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276 9. Mesure de la dépendance extrême entre deux actifs financiers E.3 Proof of lemma 2 We want to prove that, assuming ν > 0 and x0 > 1, The change of variable gives 1 ɛ ∞ 1 dx 1 x ν 1 lim ɛ→0 ɛ 1 + 1 ν ∞ 1 Cν x−x0 ɛ Consider the second integral. We have which allows us to write so that ∞ x 0 ɛ du 1 dx 1 x ν 2 ν+1 2 1 1 + 1 ν u = = (1 + u2 ) ν+1 2 (1 + ɛu x0 )ν Cν (1 + u2 ) ν+1 2 The next step of the proof is to show that Let us calculate x0 ɛ du 1−x 0 ɛ 1 (1 + ɛu x0 )ν x 0 ɛ 1−x 0 ɛ Cν du (1 + u2 ν+1 ν ) 2 1 (1 + ɛu x0 )ν − 1 = x − x0 ɛ Cν x−x0 ɛ ∞ 1−x 0 ɛ = 1 xν ∞ 1−x0 0 ɛ = 1 x ν 0 2 ν+1 2 = 1 xν . (E.117) 0 , (E.118) x 0 ɛ 1−x0 ɛ + 1 xν ∞ x0 0 ɛ 1 du (ɛu + x0) ν du du du 1 (1 + ɛu x0 )ν 1 (1 + ɛu x0 )ν 1 (1 + ɛu x0 )ν Cν (1 + u2 ν+1 ν ) 2 Cν (1 + u2 ν+1 ν ) 2 Cν (1 + u2 ν+1 ν ) 2 Cν (1 + u2 ν+1 ν ) 2 + (E.119) (E.120) . (E.121) u ≥ x0 , (E.122) ɛ ≤ Cν ν ν+1 2 ɛ ν+1 x ν+1 0 ≤ ν ν+1 2 ɛ ν+1 x ν+1 = ν ν+1 (1 + u2 ν+1 ν ) 2 x0 ɛ 0 2 ɛ ν x ν 0 , (E.123) ∞ 1 ∞ x 0 ɛ du Cν (1 + ɛu x0 )ν dv Cν (1 + v) ν (E.124) (E.125) = O(ɛ ν ). (E.126) 1−x 0 ɛ 38 du −→ 1 as ɛ −→ 0. (E.127) 1 (1 + ɛu x0 )ν Cν (1 + u2 ν+1 ν ) 2 −
9.1. Les différentes mesures de dépendances extrêmes 277 ∞ Cν − du −∞ (1 + u2 ν+1 (E.128) ν ) 2 x 0 ɛ 1 = du 1−x0 (1 + ɛ ɛu Cν − 1 )ν x0 (1 + u2 ν+1 − ν ) 2 1−x0 ɛ Cν − du −∞ (1 + u2 ∞ Cν ν+1 − du x ν ) 2 0 ɛ (1 + u2 ν+1 ν ) 2 (E.129) x 0 ɛ 1 ≤ du 1−x0 (1 + ɛ ɛu Cν − 1 )ν x0 (1 + u2 ν+1 ν ) 2 + 1−x0 ɛ Cν + du −∞ (1 + u2 ν+1 ν ) 2 + ∞ Cν du x0 ɛ (1 + u2 ν+1 ν ) 2 . (E.130) The second and third integrals obviously behave like O(ɛν ) when ɛ goes to zero since we have assumed x0 → −∞ and ɛ → ∞ when ɛ → 0. For the first integral, we have x0 ɛ 1 du (1 + ɛu − 1 )ν x0 Cν ≤ x0 ɛ 1 du (1 + ɛu − 1 )ν x0 Cν . (E.131) x0 > 1 what ensures that 1−x0 ɛ 1−x 0 ɛ (1 + u2 ν+1 ν ) 2 1−x 0 ɛ The function 1 (1 + ɛu − 1 )ν x0 (1 + u2 ν+1 ν ) 2 (E.132) vanishes at u = 0, is convex for u ∈ [ 1−x0 ɛ , 0] and concave for u ∈ [0, x0 ɛ ] (see also figure 13), so that there are two constants A, B > 0 such that 1 (1 + ɛu − 1 )ν x0 ≤ −xν 1 (1 + 0 − 1 1 − x0 ɛ · u = −A · ɛ · u, ∀u ∈ , 0 x0 − 1 ɛ (E.133) ɛu − 1 )ν x0 ≤ νɛ u = B · ɛ · u, ∀u ∈ 0, x0 x0 . ɛ (E.134) We can thus conclude that x 0 ɛ 1 du (1 + ɛu − 1 )ν x0 1−x 0 ɛ Cν (1 + u2 ν+1 ν ) 2 0 ≤ −A · ɛ 1−x0 ɛ x0 ɛ + B · ɛ du 0 du u · Cν (1 + u2 ν+1 ν ) 2 u · Cν (1 + u2 ν+1 ν ) 2 (E.135) = O(ɛ α ), (E.136) with α = min{ν, 1}. Indeed, the two integrals can be perfomed exactly, which shows that they behave as O(1) if ν > 1 and as O(ɛν−1 ) otherwise. Thus, we finally obtain x0 ɛ 1 du (1 + ɛu x0 )ν Cν − 1 = O(ɛα ). (E.137) 1−x 0 ɛ (1 + u2 ν+1 ν ) 2 39
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UNIVERSITE DE NICE-SOPHIA ANTIPOLIS
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4 Table des matières 2.2.2 Bulles
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6 Table des matières 7.3.2 Tests n
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8 Table des matières 12.1.1 Distri
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10 Table des matières Références
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12 m’ont souvent été d’un gra
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14 Introduction pour espérer se co
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16 Introduction les distributions e
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18 Introduction
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Chapitre 1 Faits stylisés des rent
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1.1. Rappel des faits stylisés 23
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1.1. Rappel des faits stylisés 25
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1.2. De la difficulté de représen
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1.3. Modélisation des propriétés
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Chapitre 2 Modèles phénoménologi
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2.1. Bulles rationnelles multi-dime
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2.2. Des bulles rationnelles aux kr
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Chapitre 3 Distributions exponentie
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Empirical Distributions of Log-Retu
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concerning tail fatness can be form
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To fit our two data sets, section 4
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properties, stressing the problems
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Another problem lies in the determi
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In order to obtain a process with S
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and Sornette (1999) for instance. W
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1. The Pareto distribution: 79 Fu(x
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empirical analog FN(x), estimated f
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have been chosen to converge to 1 a
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5 Comparison of the descriptive pow
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ased on the fact that the quantity
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tail (positive or negative) of the
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Equation (46) depends on c only and
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B Minimum Anderson-Darling Estimato
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C Local exponent In this Appendix,
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D Testing non-nested hypotheses wit
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where α = (c ∗ ,d ∗ ). It can
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where E0[·] denotes the expectatio
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References Andersen, J.V. and D. So
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Gouriéroux C. and A. Monfort, 1994
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Rubinstein, M., 1973, The fundament
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(a) Independent Data Stretched-Expo
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(a) Dow Jones Positive Tail Negativ
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Nasdaq Dow Jones Pos. Tail Neg. Tai
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Dow Jones positive tail Dow Jones n
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Variation coefficient Variation coe
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Mean excess function Mean excess fu
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Hill‘s estimate b u Hill‘s esti
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Wilks statistic (doubled log−like
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Tail 1−F(x) and parameter b Tail
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1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 2 3 4
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Chapitre 4 Relaxation de la volatil
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Volatility Fingerprints of Large Sh
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2 Long-range memory and distinction
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Figure 2: Measuring the conditional
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the system to the accumulation of m
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Appendix C: “Conditional response
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Baillie, R.T., 1996, Long memory pr
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Chapitre 5 Approche comportementale
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5.1. Prix d’un actif et excès de
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5.2. Modèles d’opinion contre mo
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5.4. Conséquences des phénomènes
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5.5. Conclusion 157 ce qui induit u
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Chapitre 6 Comportements mimétique
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RESEARCH PAPER Q UANTITATIVE F INAN
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A Corcos et al Q UANTITATIVE F INAN
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Deuxième partie Etude des proprié
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182 7. Etude de la dépendance à l
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192 8. Tests de copule gaussienne
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194 8. Tests de copule gaussienne 1
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200 8. Tests de copule gaussienne t
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202 8. Tests de copule gaussienne T
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204 8. Tests de copule gaussienne a
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206 8. Tests de copule gaussienne c
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208 8. Tests de copule gaussienne t
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210 8. Tests de copule gaussienne (
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212 8. Tests de copule gaussienne R
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326 9. Mesure de la dépendance ext
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Chapitre 10 La mesure du risque Dan
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10.1. La théorie de l’utilité 3
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10.2. Les mesures de risque cohére
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10.3. Les mesures de fluctuations 3
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10.4. Conclusion 361 et de manière
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Chapitre 11 Portefeuilles optimaux
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11.2. Prise en compte des grands ri
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11.3. Equilibre de marché 367 s’
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11.5. Annexe 369 Collective Origin
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11.5. Annexe 371 point λ = 1. Thus
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Chapitre 12 Gestion des risques gra
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12.1. Comprendre et gérer les risq
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12.2. Minimiser l’impact des gran
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Chapitre 13 Gestion de portefeuille
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VaR-Efficient Portfolios for a Clas
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dependence: from independence to co
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given a series of return {rt}t foll
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eads cV (u1, · · · , un) = 1
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THEOREM 4 (TAIL EQUIVALENCE FOR A S
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Independent Assets Comonotonic Asse
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3.2 Typical recurrence time of larg
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and finally w ∗ i = χ−1 i j
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where the ˆwi’s are solution of
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Choosing x > y > Aɛ, and applying
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Expanding fi(xi) around x ∗ i yie
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for all h ∈ AC. Thus, integrating
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B Asymptotic distribution of the su
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This yields × h+ Sc−2 − 2 dh
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References Acerbi, A. and D. Tasche
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Malevergne, Y. and D. Sornette, 200
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ln(T/T 0 ) T Figure 2: Logarithm
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Chapitre 14 Gestion de Portefeuille
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Multi-Moments Method for Portfolio
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choosen risk measure. Section 5 pre
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The variance of the return X of an
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This property is verified for all c
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µn=α of order n = 2, 4, 6 and 8.
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these two assets or portfolios. The
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Due to the homogeneity property of
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invested in the risky fund depends
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C. Then, if g1(X1), · · · , gn(X
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Differentiating with respect to x1,
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7.2 Transformation of the modified
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In the sequel, we will first evalua
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8.3 Non-symmetric assets In the cas
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y the variance will then increase).
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A Description of the data set We ha
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thus and finally 1 λ1 n−1 = w
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C Composition of the market portfol
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D Generalized capital asset princin
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This brings us back to the problem
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F.2 General case We now consider a
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Harvey, C.R. and A. Siddique, 2000,
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µ µ2 1/2 µ4 1/4 µ6 1/6 µ8 1/8
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Mean (10 −3 ) Variance (10 −3 )
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μ (daily return) x 10−3 2.5 2 1.
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w i w i 0.2 0.15 0.1 0.05 Mean−μ
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Figure 6: Schematic representation
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Empirical Cumulative Distribution 1
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Z 2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0
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Z 2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0
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10 0 10 −1 10 0 10 −1 10 −1 1
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10 0 10 −1 10 0 10 −1 10 −1 1
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κ 22 20 18 16 14 12 10 8 6 4 2 Exc
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Return μ 0.12 0.11 0.1 0.09 0.08 0
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Conclusions et Perspectives L’obj
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Conclusion 493 en univers gaussien
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Annexe A Evaluation de la conduite
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A.2. Eléments de contexte 497 bora
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A.4. Compétences acquises et ensei
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A.5. Conclusion 501 de la recherche
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Bibliographie ACERBI, C. (2002) :
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Bibliographie 505 BOUCHAUD, J. P. E
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Bibliographie 507 DE FINETTI, B. (1
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Bibliographie 509 GRANGER, C. W. ET
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Bibliographie 511 LILLO, F. ET R. N
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Bibliographie 513 PATTON, A. (2001)
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Bibliographie 515 SUSMEL, R. (1996)
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