Ciprian A Tudor
email: tudor@math.univ-lille1.fr

The determinant of the Malliavin matrix and the determinant of the covariance matrix for multiple integrals

Keywords: 2010 AMS Classification Numbers: 60F05, 60H05, 91G70 multiple stochastic integrals, Wiener chaos, Malliavin matrix, covariance matrix, existence of density

A well-known problem in Malliavin calculus concerns the relation between the determinant of the Malliavin matrix of a random vector and the determinant of its covariance matrix. We give an explicit relation between these two determinants for couples of random vectors of multiple integrals. In particular, if the multiple integrals are of the same order and this order is at most 4, we prove that two random variables in the same Wiener chaos either admit a joint density, either are proportional and that the result is not true for random variables in Wiener chaoses of different orders.

Introduction

The original motivation of the Malliavin calculus was to study the existence and the regularity of the densities of random variables. In this research direction, the determinant of the so-callled Malliavin matrix plays a crucial role.

We give here an explicit formula that connects the determinant of the Malliavin matrix and the determinant of the covariance matrix of a couple of multiple stochastic integrals. This is related to two open problems stated in [START_REF] Nourdin | Absolute continuity and convergence of densities for random vectors on Wiener chaos[END_REF]. In this reference, the authors showed that, if F = (F 1 , .., F d ) is a random vector whose components belong to a finite sum of Wiener chaoses, then the law of F is not absolutely continuous with respect to the Lebesque measure if and only if E det Λ = 0. Here Λ denotes the Malliavin matrix of the vector F . In particular, they proved that a couple of multiple integrals of order 2 either admits a density or its components are proportional.

They stated two open questions (Questions 6.1 and 6.2 in [START_REF] Nourdin | Absolute continuity and convergence of densities for random vectors on Wiener chaos[END_REF], arXiv version): if C is the covariance matrix and Λ the Malliavin matrix of a vector of multiple stochastic integrals,

• is there true that E det Λ ≥ c det C, with c > 0 an universal constant?

• is there true that the law of a vector of multiple integrals with components in the same Wiener chaos is either absolutely continuous with respect to the Lesque measure or its components are proportional?

We make a first step in order to answer to these two open problems. Actually, we find an explicit relation that connects the two determinants. In particular, if the multiple integrals are of the same order and this order is at most 4, we prove that two random variables in the same Wiener chaos either admit a joint density, either are proportional. The basic idea is to write the Malliavin matrix as a sum of squares and to compute the dominant term of its determinant.

We organized our paper as follows. Section 2 contains some preliminaries on analysis on Wiener chaos. Section 3 is devoted to express the Malliavin matrix as the sum of the squares of some random variables and in Section 4 we derive an explicit formula for the determinant of Λ which also involves the determinant of the covariance matrix. In Section 5 we discuss the existence of the joint density of a vector of multiple integrals.

Preliminaries

We briefly describe the tools from the analysis on Wiener space that we will need in our work. For complete presentations, we refer to [START_REF] Nualart | Malliavin Calculus and Related Topics[END_REF] or [START_REF] Nourdin | Normal Approximations with Malliavin Calculus From Stein's Method to Universality[END_REF]. Let H be a real and separable Hilbert space and consider (W (h), h ∈ H) an isonormal process. That is, (W (h), h ∈ H) is a family of centered Gaussian random variables on the probability space (Ω, F , P ) such that EW (h)W (g) = f, g H for every h, g ∈ H. Assume that the σ-algebra F is generated by W .

Denote, for n ≥ 0, by H n the nth Wiener chaos generated by W . That is, H n is the vector subspace of L 2 (Ω) generated by (H n (W (h)), h ∈ H, h = 1) where H n the Hermite polynomial of degree n. For any n ≥ 1, the mapping I n (h ⊗n ) = H n (W (h)) can be extended to an isometry between the Hilbert space H ⊗n endowed with the norm √ n! • H ⊗n and the nth Wiener chaos H n . The random variable I n (f ) is called the multiple Wiener Itô integral of f with respect to W .

Consider (e j ) j≥1 a complete orthonormal system in H and let f ∈ H ⊗n , g ∈ H ⊗m be two symmetric functions with n, m ≥ 1. Then

f = j 1 ,..,jn≥1 λ j 1 ,..,jn e j 1 ⊗ ... ⊗ e jn (1)
and

g = k 1 ,..,km≥1 β k 1 ,..,km e k 1 ⊗ .. ⊗ e km (2)
where the coefficients λ i and β j satisfy λ j σ(1) ,...j σ(n) = λ j 1 ,..,jn and β k π(1) ,...,k π(m) = β k 1 ,..,km for every permutation σ of the set {1, ..., n} and for every permutation π of the set {1, .., m}. Actually λ j 1 ,..,jn = f, e j 1 ⊗ ... ⊗ e jn and β k 1 ,..,km = g, e k 1 ⊗ .. ⊗ e km in (1) and [START_REF] Nourdin | Normal Approximations with Malliavin Calculus From Stein's Method to Universality[END_REF]. Note that, throughout the paper we will use the notation •, • to indicate the scalar product in H ⊗k , independently of k.

If f ∈ H ⊗n , g ∈ H ⊗m are symmetric given by (1), (2) respectively, then the contraction of order r of F and g is given by

f ⊗ r g = i 1 ,..,ir≥1 j 1 ,..,j n-r ≥1 k 1 ,..,k m-r ≥1 λ i 1 ,..,ir,j 1 ,..,j n-r β i 1 ,..,ir,k 1 ,..,k m-r × e j 1 ⊗ .. ⊗ e j n-r ⊗ e k 1 ⊗ .. ⊗ e k m-r (3)
for every r = 0, .., m ∧ n. In particular f ⊗ 0 g = f ⊗ g. Note that f ⊗ r g belongs to H ⊗(m+n-2r) for every r = 0, .., m∧n and it is not in general symmetric. We will denote by f ⊗r g the symmetrization of f ⊗ r g. In the particular case when

H = L 2 (T, B, µ)
where µ is a sigma-finite measure without atoms, (3) becomes

(f ⊗ r g)(t 1 , .., t m+n-2r ) = T r dµ(u 1 )..dµ(u r )f (u 1 , .., u r , t 1 , .., t n-r )g(u 1 , .., u r , t n-r+1 , .., t m+n-2r ) (4)
An important role will be played by the following product formula for multiple Wiener-Itô integrals: if f ∈ H ⊗n , g ∈ H ⊗m are symmetric, then

I n (f )I m (g) = m∧n r=0 r!C r m C r n I m+n-2r f ⊗r g . (5)
We will need the concept of Malliavin derivative D with respect to W , but we will use only its action on Wiener chaos. In order to avoid too many details, we will just say that, if f is given by ( 1) and I n (f ) denotes its multiple integral of order n with respect to W , then

DI n (f ) = n j 1 ,..,jn≥1 λ j 1 ,..,jn I n-1 (e j 2 ⊗ .. ⊗ e jn ) e j 1 .
If F, G are two random variables which are differentiable in the Malliavin sense, we will denote throughout the paper by C the covariance matrix and by Λ the Malliavin matrix of the random vector (F, G). That is,

Λ = DF 2 H DF, DG H DF, DG H DF 2 H .
3 The Malliavin matrix as a sum of squares

In this section we will express the determinant of the Malliavin matrix of a random couple as a sum of squares of certain random variables. This will be useful in order to derive the exact formula for the determinant of the Malliavin matrix and its connection with the determinant of the covariance matrix for a given random vector of dimension 2.

Let f ∈ H ⊗n and g ∈ H ⊗m be given by ( 1) and ( 2) respectively, with n, m ≥ 1. Let F = I n (f ), G = I m (g) denote the multiple Wiener-Itô integrals of f and g with respect to W respectively. Then

I n (f ) = j 1 ,..,jn≥1 λ j 1 ,..,jn I n (e j 1 ⊗ ... ⊗ e jn ) (6)
and

I m (g) = k 1 ,..,km≥1 β k 1 ,..,km I m (e k 1 ⊗ .. ⊗ e km ) . (7)
From ( 6) and (7) we have

DF = n j 1 ,..,jn≥1
λ j 1 ,..,jn I n-1 (e j 2 ⊗ .. ⊗ e jn ) e j 1

and

DG = m k 1 ,..,km≥1 β k 1 ,..,km I m-1 (e k 2 ⊗ .. ⊗ e km ) e k 1 .
This implies Let us make the following notation. For every i ≥ 1, let

DF 2 H = n 2 i≥1 j 2 ,..,
S i,f = n i≥1 j 2 ,..,jn≥1 λ i,j 2 ,..,jn I n-1 (e j 2 ⊗ .. ⊗ e jn ) (8)
and

S i,g = m i≥1 i,k 2 ,..,km≥1 β i,k 2 ,..,km I m-1 (e k 2 ⊗ .. ⊗ e km ) . (9)
We can write

DF 2 H = i≥1 S 2 i,f , DG 2 H = l≥1 S 2 l,g , DF, DG = i≥1 S i,f S i,g and
det(Λ) = DF 2 H DG 2 H -DF, DG 2 H = i,l≥1 S 2 i,f S 2 l,g - i≥1 S i,f S i,g 2 .
A key observation is that i,l≥1

S 2 i,f S 2 l,g - i≥1 S i,f S i,g 2 = 1 2 i,l≥1 (S i,f S l,g -S l,f S i,g ) 2 . ( 10
)
We obtained

Proposition 1 The determinant of the Malliavin matrix Λ of the vector (F, G) = (I n (f ), I m (g)) can be expressed as

detΛ = 1 2 i,l≥1 (S i,f S l,g -S l,f S i,g ) 2
where S i,f , S i,g are given by ( 8) and ( 9) respectively.

Corollary 1

The determinant of the Malliavin matrix Λ of the vector (F, G) = (I n (f ), I m (g)) can be expressed as

detΛ = 1 2 i,l≥1
( DF, e i DG, e l -DF, e l DG, e i ) 2

Proof: This comes from Proposition 1 and the relations S i,f = DF, e i , S i,g = DG, e i for every i ≥ 1.

The determinant of the Malliavin matrix on Wiener chaos

Fix n, m ≥ 1 and f, g in H ⊗n , H ⊗m respectively defined by ( 1) and (2). Consider the random vector (F, G) = (I n (f ), I m (g)) and denote by Λ its Malliavin matrix and by C its covariance matrix.

Let us compute E det Λ. Denote, for every i, l ≥ 1

s i,f = n j 2 ,..,jm≥1 λ i,j 2 ,..,jn e j 2 ⊗ .. ⊗ e jn (11)
and

s l,g = m k 2 ,..,km≥1 β l.k 2 ,..,km e k 2 ⊗ .. ⊗ e km . (12)
Clearly, for every i, l ≥ 1

S i,f = I n-1 (s i,f ), S i,g = I m-1 (s i,g ). ( 13
)
The following lemma plays a key role in our construction.

Lemma 1 If f ∈ H ⊗n and g ∈ H ⊗m are given by ( 1) and (2) respectively and s i,f , s i,g by ( 11), (12) respectively, then for every r = 0, .., n -1

f ⊗ r+1 g = 1 nm i≥1 (s i,f ⊗ r s i,g ) .
Proof: Consider first r = 0. Clearly, by ( 3)

f ⊗ 1 g = i≥1 j 2 ,..,
(s i,f ⊗ s i,g ) .
The same argument applies for every r = 1, .., n -1. Indeed, We make a first step to compute E det Λ.

f ⊗ r+1 g = j 1 ,..,
Lemma 2 Let f ∈ H ⊗n , g ∈ H ⊗m be symmetric and denote by Λ the Malliavin matrix of the vector (F, G) = (I n (f ), I m (g)). Then we have

E det Λ = (n-1)∧(m-1) k=0 T k
where we denote, for k = 0, .., (m -1) ∧ (n -1),

T k := 1 2 i,l≥1 k! 2 C k m-1 2 C k m-1 2 (m + n -2 -2k)! s i,f ⊗k s l,g -s l,f ⊗k s i,g 2 (14)
and s i,f , s i,g are given by ( 11), (12) for i ≥ 1.

Proof: By Proposition 1 and relation (13)

2 det Λ = i,l≥1 (I n-1 (s i,f )I m-1 (s l,g ) -I n-1 (s l,f )I m-1 (s i,g )) 2 = i,l≥1   (m-1)∧(n-1) k=0 k!C k m-1 C k n-1 I m+n-2-2k s i,f ⊗k s l,g -s l,f ⊗k s i,g   2
where we used the the product formula (5). Consequently, from the isometry of multiple stochastic integrals,

E det Λ = 1 2 i,l≥1 (n-1)∧(m-1) k=0 k! 2 C k m-1 2 C k n-1 2 (m + n -2 -2k)! s i,f ⊗k s l,g -s l,f ⊗k s i,g 2 = (n-1)∧(m-1) k=0 T k .
For every n, m ≥ 1 let us denote by

R n,m := (n-1)∧(m-1) k=1 T k , R n := R n,n . ( 15
)
Remark 1 Obviously all the terms T k above are positive, for k = 0, .., (n-1)∧(n-1).

We will need two more auxiliary lemmas.

Lemma 3 Assume f 1 , f 3 ∈ H ⊗n and f 2 , f 4 ∈ H ⊗m are symmetric functions. Then for every r = 0, .., (m -1) ∧ (n -1) we have

f 1 ⊗ n-r f 3 , f 2 ⊗ m-r f 4 = f 1 ⊗ r f 2 , f 3 ⊗ r f 4 .
Proof: The case r = 0 is trivial, so assume r ≥ 1. Without any loss of the generality, assume that H is L 2 (T ; µ) where µ is a sigma-finite measure without atoms. Then, by ( 4)

f 1 ⊗ n-r f 3 , f 2 ⊗ m-r f 4 T r dµ r (t 1 , .., t r ) T r dµ r (s 1 , .., s r ) T n-r dµ n-r (u 1 , .., u n-r )f 1 (u 1 , .., u n-r , t 1 , .., t r )f 3 (u 1 , .., u n-r , s 1 , .., s r ) T m-r dµ m-r (v 1 , .., v m-r )f 2 (v 1 , .., v m-r , t 1 , .., t r )f 4 (v 1 , .., v m-r , s 1 , .., s r ) = T n-r dµ n-r (u 1 , .., u n-r ) T m-r dµ m-r (v 1 , .., v m-r ) (f 1 ⊗ r f 2 )(u 1 , .., u n-r , v 1 , .., v m-r )(f 3 ⊗ r f 4 )(u 1 , .., u n-r , v 1 , .., v m-r ) = f 1 ⊗ r f 2 , f 3 ⊗ r f 4 . Lemma 4 Suppose f 1 , f 4 ∈ H ⊗n , f 2 , f 3 ∈ H ⊗m are symmetric functions. Then f 1 ⊗f 2 , f 3 ⊗f 4 = m!n! (m + n)! m∧n r=0 C r n C r m f 1 ⊗ r f 3 , f 4 ⊗ r f 2 .
Proof: This has been stated and proven in [START_REF] Nourdin | Asymptotic independence and limit law for multiple Wiener-Itô integrals[END_REF] in the case m = n. Exactly the same lines of the proofs apply for m = n.

We first compute the term T 0 obtained for k = 0 in (14).

Proposition 2 Let T 0 be given by ( 14) with k = 0.

T 0 = (n-1)∧(m-1) r=0 mnm!n!C r n-1 C r m-1 f ⊗ r g 2 -f ⊗ r+1 g 2 .
Proof: From (14),

T 0 = 1 2 (m + n -2)! i,l≥1 s i,f ⊗s l,g -s l,f ⊗s i,g 2 = 1 2 (m + n -2)! i,l≥1
s i,f ⊗s l,g 2 + s l,f ⊗s i,g 2 -2 s i,f ⊗s l,g , s l,f ⊗s i,g .

Let us apply Lemma 4 to compute these norms and scalar products. We obtain, by letting

f 1 = s i,f = f 4 and f 2 = s l,g = f 3 (note that s i,f , s i,g are symmetric functions in H ⊗n , H ⊗m respectively) (m + n -2)! s i,f ⊗s l,g , s i,f ⊗s l,g = (m + n -2)! s i,f ⊗s l,g , s l,g ⊗s i,f = (m -1)!(n -1)! (n-1)∧(m-1) r=0 C r n-1 C r m-1 s i,f ⊗ r s l,g , s i,f ⊗ r s l,g = (m -1)!(n -1)! (n-1)∧(m-1) r=0 C r n-1 C r m-1 s i,f ⊗ r s l,g 2 .
Analogously, for

f 1 = s l,f = f 4 and f 2 = s i,g = f 3 in Lemma 4 we get (m + n -2)! s l,f ⊗s i,g , s l,f ⊗s i,g = (n-1)∧(m-1) r=0 (n -1)!(m -1)!C r n-1 C r m-1 s l,f ⊗ r s i,g 2 .
Next, with

f 1 = s i,f , f 2 = s l,g , f 4 = s l,f , f 3 = s i,g (m + n -2)! s i,f ⊗s l,g , s l,f ⊗s i,g = (m + n -2)! s i,f ⊗s l,g , s i,g ⊗s l,f = (n-1)∧(m-1) r=0 (n -1)!(m -1)!C r n-1 C r m-1 s i,f ⊗ r s i,g , s l,f ⊗ r s l,g . Then (m + n -2)! i,l≥1 s i,f ⊗s l,g -s l,f ⊗s i,g 2 = (n-1)∧(m-1) r=0 (n -1)!(m -1)!C r n-1 C r m-1 i,l≥1 s l,f ⊗ r s i,g 2 + s i,f ⊗ r s l,g 2 -2 s i,f ⊗ r s i,g , s l,f ⊗ r s l,g = 2 (n-1)∧(m-1) r=0 (n -1)!(m -1)!C r n-1 C r m-1 i,l≥1 s i,f ⊗ r s l,g 2 -s i,f ⊗ r s i,g , s l,f ⊗ r s l,g = 2 (n-1)∧(m-1) r=0 (n -1)!(m -1)!C r n-1 C r m-1 × i,l≥1 s i,f ⊗ r s l,g 2 - i≥1 s i,f ⊗ r s i,g , l≥1 s l,f ⊗ r s l,g = 2 (n-1)∧(m-1) r=0 (n -1)!(m -1)!C r n-1 C r m-1 i,l≥1 s i,f ⊗ r s l,g 2 - i≥1 s i,f ⊗ r s i,g 2 .( 16
)
Notice that, by Lemma 1, for every r = 0, .., n -1

i≥1 s i,f ⊗ r s i,g 2 = n 2 m 2 f ⊗ r+1 g 2 . ( 17
)
We apply now Lemma 3 and we get i,l≥1

s i,f ⊗ r s l,g 2 = i,l≥1 s i,f ⊗ r s l,g , s i,f ⊗ r s l,g = i,l≥1 s i,f ⊗ n-1-r s i,f , s l,g ⊗ m-1-r s l,g = i≥1 s i,f ⊗ n-1-r s i,f , l≥1 s l,g ⊗ m-r-1 s l,g
and by Lemma 1 and Lemma 3, this equals i,l≥1

s i,f ⊗ r s l,g 2 = n 2 m 2 f ⊗ n-r f, g ⊗ m-r g = n 2 m 2 f ⊗ r g 2 . ( 18
)
By replacing (17) and ( 18) in ( 16) we obtain

T 0 = 1 2 (m + n -2)! i,l≥1 s i,f ⊗s l,g -s l,f ⊗s i,g 2 = (n-1)∧(m-1) r=0 mnm!n!C r n-1 C r m-1 f ⊗ r g 2 -f ⊗ r+1 g 2 .
As a consequence of the above proof, we obtain Corollary 2 For every r = 0, .., (m -1) ∧ (n -1) and if s i,f , s i,g are given by ( 11), (12), it holds that

n 2 m 2 n-1 r=0 C r m-1 C r m-1 f ⊗ r g 2 -f ⊗ r+1 g 2 = i,l≥1 s i,f ⊗s l,g -s l,f ⊗s i,g 2 .
As a consequence, for every r = 0, .., (m -1) ∧ (n -1) we have

n-1 r=0 C r m-1 C r m-1 f ⊗ r g 2 -f ⊗ r+1 g 2 ≥ 0. (19)
Proof: It is a consequence of the proof of Proposition 2.

Let us state the main results of this section.

Theorem 1 Let f ∈ H ⊗n , g ∈ H ⊗m (n, m ≥ 1) be symmetric and denote by Λ the Malliavin matrix of the vector (F, G) = (I n (f ), I m (g)).

Then

det Λ = (n-1)∧(m-1) r=0 mnm!n!C r n-1 C r m-1 f ⊗ r g 2 -f ⊗ r+1 g 2 + R n,m
where for every n, m ≥ 1, R n,m is given by (15). Note that R n,m ≥ 0 for every n, m ≥ 1.

Proof: It follows from Proposition 2 and Lemma 2.

In the case when the two multiple integrals live in the same Wiener chaos, we have a nicer expression.

Theorem 2 Under the same assumptions as in Theorem 1 but with m = n, we have

det Λ = m 2 det C+(mm!) 2 [ m-1 2 ] r=1 (C r m-1 ) 2 -(C r-1 m-1 ) 2 f ⊗ r g 2 -f ⊗ n-r g 2 +R m
with R m given by (15). Here [x] denotes the integer part of x.

Proof: Suppose n ≤ m and that m is odd. The case m even is similar. From Theorem 1 we have

det Λ = (mm!) 2   (m-1) r=0 C r m-1 2 f ⊗ r g 2 - (m-1) r=0 C r m-1 2 f ⊗ r+1 g 2   = (mm!) 2   m-1 2 r=0 C r m-1 2 f ⊗ r g 2 - (m-1) r= m-1 2 C r m-1 2 f ⊗ r+1 g 2   +(mm!) 2   m-1 m-1 2 C r m-1 2 f ⊗ r g 2 - m-1 2 r=0 C r m-1 2 f ⊗ r+1 g 2   = (mm!) 2 m-1 2 r=0 C r m-1 2 f ⊗ r g 2 -f ⊗ n-r g +(mm!) 2 m-1 2 r=1 C r m-1 2 f ⊗ n-r g 2 -f ⊗ r g 2
where we made the change of index r ′ = n -1r in the second and third sum above. Finally, noticing that for r = 0 we have

m 2 m! 2 C 0 m-1 2 f ⊗ 0 g 2 -f ⊗ n g = m 2 det C
we obtain the conclusion.

Example 1 Suppose m = n = 2. Then det Λ = 16 f ⊗ g 2 -f ⊗ 2 g 2 + R 2 = 4 det C + R 2 .
We retrieve the formula in [START_REF] Nourdin | Absolute continuity and convergence of densities for random vectors on Wiener chaos[END_REF] with R 2 = 32 f ⊗ 1 g 2f ⊗1 g 2 .

Assume m = n = 3. Then

det Λ = 9 × 36 f ⊗ g 2 -f ⊗ 3 g 2 + 9 × 36 × (C 1 2 ) 2 -1 f ⊗ 1 g 2 -f ⊗ 2 g 2 + R 3 = 9 det C + 9 × 36 × 3 f ⊗ 1 g 2 -f ⊗ 2 g 2 + R 3 . Suppose m = n = 4. Then det Λ = 16 det C + 16 × 4! × 4! (C 1 3 ) 2 -1 f ⊗ 1 g 2 -f ⊗ 3 g 2 + R 4 .

Densities of vectors of multiple integrals

Let us discuss when a couple of multiple stochastic integrals has a law which is absolutely continuous with respect to the Lebesque measure. The situations when the components of the vector are in the same chaos of in chaoses of different orders need to be separated. Let us first discuss the case of variables in the same chaos. In order better understand the relation between det Λ and det C we need more information on the terms R m in Theorem 2. It is actually possible to compute the last term T m-1 in (14). (m -1)! 2 [ s i,f , s l,gs l,f , s i,g ] 2 = (m -1)! 2 i,l≥1

s i,f , s l,g 2 - i,l≥1
s i,f , s l,g s l,f , s i,g = (m -1)! 2 i,l≥1

s i,f ⊗ s i,f , s l,g ⊗ s l,gi,l≥1

s i,f ⊗ s i,g , s l,g ⊗ s l,f

= (m -1)! 2 m 4 [ f ⊗ 1 f, g ⊗ 1 g -f ⊗ 1 g, g ⊗ 1 f ] = m 2 m! 2 f ⊗ m-1 g 2 -f ⊗ 1 g, g ⊗ 1 f

Proposition 3 (m - 1 )! 2

 312 Suppose m = n and let T m-1 be the term obtained in (14) for r = m -1. Then T m-1 = m 2 m! 2 f ⊗ m-1 g 2f ⊗ 1 g, g ⊗ 1 f . |s i,f ⊗m-1 s l,gs l,f ⊗m-1 s i,g 1)! 2 |s i,f ⊗ m-1 s l,gs l,f ⊗ m-1 s i,g

  jn≥1 k 1 ,..,kn≥1 λ i,j 2 ,..,jn λ i,k 2 ,..,kn I n-1 (e j 2 ⊗ .. ⊗ e jn ) I n-1 (e k 2 ⊗ .. ⊗ e kn ) ,..,jn≥1 l,k 2 ,..,kn≥1 β l,j 2 ,..,jm β l,k 2 ,..,km I m-1 (e j 2 ⊗ .. ⊗ e jm ) I m-1 (e k 2 ⊗ .. ⊗ e km )

	and
	DG 2 H = m 2
	l≥1 l,j 2

and DF, DG H = nm i≥1 j 2 ,..,jn≥1 k 1 ,..,km≥1 λ i,j 2 ,..,jn β i,j 1 ,..,jm I n-1 (e j 2 ⊗ .. ⊗ e jn ) I m-1 (e k 2 ⊗ .. ⊗ e km ) .

  jn≥1 k 2 ,..,km≥1 λ i,j 2 ,..,jn β i,k 2 ,..,km e j 2 ⊗ .. ⊗ e jn ⊗ e k 2 ⊗ ..e km

	=	1 nm	i≥1

  1 ,..,i r+1 j r+2 ,..,jn k r+2 ,..,km λ i 1 ,..,i r+1 ,j r+2 ,..,jn β i 1 ,..,i r+1 ,k r+2 ,..,km e j r+2 ⊗ .. ⊗ e jn ⊗ e k r+2 , .., e km

jn≥1 λ j 1 ,..,jn e j 1 ⊗ .. ⊗ e jn ⊗ r k 1 ,..,km≥1

β k 1 ,..,km e k 1 ⊗ .. ⊗ e km = i 1 ,..,i r+1 j r+2 ,..,jn k r+2 ,..,km λ i 1 ,..,i r+1 ,j r+2 ,..,jn β i 1 ,..,i r+1 ,k r+2 ,..,

km e j r+2 ⊗ .. ⊗ e jn ⊗ e k r+2 , .., e km and by (3) again i≥1 s i,j ⊗ r s i,g = nm i≥1 j 2 ,..,jn≥1 λ i,j 2 ,..,jn e j 2 ⊗ .. ⊗ e jn ⊗ r k 2 ,..,km≥1 β i,k 2 ,..,km e k 2 ⊗ .. ⊗ e km = nm i≥1 i 2 ,..,i r+1 j r+2 ,..,jn k r+2 ,..,km λ i,i 2 ,..,i r+1 ,j r+2 ,..,jn β i,i 2 ,..,i r+1 ,k r+2 ,..,km × e j r+2 ⊗ .. ⊗ e jn ⊗ e k r+2 , .., e km = nm i = f ⊗ r+1 g.

* Supported by the CNCS grant PN-II-ID-PCCE-2011-2-0015.

1

where we applied Lemmas 3 and 1.

. We first answer the open problem 6.2 in [START_REF] Nourdin | Absolute continuity and convergence of densities for random vectors on Wiener chaos[END_REF] for chaoses of order lesser than five.

Theorem 3 Let m ≤ 4 and let f, g ∈ H ⊗m be symmetric. Then the random vector (F, G) = (I m (f ), I m (g)) does not admit a density if and only if

In other words, the vector (F, G) does not admit a density if and only if its components are proportional.

Proof:

The case m = n = 1 is obvious and the case m = n = 2 follows from [START_REF] Nourdin | Absolute continuity and convergence of densities for random vectors on Wiener chaos[END_REF] (it also follows from Example 1). Suppose m = n = 3. Then

Suppose det Λ = 0. Then T 0 , T 1 , T 2 from (14) vanish. In particular T 2 = 0 in (14) and so

This implies

where R ′ 4 is the sum of terms obtained for k = 1 and k = 2 in (14

Assume det Λ = 0. Then in particular T 3 from 14) vanishes. So

If det Λ = 0 then, since T 4 vanishes, we get that f ⊗ 4 g 2f ⊗ 4 g, g ⊗ 4 f vanishes. But this is not enough. We need some additional information in order to handle the difference f ⊗ 2 g 2f ⊗ 3 g 2 . One possibility is to look to the terms T 1 , T 2 , T 3 in ( 14) but these terms cannot be written in a closed form, since they involve more complicated contractions (some "contractions of contractions").

Let us finish by some comments concerning the case of variables in chaoses of different orders. Consider (F, G) = (I n (f ), I m (g)) with n = m. First, let us note that E det Λ = 0 does not imply det C = 0. This can be viewed by considering the following example. In the case (I n (f ), I 1 (g)) there is only one term in (14) obtained for k = 0. It reads

and therefore the condition for the existence of the joint density is f ⊗

The case (I n (f ), I 2 (g)) is more complicated and needs new ideas in order to obtain the if and only if condition for the existence of the density of the vector. Even the "last term"in (14) (that is, the term obtained for k = (m -1) ∧ (n -1) cannot be written is a nice form.


Bertrand K Hassani
email: bertrand.hassani@santander.co.uk

Alexis Renaudin
email: alexis.renaudin@aon.co.uk

The Cascade Bayesian Approach for a controlled integration of internal data, external data and scenarios

Keywords: Operational Risk, Loss Distribution Approach, Bayesian Inference, Markov Chain Monte Carlo, Extreme Value Theory, Non-parametric statistics, Risk Measures

According to the last proposals of the Basel Committee on Banking Supervision, banks under the Advanced Measurement Approach (AMA) must use four different sources of information to assess their Operational Risk capital requirement. The fourth including "business environment and internal control factors", i.e. qualitative criteria, the three main quantitative sources available to banks to build the Loss Distribution are Internal Loss Data, External Loss Data, and Scenario Analysis. This paper proposes an innovative methodology to bring together these three different sources in the Loss Distribution Approach (LDA) framework through a Bayesian strategy. The integration of the different elements is performed in two different steps to ensure an internal data driven model is obtained. In a first step, scenarios are used to inform the prior distributions and external data informs the likelihood component of the posterior function. In the second step, the initial posterior function is used as the prior distribution and the internal loss data inform the likelihood component of the second posterior. This latter posterior function enables the estimation of the parameters of the severity distribution selected to represent the Operational Risk event types.

Introduction

Basel II requires bank to access the capital charge associated to operational risks [START_REF] Bcbs | Working paper on the regulatory treatment of operational risk[END_REF][START_REF] Bcbs | Basel iii: A global regulatory framework for more resilient banks and banking systems[END_REF]). There are three different approaches to measure operational risks, the basic, the standard and the advanced measurement approach (AMA), representing increasing levels of control and difficulty of implementation. The AMA requires a better understanding of the exposure to implement an internal model.

Basel II capital accord defines the capital charge as a risk measure obtained on an annual basis, at a given confidence level on a loss distribution that integrates four sources of information: internal data, external data, scenario analysis, business environment and internal control factors. The regulatory capital is given by the 99.9% percentile of the Loss Distribution and the economical capital by a higher percentile related to the rating of the financial institution which is usually between 99.95% and 99.98%.

The purpose of using multiple sources of information is to build an internal model based on the largest set of data possible, in order to increase the robustness, the stability and the conservatism of the final capital evaluation. However, the different sources of information have different characteristics which taken in isolation can be misleading. Internal loss data represent the entity risk profile, external loss data characterize the industry risk profile and scenarios offer a forward looking perspective and the unexpected loss from an internal point of view (Guégan and Hassani (2012a)). Figure 1 illustrates this point by assuming that the internal loss data tends to represent the body of the severity distribution, the scenarios the extreme tail and the external data the section in between 1 . This paper expands on the Frequency × Severity framework (Loss Distribution Approach), where the Loss Distribution Function [START_REF] Frachot | Loss distribution approach for operational risk[END_REF], [START_REF] Cruz | Operational Risk Modelling and Analysis[END_REF] and [START_REF] Chernobai | Operational Risk: A Guide to Basel II Capital Requirements, Models, and Analysis[END_REF])

G is a weighted sum of γ-fold convoluted severity F distribution where γ represents the order of convolution. The weight is provided by the frequency distribution p. Mathematically this 1 External loss data usually overlap both internal data and scenario analysis and is therefore represented as a link between the two previous components.

G = ∞ γ=1 p(k; •)F ⊗γ (x; •), x > 0, (1.1) with G = 0, x = 0.
⊗ denotes the convolution operator. Denoting g the density of G, we have

g = ∞ k=γ p(k; •)f ⊗γ (x; •), x > 0. (1.2)
Assuming a Poisson distribution to model the frequencies, the parameter is estimated by maximum likelihood on the internal loss data. However, as collection thresholds are being set up, the frequency distribution parameter is adjusted using the parameterised severity distribution.

Consequently, the frequency distribution will indirectly be informed by the three different components.

The focal point of this paper is to construct severity distributions by combining the three data sources presented above. As the level of granularity of the risk taxonomy increases, the quantity of data available per risk category tends to decrease. As a result, the traditional estimation methods such as the maximum likelihood or the method of moments tend to be less reliable.

Consequently, we opted to bring together the three components within the Bayesian Inference theoretical framework (BI) [START_REF] Box | Bayesian Inference in Statistical Analysis[END_REF], [START_REF] Shevchenko | Modelling OPerational Risk Using Bayesian Inference[END_REF]). Despite the numerous hypotheses surrounding the BI, it is known to be efficient in situations where there are only a few data points available.

In statistics, the Bayesian inference is a statistical method of inference in which Bayes' theorem [START_REF] Bayes | An essay towards solving a problem in the doctrine of chance. by the late rev. mr. bayes, communicated by mr. price, in a letter to john canton, m.a. and f.r.s[END_REF]) is used to update the probability estimate of a proposition as additional information becomes available. The initial degree of confidence is called the prior and the updated degree of confidence, the posterior.

Consider a random vector of loss data X = (X 1 , ..., X n ) whose joint density for a given vector of parameters φ = (φ 1 , ..., φ K ), is h(x|φ). In the Bayesian approach, both observations and parameters are considered to be random. Then the joint density is

h(x, φ) = h(x|φ)π(φ) = π(φ|x)h(x), (1.3)
where, π(φ) is the probability density of the parameters, known as the prior density function.

Typically, π(φ) depends on a set of further parameters that are usually called "hyper" parameters2 . π(φ|x) is the density of parameters given data X, known as the posterior density, h(x, φ) is the joint density of observed data and parameters and h(x|φ) is the density of observations for given parameters. This is the same as a likelihood function if considered as a function of φ, i.e. l X(φ) = h(x|φ). h(x) is the marginal density of X that can be written as

h(x) = ´h(x|φ)π(φ)d|φ.
The Bayesian inference approach permits a reliable estimation of distributions' parameters even if the quantity of data denoted n is limited. And as n becomes larger, the weight of the likelihood component increases such that if n → ∞, the posterior distribution tends to the likelihood function, and consequently parameters obtained from both approaches converge. As a result, the data selected to inform the likelihood component may lead the model and, as a consequence the capital charge. Therefore, in our two-step approach, there is an opportunity for operational risk managers and modelers to integrate all the aforementioned components in a way they do not have to justify a capital charge increase due to an extreme loss an another entity would have suffered.

This paper proposes applying two Bayesian inference approaches sequentially in order to obtain the parameter of the statistical distribution used to characterise the severity. Scenarios are used to build the prior distributions of the parameters denoted π(φ), which is refined using external data to inform the likelihood component. This results in an initial posterior function (π(φ|Y ))

which is then used as a prior distribution and the likelihood part is informed by the internal data. This leads to a second posterior distribution which allows the estimation of the parameters of the severity distribution used to build the loss distribution function in the LDA approach.

The next section presents the Cascade strategy and the underlying assumption allowing a data driven model. The third section deals with the implementation in practice using real data sets.

A fourth section presents the results and a fifth section concludes.

This section outlines the two step approach to parametrize the severity distribution. The cascade implementation of the Bayesian inference approach is justified by the following property. The Bayesian posterior distribution implies that the larger the quantity of data used, the larger the weight of the likelihood component. Consequently,

π(φ; x 1 , ..., x k ) ∝ π(φ) k i=1 f i (x i |φ) k→∞ → k i=1 f i (x i |φ), (2.1)
where, f is the density characterizing the severity distribution, x i a data point, and φ the set of parameters to be estimated. As a result, the order of the Bayesian integration of the components is significant.

This strategy is based on the construction of two successive posterior distribution functions.

The first uses the scenario values to inform the prior and the external loss data to inform the likelihood function. Using a Markov Chain Monte Carlo algorithm [START_REF] Gilks | Markov Chain Monte Carlo in Practice[END_REF]), a first posterior empirical distribution is obtained. This is used as prior distribution in a second Bayesian approach for which the likelihood component is informed by internal loss data. Due to the Bayesian Approach property presented above, in the worst case, the final posterior distribution is entirely driven by internal data. The method may be formalised as follows. f is the density function of the severity distribution, y i represent the internal data, x i the external data and φ the set of parameters to be estimated.

1. Prior π 0 is informed by the scenarios and the likelihood component by external data:

π 1 (φ; x 1 , ..., x k ) ∝ π 0 (φ) k i=1 f i (y i |φ). (2.2)
2. The aforementioned posterior π 1 used as prior and the likelihood component is informed by internal data:

π 2 (φ; x 1 , ..., x k ) ∝ π 1 (φ) k i=1 f i (x i |φ). (2.3)
As a result, the first posterior function becomes a prior, and only a prior, limiting the impact of the first two components on the parameters.

Carrying out the Cascade Approach In Practice

This section presents how the methodology may be carried out in practice. Firstly we will introduce the data, secondly we will detail the approach and the estimation, and finally we will present the resulting capital charges.

The Data Sets

The results presented were obtained using OpBase, the external data base developed by Aon Limited. It captures the losses corresponding to the Operational risk claims and the losses in the public domain(PKM). The scenarios correspond to those we obtained through a scenario workshop process. There are many different scenario approaches that are compatible with our approach, therefore this will not be discussed any further (Guégan and Hassani (2012a)) in this paper. the internal loss data were provided by a first tier European bank. Instruments" and "Payments".

The Priors

This approach is applicable to any type of the traditional statistical distribution. In this paper, Internal Fraud has been modeled using a lognormal distribution, External Two criteria drive the selection of the prior distributions. They are either chosen so that their supports are consistent with the acceptable values for the parameters, e.g. the shape parameter of a Generalized Pareto distribution cannot be greater than 1, otherwise the infinite mean model obtained leads to unrealistic capital values. Therefore, a beta distribution defined on a finite support has been selected. Alternatively, the priors are selected such that the joint prior is a conjugate distribution i.e. the posterior distribution belongs to the same family as the prior distribution, with a different set of parameters.

In the general case, the prior π on the severity parameters can be written as:

π(φ) = π(φ 1 , φ 2 ) = π(φ 1 |φ 2 )π(φ 2 ) = π(φ 2 |φ 1 )π(φ 1 )
If the priors are independent then the previous function becomes the Gaussian distribution for µ and the Gamma distribution for σ, 2 -the Weibull distribution, i.e. two Gamma distributions for the shape and the scale, 3 -the GPD, i.e. a Beta distribution for the shape and a Gamma for the scale. These are informed by the scenarios.

π(φ) = π(φ 1 , φ 2 ) = π(φ 1 )π(φ 2 ) Label Density Parameters Beta Γ(α+β) Γ(α)Γ(β) x α-1 (1 -x) β-1 α = shape, β = scale Gamma 1 Γ(α)β α x α-1 e -x β α = shape, β = scale Gaussian 1 b √ 2π e -(x-a) 2 2b 2 a = location, b = standard Deviation
Thereafter, priors are assumed to be independent.

For each distributions parameters, a set of prior distributions have been selected (Table 4 lists the corresponding prior density functions):

• Lognormal -a Gaussian and a Gamma distribution • Weibull -Gamma distributions for both the scale and the shape • Generalized Pareto Distribution -A Beta distribution for the shape and a Gamma distribution on the scale

Estimation

To obtain the global severity parameters 3 of the selected distributions, three different approaches may be carried out. These are listed below in order of complexity.

1. One can choose conjugate priors for the parameters in the first step of the Bayesian Inference estimation. In this case, the (joint) distribution of the posterior parameters is directly known and it is possible to sample directly from this distribution to recreate the marginal posterior distributions of each severity parameter. These may then be used to generate the posterior empirical densities required to compute a Bayesian point estimator of the parameters. Admissible estimators are the median, mean, and mode of the posterior distribution. The mode (also called MAP, Maximum A Posteriori) can be seen as the 'most probable estimator' and ultimately coincides with the Maximum Likelihood Estimator [START_REF] Lehmann | Theory of Point Estimation[END_REF]). Despite having good asymptotic properties, finding the mode of an empirical distribution is not a trivial matter and often requires some additional techniques and hypothesis (e.g. smoothing). This paper therefore uses posterior means is used as point estimators.

To our knowledge, the only conjugate priors for continuous distributions were studied by [START_REF] Shevchenko | Modelling OPerational Risk Using Bayesian Inference[END_REF], for the lognormal severity case . Conjugate approach requires some assumptions that may not be sustainable in practice, particularly for priors that are modelled with 'uncommon' distributions (e.g. Inverse Chi Square). This might lead to difficulties in the step known as "elicitation", i.e. calibrating the prior hyper-parameters from the chosen scenario values.

Another solution is to release the conjugate prior assumption, and to use a Markov Chain

Monte Carlo approach in the first step to sample from the first posterior. One can then use a parametric or non-parametric method to compute the corresponding densities. This enables the posterior function π 2 to be evaluated (see equation 2.3). Maximizing this function directly gives the MAP estimators of the severity parameters (see above). Even if this method is sufficient to compute values for the global parameters, it misses the purpose of the Bayesian Inference, which is to provide a distribution as a final result instead of a single value. Additionally, it may also suffer from all the drawbacks that an optimization algorithm may suffer (e.g. non-convex functions, sensitivity to starting values, etc.) 3. The final alternative is to use a MCMC approach at each step of the aforementioned Cascade Bayesian Inference. This method is more challenging to implement but is the most powerful as it generates the entire distribution of the final severity parameters, from which any credibility intervals and/or other statistics may be evaluated.

In this paper, the third alternative has been implemented using the Metropolis-Hasting algorithm. This allows sequential sampling from the two posterior distributions. This algorithm then enables us to build the distributions for the parameters contained in the φ vector. However, these distributions are used as priors in the next step of the cascade and as mentioned in the second point above, their densities are required from the sample obtained from the MCMC. Two solutions may be carried out, a parametric and a non parametric one. The parametric solution requires fitting statistical to empirical distributions, which may bias the construction.

In order to remain as close as possible to the empirical distributions generated, and therefore to the data, a non parametric approach based on a Kernel density estimation has been chosen.

Using an Epanechnikov kernel and a cross validation method to calibrate the bandwidth value, we obtain the parametric density representing the parameter's distributions (Appendix A and B).

As a result, each of the different elements of the new posterior distribution have been built. The new prior densities in the second step result from the first posterior. The new likelihood function is informed by another data set, for instance the Internal Loss data to guarantee an internal loss data driven model.

Remark 3.1. The innovation of this paper lies in the Cascade Bayesian Approach and its use to combine different sources of information; consequently little emphasis has been placed on our implementation of the MCMC with the Metropolis-Hasting algorithm, which is a known and

widespread topic in literature. The interested reader could for instance refer to [START_REF] Gilks | Markov Chain Monte Carlo in Practice[END_REF].

Results

The Cascade Bayesian approach enables the updating of the parameters of the distributions considered to model particular risk events. Table 7 presents the parameters estimated considering the different pieces of information used to build the LDF, i.e. the parameters of the selected distributions -for instance, the lognormal, the Weibull and the Generalized Pareto distribution -are estimated on each of these pieces of information without considering the information brought by the others. It results in a large variance in the parameters, which once translated into financial values may be inconsistent with the observations. For example, considering the lognormal distribution for Internal Fraud, the median is equal to 40, 783.42 when considering only the scenarios, 6, 047.53 using the external data and 11, 472.33 considering internal data.

The parameter implied variance of the losses is also quite different across the different elements.

The variation is similar for the Weibull distribution that models external fraud. It may be even worse in the case of the GPD, as the risk measures are extremely sensitive to the shape parameter.

Table 8 shows the evolution of the parameters as they are updated with respect to the different pieces of information.

• Scenarios Severity is derived from the calibration of the priors distributions to the scenario values. The theoretical means obtained from the calibrated priors provide scenario severity estimates.

• Intermediate Severity refers to the estimation of the severity on the first obtained posterior, i.e. the mean of the posterior distribution obtained from scenario values updated with external loss data.

• Similarly, final severity represents the severity estimation on the second and last posterior distribution, which includes scenarios, external and internal loss data. A 95% Bayesian Confidence Interval (also known as a Credible Interval) derived from this final posterior distribution is also provided. It is worth noticing that this interval -formally defined as containing 95% of the distribution mass -is not unique and is chosen here as the narrowest possible interval. It is therefore not necessarily symmetric around the posterior mean estimator.

From implementing the Cascade approach, it results in final parameters located within the range of values obtained from the different elements taken independently. As the dispersion of the information increases, the variance of the theoretical losses tends to increase, as does the theoretical kurtosis. For example, the evolution of the lognormal parameters is significant, as the introduction of data belonging to the body (Table 7, first line) tends to decrease the mean (the µ parameter is the mean of the log-transformed of the data), but naturally the dispersion increases, and therefore, the variance tends to increase (the σ parameter is a representation of the standard deviation of the log-transform of the data) (Table 8, first line). The impact of the severity on frequency is given in Table 5. The conditional frequencies are naturally increasing with respect to the severity distributions. As a final step, these results may be illustrated for the whole Bayesian Cascade estimation in the simple lognormal case. Figure 3 shows the final posterior distribution obtained for µ and σ.

This is used to derive the 95% Bayesian Credible Interval given in Table 8. We also show the results of the convergence of the posterior mean as a function of the number of MCMC iterations for µ and σ. One can see that the values stabilize after 100 to 500 iterations in this example.

In the general case we chose to sample 3000 values and discard the first 1000 MCMC generated values (burn-in period) to ensure the stability of our estimates.

Conclusion

This paper presents an intuitive approach to building the loss distribution function using the Bayesian Inference Framework and combining the different regulatory components.

This approach enables a controlled integration of the different elements through the Bayesian Inference. The prior functions have the same role as a penalization function on the parameters, and therefore behave as constraint during the estimation procedure. This results in capital charges driven by internal data (as shown in Figure 2), that are not dramatically influenced by external data or extreme scenarios.

A Appendix: Epanechnikov kernel

K(t) =      3 4 (1-1 5 t 2 ) √ 5 , for |t| < √ 5 0, otherwise (A.1)
The efficiency is equal to 1, and the canonical bandwidth is equal to 15 1 5 ≈ 1.3510

B Appendix: Least Square Cross Validation

One of the most famous method to estimate the Kernel bandwidth is known as the least square cross validation [START_REF] Rudemo | Empirical choice of histograms and kernel density estimators[END_REF] and [START_REF] Bowman | An alternative method of cross-validation for the smoothing of density estimates[END_REF]). The structural idea is that the MISE might be written has:

ζ( f (x, h) -f (x)) = ζ( f (x; h)) -2 ˆ+∞ -∞ f (x; h)f (x)dx + ζ(f (x)) (B.1)
Obviously, the last term of the equation does not depend on f therefore, we shall have the same h minimizing the full MISE or only the first part,

ζ( f (x; h)) -2 ˆ+∞ -∞ f (x; h)f (x). (B.2)
An unbiased estimator for (B.1) is given by,

h LSCV = ζ( f (x, h)) -2 1 n n i=1 f-i (X i ; h), (B.3)
where f-1 (x) is the density estimate constructed from all the data points except X i : only. Given a density π(φ|x), known up to a normalization constant, and a conditional density q(φ * |φ), the method generates the chain φ (1) , φ (2) , ... using the following algorithm:

f-1 (x) = 1 h(n -1) n j =i K (x -X j ) h . (B.4)

C Appendix: The Metropolis-Hastings algorithm

1. Initialise φ l=0 with any value within a support of π(φ|x);

2. For l = 1, ..., L (a) Set φ l = φ l-1 (b) Generate a proposal φ * from q(φ * |φ (l) )
(c) Accept proposal with the acceptance probability:

p(φ (l) , φ ( * ) ) = min 1, π(φ (l) |x)q(φ ( * ) |φ (l) ) π(φ ( * ) |x)q(φ (l) |φ ( * ) ) . (C.1)
i.e. simulate U from the uniform distribution function U(0, 1) and set φ (l) = φ ( * )

if U < p(φ (l) , φ ( * ) ). Note that the normalization constant of the posterior does not contribute 3. Next l (i.e. do an increment, l = l + 1 , and return to step 2).

D Appendix: Risk Measure evaluation

For financial institutions, the capital requirement pertaining to operational risks is related to a VaR at 99.9%. It may be defined as follows:

Given a confidence level α ∈ [0, 1], the VaR associated with a random variable X is given by the smallest number x such that the probability that X exceeds x is not larger than

(1 -α) V aR (1-α)% = inf(x ∈ R : P (X > x) ≤ (1 -α)). (D.1)
And we can compare these results to those obtained based on the Expected Shortfall (ES) defined as follows:

For a given α in [0, 1], η the V aR (1-α)% , and X a random variable which represents losses during a prespecified period (such as a day, a week, or some other chosen time period) then,

Figure 1 :

 1 Figure 1: Combination of internal loss data, external loss data and scenario analysis: depending on the risk profile or the quantity of data available, the representation may be slightly different.

  Fraud/Payment via by a Weibull distribution and Execution, Delivery & Process Management / Other than Payments is represented by a mixture model using a lognormal distribution in the body and a Generalized Pareto Distribution at the tail (Guégan et al. (2011)).

3

  These take into account the 3 different data sources 10 Documents de Travail du Centre d'Economie de la Sorbonne -2013.09

13Figure 2 :

 2 Figure 2: Comparison of Capital charge value obtained from the three components on a standalone basis and in combination.

15

  Documents de Travail du Centre d'Economie de la Sorbonne -2013.09

  (a) Obtained Posterior Distribution for each severity parameter (µ and σ) (b) Convergence of the Severity Estimation as the Posterior Mean in the MCMC Sampling

Figure 3 :

 3 Figure 3: Posterior Distributions and Convergence of the estimations obtained on Internal Fraud (lognormal case)

Fraud

  severities are modelled using a lognormal distribution, External fraud / Payments using a Weibull and Execution, Delivery, and Product Management considering a mixture model combining a lognormal distribution in the body and a Generalized Pareto distribution in the tail. The first column presents the initial parameters estimated for the scenarios. φ 1 and φ 2 represent the severity parameters of the chosen distribution, i.e. (resp.) µ and σ for the lognormal and shape and scale for the Weibull or GPD distribution. The second column shows the values obtained for the external data. The third column shows the parameters obtained for the internal data. 23 Documents de Travail du Centre d'Economie de la Sorbonne -2013.09

Table 1 :

 1 The table presents the statistical moments of the internal loss data used in this paper, as well as other statistics such as the minimum value, the maximum value and the number of data point available. NB: Level 2 "Other" gathers "Execution, Delivery & Process Management" losses other than "Financial Instruments" and "Payments".

	Level 1	Level 2	Nb used min median	mean	max	sd	skewness kurtosis
	Internal Fraud	Global	665	4 137 28 165 261 475 46 779 130 1,9E+06	21,4	407,1
	External Fraud	Payments	1 567	4 091 12 358	36 133	1 925 000 9,2E+04	11,7	185,2
	Execution, Delivery & Process Management	Other	3 602	4 084 10 789	96 620 30 435 400 9,5E+05	24,8	653,8
	Level 1	Level 2	Nb used	min	median	mean	max	sd	skewness kurtosis
	Internal Fraud	Global	2 956	20 001 88 691 697 005 130 715 800 4,5E+06	17,6	387,6
	External Fraud	Payments	1 085	20 006 36 464 326 127 106 772 200 4,3E+06	20,8	461,2
	Execution, Delivery & Process Management	Other	31 126 20 004 47 428 271 974 585 000 000 4,1E+06	107,2	14068,6

Table 2 :

 2 The table presents the statistical moments of the external loss data used in this paper,

as well as other statistics such as the minimum value, the maximum value and the number of data available. NB: Level 2 "Other" gathers "Execution, Delivery & Process Management" losses other than "Financial Instruments" and "Payments".

Table 3 :

 3 The table presents the scenario values used in this paper. 1 in 10 and 1 in 40 respectively denote the biggest loss that may occur in the next 10 and 40 years. NB: Level 2 "Other" gathers "Execution, Delivery & Process Management" losses other than "Financial

Table 4 :

 4 This table presents the priors used to parametrize: 1 -the Lognormal distribution, i.e.

  The Metropolis-Hastings algorithm is almost a universal algorithm used to generate a Markov chain with a stationary distribution π(φ|x). It has been developed by Metropolis et al. in mechanical physics and generalised by Hastings in a statistical setting. It can be applied to a variety of problems since it requires the knowledge of the distribution of interest up to a constant

Table 5 :

 5 Frequency distribution parameters used in the Capital charge calculations. The standard deviations are given in brackets.

	Event Type	µ	σ
	Execution, Delivery,	8.092849 1.882122
	and Product Management
	(Financial Instruments) -lognormal body

Table 6 :

 6 Body distribution (lognormal) parameters for Execution, Delivery, and Product Management / Financial Instruments. The standard deviations are provided in brackets.

		Scenarios	External Data		Internal Data
	Label
		φ 1	φ 2	φ 1	φ 2	Size	φ 1	φ 2	Size
	Internal Fraud	10.616031	2.024592	8.707405	2.693001	2956	9.347693	2.485755	665
	(Global) -lognormal
	External Fraud	0.37642	4.6951E+04 1.644924 5.4293E+04 1085	0.292442 976.922586 1567
	(Payments) -Weibull
	Execution, Delivery,	0.4705893 1.5309E+06 0.732526 1.1532E+06 31126	0.82231 7.0388E+05 3602
	and Product Management
	(Financial Instruments) -GPD

22 Documents de Travail du Centre d'Economie de la Sorbonne -2013.09

Table 7 :

 7 This table presents the standalone parameters estimated for each components. Internal

Documents de Travail du Centre d'Economie de la Sorbonne -2013.09

These parameters are omitted here for simplicity of notation, these are the parameters of the densities used as a prior distribution (e.g. a Gamma or Beta distribution). 6 Documents de Travail du Centre d'Economie de la Sorbonne -2013.09

Hence, with our approach, the capital requirement calculation is inherently related to the risk profile of the target financial institution and therefore provides senior management greater assurance of the validity of the results.

The next step to evaluate the financial institution global operational risk capital requirement would be the construction of the multivariate distribution function linking the different loss distributions which characterize the various event types through a copula. In the case, the approach developed by [START_REF] Guégan | Multivariate vars for operational risk capital computation : a vine structure approach[END_REF]