Slides dauphine

Arthur CHARPENTIER - tails of Archimedean copulas

Tails of Archimedean Copulas

tail dependence in risk management

Arthur Charpentier

CREM-Universite Rennes 1

(joint work with Johan Segers, UCLN)

http ://perso.univ-rennes1.fr/arthur.charpentier/

Colloque Evaluation et couverture des risques extremes

Universite Paris-Dauphine & Chaire AXA de la Fondation du Risque, Juin 2008

1

Arthur CHARPENTIER - tails of Archimedean copulas

Tail behavior and risk management

In reinsurance (XS) pricing, use of Pickands-Balkema-de Haan’s theorem

Theorem 1. F ∈MDA (Gξ) if and only if

limu→xF

sup0<x<xF

{∣∣Pr (X − u ≤ x|X > u)−Hξ,σ(u) (≤ x)∣∣} = 0,

for some positive function σ (·), where Hξ,σ (x) =

1− (1 + ξx/σ)−1/ξ , ξ 6= 0

1− exp (−x/σ) , ξ = 0.

1− F (x) ≈ (1− F (u))[1−Hξ,σ(u) (x− u)

], for all x > u.

So, if u = Xk:n, then

1− F (x) ≈ (1− F (Xk:n))︸︷︷︸≈1−Fn(Xk:n)=k/n

[1−Hξ,σ(Xk:n) (x−Xk:n)

], for all x > Xk:n,

2

Arthur CHARPENTIER - tails of Archimedean copulas

Pure premium of XS contract

Recall that πd = E((X − d)+) with d large, thus,

πd =1

P(X > d)

∫ ∞d

1− F (x)dx

≈ k

n

σ

1− ξ

(1 + ξ

d−Xn−k:n

σ

)1− 1ξ

,

i.e.

πd =k

n

σk

1− ξk

(1 + ξk

d−Xn−k:n

σk

)1− 1ξk

(see e.g. Beirlant et al. (2005).

Possible to derive explicit formulas for any tail risk measure (VaR, TVaR...).

3

Arthur CHARPENTIER - tails of Archimedean copulas

Extending extreme value theory in higher dimension

univariate case bivariate case

limiting distribution dependence structure of

of Xn:n (G.E.V.) componentwise maximum

when n→∞, i.e. Hξ (Xn:n, Yn:n)

(Fisher-Tippet)

dependence structure of

limiting distribution (X,Y ) |X > x, Y > y

of X|X > x (G.P.D.) when x, y →∞when x→∞, i.e. Gξ,σ dependence structure of

(Balkema-de Haan-Pickands) (X,Y ) |X > x

when x→∞

4

Arthur CHARPENTIER - tails of Archimedean copulas

Tail dependence in risk management

●

●●

●

●●

●

●●

●

●●●●●●●●●●●●●●●●

●●

●●●

●

●●

●

●●●

●

●●●●●

●●●●●●

●●●●●●●●●●●●●●●●●●●●●●

●

●●

●

●●●●

●

●●●●

●

●●

●

●●

●

●●●

●●●●●●●●

●●●●●●●

●

●●

●

●●

●

●●

●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●●

●

●●

●

●●●

●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●●

●

●●●●●●●●●●●●●●●

●●●●●●●●●

●

●●

●

●●●

●●●●

●●

●

●●

●●●●

●

●●

●●●

●●●●●●●●

●

●●

●

●●

●●●●●

●

●●

●

●●●●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●●

●

●●

●

●●●

●

●●

●●●●●●●●●●

●

●●●

●

●●

●

●●●

●●●●

●

●●

●●●●●

●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●●●●●

●●●

●

●●●

●

●●

●

●●●

●

●●

●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●

●

●●

●

●●●●●●

●

●●

●

●●

●●●

●

●●

●

●●●●●●●●●●●●●●

●

●●

●

●●

●

●●●

●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●

●●

●

●●

●

●●

●

●●●●

●●

●

●●●

●●

●

●●

●

●●

●●●

●

●●

●

●●

●

●●●

●

●●

●

●●●

●

●●●●

●●●●●●●

●●●●●●●●●●●●●●●●●●

●

●●●

●

●●●

●

●●

●

●●

●

●●

●●●

●

●●

●

●●

●

●●

●

●●

●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●

●

●●●

●

●●

●

●●●●●●●●●●●●●●

●●●

●●

●

●●

●

●●

●

●●

●

●●

●

●●●●

●

●●●●

●

●●

●

●●

●●●

●

●●●

●

●●●●

●

●●

●

●●

●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●

●

●●

●

●●●●

●

●●

●

●●●●●

●

●●

●

●●

●

●●

●

●●

●

●●●

●

●●

●

●●●

●●●●●●●●●●●

●

●●

●

●●

●

●●

●

●●●●●●●●●●●●●

●

●●●●●

●

●●

●

●●●

●

●●●

●●

●

●●

●

●●●

●

●●

●

●●

●

●●

●

●●●●●

●

●●

●

●●

●

●●●●

●

●●

●

● ●

1e+01 1e+03 1e+05

1e

+0

11

e+

02

1e

+0

31

e+

04

1e

+0

5

Loss (log scale)

Allo

cate

d E

xpe

nse

s

●

●●

●

●●

●

●● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●●

●

● ●

●

●●

●

● ● ●●

● ●

●

● ●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●●

●

●●

●

●●

●

● ●●

●

●●

●

● ●

●

● ●

●

●●

●

● ●

●●

●

●●

●

●●

●

●●●

●●

●

●●

● ●

●

● ●

●

●●

●

● ●

●●

●

●●

●

●●●

●

●●

●

●●

●

●●

●

● ●

●

● ●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

1 10 100 1000 100001

e+

00

1e

+0

21

e+

04

1e

+0

6

Car claims (log scale)

Ho

use

ho

ld c

laim

s

Fig. 1 – Multiple risks issues.

5

Arthur CHARPENTIER - tails of Archimedean copulas

Motivations : dependence and copulas

Definition 2. A copula C is a joint distribution function on [0, 1]d, withuniform margins on [0, 1].

Theorem 3. (Sklar) Let C be a copula, and F1, . . . , Fd be d marginaldistributions, then F (x) = C(F1(x1), . . . , Fd(xd)) is a distribution function, withF ∈ F(F1, . . . , Fd).

Conversely, if F ∈ F(F1, . . . , Fd), there exists C such thatF (x) = C(F1(x1), . . . , Fd(xd)). Further, if the Fi’s are continuous, then C isunique, and given by

C(u) = F (F−11 (u1), . . . , F−1

d (ud)) for all ui ∈ [0, 1]

We will then define the copula of F , or the copula of X.

6

Arthur CHARPENTIER - tails of Archimedean copulas

XY

Z

Fonction de répartition à marges uniformes

Fig. 2 – Graphical representation of a copula, C(u, v) = P(U ≤ u, V ≤ v).

7

Arthur CHARPENTIER - tails of Archimedean copulas

xx

z

Densité d’une loi à marges uniformes

Fig. 3 – Density of a copula, c(u, v) =∂2C(u, v)∂u∂v

.

8

Arthur CHARPENTIER - tails of Archimedean copulas

Strong tail dependence

Joe (1993) defined, in the bivariate case a tail dependence measure.

Definition 4. Let (X,Y ) denote a random pair, the upper and lower taildependence parameters are defined, if the limit exist, as

λL = limu→0

P(X ≤ F−1

X (u) |Y ≤ F−1Y (u)

),

= limu→0

P (U ≤ u|V ≤ u) = limu→0

C(u, u)u

,

and

λU = limu→1

P(X > F−1

X (u) |Y > F−1Y (u)

)= lim

u→0P (U > 1− u|V ≤ 1− u) = lim

u→0

C?(u, u)u

.

9

Arthur CHARPENTIER - tails of Archimedean copulas

Gaussian copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

● ●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

GAUSSIAN

●

Fig. 4 – L and R cumulative curves.

10

Arthur CHARPENTIER - tails of Archimedean copulas

Gumbel copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

GUMBEL

●

Fig. 5 – L and R cumulative curves.

11

Arthur CHARPENTIER - tails of Archimedean copulas

Clayton copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

● ●

●

● ●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

CLAYTON

●

Fig. 6 – L and R cumulative curves.

12

Arthur CHARPENTIER - tails of Archimedean copulas

Student t copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

● ●

●

●●

●

●●

●

●●

●

●●

● ●

●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

STUDENT (df=5)

●

Fig. 7 – L and R cumulative curves.

13

Arthur CHARPENTIER - tails of Archimedean copulas

Student t copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

●

●●

●

●● ●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

● ●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

STUDENT (df=3)

●

Fig. 8 – L and R cumulative curves.

14

Arthur CHARPENTIER - tails of Archimedean copulas

Weak tail dependence

If X and Y are independent (in tails), for u large enough

P(X > F−1X (u), Y > F−1

Y (u)) = P(X > F−1X (u)) · P(Y > F−1

Y (u)) = (1− u)2,

or equivalently, log P(X > F−1X (u), Y > F−1

Y (u)) = 2 · log(1− u). Further, if Xand Y are comonotonic (in tails), for u large enough

P(X > F−1X (u), Y > F−1

Y (u)) = P(X > F−1X (u)) = (1− u)1,

or equivalently, log P(X > F−1X (u), Y > F−1

Y (u)) = 1 · log(1− u).

=⇒ limit of the ratiolog(1− u)

log P(Z1 > F−11 (u), Z2 > F−1

2 (u)).

15

Arthur CHARPENTIER - tails of Archimedean copulas

Weak tail dependence

Coles, Heffernan & Tawn (1999) defined

Definition 5. Let (X,Y ) denote a random pair, the upper and lower taildependence parameters are defined, if the limit exist, as

ηL = limu→0

log(u)log P(Z1 ≤ F−1

1 (u), Z2 ≤ F−12 (u))

= limu→0

log(u)logC(u, u)

,

and

ηU = limu→1

log(1− u)log P(Z1 > F−1

1 (u), Z2 > F−12 (u))

= limu→0

log(u)logC?(u, u)

.

16

Arthur CHARPENTIER - tails of Archimedean copulas

Gaussian copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

● ●

●

●●

●

● ●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

● ●

●●

●

●●

●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Chi dependence functions

lower tails upper tails

GAUSSIAN

●●

Fig. 9 – χ functions.

17

Arthur CHARPENTIER - tails of Archimedean copulas

Gumbel copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

● ●

●

●●

●

●●

●● ●

●

●●

●

●●

●

● ●

●

●●

●

● ●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

● ●

●

●●

●

● ●

●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Chi dependence functions

lower tails upper tails

GUMBEL

●

Fig. 10 – χ functions.

18

Arthur CHARPENTIER - tails of Archimedean copulas

Clayton copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

● ●

●

●●

●

● ●

●

● ●

●

●●

●

●●

●

● ●

●

●●

●

● ●

●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Chi dependence functions

lower tails upper tails

CLAYTON

●

Fig. 11 – χ functions.

19

Arthur CHARPENTIER - tails of Archimedean copulas

Student t copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

●

● ●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

● ●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

● ●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

● ●

●●

●

●●

●

● ●

●

● ●

●

●●

●

● ●

●

●●

●

● ●●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Chi dependence functions

lower tails upper tails

STUDENT (df=3)

●

Fig. 12 – χ functions.

20

Arthur CHARPENTIER - tails of Archimedean copulas

Application in risk management : Loss-ALAE

●

●●

●

●●

●

●●●●

●●●

●

●●

●

●●

●

●●

●

●●●●●

●●●●●●

●●

●●●●●●

●●

●

●●●●●

●

●●

●

●●

●

●●●●

●

●●●●●

●

●●●●

●

●●

●

●●

●

●●●

●●

●●●

●

●●

●

●●

●

●●

●●●●

●

●●

●●●●

●●

●●●●●●

●●

●

●●●

●

●●

●

●●●●

●●●●●●●●●●

●●●●●●●

●●

●●●

●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●

●●

●

●●●●●●●●●

●

●●●●●

●

●●●●

●

●●●●●

●●

●

●●

●

●●

●

●●●●

●

●●

●

●●●●●

●

●●

●

●●

●

●●

●

●●●●●●●●●●●●●●●●●●●

●●●●●●●●

●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●

●●●●●●

●

●●

●

●●●

●

●●

●

●●

●

●●

●

●●

●●●●●●●●

●●●

●●

●

●●●

●●●●●●

●●●●●●●

●

●●●

●

●●

●

●●

●

●●

●

●●●

●

●●

●

●●●●●

●●●

●●●●●●●●●●

●●●

●●●●●●●

●●●●●

●●

●●●●●●

●●●●●●●●●●●●●●

●●●●●

●●●●●●

●

●●●●●

●

●●●

●

●●

●●●

●

●●

●●●

●

●●

●●●●

●

●●

●

●●

●

●●

●

●●●●●

●●●

●●●●●●●●●●

●●

●●●●

●●●●●●●●●●●●●

●●●

●

●●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●●

●●●●

●●

●

●●●●●●●●●●●●

●●

●●●

●

●●●

●

●●●

●

●●

●

●●

●

●●

●

●●●●

●●

●

●●●

●●●●●

●

●●●●

●●●●●●●●●●●●●

●●

●

●●

●

●●●

●●

●

●●●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●●●

●

●●

●

●●

●

●●

●

●●

●●●●●●●●●

●

●●●●●●●●

●●●

●●

●

●●●

●

●●

●

●●●

●

●●

●

●●

●●●●

●●●●●

●

●●

●

●●●

●

●●●●●●●●

●●●●

●

●●

●

●●

●

●●●

●●

●

●●

●

●●●

●

●●

●

●●●●●

●

●●●

●

●●

●

●●●

●

●●●●

●●

●

●●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Loss

Allo

cate

d E

xpe

nse

s

Fig. 13 – Losses and allocated expenses.

21

Arthur CHARPENTIER - tails of Archimedean copulas

Application in risk management : Loss-ALAE

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

●●●

●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●

●●●●●●●●●●●●●●●●●●

●●●

Gumbel copula

●

0.0 0.2 0.4 0.6 0.8 1.00

.00

.20

.40

.60

.81

.0

Chi dependence functions

lower tails upper tails

●●●

●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●

●●●

Gumbel copula

●

Fig. 14 – L and R cumulative curves, and χ functions.

22

Arthur CHARPENTIER - tails of Archimedean copulas

Application in risk management : car-household

●

● ●

●

● ●

●

●●●

●

●●

●

●●

●

●● ●

●●

●

●●

●

●●

●

●●

●

● ●

●

● ●

●

●●●

●

●●

●

● ●●

●

●●

●

● ●

●

●●

●

●●

●

● ●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

● ●

●

●●

●

● ●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

● ●

●

●●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

● ●

●

● ●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Car claims

Ho

use

ho

ld c

laim

s

Fig. 15 – Motor and Household claims.

23

Arthur CHARPENTIER - tails of Archimedean copulas

Application in risk management : car-household

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●

●

●●●●●●●

●●●●●●

●

●●

●

Gumbel copula

●

0.0 0.2 0.4 0.6 0.8 1.00

.00

.20

.40

.60

.81

.0

Chi dependence functions

lower tails upper tails

●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●

●

Gumbel copula

●

Fig. 16 – L and R cumulative curves, and χ functions.

24

Arthur CHARPENTIER - tails of Archimedean copulas

Archimedean copulas

Definition 6. A copula C is called Archimedean if it is of the form

C(u1, · · · , ud) = φ−1 (φ(u1) + · · ·+ φ(ud)) ,

where the generator φ : [0, 1]→ [0,∞] is convex, decreasing and satisfies φ(1) = 0.

A necessary and sufficient condition is that φ−1 is d-monotone.

25

Arthur CHARPENTIER - tails of Archimedean copulas

Some examples of Archimedean copulas

φ(t) range θ

(1) 1θ

(t−θ − 1) [−1, 0) ∪ (0,∞) Clayton, Clayton (1978)

(2) (1 − t)θ [1,∞)

(3) log 1−θ(1−t)t

[−1, 1) Ali-Mikhail-Haq

(4) (− log t)θ [1,∞) Gumbel, Gumbel (1960), Hougaard (1986)

(5) − log e−θt−1e−θ−1

(−∞, 0) ∪ (0,∞) Frank, Frank (1979), Nelsen (1987)

(6) − log{1 − (1 − t)θ} [1,∞) Joe, Frank (1981), Joe (1993)

(7) − log{θt + (1 − θ)} (0, 1]

(8) 1−t1+(θ−1)t [1,∞)

(9) log(1 − θ log t) (0, 1] Barnett (1980), Gumbel (1960)

(10) log(2t−θ − 1) (0, 1]

(11) log(2 − tθ) (0, 1/2]

(12) ( 1t− 1)θ [1,∞)

(13) (1 − log t)θ − 1 (0,∞)

(14) (t−1/θ − 1)θ [1,∞)

(15) (1 − t1/θ)θ [1,∞) Genest & Ghoudi (1994)

(16) ( θt

+ 1)(1 − t) [0,∞)

26

Arthur CHARPENTIER - tails of Archimedean copulas

Why Archimedean copulas ?

Assume that X and Y are conditionally independent, given the value of anheterogeneous component Θ. Assume further that

P(X ≤ x|Θ = θ) = (GX(x))θ and P(Y ≤ y|Θ = θ) = (GY (y))θ

for some baseline distribution functions GX and GY . Then

F (x, y) = P(X ≤ x, Y ≤ y) = E(P(X ≤ x, Y ≤ y|Θ = θ))

= E(P(X ≤ x|Θ = θ)× P(Y ≤ y|Θ = θ))

= E((GX(x))Θ × (GY (y))Θ

)= ψ(− logGX(x)− logGY (y))

where ψ denotes the Laplace transform of Θ, i.e. ψ(t) = E(e−tΘ). Since

FX(x) = ψ(− logGX(x)) and FY (y) = ψ(− logGY (y))

and thus, the joint distribution of (X,Y ) satisfies

F (x, y) = ψ(ψ−1(FX(x)) + ψ−1(FY (y))).

27

Arthur CHARPENTIER - tails of Archimedean copulas

0 5 10 15

05

1015

20

Conditional independence, two classes

!3 !2 !1 0 1 2 3

!3

!2

!1

01

23

Conditional independence, two classes

Fig. 17 – Two classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).

28

Arthur CHARPENTIER - tails of Archimedean copulas

0 5 10 15 20 25 30

010

2030

40

Conditional independence, three classes

!3 !2 !1 0 1 2 3

!3

!2

!1

01

23

Conditional independence, three classes

Fig. 18 – Three classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).

29

Arthur CHARPENTIER - tails of Archimedean copulas

0 20 40 60 80 100

020

4060

80100

Conditional independence, continuous risk factor

!3 !2 !1 0 1 2 3

!3

!2

!1

01

23

Conditional independence, continuous risk factor

Fig. 19 – Continuous classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).

30

Arthur CHARPENTIER - tails of Archimedean copulas

Properties of Archimedean copulas

• the countercomonotonic copula C− is Archimedean, φ(t) = 1− t,• the independent copula C⊥ is Archimedean, φ(t) = − log(t),• the comonotonic copula is not Archimedean (but can be a limit of

Archimedean copulas).

0.2

0.40.6

0.8

u_10.2

0.4

0.6

0.8

u_2

00.

20.

40.

60.

81

Frec

het l

ower

bou

nd

0.2

0.4

0.6

0.8

u_10.2

0.4

0.6

0.8

u_2

00.

20.

40.

60.

81

Inde

pend

ence

cop

ula

0.2

0.40.6

0.8

u_10.2

0.4

0.6

0.8

u_2

00.

20.

40.

60.

81

Frec

het u

pper

bou

nd

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

Scatterplot, Lower Fréchet!Hoeffding bound

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

Scatterplot, Indepedent copula random generation

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

Scatterplot, Upper Fréchet!Hoeffding bound

31

Arthur CHARPENTIER - tails of Archimedean copulas

Properties of Archimedean copulas

• Frank copula is the only Archimedean such that (U, V ) L= (1− U, 1− V )(stability by symmetry),

• Gumbel copula is the only Archimedean such that (U, V ) has the same copulaas (max{U1, ..., Un},max{V1, ..., Vn}) for all n ≥ 1 (max-stability),

• Clayton copula is the only Archimedean such that (U, V ) has the same copulaas (U, V ) given (U ≤ u, V ≤ v) (stability by truncature).

32

Arthur CHARPENTIER - tails of Archimedean copulas

Lower tails of Archimedean copulas

Study regular variation property of φ at 0,

lims→0

φ(st)φ(s)

= t−θ0 , t ∈ (0,∞)⇐⇒ θ0 = − lims→0

sφ′(s)φ(s)

.

If θ0 > 0 : asymptotic dependence

Proposition 7. If 0 < θ0 <∞, then for every ∅ 6= I ⊂ {1, . . . , d}, every(xi)i∈I ∈ (0,∞)|I| and every (y1, . . . , yd) ∈ (0,∞)d,

lims↓0

Pr[∀i = 1, . . . , d : Ui ≤ syi | ∀i ∈ I : Ui ≤ sxi]

=(∑

i∈Ic y−θ0i +

∑i∈I(xi ∧ yi)−θ0∑

i∈I x−θ0i

)−1/θ0

This is Clayton’s copula.

33

Arthur CHARPENTIER - tails of Archimedean copulas

Lower tails of Archimedean copulas

Study regular variation property of φ at 0,

lims→0

φ(st)φ(s)

= t−θ0 , t ∈ (0,∞)⇐⇒ θ0 = − lims→0

sφ′(s)φ(s)

.

If θ0 = 0 : asymptotic independence (dependence in independence) for strictgenerators (φ(0) =∞)Proposition 8. If θ0 = 0 and φ(0) =∞, for every ∅ 6= I ⊂ {1, . . . , d}, every(xi)i∈I ∈ (0,∞)|I| and every (y1, . . . , yd) ∈ (0,∞)d,

lims↓0

Pr[∀i ∈ I : Ui ≤ syi;∀i ∈ Ic : Ui ≤ χs(yi) | ∀i ∈ I : Ui ≤ sxi]

=∏i∈I

(yjxj∧ 1)|I|−κ ∏

i∈Icexp

(−|I|−κy−1

i

),

where χs(·) = φ−1 (−sφ′(s)/·), and κ is the index of regular variation of ψ, withψ(·) = −φ−1(·)φ′(φ−1(·)).

34

Arthur CHARPENTIER - tails of Archimedean copulas

Upper tails of Archimedean copulas

Study regular variation property of φ at 1,

lims→0

φ(1− st)φ(1− s)

= tθ1 , t ∈ (1,∞)⇐⇒ θ1 = − lims→0

sφ′(1− s)φ(1− s)

.

If θ1 > 1 : asymptotic dependence

Proposition 9. If 1 < θ0 <∞, then for every ∅ 6= I ⊂ {1, . . . , d}, every(xi)i∈I ∈ (0,∞)|I| and every (y1, . . . , yd) ∈ (0,∞)d,

lims↓0

Pr[∀i = 1, . . . , d : Ui ≥ 1− syi | ∀i ∈ I : Ui ≥ 1− sxi] =rd(z1, . . . , zd; θ1)r|I|((xi)i∈I ; θ1)

where zi = xi ∧ yi for i ∈ I and zi = yi for i ∈ Ic and

rk(u1, . . . , uk; θ1) =∑

∅ 6=J⊂{1,...,k}

(−1)|J|−1(∑i∈J

uθ1j)1/θ1

for integer k ≥ 1 and (u1, . . . , uk) ∈ (0,∞)k.

35

Arthur CHARPENTIER - tails of Archimedean copulas

Upper tails of Archimedean copulas

Study regular variation property of φ at 1,

lims→0

φ(1− st)φ(1− s)

= tθ1 , t ∈ (1,∞)⇐⇒ θ1 = − lims→0

sφ′(1− s)φ(1− s)

.

If θ1 > 1 and φ′(1) < 0 : asymptotic independence, or near independence

Proposition 10. If 1 < θ1 = 1 and φ′(1) < 0, then for all (xi)i∈I ∈ (0,∞)|I| and(y1, . . . , yd) ∈ (0, 1]d ,

lims↓0

Pr[∀i ∈ I : Ui ≥ 1− syi;∀i ∈ Ic : Ui ≤ yi | ∀i ∈ I : Ui ≥ 1− sxi]

=∏i∈I

yj ·(−D)|I|φ−1(

∑i∈Ic φ(yi))

(−D)|I|φ−1(0).

36

Arthur CHARPENTIER - tails of Archimedean copulas

Upper tails of Archimedean copulas

If θ > 1 and φ′(1) = 0 : asymptotic independence, dependence in independence

Proposition 11. If 1 < θ1 = 1 and φ′(1) = 0, if I ⊂ {1, . . . , d} and |I| ≥ 2, thenfor every (xi)i∈I ∈ (0,∞)|I| and every (y1, . . . , yd) ∈ (0,∞)d,

lims↓0

Pr[∀i = 1, . . . , d : Ui ≥ 1− syi | ∀i ∈ I : Ui ≥ 1− sxi] =rd(z1, . . . , zd)r|I|((xi)i∈I)

where zi = xi ∧ yi for i ∈ I and zi = yi for i ∈ Ic and

rk(u1, . . . , uk) :=∑

∅ 6=J⊂{1,...,k}

(−1)|J|(∑J

uj) log(∑J

uj)

= (k − 2)!∫ u1

0

· · ·∫ uk

0

(t1 + · · ·+ tk)−(k−1)dt1 · · · dtk

for integer k ≥ 2 and (u1, . . . , uk) ∈ (0,∞)k.

37

Arthur CHARPENTIER - tails of Archimedean copulas

Tails of Archimedean copulas

• upper tail : calculate φ′(1) and θ1 = − lims→0

sφ′(1− s)φ(1− s)

,

◦ φ′(1) < 0 : asymptotic independence

◦ φ′(1) = 0 et θ1 = 1 : dependence in independence

◦ φ′(1) = 0 et θ1 > 1 : asymptotic dependence

• lower tail : calculate φ(0) and θ0 = − lims→0

sφ′(s)φ(s)

,

◦ φ(0) <∞ : asymptotic independence

◦ φ(0) =∞ et θ0 = 0 : dependence in independence

◦ φ(0) =∞ et θ0 > 0 : asymptotic dependence

38

Arthur CHARPENTIER - tails of Archimedean copulas

upper tail lower tail

φ(t) range θ −φ′(1) θ1 φ(0) θ0 κ

(1) 1θ

(t−θ − 1) [−1,∞) 1 1 1(−θ)∨0 θ ∨ 0 ·

(2) (1 − t)θ [1,∞) 1(θ = 1) θ 1 0 ·

(3) log 1−θ(1−t)t

[−1, 1) 1 − θ 1 ∞ 0 0

(4) (− log t)θ [1,∞) 1(θ = 1) θ ∞ 0 1 − 1θ

(5) − log e−θt−1e−θ−1

θeθ−1

1 ∞ 0 0

(6) − log{1 − (1 − t)θ} [1,∞) 1(θ = 1) θ ∞ 0 0

(7) − log{θt + (1 − θ)} (0, 1] θ 1 − log(1 − θ) 0 ·(8) 1−t

1+(θ−1)t [1,∞) 1θ

1 1 0 ·

(9) log(1 − θ log t) (0, 1] θ 1 ∞ 0 −∞(10) log(2t−θ − 1) (0, 1] 2θ 1 ∞ 0 0

(11) log(2 − tθ) (0, 1/2] θ 1 log 2 0 ·(12) ( 1

t− 1)θ [1,∞) 1(θ = 1) θ ∞ θ ·

(13) (1 − log t)θ − 1 (0,∞) θ 0 ∞ 0 1 − 1θ

(14) (t−1/θ − 1)θ [1,∞) 1(θ = 1) θ ∞ 1 ·(15) (1 − t1/θ)θ [1,∞) 1(θ = 1) θ 1 0 ·(16) ( θ

t+ 1)(1 − t) [0,∞) 1 + θ 1 ∞ 1 ·

(17) − log (1+t)−θ−12−θ−1

θ2(2θ−1)

1 ∞ 0 0

(18) eθ/(t−1) [2,∞) 0 ∞ e−θ 0 ·(19) eθ/t − eθ (0,∞) θeθ 1 ∞ ∞ ·

(20) et−θ− e (0,∞) θe 1 ∞ ∞ ·

(21) 1 − {1 − (1 − t)θ}1/θ [1,∞) 1(θ = 1) θ 1 0 ·(22) arcsin(1 − tθ) (0, 1] θ 1 π/2 0 ·

39

Arthur CHARPENTIER - tails of Archimedean copulas

How to extend to more general dependence structures ?

• mixtures of generators, since convex sums of generators defines a generator,• the α− β transformations in Nelsen (1999), i.e.

φα(t) = φ(tα) and φβ(t) = [φ(t)]β , where α ∈ (0, 1) and β ∈ (1,∞).

• other transformations, e.g.◦ exp(αφ(t))− 1, α ∈ (0,∞),◦ φ(1− [1− t]α), α ∈ (1,∞),◦ φ(αt)− φ(α), α ∈ (0, 1),

=⇒ can be related to distortion of Archimedean copulas.

40

Arthur CHARPENTIER - tails of Archimedean copulas

upper tail lower tail

φα(t) range α φ′α(1) θ1(α) φα(0) θ0(α) κ(α)

(1) (φ(t))α (1,∞) 0 αθ1 (φ(0))α αθ0κα

+ 1 − 1α

(2) eαφ(t)−1α

(0,∞) αφ′(1) θ1αφ(0)−1

α∗ ∗

(3) φ(tα) (0, 1) αφ′(1) θ1 φ(0) αθ0 κ

(4) φ(1 − (1 − t)α) (1,∞) 0 αθ1 φ(0) θ0 κ

(5) φ(αt) − φ(α) (0, 1) αφ′(α) 1 φ(0) − φ(α) θ0 κ

41

qq qq q q q q q q q

qqqq qq qq q qq

n x xk

distribution x

thatf x

then1 f x

iflimsuppr x u

haanpickands x

Documents

fougere, somerford, dauphine

introductionauxsériestemporellesroche/enseignement/series_tem… ·...

hotel dauphine saint germain w 36, rue dauphine 75006...

lamsade - université paris-dauphine

fact sheet 2017-2018 - université paris-dauphine€¦ ·...

#dauphine talk as240614

dauphine en mains

#dauphine talk 2014

routebook dauphine

sommaire - fondation dauphine

llbg catalogo dauphine 2015

mba dauphine...du savoir-faire de l’université...

presentation slides for my simulation course at dauphine

variables aléatoires -...

catalogue dauphine ndl-fr

les vierges du dauphine

guide dauphine en mains bd

thÈse - paris dauphine university

club dauphine rh

master actuariat de dauphine mémoire présenté devant...