-Pontoise Rapporteurs: Monsieur Jameleddine CHICHTI Professeur à l ESC de Tunis Monsieur Fredj JAWADI Maître de Conférences à l Université d Evry Examinateurs: Monsieur Mondher BELLALAH Professeur à l Université de Cergy-Pontoise Monsieur Olivier SCAILLET Professeur à l Université de Genève 1

3 Je dédie cette thèse à mon père, ma mère, mon mari et mon fils

4 Remerciements Au terme de ce travail, c est avec beaucoup d émotion que je tiens à remercier tous ceux qui, de près ou de loin, m ont aidé à réaliser cette thèse. Je tiens tout d abord à remercier mes deux directeurs de thèse, Mr Adel Karaa et Mr Jean Luc Prigent, pour m avoir dirigé. J ai beaucoup appris à leurs côtés et je suis très honorée de les avoir eu comme encadreurs. Je tiens à les remercier pour leur patience, mais aussi pour leurs conseils et leur soutien. Je remercie tous les membres du jury pour avoir accepté de participer à mon jury de thèse : Messieurs Fredj Jawadi, Jameleddine Chichti, Mondher Bellalah et Olivier Scaillet. J adresse également toute ma gratitude à toute ma famille, tous mes ami(e)s et à toutes les personnes qui m ont aidé dans la réalisation de ce travail. Je remercie profondément mon mari, Anis Ghamgui pour tous ses encouragements, son aide précieuse pour la collecte des données et son soutien. La thèse a été parfois un moment difficile, et très preneuse de temps, mais tu as toujours su être patient et compréhensif. Je tiens également à remercier mon frère Mohamed qui a toujours été auprès de moi. Enfin, les mots les plus simples étant les plus forts, j adresse toute mon affection à mes parents. Malgré mon éloignement depuis de nombreuses années, leur confiance, leur tendresse et leur amour me portent et me guident tous les jours. Merci pour avoir contribué à faire de moi ce que je suis aujourd hui.

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48 & D55D /# (5#22>2 5D># '&(L>I " [0,t] B N t = T n t1, :; Z t = N t k=1 X k. : ; D. =(ω;ds;dx) Z A B(R), (ω;(0,t] A) = T n ti Xn A3. & " 9 ) - " ν = ν(ω;ds;dx) " " ) ( ν) " " : 9 A B(R {0})9 ((ω;(0,t] A) ν(ω;(0,t] A) t " ; :) " U 2<) :..; ; P. #G n (dt,dx)! (T n+1,x n+1 ) + F Tn #, -./01 ν(dt,dx) = I{T n <t T n+1 } G n(dt,dx) G n 0 n ([t, [ R). :; )F n! T n+1 + F Tn F n (dt) =G n (dt R). "1 &K K(T n+1,dx)2 X n+1 +F Tn T n+1! L. F (T n,x n )1 H 3 H(ω,t,x) = T n t1 ]]Tn,T n ]]h n (x) h n (x) =h(n;(t 1,X 1 );...;(T n,x n ))(x). :; $ H + v H ν t = + t 0 H(ω,t,x)ν(ds,dx) = :%;

49 + n 1 + k 0 + t Tk T k #7#'#2 =5D/&'#D&>7 % G k (ds,dx) h(n;(t 1,X 1 );...;(T k,x k ))(x) G k ([s, [ R) G n (ds,dx) h(n;(t 1,X 1 );...;(T n,x n ))(x) T n G n ([s, [ R) I {T n<t T n }. / 9 "9 " " ) D! & ' E!. - "* B " ) /) " $ T n = t n : "; D F Tn = F n R F n -$" "t n D 9 ) B φ(x)ν(t n,dx) =E[φ(X n ) F n 1 ]. E!. < "* " -) " )< D"9 " ) ) 2 Φ = (T n ) n R Λ " -(X n ) n X )") G Φ " Φ = ((T n,x n )) n G?" Φ )" $ ) Φ " - Λ G &9 B ν(dt,dx) = F n (dt) {Tn <t T n } n 01 F n ([t, [) G(dx). " (? - " * 2 Φ = (T n ) n " * D ) B?; " N(A 1 ),...,N(A n ) -? " " 6 4 A 1,...,A n :? " ;?; 2A B + &9 B Λ(A) P[N(A) =k] = k e Λ(A), k!

50 #7#'#2 =5D/&'#D&>7 R Λ(A) = λ A, A " A " "? " " " * " )< " "* " " -) $ B - -J " ) ) "* # 9 ) ) " -) $ ) J Y E!. > " Φ = ((T n,x n )) n " Φ = (T n ) n X 1,...,X n... " Φ P[X n Φ] =G(T n,dx) / "? Φ B ν(dt, dx) = G(t, dx)ν(dt). " B? Φ ) " - Λ ν(dt,dx) =G(t,dx) Λ(dt) Z t = T n t X n " "? : ; / 9 " Φ ) "? X 4 F t 4 DN t (B) " ) " B X " [0,t] Φ F t " "? :G/ 2 G /2; - " ν [0,T] X9F 0 " B P[(N t N s )(B) =k F s ] = (ν(]s,t] B))k e (ν(]s,t] B)), p.s. k! R#$. 2 B 41N(. B) 1 N(. B) F t 2 ) 1 $ $& 4

51 #7#'#2 =5D/&'#D&>7, ν({t}) = 0 4 ν 3 ν(dt,dx) =E[N(dt,dx) F 0 ] " " "8) " " B - '8) Y "* ) B Z '8) Y = (Y t ) t ) " ) 2{λ i :,i Y} $" " $ (T n ) n " "8) P[N. Y] P? 4 - ) $ -t λ Yt & -) " " 9 F 0 F Y D ) B P[(N t N s ) =k Y] = ( t s λ Y u du) k e t s λ Yu du. k! &9 " Φ B ν(dt) =λ Yt dt. J "" )" Y - " )?? " D ) B ν(dt,dx) =λ Yt dtg(t,dx) R#$. #Y $ 1F t 3 ν ν(dt,dx) =E[λ Yt F t ]dtg(t,dx), 5 E[λ Yt F t ] 6 F λ Y : 4 + F * 2

52 #7#'#2 =5D/&'#D&>7 + E!. D " "* R ) " ) "8) 2 ((U n,x n )) n ) 4 ) ]0, ] X RX " " " 9 X?" N 2 B P[U 0 dt,x 1 =j X 0 ] =Q 0 X j (dt) :P ps. P[U n dt,x n+1 =i X 0,U 0,...,U n 1,X n ] =Q Xni(dt) RQ 0 ij Q ij " ]0, ] (i,j) X X B Q 0 ij(]0, [) = 1 Q ij (]0, [) = 1. j j / )9 (T n+1,x n+1 ) F Tn ) X n &9 ((T n,x n )) n )T n = n 1 i=0 U i "?"8)#% )- X9$"Q 0 ={Q 0 ij/(i,j) X X} < :G8 <G; $" Q = {Q ij /(i,j) X X} < "?"8) :2'; ( j X Q0 ij =F 0 j X Q ij =F i "?"8) )" : -U i )-F F 0 ; $9 Q 0 ij(dt) Q ij (dt) " 4 " σ γ R ) B Q 0 ij(dt) =q 0 ij(t)γ(dt) Q ij (dt) =q ij (t)γ(dt) D Q 0 j = j Q0 ij Q j = j Q ij / r 0 ij(t) r ij (t) q ij (t) Q i [t, [ q ij(t) Q i [t, [ j "9 " $ r(i,j) ) r(i) = r(i,j), ) B Q ij(dt) =r(i,j)e r(i)t dt Q 0 = Q X Z '8) "? * / 9 Hr(i,j) p(i,j) = r(i,j) r(i) $

53 #7#'#2 =5D/&'#D&>7! / "ν B ν(dt,dx) = G n (dt,dx) {Tn<t T n } n 01 G n ([t, [ X) J " " ν " B ν(dt,{j}) = 1 t T Q 0 X i (dt) Q 0 X ([t, [) + n 1 # 1 {Tn<t T n } Q 0 ij(dt) =q 0 ijγ(dt) Q ij (dt) =q ij γ(dt) Q Xnj(dt T n ) j Q X nj([t T n, [) ν(dt,{j}) = 1 t T r 0 X j(t)γ(dt)+ n 11 {Tn<t T n }r(x n,j)(t T n )γ(dt T n ) ) B Q 0 ij(dt) =Q ij (dt) =r(i,j)e r(i)t dt, ν(dt,{j}) = 1 t T r 0 X,j (t)dt+ n 11 {Tn<t T n }r(x n,j)(dt T n ), < B K(T n,j) = Q X n j(dt+t n ) k Q X n k(dt+t n ). "" )9 $" -? " " ν9 *" "9 < " * T"9 " J " 4 F t : " E; B # (F t ) t % Φ = (T n,x n ) n (Y t ) t 1 ν(dt,dx) = G n (dt,dx) {Tn <t T n } n 01 G n ([t, [ R) 5 G n (dt,dx) (T n+1,x n+1 ) + Φ1 (F t ) t

54 E! %. #7#'#2 =5D/&'#D&>7 %. "* 0? " -< ")? "" " " - ) " ) )- " 9 ) 9 "? ) 9 "9 4 " ) D ) " R )? " -) - )9 "* - "* &/:m9q; # ( :!!,9!!+; "* 4 d j+1 =T j+1 T j B d j+1 =ψ j+1 ξ j+1, Rξ j ) B m ψ j =ω+ α k d j k + k=0 q β k ψ j k. $ A - "* &/ t ]]T j,t j+1 ]] 4 B λ t =ψ 1 t Tj j+1λ 0, ψ j+1 R λ 0 $ ξ : $ ) ξ; / " ξ B - S9 B λ t =ψ 1 j+1, B λ t = Γ(1+1/γ)ψ 1 j+1 γ(t Tj ) γ 1 γ, R Γ $ "" γ "* S # "9? - - :!,.9!+; ( # :!!+; D " " ) B υ(dt,dx) =λ t K(t,dx), R t ]]T j,tj+1]]9 -λ t < K(t,dx) $ 4 - "? * &/ k=0

55 #7#'#2 =5D/&'#D&>7 % E!. D?* " $" T. 1. #" / "* = ) = # λ > 0 exp( λt) λexp( λt) S α>0, λ>0 exp( λt α ) αλt α 1 exp( λt α ) "" "" γ> 0, λ>0 α>0, λ>0, γ > 0 t t 1 Γ(γ) λ(λs) γ 1 exp( λs)ds 1 Γ(γ) λα(λs) αγ 1 exp( (λs) α )ds?" σ,>0 1 Φ lnt σ α>0,? 2?' 1 λ>0 α>0, λ>0, γ > 0 α>0, λ>0, σ> 0 λt γ exp( λt) Γ(γ) λαλt αγ exp( (λt) α ) Γ(γ) 1 σt ϕ lnt σ [1+(λt) γ ] 1 (γλ γ t γ 1 ) [1+(λt) γ ] 2 [1+(λt) γ ] ( α λ) γ (γα γ λ γ t γ 1 ) [1+(λt) γ ] 2 1 σ [1+σ 2 (λt) γ ] (1/σ ) (γλ γ t γ 1 ) [1+σ 2 (λt) γ ] (1+1/σ )

56 #7#'#2 =5D/&'#D&>7 % T. 2. #" :; / = " ) $ dh(t)/dt # λ dh(t)/dt = 0 S αλt α 1 α>1 α<1 α = 1 "" ""?" λ(λt) γ exp( λt) λ(λs) γ exp( λs)ds t λα(λt) αγ exp( (λt) α ) λα(λs) αγ exp( (λs) α )ds t σt ϕ [ t σ ] 1 Φ[ t σ ] (h(t) =λ) γ> 1 γ< 1 γ= 1 (h(t) =λ) / "" ) "J λ>0 γ> 0 α = 1 / ) "J λ α = 1 γ= 1 *? γλ γ t γ 1 [1+(λt) γ ] 1 * γ> 1 γ 1 2?' γα γ t γ 1 [1+(λt) γ ] 1? :γ9λ; λ =α; S :α γ 9γ; λ = 0; :α; λ = 0 γ= 1; * γ> 1, 1 (γλ γ t γ ) σ [1+σ 2 (λt) γ ] 1 γ< 1? :γ9λ; σ=1; S :α γ 9γ; λ = 0; :γ; λ = 1 σ= 0; * γ> 1 γ< 1

57 &&5D2 #D (2L># /# (I/ % % ( "" 9 " ) " * $9 ""!!. B? - B # "* ':!,; # " ) " 2? '9 G`) "G $ $ $ " ) ) $ " 4 "? F ' $ " "" " :;9 -" $ 4 19 " * """ Z > -" 4 19 * ) " * " Z $ 4 " " 4 " " " 4 G)" G $ $ 59 ) 9 -) " " $9 $ " ) ) *" "? ")"*'4" ) D 1 -) )

58 &&5D2 #D (2L># /# (I/ % "* ' - $" 4 ) "" :) ; 2 B & GL CG F $ " " "* )? * $ - $ -" " - $ :) " & 2 :!!; # (< :!!+; R $ " "* ;? - 4 B J -)*? " $ "" * / 9? ) 9 "" UE :!!%; /H 2 :!!!; " $ "" )*" ") :G) G;9 < 4 "" "* " $" G G ) " 9? " " "" & 2 :...; "* :G 8G; ) "* " " - /0 :!!!; " "* - ) "* " - H" ) ) " 9 ""

59 &&5D2 #D (2L># /# (I/ %% /H 2 :!!!;9 " ) ) :) /09!!+; " $ "? ) "* 4 " * 9 " R "" 0 *" -) H"" "? - $ B > " -) " UE9 9 :!!,; Z '8) "9 " 9 9 " - 4 i4 t9 j 4 t+1 >? $ -) $ " ()" " " * 9 9 ^9 < :!!!;9 " " $ " - "J" 9 " 4 " "* Z '8) 4? ) D " " 9? " 4 9 ) " $"" 9 "" " * :9 4 $ ; " c) " $ "* - ) B "* ^'Q9 ) - 9 ' :!,;9 "* (8] G 2 = G "* U ' - "* "* 4 H " " 4 " ) ) $ % ( $ ) * / ) "* 9? " - < ) "* " ) J < ) 4 9

60 &&5D2 #D (2L># /# (I/ % -) J ) '? "" -) * H 2 U ' :!!,; 2 = :!!,; J " " 6 J " " $ -J " " " > "" $ ) 4? " '8) -<? * '8)9 9 $9 H? -) * ) ) # 09 -? # - "9 ) $ - " - 4 ) " ( " -) 4 A - - " 4 ) " " - ) # " -" $ -"? D "" " " AAA " AAA * ) $? $9 -?4? " $ $9 9 $ - AAA 4 -A # " ) B P 0,2 (D AAA) = P 0,1 (D AAA).1+P 0,1 (AA AAA)P 1,2 (D AA)+... +P 0,1 (CCC AAA)P 1,2 (D CCC) D ) - " - i ) j 0 2 B D P 0,2 (j i) = P 0,1 (k i)p 1,2 (j k) k=1 2 - " 4 -<*? 9 -?4? P 0,1 P 1,2 9 -? - $K ) BP 0,2 =P 2 0,1 : "P 0,1 ; 9 " " "9 " 4 - B P 0,N =P N 0,1

61 &&5D2 #D (2L># /# (I/ %, &" 9 4 A T -" $K ) B P 0,T = exp(tψ) R Ψ " D - - " X - $" - )" * B exp(x) = X n n!. n 0 - " )? - '8) 5 " " ) " $ / 9 - " 4 A B Ψ(1) = 1 N (Ψ(N)). # " 9 ) "" 5 " -) " 4 $" ) $ &9 )"J"4 "V4 " $ ) > $ " $ " - / - ") 9 $ " " ) > " : CCC "; ) $ ) 4 "?4 - ) -" / $ $E $ $ "J" - "" "" "? $ * - ) &9 "" 4 ) )? " "J" ) % # + - " " * ) $ * 5 " 4? " " ) - ) ) $ - "

62 &&5D2 #D (2L># /# (I/ %+ D 4 $"" /H2 :!!!;9 ) " - $9 $ "" )" ") D ""K )" " " $ $ 29-4-J 4 r - A? $ 4 t" B B(t,T) =B(T,T)exp( r(t t)) = exp( r(t t)). #" " - ) $? ) "T 2 - $9 )" : ) "; D ) A? 4 B RT 1 - $ B d (T,T) =I T >T, / 4 -tb B < $ )t 4 -t " )" 9 )B(t,T) 4 -t B -< $ )t 4 -tb ) B(t,T) 4 - t $" )? B B d (t,t) = exp( r(t t))e Q [B d (T,T)/F t ], R Q 2T 1 QB )λ>0 D ) B f(θ) =λe λθ B d (t,t) = exp( r(t t))e Q [I T >T F t ], ) 4 B B d (t,t) = exp( r(t t))e Q [I T >T T 1 >t]. D B B d (t,t) = exp( r(t t))q[t 1 >T T 1 >t]. " - "" 9 B

63 &&5D2 #D (2L># /# (I/ %! B d (t,t) = exp( r(t t))e λ(t t). # 9 Z sp: ; - ) $ B sp =λ 2 " )" $? )L) 0 L<1 B B d (T,T) = (1 L).I T T +I T >T. B < $ )t 4 -t " )" (1 L)9 ) B(t,T) 4 - t 4 (1 L)exp( r(t t)) B -< $ )t 4 -tb ) B(t,T) 4 - t $" )? B B d (t,t) = exp( r(t t))e Q [(1 L).I T T +I T >T/T 1 >t] :,; 2 9 Q9 L T 1 L = E Q (L). D B B d (t,t) = exp( r(t t)) (1 L)Q[T 1 T T 1 >t]+q[t 1 >T T 1 >t] 2 <* T 1 Q9 B B d (t,t) = exp( r(t t)) (1 L) 1 e λ(t t) +e λ(t t). 9 ) ) ) B exp( [r +sp](t t)) = exp( r(t t)) (1 L) 1 e λ(t t) +e λ(t t). ) sp B sp =λ 1 T t ln (1 L)(eλ(T t) 1)+1. " ) " 9 B sp λl). # 9 ) $λ" - L < ) -<" /H 2 :!!!; :;

64 Q &&5D2 #D (2L># /# (I/. " $ 9 ) D B λ s : 4 -s Q $ ) ss+1 " $"? 4 -s $ s Z s : )" " $ 4-s r s : $ $ $ 4 -t9 ) "B(t,T) ) "X t+1 $ tt+1 ) - B (t+1,t) - $ B B(t,T) =λ t e rt E Q,t (X t+1 )+(1 λ t )e rt E Q,t (B (t+1,t) ), :+; RE Q,t (.) - Q9 4 -$" ) 4 t - ) :+; -"9B(t,T) J " "* )? "" B j T 1 r tk j k B(t,T) =E Q,t λ t+j e (1 λ t+l 1 ) j=0 T +E Q,t e r tkbd k (t,t) t+j+1 Z t+j+1 l=0 T (1 λ t+j 1 ). :!; -) $" " ) $ $ 4 6 Z9 r λ ) A "* - :!; J " - )" 4 -s9 $ ""s+19 "" $ )? ) " ) "s+1 dg ) "G :('Q;e 2 <*9 Z9 19 B J=1 E Q,s (Z s+1 ) = (1 L s )E Q (B(s+1,T)). # ('Q - :+; 9 B B(t,T) = (1 λ t )e rt E Q,t (B (t+1,t) )+λ t e rt (1 L t )E Q,t (B (t+1,t) ) :.;

65 &&5D2 #D (2L># /# (I/ =E Q,t T e R tjbd j (t,t), ) B e Rt = (1 λ t )e rt +λ t e rt (1 L t ) $ 9? R t B B R t =r t +λ t L t. sp t =λ t L t. :; - :.; - J " "" ) B d (t,t) "" $9 $ $ 6R t /H 2 :!!!; " -" R t = r t +λ t L t $ 6 6 " "9 λ t 9 Z t? *9 ) "" "* -)? $9 - $ 6R t =r t +λ t L t r t "9RJ""*<""? $ 9 < " "* 9 ( :!+%; 9 "" "* T9 UE ' :!! ; :TU'; -" 4 " $ """ 4 -) $ ) $ $" ('Q " λ )" 4 ) # 9 λ t L t )? 4 " 9 " 0 "* l "" $ V -" $ D 9 )" 9 "? * -) - ) $ 6 B R t =r t +λ t L t +l : ; # 9 $ W OP $ / 9

66 &&5D2 #D (2L># /# (I/ O GrG "" OP $ #9 R t =r t +λ t L t +l, Rl " 0 $ " V? " $ 5 :,F,P)? ) - 4 t-) " W $ 6-4 s & 9 r t " $ - ) - "8)9 - "* $E TU' > $ - :W9 τ; ) W " -J τ W < D W " : " J 0 $" " ; D "" " )? Q ) r t 9 $ :W9τ; 9 B(t,T) B(t,T) = 0 t τ τ B(t,T) =E Q exp( r u du)w :; )t<τ RE Q - Q D $ ) $ "" ((X,T),(X,T )) " (X,T) - -" 4 < X 4 T *" " (X,T ) " -J T " $ K) " X " (Z, r) $ ) $ ((X,T),(X,T )) B τ = min(t,t ), W =X1 {T<T } +X 1 {T T }. t :; :%; D ) " A (X = 1;9 ) 4 T9 ) - "9 "" " -#9 )X $" " 4 -T D ) -$ ) $

67 % 5D>25D ((X,T),(X,T )) - (X,T)?"J" " $ ) $ 9 "" ) 5#) - -" ) J?"J" 0 -?6 D 6$ U -$ ) $ ((X,T),(X,T )) 5 " $T 4 -h< $ 9 Λ9 4 0 ) $ 4 1 * :-?4?9 Λ = 1 t T ;9 J $" B dλ t = (1 Λ t )λ t dt+dm t, :; RM " Q 5 Λ t "" - -)4-t:Q;-" $ / "J"9 $9 -$" F 4 - t9 - $ t9 ")" 4h t Λ t Λ t #" $" )V t < B V t =E Q,t exp T t r u du X :,; D $" ) ) GB G t = exp t r u du V t (1 Λ t ) :+; t + 0 exp s r u du (1 L s )V s dλ s. :!; 0 0, # "" 9 " $" " 0 9 " $ " " - " " -4"") " " "" 9 #9 $" 4 " -" -0 / 9 $ 4 " $" ) ) 9 ) "

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72 '5/I2&5D /#2 &(5>(2 /# D5&5D2 =D&D3(#2 /#2 =('#2 + ^" 2 :..+; "* 4 0 " # ^"9 ' :..%; ) "* )9 " 9 " $ <" ""! - D < " - 2N ) - - * " - " " "9? ) - 2N? - -"? 9 -) " -? / 9 AAA,AA,A,BBB,BB,B,NR. 2 F = 532 " " N f " " " f (f = 1,2,...,F) D B τ f n B -)?*" " "f. X f τ n ) " 4 -? "BX f τ n ERE ={AAA,AA,A,BBB,BB,B,NR}d f n =τ f n τ f n 1, 9""9-) " - 4 " 6? 9 "f (f = 1,2,...,F)9 B (y f 1,y f 2,...,y f N f ),Ry f N f = (X f τ n,d f n). D -" " "J" " X t 9t R9 ) - E 9X t < * ) 4 t =""9 X t "* ) B X t =X f τ n t [τ f n 1,τ f n[. ) X t 9 "9 " -) " "J" - 4 "* " 4 "" " :G"8 G; :) ; /- "* 9? 4 -) : " ) - ; ) :<

73 '5/I2&5D /#2 &(5>(2 /# D5&5D2 =D&D3(#2 /#2 =('#2! -); / -<9 " J K " 6 ) " "? : "?" "? " ); * :; - " 6 9 " 9 -) X t "* 6-4 -) F. 1. #" 6 -) 7 0 ) -) " "" * "9 9 " )*" $ # )9 )9 "f9 $ "" $ " B f(y f 1,y f 2,...,y f N f 1 ;Xf τ ;θ) = :.; f(y f 1/X f τ ;θ) f(y f 2/y f 1,X f τ ;θ)... f(y f N f /y f 1,y f 2,y f 3,...y f N f 1 /Xf τ ;θ) Rθ ) "* $ $ )" " " < "* 4 " - 2N

74 '5/I2&5D /#2 &(5>(2 /# D5&5D2 =D&D3(#2 /#2 =('#2,. /-* :.;9 9 ) 9 $ " n (n = 1,2,...,N f ), 6 (X f τ n,d f n) "9 9 " 4 9 * 9y f n 1 / 9 ) $":.; "* ) B N f f(y1,y f 2,... f,y f N f 1 ;Xf τ ;θ) = g X f τn (Xτ f n,d f n/in 1;θ) f : ; R g X f τn (Xτ f n,d f n/i f n 1 ;θ) $ 6 (Xτ f n,d f n) $ n ième ) $ " f9 6 n ième "? - d f n "9 - ) k ) " < ) - Xτ f n 9 : ;9 4 -$" 6 4 "f,in 1 f # -< "* 4 9 < ) 4 J "" - 9 )9 " 6 K ) B n=1 D f n,k ;k =AAA,AA,...,NR. D f n,k : " "; - -)*" (n 1)ème " -) - ) " 4 - * " ) k < k - - $9-4 ) B D f n,k D = min f n,aaa,df n,aa,...,df n,nr.? - ) "* "* (i.e.dn f =D f n,k )9 "" "* " "" 4 (i.e.dn<d f f n,j ;j=k) " )D f n,k (k =AAA,AA,...,NR) J $ B d fn/i fn 1;θ j,k. h X f τn,k ) $ " " f :? " < ) -X f τ n ;9 "*9 $ 4 :!!.9.;

75 '5/I2&5D /#2 &(5>(2 /# D5&5D2 =D&D3(#2 /#2 =('#2, -) - ) " " - ) k 9 - " -? 6 49 d f n " * " ) " - $ -*)" - ) :"? ; - ) # $ ) Dn f B d n S X f τn,k (df n/in 1;θ) f = exp h X f τn,k (u/if n 1;θ 1 )du : ; ) B θ = (θ AAA,θ AA,...,θ NR ) et X f τ n E $? 9 h X f τn,k (df n/i f n 1 ;θ k)9 ) " $ 0 6 (Xτ f n,d f n) "* ) B d f n g X f τn (d f n,k/in 1;θ) f =h X f τn,k (df n/in 1;θ f k )exp h X f τn,l ( u ;θ I f 1 )du n 1 l E ) Bθ=(θ AAA,θ AA,...,θ NR )etx f τ n E 0 l E : ; & 9 - "< -"? " 4 $ )" "" B X $ "*? < k - ) - X / $ ) * 4 - ) < l;l=k. $ )" -" " B F N f L(θ) = g X f τn (Xτ f n,d f n/in 1;θ) f : ; = F N f f=1n=1k E f=1n=1! " 1 δ f h X f τn,k (df n/in 1;θ f k ) I(Xf τn =k) s X f τn (d f n/in 1;θ f n k )! " δ f S X f τn (T τ f n 1/In 1;θ f n k ) - $" B F L f k (θ k) : %; k Ef=1

76 '5/I2&5D /#2 &(5>(2 /# D5&5D2 =D&D3(#2 /#2 =('#2, &) B L f k (θ k) = : ; 5R B N f n=1! h X f τn,k (df n/i f n 1 ;θ k) I(Xf τn =k) S X f τn (d f n/i f n 1 ;θ k)" 1 δ f n! S X f τn (T τ f n 1 /If n 1 ;θ k)" δ f n, l(x f τ n =k) = # 1siX f τn =k 0 # δ f 1 " $ 4 n = 0 " -< " -)9 - L f k (θ k) 4 BS X f τ (T τ 0 /I f 0;θ k ). I $ L f k (θ k) ) "* θ k 9 -" -" "* θ = (θ AAA,θ AA,...,θ NR )9 J 4 "" "? * $L f k (θ k) ) "* θ k,k =AAA,AA,...,NR #$h X f τn,k (df n,k /If n 1;θ k ) S X f τn (d f n,k /If n 1;θ k ) "*?&/ :/ &? );9 d f n,k ;n = 1,2,...,Nf " " 4 - -# ( :!!+; ) " / -<9 d f n,k (k =AAA,AA,...,B)9 "" " B X > "* " " B! " Ψ f αln E D f n,k /Xτ f n /If n 1 n " - - n X > *" " " ε n 9? " "" I f n 1 9 -$"? "f 4 τ f n 1

77 Ψ f n,k /X f τ n '5/I2&5D /#2 &(5>(2 /# D5&5D2 =D&D3(#2 /#2 =('#2, 2 $"9 " "f "* ) B n = 2,...,N f 1 d f n,k = exp(ψf n,k/x f τ n )ε n,k :,; Ψ f =w n,k/xτ f X f +α k ε n 1,k +β n τ n,k k Ψ f +δ n 1,k/Xτ f k n : +; n ) β k < 1. n = 1 B Ψ f =w 1,k/Xτ f X f +δ k :!; τ,k &)E D f n,k /If n 1 = k exp(ψ f )E ε n,k/xτ f n,k /I f n 1 = k n Var ε n,k /I f n 1 =σ 2 k. " - $ )" 9 k :k = A,AA,...,BBB)9 " ε n,k 1 ) $ ) B f k (ε n,k ) = γ k γ k k εγ k n,k 1+σ 2 v k γ 1+ k k εγ k σ v k n,k )γ k >σ 2 v k > 0 "< B k = Γ(1+ 1 γ k )Γ( 1 σ v k + 1 γ k ) σ 2 v k (1+ 1 γ k )Γ(1+ 1 σ v k ) :.; :; # 4 " " )9 )? " 4 In 19 f D f n,k 1 "* B k Ψ f n,k/x f τ n γk,γ k,σ 2 v k : ; < $ ) B f γ k k Ψ h X f τn (Xτ f n,d f n/i f n 1 ;θ) = n,k/x f τ n &1+σ 2vk $ $ d f n Ψ f n,k /X f τ n d f n % γk 1 % γk ' :; 4 < 1 $ 9 S? # 09 *

78 '5/I2&5D /#2 &(5>(2 /# D5&5D2 =D&D3(#2 /#2 =('#2, 19 " <* ) - $ #09 "*σ 2 v k A9 1 4 S "* B Γ(1+ 1 γ k ) γ k,γk Ψ f n,k/x f τ n )γ k > 0 " 4 19 $? - S "# " $ γ k > 19 " γ k < 19 γ k = 1 / 9 S 4 "* (Ψ f n,k/x f τ n ) 1. # )9 "*σ 2 v k 4 19 ) )? "* B Γ(1+ 1 )Γ(1 1 γ k ) γ k γ k,γk, Ψ f n,k/x f τ n )γ k > 1 / 9 $ " "9 """ J? :!!.; "" " S ""9 "< " - " 5 )"" ) 9 - "9 9 H"" W? ) - /-R -9 - $ " " - " " " v " "" D f n ) "" "< 1 ) :;σ 2 v. -" ) "* θ 0? $ - - " )9 < &9 ) : ; "* ) B g (n) j (d f n,k/i f n 1;θ j ) = 5 4 -"9-9 "9 < - 9 " 9

79 '5/I2&5D /#2 &(5>(2 /# D5&5D2 =D&D3(#2 /#2 =('#2,% h (n) j,k (df n/in 1;θ f j,k ) h (n) j (d f n/in 1;θ f j,k ) h(n) j (d f n/in 1;θ f j,k ) exp d n 0 l E j,l (u/if n 1;θ j,l )du, h (n) =π (n) j (d f n /If n 1 ;θ j,k) g (n) j (d f n,k/if n 1 ;θ j) :; :%; ) j (d f n /If n 1;θ j,k ) = h(n) j,k (df n/in 1;θ f j,k ) h (n) j (d f n/in 1;θ f j,k ), π (n) g (n) j (d f n,k/in 1;θ f j ) = h (n) j,l (u/if n 1;θ j,l ) exp l E d n 0 l E h (n) :; j,l (u/if n 1;θ j,l )du. :,; D)"n(n = 1,2,...,N f 1)9 - <j 9 ) <k (j,k E) 4 - ) B =! " π (n) j,k (θ j,k/in 1) f =E D f n π (n) j,k (u/if n 1;θ j,k ) + 0 h (n) j,k (u/if n 1;θ j,k ) s (n) j (u/i f n 1;θ j )du. :+; / "J"9 ) ) - :+; " "< " $ # 9 R - < j (j E)9 J " "* ) B E(d f n /Xf τ n =j,x f τ n =k;i f n 1 ) :!; = + 0 u l E h (n) j,l (u/if n 1;θ j,l ) s (n) j (u/i f n 1;θ j )du. 2 9 "J" ) - < k(k E)9 -:!; ) B E(d f n/x f τ n =j,x f τ n =k;i f n 1) = + 0 u h(n) j,k (u/if n 1;θ j,k ) π (n) j,k (θ j,k/in 1) f s(n) j (u/in 1;θ f j )du. :.;

80 #2 /5DDI#2, % " #9 & -&$ - 2N % ) 9 " &V!+! $) " -) $" "" - ) 4 "" D ) -" 2N9 -) 09 4 ) B?2 "?2 "?2 -)"?2 "?2 ""?2 -?2 )?2 W?2 "?2 " "?2 "?2 D "" -" < " - 2N : " =? ( ; 2N -" : ( * - 4 " 4 ) -"? * ) ) W $ $ 0 9 " -) " D 1595 ":<; -" " "" R " "" " 4 % 4.. [ " - 'E?T? "" 6 1 S8 2 -)" S9 '<-

81 #2 /5DDI#2,, - < - " ) : " * ; 4 "* A ) 0 ) 21,6% 23, 6% " - 4 AAA/AA )" 9, 2% T. 1. /"" $ < < - < - &&& && & D( &&&`&& #0$ [ 41, 80 43, 20 0, 70 14, [ 81,30 21,30 0,30 7,40 9,20 & #0$ [ 3,00 52,20 31,40 0,50 13, [ 14,70 65,20 31,40 0,70 17,00 23, #0$ [ 0,60 10,70 57,90 16,30 1,10 13, [ 2,70 13,20 56,90 21,30 1,40 17,30 22,80 11 #0$ [ 0,30 11,00 40,30 27,80 20, [ 0,30 10,30 50,20 32,50 25,10 21,60 1 #0$ [ 0,30 1,10 20,80 52,60 25, [ 1,30 1,10 27,80 66,10 33,20 23,30 #0$ [ 4,70 18,60 23,10 17,40 18,50 17, [ "J" J $ - -? ) - : " * ; /-9-4 ) AAA AA $ -6 - "9 " * $ -) * : ) ";

82 #2 /5DDI#2,+ D ) - ) -? -) " -) * - 4 NR :D ( ; *) *" "" " "" " - 4 -< < - J D ( 4 " $ 4 " - 4 $ ) 4 ) :$ ) ) -) * ; 9 ) "" " - -) "" " 4 ) 9 D f n,nr9 )AAA/AA,AA,...,BBB,...,B. T. 2. "< $ < < - < &&&`&& & D( - &&&`&& 34, , , , , 327 & 25,79 22,923 24,866 26,036 23,954 23, ,743 20,972 20,44 21,611 17,529 20,992 20, , , 61 15, , 21 16, , ,416 19,04 16,845 13,989 16,915 17,049 23,767 21,792 23,862 23,197 19,433 22,258 22,385 " "< -? "9 -) - -4 * 4 9 "< " "9 "" D ( 9 $ 4 " -? " " 4 )

83 % (I2>&2 # D#((I&5D2,! "" $ 1*)"9 ) 58% " 9 " "" T. 3. # 5 - D" , 9% , 5% , 0% , 2% , 9% , 1% , 3% , 1% 9 9 0, 6% , 4% , 1% %,. -< -" "* $ 0 < $ -) " " t - j ) k (j,k E) $ 64 - /- ) 9 ) 4 $ -<*9 9 ) "* : -<* -"?" ; $ 9 "9 -? " " "< 0

84 , ' $ % (I2>&2 # D#((I&5D2 +. -" "* "* " """ )" % :" -) g %!%; T. 4. #" "* $ :; "* &&& ` && & 111 γ k (0.1085) (0.1843) (0.1052) σ 2 v k (0.0535) (0.0752) (0.0475) w AAA/AA k (0.7732) (0.2417) (0.7552) w A k (0.6576) (0.2819) (0.1538) w BBB k (0.9465) (0.3485) (0.1502) w BB k (1.5070) (1.1193) (0.2229) w B k (0.8688) (2.0416) (0.3966) α k (0.0981) (0.0510) (0.0603) β k (0.0337) (0.0208) (0.0391) θ k (0.0150) (0.0533) (0.0249)?) ? YYY B Q )? " :; < " "* " "* γ k σ 2 v k )" 0 A NR R *" "* " J $ * $ $ "* σ 2 v k 6 " 4 < 1 ) D f n,k ;k E {NR} 9 - S " J " NR ' ) "* γ k 9? ) - )" 4 19

85 % (I2>&2 # D#((I&5D2 + T. 5. #" "* $ : ; "* 11 1 D( γ k (0.1851) (0.1428) (0.0600) σ 2 v k (0.0761) (0.0839) (0.0168) w AAA/AA k (3.1458) (1.8733) (0.2030) w A k (0.6310) (1.9243) (0.1632) w BBB k (0.3039) (0.6215) (0.1718) w BB k (0.3939) (0.4589) (0.1493) w B k (0.3265) (0.4517) (0.1288) α k (0.0687) (0.0545) (0.1190) β k (0.0239) (0.0231) (0.0362) θ k (0.0364) (0.0131) (0.0307)?) "J" 9 ) - - : " ; 4 - " 64 - / - < 19 Z " " ) 9 """?4 - < S9 " 9 " - (i.e.γ k = 1) R 6

86 % (I2>&2 # D#((I&5D2 + $ "? : ; # " ) Ψ f n,k/x f τ n 4 "< 4 - F.2. " - 6 ) 8 - " - ) ) " -" ) ) 0? 9 AAA/AA, A, BBB, BB B " 9 )"9 77,36,23,15 10 " " 9 $ )9" ) )?4? 9 -" 9 Z "* ) ) "? " *" D ) "

87 % (I2>&2 # D#((I&5D2 + ) 4 - " $ 4 " - - ) 4 $ NR9 $ " ) "? 9? - ) $ ) :" 9 $ -? ) - ; / : ;9 $ " )9 9 "? 9 ) 9? 6 9 " $ * ) " NR H4"?? $ "9 $ $ - " ) ) - - ) - *" $ - -*) ) " NR - ) A "J" " J $ ) " "*? $ 4 ) : +; " * 8,9 10 ) "" ) $) H $ $ *" 9 )* 9 4 " Z - -9? 4 -" " " " " ) - - ) H $ 9 $ $ " ) ) 4 - ) -? > - 9 $ - - ) / -9 ) 4 - ) J ) "" - &? 9-2N )"$4" 4 9 " -$"9 2 H $ 4 &&&`&& - ) $

88 % (I2>&2 # D#((I&5D2 + "9 ) 9 - "9 ) $ $ ) - $" -) ) H β k ;k =AAA/AA,A,BBB,BB... : 9 ;? )" ) D f n,k ;k =AAA/AA,AA,...,BBB,...,B. ) H? J " $ 4 1 -" "* " """ )"? D9 $9 ) -J " 9 ) " ) - "* &/ 4-6* ",, "" ""9 "* -) $ < $ ) :"9 < ;? - "< ) <" " -)? ) :9 9 %; ) # $K ) ) ) ) - B - $ - ) " " - - " 9 ) 9 9 < < $? 4 5% 9 G & / G 5 < ) ") ) "" 9 -" " 4 " " )? )" * $ : )*";, "9 ) 1E :...9..;9 /$ # :...;9 = 9 U 2 :..;

89 % (I2>&2 # D#((I&5D2 +% " $ " "* " "? F ) " :;

90 % (I2>&2 # D#((I&5D2 + F ) " : ; " -)"9? ) " - ) " "* 4 ) ) 0,70 * ) - 0, " " &9-0,38? *" 9 ) "* ) - 25% 4 - " / - W9 " 9 1 " " " "

91 % (I2>&2 # D#((I&5D2 +, F ) " :; X ) " 0, "40,28 4 *"? BBB etb, "" - &9 - * " " " ) #9 " D( "" ) " * 4 " "Z -) "

92 5D>25D ++ / # / 9 ) "* &/ "? ) 4-2N 5 " 9 - " -"9 " 9 $ < " + " # & &$ - 2N ) 9 " &V!+! $)..+ 2 " 9 ) "" "* ) 4 ) "*&/ " *"9 -< ) "*9 ) "< " " "* "* 4 J " "" - " ) 0 -) 2 9 -" "* $ 4 " """ )" " < 1 * $ " ) ) /- W9 4 ) " $ "? $ "9 D ( $ 6 " ) *" $ - -*) ) " D( 4 $ ) 4 9 &&& ) 19 1 ) && "" 9 ) ) * " $ 9 " "

93 , &DD#7# +! 0 ( F. 6. / "< " - " - 6 ) 89 -

94 T&(# -) : G" G; "" 6 W * " * *9 "" " -" * - $? "" * 6 W " " " - -4 < ) 2 W 6 -< " * - ) - ) $ )" " 4 " :O E P; " :O E P; " - " - " 9 9 ) ) " V? " - # 09 -" - " " " 9 < ) 4 ) " -" " *..,`..+ -? -" - # $ " 9 -"9 "9 9? ) -"" *" A # 09 -&'=9 " * 4 %% "..+ % ".., " <..+9 " ")4) 9*"? ) $" $ ) " / " &&&?&& $ 4 " /- ) "? " # 09 - " -!.

95 D(5/>5D! " "6 9 4 J ) ) ) 4 /9 ) J " -? "* -) D -4-9 ) - " 9 " ) " 9 ) " " 9 "* # 09 $9 "? ) " <$" " -6$ -< -" $" " " -0 D ) 4 " " ) ) -" " " - $ 4 *" 6 -" " " 46"*&/ 2 R "< " "" " )? ) -@ > "* " " 9 ) * "* "* " " """ )" "" B ) " " - $ - " "9 " " c) " % " ) -0 - $ ) "

96 (#Q># /# & I(&>(#!." "* " " $ -6 "? ) ) " * 2 " c) "* " '8)9 ) -" " " D "* $? $ " -) " " - " 64 " c) "? = 9 :..%9..%; # " " "9 1 :.. ; ) $ ) 4 - "* " #? -" "9 9 " $ )? "9 "" ) " 4 0 -" $ ) "? " "" " - $ :; " " "" $ " "*? 0 " " ) /- "* "" =<" 2" :..,;? "* '8) " " " "* -? " "* " " "* '8) " " " $ - # 09 < "J" ) ) " 0 ) " $" "* " Z '8) D8 :...; ) < " " "* -" $ " " " 4 0 & "* 9 -"? " " c) "* " " "" 4 " $ $ " " " ) -)

97 '5/3#! 4 0 < ) 4 0 < " *" "* " < -) ) " -" " " -? - 9 " - - < " & 9 "? c) "* ) -" 4 "*! - 4 " 6 ) " $ :" " " $; " 4 - " - 2N 9 )9 n (n = 1,2,...,N f 1) - " f (f = 1,2...,F)9 6 yn f = (xf τ n,d f n ) "* ) B f Y (x f n,d f n/z f 1n,z f 2n,v,I f n 1;θ) =f X (x f n/d f n,z f 1n,v;θ)f D (d f n/z f 2n,v,I f n 1;θ) :; R B :; f X (x f n/d f n,z f 1n,v;θ) 0 " 4 -?*"? " ) <" " "* ) B x f n 1 Abaissementmajeur delanote 2 Abaissementmineur delanote 3 Augmentationmineuredelanote 4 Augmentationmajeuredelanote $ " 4 "? )9 z f 1n9 ) v *" " "f,d f n : ; $ f D (d f n/z f 2n,v,I f n 1;θ) 4 $ " 4 I f n 1 : -$" 6 4 <" ) " f; 4 9 $ " D " " & -"9 && 4 &9-2 - > " " 0 - G]G ) 0 - G?G 9 4 -" &&& 4 &&&

98 '5/3#! $ - " )9 z f 2n9 ) v :; "θ ) "* 4 "! 1 4 "* " $ " $ 4 "" " 4 -) - ) / 9 "" $ $ " 2S f n9 ) " $ 4 - *" 9 ) " &9 B y f n ) B 1si S f n <c 1 Abaissementmajeur delanote 2sic 1 S f n <c 2 Abaissementmineur delanote 3sic 2 S f n <c 3 Augmentationmineuredelanote 4si S f n c 3 Augmentationmajeuredelanote S f n =Sf n Sf n 1 =β 0 +x f n β +β vv +ε : ; / $" : ;9 Sn f 0 nème " $ 0 " $ - " ) ) ) xn,: f ? ) * 9 - ;9 $ - )v " -ε " "< A )σ $ " -" ) :? " "6 9 " " 9 ) " 9 ) "6 ; D 9 4 " ) "* B (c 1,c 2,c 3,β,β v,σ) R c i 4 " " """ )" " "* β.

99 '5/3#!% " $K ) B Pr(y f n = 1/d f n,i f n 1,v) = Pr( Sf n <c 1 ) = Pr(ε<c 1 β 0 x f nβ β v v) c1 β = Φ 0 x f nβ β v v) σ Pr(y f n = 2/d f n,i f n 1,v) = Pr(c 1 S f n <c 2 ) = Pr(c 1 β 0 x f nβ β v v)<ε<c 2 β 0 x f nβ β v v) c2 β = Φ 0 x f nβ β v v) c1 β Φ 0 x f nβ β v v) σ σ Pr(y f n = 3/d f n,i f n 1,v) = Pr(c 2 S f n <c 3 ) = Pr(c 2 β 0 x f nβ β v v)<ε<c 3 β 0 x f nβ β v v) c3 β = Φ 0 x f nβ β v v) c2 β Φ 0 x f nβ β v v) σ σ Pr(y f n = 4/d f n,i f n 1,v) = Pr( S f n c 3 ) = Pr(ε c 3 β 0 x f nβ β v v) c3 β = 1 Φ 0 x f nβ β v v) σ :; :; :%; :; :,; :+; -9 ) " - ε? " 4 :σ = 1;! 1 23(#4 3 " 9 $ 4 "* 5?&/ "" < "* -) " " - "J" " " " ) " &9 d f n9 "" " B! " X > "* " " Ψ f ln E D n/in α f f n /If n 1 " -? - n X > *" " " ε n 9? " "" 9 I f n 1 -$" "f 4 τ f n 1

100 '5/3#! 2 $"9 " "f "* ) B n = 2,...,N f 1 d f n = exp(ψ f n/i f n )ε n :!; Ψ f n/i f n =α z f n +α v v :%.; avece Dn f /If n 1 E ε n /I f n 1 var ε n /I f n 1 = exp(ψ f n/i f n ) :%; = = 1 " - $ )"9 9 " 4 I f n 1 9 " ε n S ) $ ) B f k (ε n ) =γ γ exp( γ γ ε γ n) :% ; )γ> 0 "< B = Γ(1+ 1 γ ) # 4 " " )9 ) 9? " 4 I f n 1 9 Df n S "* B Γ(1+ 1 γ ) γk,γ. Ψ f n/i f n / - S9 $ " # " $ γ k > 19 " γ k < 19 γ k = 1 / 9 S 4 "* (Ψ f n/i f n ) 1!! ' " "" ) 9 ) " f X (.) f D (.) $ - " ) v " - ) " ) # 09 4 $ -?" ) -@

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