EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 3, 2021, 1057-1081
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Extremal Copulas and Tail Dependence in Modeling
Stochastic Financial Risk

Hassane Abba-Mallam1, Natatou Dodo Moutari1, Diakarya Barro2,∗,
Bisso Saley1

1 FAST, Université Abdou Moumouni, Niamey, Niger
2 UFR-SEG, Université Thomas SANKARA, Burkina Faso

Abstract. These last years the stochastic modeling became essential in financial risk management
related to the ownership and valuation of financial products such as assets, options and bonds.
This paper presents a contribution to the modeling of stochastic risks in finance by using both
extensions of tail dependence coefficients and extremal dependance structures based on copulas.
In particular, we show that when the stochastic behavior of a set of risks can be modeled by
a multivariate extremal process a corresponding form of the underlying copula describing their
dependence is determined. Moreover a new tail dependence measure is proposed and properties of
this measure are established.

2020 Mathematics Subject Classifications: 60G70, 62E10, 91B05, 91B70

Key Words and Phrases: Copulas, stochastic finance, extreme values theory, stochastic process,
tail dependence, Value at risk.

1. Introduction

Now a days, stochastic modeling in finance has become essential in risks management
related to the ownership and valuation of financial products (assets, options, bonds,etc.).
Several models exist in financial literature including the Markowitz [17] model, that of
Black-Scholes-Merton [5] and the model of Heston [15], see [17]. In the Markowitz model,
the variance (assumed to be constant) is used as a risk measure to determine the optimal
portfolio. The BSM model based on the use of stochastic differential equations, provides
a formula for valuing options and bonds. This model includes several assumptions such as
constant volatility and a constant and deterministic interest rate. The model of Heston
based on diffusion processes, gives a semi-analytical formula for some derivative products
with a certain realism . This model is part of stochastic volatility models which differs from
BSM. Despite their wide use, all of these models are built under the Gaussian hypothesis

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.4014

Email addresses: lachimer2@gmail.com (H. Abba-M.), dnatatumutari@gmail.com (N. D. Moutari),
dbarro2@gmail.com ( D. Barro), bsaley@yahoo.fr ( B. Saley)

http://www.ejpam.com 1057 c© 2021 EJPAM All rights reserved.



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1058

which does not capture the leptokurticity effect in stochastic finance and there are also
weaknesses when we want to model the joint distribution. To overcome these problems,
other models developped on the basis of on extreme values theory (EVT) and copulas
theory, are proposed. The EVT provides a framework for better modeling leptokurticity
and copulas gives better link between marginal and joint distribution, if one is confronted
with the study of several risks. Indeed, the copula makes it possible to capture the
structure of dependence which exists between several random variables. The first uses of
copulas in financial modeling are very recent (see [4]). They are more and more studied
because of their flexibility and their ease of interpretation and of implementation. For
more details, see Embrechts[11]; Malavergne[16] and the references therein.

The essentiels results in EVT are due to Fisher-Tippet (theorem of three types)
[13] and Balkema-de Haan- Pickands [1]. These results establish in particular that, if
X1; . . . ;Xn is a sequence of random variables with common distribution F , then the excess
variable {Yj = Xj − u/Xj > u}; Nu = card{k/Xk > u} which is governed by the law Fu
(conditional distribution of the unknown continuous distribution function F with respect
to the threshold u), converges asymptotically towards a non-degenerate law. According to
Pickands, Balkema and de Haan: when the threshold u tends towards the right endpoint
xF , it follows that :

lim
u→xF

|Fu(y)−H(y)| = 0.

where the function H(y) corresponds to the distribution function of the generalized
Pareto law(GPD) :

Hξ,σ(y) =


1− [1 + ξ yσ ]

− 1
ξ , ξ 6= 0

1− exp(− y
σ ), ξ = 0

; (1)

where {ξ ∈ R} is the tail index and {µ ∈ R}, {σ � 0} are localization and dispersion
parameters respectively; and y ∈ [0;xF − u] , if ξ ≥ 0 and y ∈ [0,−σ

ξ ] , if ξ < 0.†

Risk is omnipresent in any kind of investment operation on the financial market. Its
universe keeps growing due to the growing creation of new financial products and the emer-
gence of financial markets. In this context, management practicers have been strengthened
especially in matters of regulation (Basel I, II then III), recovery (see [10]) and in matters
design of risk measurement tools. These risk measurement tools are diverse (depending
on the nature of the risk) and various. The main theoretical risk measures are volatility,
correlation coefficient, beta coefficient, VaR, TVaR, CVaR, Kendall tau, Spearman rhô,
the tail dependence coefficient. For more details on the typology of risk and the applica-
tion of these measures in finance, see [11],[16].

Since its first use by JP Morgan (1990), the VaR is currently the most used in finance
(in the univariate case), due to its simplicity of interpretation and calculation. Indeed,

†the GPD H can be written in the form: H(y) = 1 + logG(y) where G corresponds to the Generalized
Extreme Value (GEV) distribution. Considering the GEV model with localization parameter µ = 0 (for
the excesses the effect of the localization parameter is taken in account in the sequence (an)n≥1).



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1059

if X is a variable modeling the gain (or loss) with distribution F then the VaR at the
threshold α is defined by:

V aRα(X) = inf [x/P (X ≥ x) ≤ α] = F−1(α). (2)

Several versions of this measure are proposed in the multivariate framework in Embrechts
[11] and Garcin et al.[14]. To better assess the risk dynamics of a portfolio (the risk linked
to holding the asset portfolio, under various conditions and over time), it is often con-
venient to model the dynamics of assets (and therefore of the portfolio) by a stochastic
process. The stochastic process thus defined will be assimilated to the stochastic risk
which we want to study and quantify the magnitude of the danger.

The objective of this article is to investigate, first, the dependence structure of mul-
tivariate extremal processes. Then, we introduced a new measure of multivariate tails
dependences (lower and upper). Many interesting properties are established. The rest of
the paper is organized as follows: in section 2, the essentiels concepts to the study are
recalled and in section 3 the main results obtained are presented and section 4 we give a
conclusion and discussion.

2. Preliminaries

In this section, we collect essentials notions, definitions and properties on copulas, tail
dependence coefficients and extremal processes, which will be necessary for our approach.
For more details, the reader are refered to authors such as Schimitz [20] which offers a
well-developed framework for the analysis of stochastic processes by copulas or Nelsen [18]
who gave an introduction to copulas and their statistical and mathematical foundations.

2.1. Survey of Copulas of Multivariate Processes

Copulas provide a natural way to construct multivariate distribtutions whose marginals
are uniform and not necessarily exchangeable.

Definition 1. [19] Let X = (X1, ..., Xn) be a random vector with multivariate continuous
distribution function (c.d.f.) F and c.d.f marginal F1, ..., Fn. The copula of X (or the c.d.f.
F respectively) is the multivariate c.d.f. C of the random vector U = [F1(X1), ..., Fn(Xn)].
Due to the continuity of {Fi, 1 ≤ i ≤ n} , each component of U is standard uniformly dis-
tributed, i.e., Ui v U(0, 1) for i = 1, ..., n.

Particularly, every n-copula must satisfy the n-increasing property. That means that,
for any rectangle B = [a, b]n ⊆ Rn, the B-volume CB of C is positive, that is,

CB =

∫
B
dC (u) =

2∑
i1=1

...

2∑
in=1

(−1)i1+...+in C (u1,i1 ; ...;u1,in) ≥ 0. (3)



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1060

for all (u1,1; ...;un,1) and (u1,2; ...;un,2) ∈ [0, 1]n with ui,1 ≤ ui,2, for i = 1, ..., n.
Moreover, from Definition 1, it yields the following parameterization of F (see [4] and

[20]), for (x1, ..., xn) ∈ R̄n (where R̄ = [−∞,+∞])

F (x1, ..., xn) = C [F1(x1), ..., Fn(xn)] . (4)

This result makes possible to join the marginal to the multivariate joint distribution.
Especially in the survival analysis (finance or biostatistics) the above relation, gives the
survival copula in function of the survival law F̄ of F by:

C̄(F̄1(x1), ..., F̄n(xn)) = F̄ (x1, ..., xn). (5)

The survival copula C̄ is linked to the copula C, for all (u1, ..., un) ∈ [0, 1]n, by:

C̄(u1, u2, ..., un) =
∑
M⊂N

(−1)mC
[
(1− u1)11∈M , (1− u2)12∈M , ..., (1− un)1n∈M] ; (6)

where N = {1, 2, ..., n}, m = |M | is the cardinal number of M , and 1i ∈ M indicates the
appartenence of i to M .

A n-dimensional stochastic process being a collection of random variables Xt; t ∈ T
defined on a probability space (Ω,F , P ) and taking values in Rd, n ∈ N, where T is the
set of the parameters (space or time).

Let C = {Ct1,...,tn ; t1 < ... < tn, n ∈ N} is a collection of copulas of stochastic process.
satisfying the consistency condition

lim
uk→1−

Ct 1,...,t n (u1, ..., un) = Ct 1,...,t k−1,t k+1...,t n (u1, ..., uk−1, uk+1, ..., un)

for all ui ∈ (0, 1), 1 ≤ k ≤ n and D = {Ft, t ∈ T} a collection of uni-dimensionnal
distribution. Then, there exists a probability espace (Ω,F , P ) and a stochastic process
{(Yt) , t ∈ T} such that

P (Yt1 < x1, ..., Ytn < xn) = Ct 1,...,t n (Ft1 (x1) , ..., Ftn (xn)) . (7)

for all xi ∈ R, ti ∈ T ⊂ R, 1 ≤ i ≤ n and Yt is Ft-mesurable for all t ∈ T.
Next, we present the notions of extremal processes which are processes obtained as

limits of a normalized maxima processes of a sequence of random variables independent
identically distributed (i.i.d) (or independent sequences not identically distributed). For
more details, the reader can consult Resnick[19] and references therein.

Definition 2. Suppose that X1, X2, ...Xn is a sequence of independent and identically
distributed random variables, and Mn = max{X1, X2, ..., Xn}.
Let {Y (t), t ≥ 0} the stochastic process which is the natural limit when n → ∞, of the
process

Yn(t) = (M[nt] − an)/bn, t ≥ 1/n. (8)



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1061

The joint distribution of vector Yt1, Yt2,..., Ytk is same as the distribution of

U1,max(U1, U2), ...,max(U1, U2, ..., Uk),

where U1,U2,...,Uk are independantes variables whose distribution is

Ftj−tj−1 = Gtj−tj−1 , j = 1, ..., k with t0 = 0. (9)

where G is GEV type. Such a process is called extremal process.

The joint finite-dimensionnal distrbution is defined by

Gt1,t2,...,tk(y1, y2, ..., yk) = P [Y (t1) < y1, Y (t2) < y2, ..., Y (tk) < yk].

which gives, taking into account the independance

Gt1,t2,...,tk(y1, y2, ..., yk) =
[
G(∧ki=1yi)

]t1
×
[
G(∧ki=2yi)

]t2−t1
× ...× [G(yk)]

tk−tk−1 . (10)

Let’s remark that the advantage of these types of processes lies in the fact that they
allow us to introduce a certain dynamic (evolution over time) in the modeling of extreme
risks.

2.2. Overview of the tail multivariate dependence

The concept of tail dependence is widely used in multivariate analysis mainly in the
bivariate case, see[10], [8] and [3]. However, their study in larger dimension is expanding.
The generalization of tail dependence in dimension d > 2 consists in choosing (h < d)
variables and quantifying the conditional probability that each variables h take values in
the tail knowing that the d− h variables take this value too (see Barro [2]).

Let X = (X1, ..., Xd) be a random vector of Rd of joint distribution F and of copula
C. We note, in the rest of this study, for all h ≤ d, X(h) = (X1, ..., Xh); X(d−h) =
(Xh+1, ..., Xd); and by Ch and Cd−h their respective copulas. The generalization of the
coefficients of upper tail dependencie λU,h, and lower tail dependencie λL,h, are given by:

λU,h = lim
u→1−

P{X1 > F−1
1 (u), ..., Xh > F−1

h (u)/Xh+1 > F−1
h+1(u), ..., Xd > F−1

d (u)}, (11)

and

λL,h = lim
u→0+

P{X1 ≤ F−1
1 (u), ..., Xh ≤ F−1

h (u)/Xh+1 ≤ F−1
h+1(u), ..., Xd ≤ F−1

d (u)}. (12)

In terms of copulas, (11) and (12) take respectively the form :

λU,h = lim
u→1−

−
C(1− u, ..., 1− u)
−
Cd−h(1− u, ..., 1− u)

, and λL,h = lim
u→0+

C(u, ..., u)

Cd−h(u, ..., u)
,

where Cd−h is the marginal copula of C associeted to vector X(d−h) and C̄ is the survival
copula associeted to C.

The four following sections present our main contribution in stochastic risk modeling.



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1062

3. Stochastic Risk modeling via extremal copulas

In this part of the paper, we propose some results on the multivariate extremal pro-
cesses De-Haan [9] for more details. Let Y (t) = {(Y1(t), ..., Yd(t)} be a vector of extremal
process with joint distribution Ft = (F1,t, ..., Fd,t), where Fi,t is given by the formula (9).
Y (t) can be seen as the ”limit” of Yn(t) = {(Y1,n(t), ..., Yd,n(t))} where Yi,n(t), i = 1, .., d
are defined as in (8) :

P [Y1,n(t) ≤ y1, ..., Yd,n(t) ≤ yd]→ Ft(y1, ..., yd), n→∞. (13)

and according to De-Haan[9], Ft(y1, ..., yd) = Gt∗(y1, ..., yd) where G∗ is the multivariate
extreme value distribution.

Proposition 1. Let Y (t) be a vector of extremal processes in Rd, with joint distribution
Ft and marginal Fi,t, i = 1, ..., d. Then, there exists a unique convex function Bt : [0,∞]×
Sd → [0;∞[ such that a copula associeted to Ft be defined by,

C∗t (u1, ..., ud) = exp

{
−

(
d∑
i=1

ũi,t

)
Bt

(
ũ1,t∑d
i=1 ũi,t

, ...,
ũd,t∑d
i=1 ũi,t

)}
, (14)

where Sd = {(x1, .., xd) ∈ Rd/
∑d

i=1 xi ≤ 1} is the simplex defined on Rd and ũi,t =

µi −
σi
ξi

[
1− (− lnui

t
)−ξi

]
, i = 1, 2, ..., d.

Proof. According to Pickands, the function G∗ is given (see Falk[12] and Resnick[19])
by:

G∗(x1, ..., xd) = exp

{
−

(
d∑
i=1

xi

)
A

(
x1∑d
i=1 xi

, ...,
xd∑d
i=1 xi

)}
, (15)

where the convex function A : Sd → [
1

d
, 1], is the Pickands dependence function.

Since Ft(x1, ..., xd) = Gt∗(x1, x2, ..., xd), then

Ft(x1, ..., xd) = exp

{
−t

(
d∑
i=1

xi

)
A

(
x1∑d
i=1 xi

, ...,
xd∑d
i=1 xi

)}
(16)

the marginal distribution Fi,t of Ft being continuous, according to Sklar’s theorem, so,
there exists a unique copula C∗t such that,

Ft(x1, ..., xd) = C∗t (F1,t(x1), ..., Fd,t(xd)), (17)

which can be write, by taking ui = Fi,t(xi) for all i, such as

C∗t (u1, ..., ud) = Ft(F
−1
1,t (u1), ..., F−1

d,t (ud)). (18)



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1063

Combining the relations (16) and (18) give us,

C∗t (u1, ..., ud) = exp

{
−t

(
d∑
i=1

F−1
i,t (ui)

)
A

(
F−1

1,t (u1)∑d
i=1 F

−1
i,t (ui)

, ...,
F−1
d,t (ud)∑d

i=1 F
−1
i,t (ui)

)}
,

which gives,

C∗t (u1, ..., ud) = exp

{
−

(
d∑
i=1

ũi,t

)
Bt

(
ũ1,t∑d
i=1 ũi,t

, ...,
ũd,t∑d
i=1 ũi,t

)}
, (19)

with Bt(x1, ..., xd) = tA(x1, ..., xd) and ũi,t = F−1
i,t (ui), for all i = 1, 2, .., d and ui ∈ [0, 1].

The copula C∗t being unique, so is the function Bt. The latter function is convex, being
the product of a positive real and a convex function (function A)

The following result proposes a new measure of stochastic multivariate dependence.

Proposition 2. Let Y (t) be a vector of extremal processes in Rd, with joint distribution Ft
and marginal Fi,t, i = 1, ..., d. Let consider k random vector {Y (tj) = (Y1,tj , ..., Yd,tj ), j =

1, 2, .., k} and yj = (yj1, ..., y
j
d) ∈ Rd. Then, the joint finite-dimensionnal copula C∗t1,t2,...,tk

of Y (t1), Y (t2), ..., Y (tk) is defined, for 0 = t0 < t1 < ... < tk by:

C∗t1,t2,...,tk [(uj1, ..., u
j
d)] = exp

{
−

k∑
m=1

Ltm(ũ1,tm , ũ2,tm , ..., ũd,tm)

}
; (20)

where (uj1, ..., u
j
d) ∈ [0; 1]d, for j = 1; ..., k and Ltm is a suitable convex funtion.

Proof. The joint finite-dimensionnal distribution of {Y (tj) = (Y1,tj , Y2,tj , ..., Yd,tj )}, for

all yj = (y1
j , ..., y

d
j ) and j = 1, 2, .., k is given by :

P [(Y1,tj ≤ y
j
1, Y2,tj ≤ y

j
2; ...;Yd,tj ≤ y

j
d)] = Ft1,t2,...,tk((yj1, y

j
2, ..., y

j
d),

according to de Haan[9], Which gives

P [(Y1,tj ≤ y
j
1; ...;Yd,tj ≤ y

j
d)]

=Ft1

(∧k
j=1 y

j
1, ...,

∧k
j=1 y

j
d

)
× Ft2−t1

(∧k
j=2 y

j
1; ...;

∧k
j=2 y

j
d

)
× ...× Ftk−tk−1

(yk1 ; ...; ykd)

(21)
Let’s take uji = Fi,tj (y

j
i ) for all i = 1, ..., d and j = 1, ..., k, we have

C∗t1,t2,...,tk [(uj1, u
j
2, ..., u

j
d)] = Ft1,t2,...,tk((F−1

1,tj
(uj1), ..., F−1

d,tj
(ujd)))

That leads to,

C∗t1,t2,...,tk [(uj1, ..., u
j
d)] = Ft1

(
k∧
i=1

F−1
t1

(ui1),
k∧
i=1

F−1
t1

(ui2), ...,
k∧
i=1

F−1
t1

(uid)

)
× (22)



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1064

×Ft2−t2

(
k∧
i=2

F−1
t2−t1(ui1),

k∧
i=2

F−1
t2−t1(ui2), ...,

k∧
i=2

F−1
t2−t1(uid)

)
×Ftk−tk−1

(
F−1
tk−tk−1

(uk1), F−1
tk−tk−1

(uk2), ..., F−1
tk−tk−1

(ukd)
)
.

But according to Proposition 1, C∗t1,t2,...,tk((uj1, ..., u
j
d) for j = 1, ..., k) can be written

as

C∗t1,t2,...,tk [(uj1, ..., u
j
d)] = exp

−
 d∑
i=1

k∧
j=1

ũji,t1

Bt1

( ∧k
j=1 ũ

j
1,t1∑d

i=1

∧k
j=1 ũ

j
i,t1

, ...,

∧k
j=1 ũ

j
d−1,t1∑d

i=1

∧k
j=1 ũ

j
i,t1

)×
× exp

−
 d∑
i=1

k∧
j=2

ũji,t2

Bt2−t1

( ∧k
j=2 ũ

j
1,t2∑d

i=1

∧k
j=2 ũ

j
i,t2

, ...,

∧k
j=2 ũ

j
d−1,t2∑d

i=1

∧k
j=2 ũ

j
i,t2

)×
× ...× exp

{
−

(
d∑
i=1

ũki,tk

)
Btk−tk−1

(
ũk1,tk∑d
i=1 ũ

k
i,tk

, ...,
ũkd−1,tk∑d
i=1 ũ

k
i,tk

)}
.

So, it comes that

C∗t1,t2,...,tk [(uj1, ..., u
j
d)] = exp

{
−

k∑
m=1

Ltm(ũ1,tm , ũ2,tm , ..., ũd,tm)

}
(23)

where the dependence function Ltm is given by

Ltm(ũ1,tm , ..., ũd,tm) =

 d∑
i=1

k∧
j=m

ũji,tj

Btm−tm−1

( ∧k
j=m ũ

j
1,tj∑d

i=1

∧k
j=m ũ

j
i,tj

, ...,

∧k
j=m ũ

j
d−1,tj∑d

i=1

∧k
j=m ũ

j
i,tj

)

is a suitable convex function defined in the simplex of Rd while, for all i = 1, ..., d and
j = 1, ..., k, one have ũi,tm =

∧k
j=m ũ

j
i,tj

, and

ũji,tj = µi −
σi
ξi

1−

(
−

lnuji
tj − tj−1

)−ξi

4. The Expected Tail Dependence Coefficients

In this section, we first introduce the notion of expected tail dependence coefficient
then we present essential properties related to the concept. The notion is a generalization
of the notion of multivariate tail dependence coefficients (relations (11) and (12)) in the
sense that they give us information on dependence average in the tails of multivariate
distribution. In the bivariate case, it is close to the proposed risk measure by Brahim et
al.[6].



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1065

Definition 3. Let X = (X1, ..., Xd) a random vector in Rd, with joint distribution F ,
marginal Fi, i = 1, ..., d and copula C. We call the expected or average tail function of
lower tail ζLh and of upper tail ζUh , the quantities respectively defined by:

ζLh (u) = E{X(h)/Xh+1 ≤ F−1
h+1(u), ..., Xd ≤ F−1

d (u)},

respectively

ζUh (u) = E{X(h)/Xh+1 > F−1
h+1(u), ..., Xd > F−1

d (u)}.

A vectorial approach of these coefficients can be written:

ζUh (u) =


E{X1/Xh+1 > F−1

h+1(u), ..., Xd > F−1
d (u)}

E{X2/Xh+1 > F−1
h+1(u), ..., Xd > F−1

d (u)}
...
...

E{Xh/Xh+1 > F−1
h+1(u), ..., Xd > F−1

d (u)}

 ;

for the lower coefficient, and

ζLh (u) =


E{X1/Xh+1 ≤ F−1

h+1(u), ..., Xd ≤ F−1
d (u)}

E{X2/Xh+1 ≤ F−1
h+1(u), ..., Xd ≤ F−1

d (u)}
...
...

E{Xh/Xh+1 ≤ F−1
h+1(u), ..., Xd ≤ F−1

d (u)}

 .

for the right one.
It’s abouts determining on average, having a vector X = (X1, ..., Xd), what is happen-

ing in the distribution tails of the h ≤ d random variables X1, ..., Xh knowing that the
(d− h) remaining variables each exceed its VaR at a threshold u ∈ [0, 1] chosen.

Definition 4. We call, respectively, expected (average) tail dependence of lower tail and
upper tail respectively, the limits;

ζLh = lim
u→0+

ζLh (u), and ζUh = lim
u→1−

ζUh (u). (24)

The result below gives us the expressions of the marginal of the average dependence
coefficients of lower and upper tail in terms of copulas.

Proposition 3. Let X = (X1, ..., Xd) a random vector with copula C and suppose that
the conditionnals densities fXi/X(d−h) and f̆Xi/X(d−h) of {Xi/Xh+1, ..., Xd}, respectively for
lower and upper tail, exist. Then,

i) the marginal expected tail dependence coefficient of lower tail of X is given by:

ζLh,i = lim
u→0+

∫ 1

u
V aRXi(α).cXi/X(d−h)(α, u, ..., u)dα, (25)



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1066

where

cXi/X(d−h)(α, u, ..., u) = ∂h+1,...,d
∂iC(ui, uh+1, ..., ud)

Cd−h(uh+1, ..., ud)
|(ui,uh+1,...,ud)=(α,u,...,u) and

C(ui, uh+1, ..., ud) = C(1, ..., 1, ui, 1, ...1, uh+1, ..., ud).

ii) the marginal expected tail dependence coefficient of upper tail of X is given by:

ζUh,i = lim
u→1−

∫ 1−u

0
V aRXi(1− α).c̆Xi/X(d−h)(α, u, ..., u)dα, (26)

where

c̆Xi/X(d−h)(1−α, 1−u, ..., 1−u) = ∂h+1,...,d
∂iC̄(1− ui, 1− uh+1, ..., 1− ud))

C̄d−h(1− uh+1, ..., 1− ud)
|(ui,uh+1,...,ud)=(α,u,...,u)

and
C̄(1− ui, 1− uh+1, ..., 1− ud) = C̄(1, ..., 1, 1− ui, 1, ...1, 1− uh+1, ..., 1− ud).

Proof.

i) Each marginal coefficient for lower tail ζLh,i, i = 1, .., h is defined by,

ζLh,i = lim
u→0+

E{Xi/Xh+1 ≤ F−1
h+1(u), ..., Xd ≤ F−1

d (u)}

which gives

ζLh,i = lim
u→0+

∫ +∞

φXi (u)
xfXi/X(d−h)dx; (27)

where φXi(u) designates the u level quantile associeted to the variable Xi and
fXi/X(d−h) is the conditional density associeted to {Xi/Xh+1, ..., Xd}, that we sup-
pose the existence.
The associeted distribution function FXi/X(d−h) of fXi/X(d−h) is equal to,

FXi/X(d−h)(xi, xh+1, ..., xd) = P [Xi ≤ xi/Xh+1 ≤ xh+1, ..., Xd ≤ xd]

Furthermore, we have

FXi/X(d−h)(xi, xh+1, ..., xd) =
P [Xi ≤ xi, Xh+1 ≤ xh+1, ..., Xd ≤ xd]

P [Xh+1 ≤ xh+1, ..., Xd ≤ xd]

and by using Sklar’s theorem, it follows that

FXi/X(d−h)(xi, xh+1, ..., xd) =
C(Fi(xi), Fh+1(xh+1), ..., Fd(xd))

Cd−h(Fh+1(xh+1), ..., Fd(xd))
(28)

where

C(Fi(xi), Fh+1(xh+1), ..., Fd(xd)) = C(1, 1, ...1, Fi(xi), 1, ..1, Fh+1(xh+1), ..., Fd(xd)).



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1067

Then the corresponding density fXi/X(d−h) is given by,

fXi/X(d−h)(xi, xh+1, ..., xd) = ∂i,h+1,...,d
C(Fi(xi), Fh+1(xh+1), ..., Fd(xd))

Cd−h(Fh+1(xh+1), ..., Fd(xd))
.

which gives

fXi/X(d−h)(xi, xh+1, ..., xd) = fi(xi)∂h+1,...,d

(
∂iC(Fi(xi), Fh+1(xh+1), ..., Fd(xd))

Cd−h(Fh+1(xh+1), ..., Fd(xd))

)
(29)

for all (ui, uh+1, ..., ud) ∈ [0, 1]× [0, 1]d−h,

cXi/X(d−h)(ui, uh+1, ..., ud) = ∂h+1,...,d
∂iC(ui, uh+1, ..., ud)

Cd−h(uh+1, ..., ud)

it follows that,

cXi/X(d−h)(α, u, ..., u) = ∂h+1,...,d
∂iC(ui, uh+1, ..., ud)

Cd−h(uh+1, ..., ud)
|(ui,uh+1,...,ud)=(α,u,...,u)

where C(ui, uh+1, ..., ud) = C(1, ...1, ui, 1...1, uh+1, ..., ud).

Finally,

ζLh,i = lim
u→0+

∫ 1

u
V aRXi(α)cXi/X(d−h)(α, u, ..., u)dα.

ii) In the same way, for each marginal coefficient of upper tail ζLh,i, i = 1, .., h; the
conditional distribution of {Xi > ./Xh+1 > ., ...,Xd > .} is given by;

F̆ (xi, xh+1, ..., xd) =
C̄(1− Fi(xi), 1− Fh+1(xh+1), ..., Fd(xd))

C̄d−h(1− Fh+1(xh+1), ..., Fd(xd))

so, the corresponding density f̆Xi/X(d−h) is obtained,

f̆Xi/X(d−h)(xi, xh+1, ..., xd) = −fi(xi)∂h+1,...,d
∂iC̄d−h(1− Fi(xi), 1− Fh+1(xh+1), ..., Fd(xd))

C̄(1− Fh+1(xh+1), ..., Fd(xd))
,

it follows that, for all (ui, uh+1, ..., ud) ∈ [0, 1]× [0, 1]d−h,

c̆Xi/X(d−h)(ui, uh+1, ..., ud) = ∂h+1,...,d
∂iC̄(1− ui, 1− uh+1, ..., 1− ud))

C̄d−h(1− uh+1, ..., 1− ud)
,

which implies,

c̆Xi/X(d−h)(α, u, ..., u) = ∂h+1,...,d
∂iC̄(1− ui, 1− uh+1, ..., 1− ud))

C̄d−h(1− uh+1, ..., 1− ud)
|(ui,uh+1,...,ud)=(α,u,...,u),

with C̄(1− ui, 1− uh+1, ..., 1− ud) = C̄(1, ...1, 1− ui, 1...1, 1− uh+1, ..., 1− ud).
and finaly,

ζUh,i = lim
u→1−

∫ 1−u

0
V aRXi(1− α)c̆Xi/X(d−h)(α, u, ..., u)dα.



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1068

It is easy to remark that for the particular case where the copula associeted to the
vector X = (X1, ..., Xd) is the independant one, that is : C(u1, ..., un) = u1 × ... × un,
then,

CXi/X(d−h)(ui, uh+1, ..., un) = ui, ⇒ cXi/X(d−h)(ui, uh+1, ..., un) = 0

and
C̆Xi/X(d−h)(ui, uh+1, ..., un) = 1− ui, ⇒ c̆Xi/X(d−h)(ui, uh+1, ..., un) = 0

from where ζUh = 0 and ζLh = 0.

Belonging both to the max-stable and the Archimax families, the logistic family plays
a key role in extremal modeling.

Corollary 1. The classic copula associated to the logistic family (Gumbel’s copula) does
not admit coefficient of average dependence of lower tail, while that of upper tail is asymp-
totically negligible.

Proof. Let (X,Y ) a bivariate vector with joint distribution F (x, y) = (1−e−x−e−y)−1.
The marginal distributions are equal to FX(t) = FY (t) = (1 − e−t)−1 and the associeted

copula is C(u, v) =
uv

u+ v − uv
. The conditional copula for lower tail CX/Y is given by

CX/Y (u, v) =
u

u+ v − uv
,

i) For the upper tail C̆X/Y it comes that

C̆X/Y (u, v) =
(1− u− v)(u+ v − uv) + uv

(1− v)(u+ v − uv)
= 1− u2

u+ v − uv
.

The average dependence functions of lower and upper tail are calculated such as:

ζL(v) =

∫ 1

v

[
x− v + xv

(x+ v − xv)3
ln

(
x

1− x

)]
dx.

Dealing with integration once by parts, it follows that

ζL(v) =
1− v

(2− v)2
ln(

v

1− v
)−

∫ 1

v

1

x(1− x)

−x(1− x)

(x+ v − xv)2
dx

which gives, twice by parts

ζL(v) =
(1− v)

(2− v)2
ln(

v

1− v
) +

1− v
2v − v2

(30)

and finally, we have

ζL = lim
v→0+

(1− v)

(2− v)2
ln

(
v

1− v

)
+

1− v
2v − v2

= +∞.



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1069

Figure 1: Expected (average) dependence function of lower tail and the VaR of lower tail for the copula
associated with the bivariate logistics distribution of Gumbel.

The figure1, gives us the evolution of the average dependence function of the lower
tail, with the corresponding VaR and the bound defined in relation (52).

ii) For the upper tail, it comes that

ζU (v) =

∫ v

0

[
−x4(1− v)− xv(−x2 − 2x+ 2)

(x+ v − xv)3
ln

(
x

1− x

)]
dx

which gives, once by parts

ζU (v) = ln(
v

1− v
)
v2(1− v)

(2v − v2)
−
∫ v

0

x

(x+ v − xv)2
dx

and twice by parts

ζU (v) =
v(1− v)

(2− v)
ln(

v

1− v
)− 1− v

2− v
+

∫ v

0

1− x
x(1− v) + v

dx

By taking

R (v) =
v(1− v)

(2− v)
ln(

v

1− v
)− 1− v

2− v
one obtains

ζU (v) = R (v) +

[
1− x
1− v

ln(x(1− v) + v)

]v
0

− 1

1− v

∫ v

0
ln(x(1− v) + v)dx



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1070

So, it comes that:

ζU (v) = R (v)+ln(v(1−v)+v)− 1

1− v

([
x(1− v) + v

1− v
ln(x(1− v) + v)

]v
0

−
∫
dx

)
.

And finally:

ζU (v) =
v(1− v)

(2− v)
ln

(
v

1− v

)
+

(
−1− v

2− v
+

2v − v2

(1− v)2

)
ln(2v − v2) + 1. (31)

Therefore, it comes that

ζU = lim
u→1−

v(1− v)

(2− v)
ln

(
v

1− v

)
+

(
−1− v

2− v
+

2v − v2

(1− v)2

)
ln(2v − v2) + 1 = 0.

The figure2, gives us the evolution of the expected dependence function of upper
tail, with the corresponding VaR and the bound defined in relation (53).

Figure 2: Expected dependence function of the upper tail and VaR of upper tail of the copula associated with
the bivariate logistics distribution of Gumbel.

Corollary 2. Let (X,Y ) be a bivariate random variable which marginal distribution is
standard Fréchet one FX(x) = exp{−x−α} and that the copula which determines their
dependence structure is Gumbel-Hougaard bivariate copula. Then,

ζL = +∞, and ζU = +∞. (32)



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1071

Proof. The bivariate Gumbel-Hougaard copula (see [18]) is defined ∀u, v ∈ [0, 1] and
θ ≥ 1, by

Cθ(u, v) = exp

{
−
[
(− ln(u))θ + (− ln(v)))θ

]1/θ}
. (33)

The conditional copulas CX/Y and C̆X/Y are given by

CX/Y (u, v) =
Cθ(u, v)

v
. (34)

and

C̆X/Y (u, v) =
1− u− v + Cθ(u, v)

1− v
. (35)

So, we obtain the densities cX/Y and c̆X/Y , respectively by

c̆X/Y (u, v) =
Cθ(u, v)

uv(1− v)

[
(1− θ)

[
(− lnu)θ + (− ln v)θ

] [
1−2θ
θ (− lnu)(− ln v)

]θ−1
+

+

[
(− ln v)

[
θ−1(− lnu)θ + (− ln v)θ

] 1−θ
θ +

v

1− v

]
×[

(− lnu)θ−1
[
(− lnu)θ + (− ln v)θ

]
 . (36)

The expected tail dependence functions are calculated, ζL(u) =
∫ 1
u (− ln t)−1/αcX/Y (t, u)dt

which gives :

ζL(u) =

[
−(− ln t)−1/αCθ(t, u)

tu2

[
(− lnu)θ−1(− ln t)θ + (− lnu)θ

] 1−θ
θ

+ 1

]1

u

+

+

∫ 1

u

[
(− ln t)

−1−α
α

α

Cθ(t, u)

tu2

[
(− lnu)θ−1(− ln t)θ + (− lnu)θ

] 1−θ
θ

+ 1

]
dt

and finally

ζL(u) = (− lnu)−1/αu(21/θ−2)(2
1−θ
θ + 1) +

(− lnu)θ−1

αu2
K1(u), (37)

where

K1(u) =

∫ 1

u

[
(− ln t)

−1−α
α

Cθ(t, u)

t

[[
(− ln t)θ + (− lnu)θ

] 1−θ
θ + (− lnu)1−θ]] dt.

The figure 3 shows the behavior of average dependence function of lower tail for dif-
ferent values of the parameter (θ = 1.5 and θ = 1.7) of the copula. We see that it is a
increasing function of the parameter.

ζU (u) =

∫ u

0
(− ln(1− t))−1/αc̆X/Y (t, u)dt

which gives



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1072

Figure 3: Lower tail average dependence function and lower tail VaR for the Gumbel-Hougaard copula with
standard frechet margin with parameter α.

ζU (u) =

⌊
(− ln(1− t))−1/α Cθ(t, u)

(1− u)2

[
1− u
u

{
− (− lnu)θ−1

[
(− ln t)θ + (− lnu)θ

] 1−θ
θ

}
+ 1

]⌋u
0

+

+

∫ u

0

[
(− ln(1− t))

−1−α
α

αt

Cθ(t, u)

(1− u)2

[
1− u
u

{
− (− lnu)θ−1

[
(− ln t)θ + (− lnu)θ

] 1−θ
θ

}
+ 1

] ]
dt

and finally

ζU (u) =
[1− u

u
2( 1−θ

θ
) + 1

]u21/θ(− ln(1− u))−1/α

(1− u)2
+

(− lnu)θ−1

αu(1− u)
K2(u), (38)

where

K2(u) =
∫ u

0

(− ln(1− t))
−1−α
α

t
Cθ(t, u)

[{
−
[
(− ln t)θ+(− lnu)θ

] 1−θ
θ

}
+

u

(1− u)(− lnu)θ−1

]
dt.

The figure 4 shows the behavior of average dependence function of upper tail for
different values of the parameter (θ = 1.5 and θ = 1.7) of the copula. We see that it’s a
increasing function of the parameter.

5. Density of Conditional Extremal copulas

The resultat below gives us the form of the density cXi/X(d−h) in the case of conditional
extremal copula.



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1073

Figure 4: Lower tail average dependence function and upperr tail VaR for the Gumbel-Hougaard copula with
standard frechet margin with parameter α.

Proposition 4. Let X = (X1, ..., Xd) a random vector with extremal copula C. Let note
u = (ui, uh+1, ..., ud) ∈ [0, 1] × [0, 1]d−h and l stable tail dependence function. Then the
density function cXi/X(d−h) of the copula CXi/X(d−h) associeted to the conditionnal vector

{Xi/X(d−h)} is given by;

cXi/X(d−h)(u) =
CXi/X(d−h)(u)

ui
∏d
h+1 uj

[∑
κ∈E

∂κ (∂il(ũ)) .

(∑
π∈Π

(−1)|π|
∏
B∈π

∂BL(ũ)

)]
(39)

where ũ = − log(u); ∂i(.) indicates the partial derivative with respect to the i-th vari-
able;

∏
u = ui.uh+1...ud, E is the set of all parts of E = {h + 1, h + 2, ..., d} , π

runs through the set Π of partitions of κc, the complement of κ in E, B ∈ π sig-
nificates that B runs through the set π; |π| designates the cardinal number of π and
L(xi, xh+1, ..., xd) = l(xi, xh+1, ..., xd)− ld−h(xh+1, ..., xd).

For prove Proposition 10 we need the following Lemma which establishes a property
of two partially devative functions.

Lemma 1. Let f and g two functions defined on Rd such that their partial derivatives of
all order exist and are continuous. Then, for all x = (x1, x2, ...., xd) ∈ Rd

∂1,2,...,d(f(x).g(x)) =
∑
κ∈E

∂κf(x).∂κcg(x), (40)

where E is the set of all parts of E = {1, 2, ..., d} and κc is the complement of κ in E.



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1074

The proof of this lemma is given by induction for recurrence relation.
Proof.

i) At first order, we have :

If E = {1} then E = {∅, {1}} therefore κ = ∅ or κ = {1},

∂1(f(x).g(x)) =
∑

κ=∅ ,{1}

∂κf(x).∂κcg(x) = ∂∅f(x).∂{1}g(x) + ∂{1}f(x).∂∅g(x)

which gives
∂1(f(x).g(x)) = f(x)∂1g(x) + g(x)∂1f(x).

So, the formula is true at first order.

ii) Suppose that the formula is true at order for p ≥ 1, that is

∂1,2,...,p(f(x).g(x)) =
∑
κ∈E

∂κf(x).∂κcg(x), (41)

where E is the set of all parts of E = {1, 2, ..., p} and κc is the complement of κ in
E.

iii) Let’s prove that the formula is true at order p+ 1,

∂1,2,...,p,p+1(f(x).g(x)) = ∂p+1(∂1,2,...,p(f(x).g(x)))

= ∂p+1

(∑
κ∈E

∂κf(x).∂κcg(x)

)
=

∑
κ∈E

∂p+1

(
∂κf(x).∂κcg(x)

)
=

∑
κ∈E

[ ∑
κ′=∅ ,{p+1}

∂κ′
[
∂κf(x)

]
.∂κ′c

[
∂κcg(x)

]]
=

∑
κ∈E

∑
κ′=∅ ,{p+1}

∂κ′∪κf(x).∂κ′c∪κcg(x). (42)

So, finally, we get

∂1,2,...,p,p+1(f(x).g(x)) =
∑
κ1∈E1

∂κ1f(x).∂κc1g(x). (43)

In (42), note that for κ′ = ∅ , {p+ 1} ( κ′c = {p+ 1} , ∅) and ∀κ ∈ E the set of parts
of E, then:

(κ′ ∪ κ) ∩ (κ′c ∪ κc) = ∅
(κ′ ∪ κ) ∪ (κ′c ∪ κc) = {1, 2, ..., p, p+ 1}, (44)

and also κ′∪κ runs through the set E1 of all parts of E1 = {1, 2, ..., p, p+1}. Therefore
(40) holds



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1075

Now, we are able to proove Proposition 4
Proof. (of Proposition 4) Since the copula C is extreme, then it have the representa-

tion(see Falk[12]):

C(u1, u2, ..., ud) = exp
{
− l(− log u1,− log u2, ...,− log ud)

}
, (45)

where the function l(.) is the stable tail dependence function associeted to C.
Then, for u = (ui, uh+1, ..., ud) ∈ [0, 1]× [0, 1]d−h,

CXi/X(d−h)(ui, uh+1, ..., ud) =
exp {−l(− log ui,− log uh+1, ...,− log ud)}

exp {−ld−h(− log uh+1, ...,− log ud)}
.

Then the density is given by,

cXi/Xd−h(ui, uh+1, ..., ud) = ∂h+1,...,d
∂i exp

{
− l(− log ui,− log uh+1, ...,− log ud)

}
exp

{
− ld−h(− log uh+1, ...,− log ud)

} .

That gives

cXi/Xd−h(ui, uh+1, ..., ud) =
1

ui
∂h+1,...,d

[
∂ui l(− log u). exp{−L(− log u)}

]
(46)

That is :

cXi/Xd−h(ui, uh+1, ..., ud) =
1

ui

∑
κ∈E

∂κ
(
∂ui l(− log u)

)
.∂κc exp{−L(− log u)}. (47)

with
L(xi, xh+1, ..., xd) = l(xi, xh+1, ..., xd)− ld−h(xh+1, ..., xd)

; E being the set of all parts of E = {h+ 1, ..., d} and κc the complement of κ in E.
Using the formula of Fàa di Bruno, it follows for all x = (x1, ..., xd) ∈ Rd;

∂d

∂x1...∂xd
f(g(x)) =

∑
π∈Π

f (|π|)(g(x)).
∏
B∈π

∂(|B|)g(x)∏
j∈B ∂xj

; (48)

where π runs through Π the set of partitions of {1, 2, ..., d} and B ∈ π significates that it
runs through the elements of π.

Furthermore, by taking x = − log(u), one can remark that the expression exp{−L(x)}
is of the form f(g(x)). Then the expression

∂κc exp{−L(− log u)} =
∑
π∈Π

(−1)(|π|) exp{−L(x)}.
∏
B∈π

∂(|B|)L(x)∏
j∈B ∂uj

;

where Π is the set of partitions of κc.
By noticing that

t
∂(|B|)L(x)∏
j∈B ∂uj

=
∂(|B|)L(x)∏
j∈B ∂xj

.
∏
j∈B

∂xj
∂uj

,



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1076

on obtains,

exp{−L(x)}
∑
π∈Π

(−1)|π|
∏
B∈π

∂(|B|)L(x)∏
j∈B ∂xj

.
∏
j∈B

∂xj
∂uj

= exp{−L(x)}
∑
π∈Π

(−1)(|π|)
∏
B∈π

∂(|B|)L(x)∏
j∈B ∂xj

.
∏
j∈B

−1

uj

 .

which gives

exp{−L(x)}
∑
π∈Π

(−1)|π|
∏
B∈π

∂(|B|)L(x)∏
j∈B ∂xj

.
∏
j∈B

∂xj
∂uj

= exp{−L(x)}
∑
π∈Π

(−1)|π|

∏
B∈π

∏
j∈B

−1

uj

 ∏
B∈π

∂(|B|)L(x)∏
j∈B ∂xj

.

And finally,

exp{−L(x)}
∑
π∈Π

(−1)|π|
∏
B∈π

∂(|B|)L(x)∏
j∈B ∂xj

.
∏
j∈B

∂xj
∂uj

=
− exp{−L(x)}∏

j∈κc Uj

∑
π∈Π

(−1)|π|
∏
B∈π

∂BL(x).

(49)
In other hand, we have

∂κ
(
∂il(− log u)

)
=
∂(|κ|)(∂il(x))∏

j∈κ ∂xj
.
−1∏
j∈κ uj

That is:

∂κ
(
∂il(− log u)

)
=

−1∏
j∈κ uj

.∂κ(∂il(x)) (50)

by using the relations (49) and (50), we obtain (39) as disserted.

6. A relationship between VaR and TVaR

It is convenient in risk analysis to compare each risk measure to VaR (which is a
reference) or to its derivative measures, the tail value-at-risk of risk X at the α ∈ [0, 1]
level defined by,

TV aR(α) =
1

1− α

∫ 1

α
V aRX(t)dt. (51)

For detailed statements , see [16].

Corollary 3. Let X = (X1, ..., Xd) be a random vector with extremal copula C .
The marginal of the expected tail dependence is defined, for all u ∈ [0, 1], by

ζLh,i = lim
u→0+

1

(u)d−h

∫ 1

u

V aRXi(α)

α
CXi/X(d−h)(α, u, ..., u)Hi(α, u)dα,

where

Hi(α, u) =

⌊∑
κ∈E ∂κ

(
∂il(x)

)
.
(∑

π∈Π(−1)|π|
∏
B∈π ∂BL(x)

)⌋
x=(− logα,− log u,− log u...,− log u)

.



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1077

Proof. This result is immediat by using relations (25) and (39) and taking ui =
α, uh+1 = u, ..., ud = u for all i = 1, ..., h and h ≤ d.

Corollary 4. Let X = (X1, ..., Xd) be a random vector, with distribution F and associeted
copula C. for all i = 1, ..., h with h ≤ d, then

i) the marginal of the expected tail dependence function of lower tail verify, for all u
near 0

ζLh,i(u) ≥ γ(u).V aRXi(u), (52)

where γ(u) =
∫ 1
u cXi/X(d−h)(α, u, ..., u)dα.

ii) the marginal of the expected tail dependence function of upper tail verify, for all u
near 1

ζUh,i(u) ≥ β(u).V aRXi(1− u), (53)

where β(u) =
∫ u

0 c̆Xi/X(d−h)(α, u, ..., u)dα.

Proof. For all u, α ∈ [0, 1[ such that u ≤ α, it follows that

V aRXi(α) ≥ V aRXi(u), (54)

since ∂h+1,...,d
∂iC(ui, uh+1, ..., ud)

Cd−h(uh+1, ..., ud)
is a density, we have

∂h+1,...,d
∂iC(ui, uh+1, ..., ud)

Cd−h(uh+1, ..., ud)
≥ 0, (55)

which implies,
cXi/X(d−h)(α, u, ..., u) ≥ 0, (56)

so,
V aRXi(α)cXi/X(d−h)(α, u, ..., u) ≥ V aRXi(u)cXi/X(d−h)(α, u, ..., u).

Moreover,

ζLh,i(u) =

∫ 1

u
V aRXi(α)cXi/X(d−h)(α, u, ..., u)dα

≥ V aRXi(u)

∫ 1

u
cXi/X(d−h)(α, u, ..., u)dα

= V aRXi(u).γ(u). (57)

Finally,
ζLh,i(u) ≥ γ(u).V aRXi(u), i = 1, .., h. (58)

For all u, α ∈ [0, 1[ such that α ≤ u, then

V aRXi(1− α) ≥ V aRXi(1− u), (59)



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1078

Since ∂h+1,...,d
∂iC̄(1− ui, 1− uh+1, ..., 1− ud))

C̄d−h(1− uh+1, ..., 1− ud)
is a density function, it follows that

∂h+1,...,d
∂iC̄(1− ui, 1− uh+1, ..., 1− ud))

C̄d−h(1− uh+1, ..., 1− ud)
≥ 0, (60)

which implies,
c̆Xi/X(d−h)(α, u, ..., u) ≥ 0, (61)

Finally,

ζUh,i(u) ≥ V aRXi(1− u)

∫ u

0
c̆Xi/X(d−h)(α, u, ..., u)dα = β(u).V aRXi(1− u) (62)

Remark 1. One can find a relation between the marginal function of dependence tail
mean and marginal of the multivariate VaR proposed by Cousin[7], by using the previous
corollary and proposition 2.4 in Cousin[7].

Corollary 5. Let X be a random vector, with multivariate distribution F and associeted
copula C. Then, for all u ∈ (0, 1), i = 1, ..., h

i) the marginal function of expected dependence of lower tail verify,

ζLh,i(u) ≤ (1− u)TV aRXi(u),

ii) the marginal function of expected dependence of upper tail verify,

ζUh,i(u) ≤ (1− u)TV aRXi(1− u), (63)

Proof.

i) We have,

ζLh (u) =

∫ 1

u
V aRXi(α)cXi/X(d−h)(α, u, ..., u)dα

(64)

by using Hölder inequality we obtain,

ζLh (u) ≤
(∫ 1

u
V aRXi(α)dα

)
‖cXi/X(d−h)(α, u, ..., u)‖∞.

In other hand, we have

TV aRXi(u) =
1

1− u

∫ 1

u
V aRXi(α)dα. (65)

So, we obtain

ζLh (u) ≤ (1− u)TV aRXi(u).‖cXi/X(d−h)(α, u, ..., u)‖∞
≤ (1− u)TV aRXi(u). (66)

since 0 ≤ ‖cXi/X(d−h)(α, u, ..., u)‖∞ ≤ 1, with ‖f‖∞ = Supx{
∫
|f(x)|dx}.



D. Barro et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1057-1081 1079

ii) we have,

ζUh,i(u) =

∫ 1−u

0
V aRXi(1− α).c̆Xi/X(d−h)(α, u, ..., u)dα, (67)

by using Hölder inequality we obtain.

ζUh,i(u) ≤
(∫ 1−u

0
V aRXi(1− α)dα

)
‖c̆Xi/X(d−h)(α, u, ..., u)‖∞,

in one other hand, we have

TV aRXi(1− u) =
1

1− u

∫ 1−u

0
V aRXi(1− α)dα, (68)

we obtain

ζUh (u) ≤ (1− u)TV aRXi(1− u).‖c̆Xi/X(d−h)(α, u, ..., u)‖∞
≤ (1− u)TV aRXi(1− u). (69)

since 0 ≤ ‖c̆Xi/X(d−h)(α, u, ..., u)‖∞ ≤ 1

7. Conclusion and Discussion

The diversity of financial products and the interconnections between financial markets
make investments increasingly risky. To avoid extreme losses or at least reduce their
magnitude, it is necessary to acquire tools to properly model it. Through this present
work, we have provided a contribution on this theme. In particular, the results offer us
the possibility of modeling and describing the joint extreme behavior of several stochastic
risks simultaneously.However, the applicability of some results, such as those on extremal
processes, may seem problematic given their rare use in practice. There are works making
point process applications and records ‡ in finance for example. Some researchers, as
Resnick [19], give the way to construct extremal process based on Poisson point process,
which can be helpful for application.

Moreover, extremal processes give us a time frame to model extreme events. The in-
teresting fact is that they check the property of max-stability, thus making them strongly
related to max-stable processes. Fortunately, the latter are widely used in stochastic mod-
eling in various fields (hydrology, meteorology, geography, finance, etc.). Consequently,
there are several models (temporal, spatial, spatio-temporal) of these processes with in-
teresting results of applications. The particular advantage of extremal processes over
max-stable ones is that they have distributions that can be expressed as a function of the
time parameter. Therefore, for example, to make a spatio-temporal study of an extreme

‡”For the study of the stochastic behavior of maxima and records, extremal processes are a useful tool.”
Resnick [19] (section 4.3 P.179)



REFERENCES 1080

phenomenon with these processes, it suffices to integrate a spatial parameter.

The second important property of extremal processes is that of Markov. This prop-
erty offers us the possibility of making predictions about the future values of the study
variable. For example, This would make it possible to forecast the VaR, in particular the
multivariate case, in the purely extreme setting (alternative to the mixed EVT-GARCH
methods, etc.).

The Markov processes and the max-stable processes being popularized modeling ob-
jects, various estimation and inference methods exist in the literature and those in multi-
tudes of fields including finance. Based on the aspects common to these types of processes,
it would then be easy to adapt the extremal processes to applications. Extremal processes
are better suited ”naturally” to model the dynamics of extreme events as shown by the
results of applications of simple EVT models (”deterministic”) compared to non-extreme
methods (Gaussian for example). It would be judicious to explore all of these possibilities
through further practical studies to compare these approaches.

We have also defined two multivariate risk measures whose similar versions exist in
the bivariate framework. Thus, by using of these two new measures, we can now measure
the average occurrence of certain risks compared to others in the tails of the distributions.
As example, if we consider a vector of two random variables Z = (X,Y ), modelizing the
profits/losses of two assets, with Frchet margin and with dependance structure described
by Gumbel-Hougaard’s copula then the profits/losses average of one asset in the tails
(lower and upper tails) is very important when the profits/losses of the second asset is
superior or inferior of the Value at Risk (corrolary2).

Note also that the limits ζLh and ζUh can diverge since in Corollary 13, the functions
ζLh (u) and ζUh (u) are greater than quantities which are expressed as the product of a finite
quantity and the VaR. The latter can take infinite limit values and thus make these lower
bounds quite wide. In perspective, it would be interesting to build consistent estimators
for these risk measures in the context of extreme values for possible applications to real
data.

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