P G Grinevich

R G Novikov
email: novikov@cmap.polytechnique.fr

Moutard transform for the generalized analytic functions *

We construct a Moutard-type transform for the generalized analytic functions. The first theorems and the first explicit examples in this connection are given.

Introduction

We consider the equation

∂ z ψ = u ψ in D ⊆ C, (1)
where u = u(z) is a given function in D, D is open in C. The functions ψ = ψ(z) satisfying (1) are known as generalized analytic functions in D. In 1 this article the notation f = f (z) does not mean that f is holomorphic. In the literature it is usually assumed that

u ∈ L p (D), p > 2, if D is bounded, (2) u ∈ L p,2 (C), p > 2, if D = C, (3)
where L p,ν (C) denotes complex-valued functions u such that

u ∈ L p (D 1 ), u ν ∈ L p (D 1 ), where u ν (z) = 1 |z| ν u ( 1 z ) , ( 4
)
D 1 = {z ∈ C : |z| ≤ 1}.
The theory of generalized analytic functions is presented in [START_REF] Bers | Theory of pseudo-analytic functions[END_REF], [START_REF] Vekua | Generalized Analytic Functions[END_REF]. However, to the best of our knowledge, algebraic Moutrad-type transforms, going back to [START_REF] Moutard | Sur la construction des équations de la forme 1 z ∂ 2 z ∂x∂y = λ(x, y) qui admettenent une intégrale générale explicite[END_REF] were not yet considered in the framework of this theory. On the other hand, the Moutard-type transforms are successfully used in studies of integrable models in dimension 2+1 and in the spectral theory in dimension 2; see [START_REF] Doliwa | Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme[END_REF], [START_REF] Matveev | Darboux transformations and solitons[END_REF], [START_REF] Nimmo | Superposition principles associated with the Moutard transformation: an integrable discretization of a 2+1dimensional sine-Gordon system[END_REF], [START_REF] Novikov | Two-dimensional von Neumann-Wigner potentials with a multiple positive eigenvalue[END_REF]- [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and Soliton equations[END_REF] and references therein. The Moutard-type transforms correspond in quadratures to coefficients and related solutions of an appropriate linear PDE (and its conjugate) on the plane new coefficients and related solutions for this PDE (and its conjugate). In this article we show that this approach is applicable to equation [START_REF] Adams | Sobolev spaces[END_REF]. In particular, it allows to study equation [START_REF] Adams | Sobolev spaces[END_REF] in some important cases when u has strong singularities (e.g. contour poles), and, as a corollary, assumptions [START_REF] Bers | Theory of pseudo-analytic functions[END_REF], [START_REF] Doliwa | Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme[END_REF] are not valid at all. It is quite likely that the known methods of the generalized analytic functions theory dot not admit appropriate generalizations for the aforementioned cases.

In addition, our interest to the generalized analytic functions with strong singularities was, in particular, strongly motivated by studies going back to [START_REF] Grinevich | Two-dimensional 'inverse scattering problem' for negative energies and generalized-analytic functions. 1. Energies below the ground state[END_REF].

Note that the present work was stimulated by recent articles [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF], [10] by I.A. Taimanov on Moutard-type transforms for two-dimensional Dirac operators with applications to integrable systems in dimension 2+1 and to differential geometry.

The results of the present work can be summarized as follows:

• We construct Moutard-type transforms for equation [START_REF] Adams | Sobolev spaces[END_REF] considering this equation as a particular case of the two-dimensional Dirac system

∂ z ψ 1 = uψ 2 , ∂ z ψ 2 = vψ 1 in D ⊆ C, ( 5
)
where D is the same that in [START_REF] Adams | Sobolev spaces[END_REF]. Related algebraic and analytic results in this connection are presented in Section 2.

• Explicit examples of generalized analytic functions obtained by the Moutard-type transforms from usual holomorphic functions are given in Section 3. These examples include generalized analytic functions with contour poles.

Moutard transform

We consider the two-dimensional Dirac system (5) and the conjugate system

∂ z ψ + 1 = -vψ + 2 , ∂ z ψ + 2 = -uψ + 1 in D ⊆ C. ( 6
) If v = ū, (7)
then system ( 5) is reduces to [START_REF] Adams | Sobolev spaces[END_REF]. More precisely, in this case:

if ψ satisfies (1), then [ ψ 1 ψ 2 ] = [ ψ ψ ] satisfies (5); (8)
if [START_REF] Matveev | Darboux transformations and solitons[END_REF], then ψ = 1 2 (ψ 1 + ψ2 ) and ψ = 1 2i (ψ 1 -ψ2 ) satisfy (1).

[ ψ 1 ψ 2 ] satisfies (
In addition, if [START_REF] Nimmo | Superposition principles associated with the Moutard transformation: an integrable discretization of a 2+1dimensional sine-Gordon system[END_REF] holds, then the conjugate system (6) is reduced to the equation

∂ z ψ + = -ū ψ+ in D ⊆ C. (9)
In the theory of generalized analytic functions equation ( 9) is known as the conjugate equation to the initial equation [START_REF] Adams | Sobolev spaces[END_REF]; see [START_REF] Vekua | Generalized Analytic Functions[END_REF].

Let

ψ(j) = [ ψ 1 (z, j) ψ 2 (z, j) ] , ψ + (j) = [ ψ + 1 (z, j) ψ + 2 (z, j) ] , j = 1, . . . , N, (10)
denote a set of fixed solutions of systems ( 5) and (6), respectively. In addition, for systems (5) and ( 6), we consider the potentials ω j,k = ω j,k (z) defined as follows:

∂ z ω j,k = ψ 1 (j)ψ + 1 (k), ∂ z ω j,k = -ψ 2 (j)ψ + 2 (k) in D ⊆ C, j, k = 1, . . . , N, (11)
under the assumption that D is simply connected. Note that

∂ z ∂ z ω j,k = (∂ z ψ 1 (j)) ψ + 1 (k)+ψ 1 (j) (∂ z ψ + 1 (k)) = uψ 2 (j) ψ + 1 (k)-vψ 1 (j) ψ + 2 (k), ∂ z ∂ z ω j,k = -(∂ z ψ 2 (j)) ψ + 2 (k)-ψ 2 (j) (∂ z ψ + 2 (k)) = -vψ 1 (j) ψ + 2 (k)+uψ 2 (j) ψ + 1 (k), and, thus, ∂ z ∂ z ω j,k = ∂ z ∂ z ω j,k . ( 12
)
Therefore, definitions [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and Soliton equations[END_REF] are self-consistent, and the integration constants may depend on the concrete situation.

Let us introduce N × N matrix Ω = (Ω k,j (z)) = (ω j,k (z)):

Ω =      ω 1,1 ω 2,1 . . . ω N,1 ω 1,2 ω 2,2 . . . ω N,2 . . . . . . . . . . . . ω 1,N ω 2,N . . . ω N,N      . ( 13
)
Theorem 1 Suppose that ψ(j), ψ + (j), j = 1, . . . , N , (see [START_REF] Taimanov | The Moutard transformation of two-dimensional Dirac operators and Möbius geometry[END_REF]) are formal solutions of systems ( 5), [START_REF] Moutard | Sur la construction des équations de la forme 1 z ∂ 2 z ∂x∂y = λ(x, y) qui admettenent une intégrale générale explicite[END_REF], respectively, for given coefficients u = u(z), v = v(z), and that det Ω = 0, where Ω is defined by ( 11)-( 13). Let the transform {u, v} → {ũ, ṽ}, (14)

{ ψ(0), ψ + (0)} → { ψ(0), ψ+ (0)} ( 15
)
be defined as follows:

ũ = u + [ ψ 1 (1) . . . ψ 1 (N ) ] Ω -1    ψ + 2 (1) . . . ψ + 2 (N )    , ( 16
) ṽ = v - [ ψ 2 (1) . . . ψ 2 (N ) ] Ω -1    ψ + 1 (1) . . . ψ + 1 (N )    , ( 17
) [ ψ1 (0) ψ2 (0) ] = [ ψ 1 (0) ψ 2 (0) ] - [ ψ 1 (1) . . . ψ 1 (N ) ψ 2 (1) . . . ψ 2 (N ) ] Ω -1    ω 0,1 . . . ω 0,N    , ( 18
) [ ψ+ 1 (0) ψ+ 2 (0) ] = [ ψ + 1 (0) ψ + 2 (0) ] - [ ψ + 1 (1) . . . ψ + 1 (N ) ψ + 2 (1) . . . ψ + 2 (N ) ] (Ω -1 ) t    ω 1,0 . . . ω N,0    , ( 19
)
where ψ(0), ψ + (0) are some formal solutions to systems ( 5), [START_REF] Moutard | Sur la construction des équations de la forme 1 z ∂ 2 z ∂x∂y = λ(x, y) qui admettenent une intégrale générale explicite[END_REF], respectively, ω 0,j and ω j,0 are defined as in [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and Soliton equations[END_REF], but in terms of ψ(0), ψ + (j) and ψ(j), ψ + (0), respectively, and the symbol t in ( 19) stands for the matrix transposition. Then the transformed functions ψ(0), ψ+ (0) solve the transformed Dirac equations:

∂ z ψ1 (0) = ũ ψ2 (0), ∂ z ψ2 (0) = ṽ ψ1 (0) in D, ( 20
)
∂ z ψ+ 1 (0) = -ṽ ψ+ 2 (0), ∂ z ψ+ 2 (0) = -ũ ψ+ 1 (0) in D. ( 21
)
Historically, transformations like ( 14)-( 15) go back to the paper [START_REF] Moutard | Sur la construction des équations de la forme 1 z ∂ 2 z ∂x∂y = λ(x, y) qui admettenent une intégrale générale explicite[END_REF], and are known in the literature as Moutard transforms or Moutard-type transforms. For the two-dimensional Dirac operators the Moutard transforms were considered many times in the literature, see, for example, [START_REF] Matveev | Darboux transformations and solitons[END_REF], [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF], [START_REF] Taimanov | The Moutard transformation of two-dimensional Dirac operators and Möbius geometry[END_REF] and references therein. Nevertheless, for completeness of exposition, we present the proof of Theorem 1 in Section 4.

The point is that the Moutard-type transform ( 14),(15) admits the following reduction to the case of equations ( 1), [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF], i.e. to the case of generalized analytic functions.

Let

ψ(j) = ψ(z, j), ψ + (j) = ψ + (z, j), j = 1, . . . , N, (22)
denote a set of fixed solutions of ( 1) and ( 9), respectively. Then we can define ω j,k = ω j,k (z) as imaginary-valued functions satisfying

∂ z ω j,k = ψ(j)ψ + (k), ∂ z ω j,k = -ψ(j)ψ + (k) in D, j, k = 1, . . . , N, (23)
where ( 23) is the reduction of (11) corresponding to

ψ 1 (j) = ψ(j), ψ 2 (j) = ψ(j), ψ + 1 (j) = ψ + (j), ψ + 2 (j) = ψ + (j). ( 24
)
Remark 1 It is interesting to note that real-valued function φ = ω j,k 2i , i = √ -1
, for some fixed j, k, is known in the generalized analytic functions theory as a potential for equation [START_REF] Adams | Sobolev spaces[END_REF]; see [START_REF] Vekua | Generalized Analytic Functions[END_REF].

Theorem 2 Suppose that ψ(j), ψ + (j), j = 1, . . . , N , (see ( 22)) are formal solutions of equations ( 1), [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF], respectively, for a given coefficient u = u(z), and that det Ω = 0, where Ω is defined according to ( 13), ( 23) with imaginaryvalued ω j,k . Let the transform

u → ũ, ( 25
)
{ψ(0), ψ + (0)} → { ψ(0), ψ+ (0)} ( 26
)
be defined as follows:

ũ = u + [ ψ(1) . . . ψ(N ) ] Ω -1    ψ + (1) . . . ψ + (N )    , ( 27
) ψ(0) = ψ(0) - [ ψ(1) . . . ψ(N ) ] Ω -1    ω 0,1 . . . ω 0,N    , ( 28
)
ψ+ (0) = ψ + (0) - [ ψ + (1) . . . ψ + (N ) ] (Ω -1 ) t    ω 1,0 . . . ω N,0    , ( 29
)
where ψ(0), ψ + (0) are some formal solutions to equations ( 1), ( 9), respectively, ω 0,j and ω j,0 are imaginary-valued and are defined as in ( 23), but in terms of ψ(0), ψ + (j) and ψ(j), ψ + (0), respectively, and t in (29) stands for the matrix transposition. Then the transformed functions ψ(0), ψ+ (0) solve the transformed generalized-analytic function equations:

∂ z ψ(0) = ũ ψ(0) in D, ( 30
)
∂ z ψ+ (0) = -ũ ψ+ (0) in D. ( 31
)
To our best knowledge, Moutard-type type transforms like (25), (26) were not yet considered in the generalized-analytic function theory.

Theorem 2 follows from Theorem 1 in the framework of reductions ( 7), (24). In this case using also that Ω = -Ω one can see that the transform (14), (15) preserves symmetries [START_REF] Nimmo | Superposition principles associated with the Moutard transformation: an integrable discretization of a 2+1dimensional sine-Gordon system[END_REF], (24).

The algebraic result of Theorem 2 admits, in particular, the following analytic realization: Theorem 3 Let D be an open simply-connected bounded domain in C with C 1 -boundary and u satisfy [START_REF] Bers | Theory of pseudo-analytic functions[END_REF]. Let ψ(j) ∈ W 1,p (D) and ψ + (j) ∈ W 1,p (D), j = 1, . . . , N , be solutions of equations ( 1) and ( 9), respectively, and det Ω = 0 in D∪∂D, where Ω is defined according to (13), ( 23) with imaginary-valued ω j,k . Let the Moutard transform be defined according the formulas ( 25)-( 29) as in Theorem 2.

Then the transformed coefficient ũ satisfied (2) as well as the initial u.

In Theorem 3, W 1,p denotes the standard Sobolev space. Theorem 3 follows from formula (27) and from the properties that ψ(j), ψ + (j), j = 1, . . . , N , and ω j,k , 1 ≤ j, k ≤ N , and det Ω are continuous in D ∪ ∂D. Note that, in order to obtain the aforementioned continuity of ψ(j), ψ + (j), we use that, under our assumptions on D, if f ∈ W 1,p (D) then f ∈ C α (D ∪∂D), where C α denotes the standard Hölder space, α = (p-2)/p (see [START_REF] Adams | Sobolev spaces[END_REF]). Finally, the continuity of ω j,k follows from the continuity of ψ(j), ψ + (k) and from formula [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and Soliton equations[END_REF].

Explicit examples

In this section we present explicit examples illustrating the Moutard-type transforms for generalized analytic functions, i.e. transforms (25)-(29).

Let

u ≡ 0, N = 1, ψ(1) = f (z), ψ + (1) = f + (z), (32) ψ(0) = ψ(z, 0), ψ + (0) = ψ + (z, 0),
where f , f + and ψ(0), ψ + (0) are holomorphic on D. Then:

1. For imaginary-valued ω 1,1 and ω 0,1 , ω 1,0 of ( 23) and ( 28), (29), we have

ω 1,1 (z) = F 1,1 (z) -F 1,1 (z) + c 1,1 , ( 33
)
ω 0,1 (z) = F 0,1 (z) -F 0,1 (z) + c 0,1 , ω 1,0 (z) = F 1,0 (z) -F 1,0 (z) + c 1,0 ,
where F 1,1 and F 0,1 , F 1,0 are holomorphic functions on D such that

∂ z F 1,1 (z) = f (z)f + (z), (34) ∂ z F 0,1 (z) = ψ(z, 0)f + (z), ∂ z F 1,0 (z) = f (z)ψ + (z, 0),
and c 1,1 , c 0,1 , c 1,0 are pure imaginary constants; 2. Formulas ( 27)-(29) take the form:

ũ(z) = f (z)f + (z) ω 1,1 (z) , ( 35
) ψ(z, 0) = ψ(z, 0) -f (z) ω 0,1 (z) ω 1,1 (z) , ( 36
) ψ+ (z, 0) = ψ + (z, 0) -f + (z) ω 1,0 (z) ω 1,1 (z) . ( 37
)
In addition, equations ( 30), (31) for ψ(0), ψ+ (0) are nontrivial already for

f (z) = C = 0, f + (z) = C + = 0, ( 38
)
where C, C + are complex constants. In particular, in this case

ũ(z) = CC + 2i Im(CC + z) + c 1,1 . ( 39
)
One can see that this ũ satisfies (2) if (D ∪ ∂D) ⊂ Λ + or (D ∪ ∂D) ⊂ Λ -, where

Λ + = {z ∈ C : Im ( 2CC + z + c 1,1 ) > 0}, (40)
Λ -= {z ∈ C : Im ( 2CC + z + c 1,1 ) < 0}.
However, this ũ does not satisfy (2) at all if D ∩ L = ∅, where

L = {z ∈ C : Im ( 2CC + z + c 1,1 ) = 0}. ( 41
)
The point is that already the Moutard transform ( 25)-( 29) for the case (32), (38) yields generalized analytic functions with contour poles: with poles on L in this particular case.

Next, we consider the example when

u ≡ 0, N = 2, ψ(1) = f 1 , ψ + (1) = f + 1 , ψ(2) = f 2 , ψ + (2) = f + 2 , (42) ψ(0) = ψ(z, 0), ψ + (0) = ψ + (z, 0),
where f 1 , f + 1 , f 2 , f + 2 are complex constants, and ψ(0), ψ + (0) are holomorphic functions on D. In this case:

1. For imaginary-valued ω j,k of ( 23), ( 28), (29), j, k ∈ {0, 1, 2}, we have

ω j,k (z) = f j f + k z -f j f + k z + c j,k , ( 43
)
ω 0,k (z) = Ψ(z)f + k -Ψ(z)f + k + c 0,k , ω j,0 (z) = f j Ψ + (z) -f j Ψ + (z) + c j,0 , where j, k = 1, 2, Ψ, Ψ + are holomorphic functions on D such that ∂ z Ψ(z) = ψ(z, 0), ∂ z Ψ + (z) = ψ + (z, 0), ( 44
)
and c j,k are pure imaginary constants, j, k = 0, 1, 2;

2. Formulas ( 27)-( 29) take the form:

ũ(z) = f 1 f + 1 c 2,2 -f 2 f + 1 c 1,2 -f 1 f + 2 c 2,1 + f 2 f + 2 c 1,1 det Ω , (45) ψ(z
, 0) = ψ(z, 0) -(det Ω) -1 × (46) × ([( -f 1 f 2 f + 2 + f 2 f 1 f + 2 ) z + f 1 c 2,2 -f 2 c 1,2 ] ω 0,1 (z)+ + [( f 1 f 2 f + 1 -f 2 f 1 f + 1 ) z -f 1 c 2,1 + f 2 c 1,1 ] ω 0,2 (z)
) ,

ψ+ (z, 0) = ψ + (z, 0) -(det Ω) -1 × (47) × ([( -f + 1 f 2 f + 2 + f + 2 f 2 f + 1 ) z + f + 1 c 2,2 -f + 2 c 2,1 ] ω 1,0 (z)+ + [( f + 1 f 1 f + 2 -f + 2 f 1 f + 1 ) z -f + 1 c 1,2 + f + 2 c 1,1 ] ω 2,0 (z)
)

,

where det Ω = 2 Re [( f 2 f + 1 f 1 f + 2 -f 1 f + 1 f 2 f + 2 ) z z + (48) + ( c 2,2 f 1 f + 1 + c 1,1 f 2 f + 2 -c 1,2 f 2 f + 1 -c 2,1 f 1 f + 2 ) z ] + c 1,1 c 2,2 -c 2,1 c 1,2 ,
Ω is the matrix defined according to (13

) for N = 2. Note that if c 2,2 f 1 f + 1 + c 1,1 f 2 f + 2 -c 1,2 f 2 f + 1 -c 2,1 f 1 f + 2 = 0, (49)
f 1 f + 1 c 2,2 -f 2 f + 1 c 1,2 -f 1 f + 2 c 2,1 + f 2 f + 2 c 1,1 = 0, Re ( f 2 f + 1 f 1 f + 2 -f 1 f + 1 f 2 f + 2 ) = 0,
then ũ in (45) is spherically-symmetric and non-trivial. In addition, assuming that σ ≥ 0, where

σ = - c 1,1 c 2,2 -c 2,1 c 1,2 2
Re

( f 2 f + 1 f 1 f + 2 -f 1 f + 1 f 2 f + 2 ), (50)
we have that: ũ has a pole on

S r = {z ∈ C : |z| 2 = r}, where r 2 = σ; (51) ũ(z) = O(|z| -2 ) as |z| → ∞. ( 52
)
For example, assumptions (49) are fulfilled if

f 1 = 1, f + 1 = 1, f 2 = i, f + 2 = i, c 1,1 = c 2,2 , c 1,2 = -c 2,1 , |c 1,1 | 2 +|c 1,2 | 2 = 0.
(53) One can see that the Moutard transform (25)-(29) for the case (42), ( 49), (50) yields again generalized analytic functions with contour poles: with poles on S r in this particular case.

Actually, our Moutard-type transforms (25)-( 29) give a method for developing a proper theory of generalized analytic functions with contour and point poles of some natural class. This will be developed in the subsequent work.

Proof of Theorem 1

Using [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and Soliton equations[END_REF],(13) we obtain:

∂ z Ω =    ψ + 1 (1)
. . .

ψ + 1 (N )    [ ψ 1 (1) . . . ψ 1 (N ) ] , ( 54
)
∂ z Ω = -    ψ + 2 (1)
. . .

ψ + 2 (N )    [ ψ 2 (1) . . . ψ 2 (N ) ] , ( 55
)
∂ z Ω -1 = -Ω -1    ψ + 1 (1)
. . .

ψ + 1 (N )    [ ψ 1 (1) . . . ψ 1 (N ) ] Ω -1 , ( 56
)
∂ z Ω -1 = Ω -1    ψ + 2 (1)
. . .

ψ + 2 (N )    [ ψ 2 (1) . . . ψ 2 (N ) ] Ω -1 . ( 57
)
Using ( 18) and ( 5), ( 11), ( 56), (57), we obtain:

∂ z ψ1 (0) = ∂ z ψ 1 (0) - [ ∂ z ψ 1 (1) . . . ∂ z ψ 1 (N ) ] Ω -1    ω 0,1 . . . ω 0,N    + + [ ψ 1 (1) . . . ψ 1 (N ) ] Ω -1    ψ + 2 (1)
. . .

ψ + 2 (N )    ψ 2 (0)- - [ ψ 1 (1) . . . ψ 1 (N ) ] Ω -1    ψ + 2 (1)
. . .

ψ + 2 (N )    [ ψ 2 (1) . . . ψ 2 (N ) ] Ω -1    ω 0,1 . . . ω 0,N    = =    u + [ ψ 1 (1) . . . ψ 1 (N ) ] Ω -1    ψ + 2 (1)
. . .

ψ + 2 (N )       × ×    ψ 2 (0) - [ ψ 2 (1) . . . ψ 2 (N ) ] Ω -1    ω 0,1 . . . ω 0,N       = =    u + [ ψ 1 (1) . . . ψ 1 (N ) ] Ω -1    ψ + 2 (1)
. . .

ψ + 2 (N )       ψ2 (0); (58) ∂ z ψ2 (0) = ∂ z ψ 2 (0) - [ ∂ z ψ 2 (1) . . . ∂ z ψ 2 (N ) ] Ω -1    ω 0,1 . . . ω 0,N    - - [ ψ 2 (1) . . . ψ 2 (N ) ] Ω -1    ψ + 1 (1)
. . .

ψ + 1 (N )    ψ 1 (0)+ + [ ψ 2 (1) . . . ψ 2 (N ) ] Ω -1    ψ + 1 (1)
. . .

ψ + 1 (N )    [ ψ 1 (1) . . . ψ 1 (N ) ] Ω -1    ω 0,1 . . . ω 0,N    = =    v - [ ψ 2 (1) . . . ψ 2 (N ) ] Ω -1    ψ + 1 (1)
. . .

ψ + 1 (N )       × ×    ψ 1 (0) - [ ψ 1 (1) . . . ψ 1 (N ) ] Ω -1    ω 0,1 . . . ω 0,N       = =    v - [ ψ 2 (1) . . . ψ 2 (N ) ] Ω -1    ψ + 1 (1)
. . .

ψ + 1 (N )       ψ1 (0). (59)
Formulas ( 58), ( 59) and ( 16), ( 17) imply (20). Using ( 19) and ( 5), ( 11), ( 56), (57), and the formulas

∂ z (Ω -1 ) t = (∂ z Ω -1 ) t , ∂ z (Ω -1 ) t = (∂ z Ω -1 ) t ,
we obtain:

∂ z ψ+ 1 (0) = ∂ z ψ + 1 (0) - [ ∂ z ψ + 1 (1) . . . ∂ z ψ + 1 (N ) ] (Ω -1 ) t    ω 1,0 . . . ω N,0    + + [ ψ + 1 (1) . . . ψ + 1 (N ) ] (Ω -1 ) t    ψ 2 (1) . . . ψ 2 (N )    ψ + 2 (0)- - [ ψ + 1 (1) . . . ψ + 1 (N ) ] (Ω -1 ) t    ψ 2 (1) . . .

  ) = ∂ z ψ + 2 (λ) -[ ∂ z ψ + 2 (1) . . . ∂ z ψ + 2 ), (61) and (16), (17) imply (21). This completes the proof of Theorem 1.

The main part of the work was fulfilled during the visit of the first author to the Centre de Mathématiques Appliquées of École Polytechnique in October 2015. The first author was partially supported by the Russian Foundation for Basic Research, grant 13-01-12469 ofi-m2, by the program "Leading scientific schools" (grant NSh-4833.2014.1), by the program "Fundamental problems of nonlinear dynamics".


Guillaume Arnould
email: guillaume.arnould@univ-paris1.fr

Salim Dehmej
email: salim.dehmej@univ-paris1.fr

Thibaut Duprey

Faith Witryol

Michel Crinetz

Nicolas Véron

Martien Lubberink

Christophe Boucher

Stefanie Schulte

Is the European banking system more robust? An evaluation through the lens of the ECB's Comprehensive Assessment

Keywords: G21, G28 financial stability, stress tests, banking, financial regulation, Basel III

The results of the Comprehensive Assessment (CA) conducted by the ECB seem to attest the soundness of the European banking system since only 8 of 130 assessed banks still need to raise €6 billion. However it would be a mistake to conclude that non failing banks are completely healthy. Using data provided by the ECB and the EBA after the CA, we assess the capital shortfalls for each banks by considering the transitional arrangements, an implementation of Basel III sovereign debt requirements and an enhancement of the leverage ratio. In addition we show, that if the CA has been a very complex exercise, it is not the best lens through which the soundness of the eurozone banking system should be evaluated. The assumptions used for the Asset Quality Review (AQR) and the stress-tests lead to week scenarios and requirements that undermine the reliability of the results. Finally we show that the low profitability, the massive dividend distribution and the incurred fines, give rise to concern on the ability of eurozone banks to meet the incoming capital requirements.

Section 1. Introduction

On 4 November 2014, the European Central Bank became the new micro-prudential supervisor within the framework of the Single Supervisory Mechanism (SSM), the first pillar of the banking union3 . A few days earlier, on 26 October, the results of a comprehensive assessment of 130 eurozone banks were published, constituting a preliminary step in the ECB's new mission. This assessment4 was based on an Assets Quality Review (AQR), backed-up by a stress test conducted jointly with the European Banking Authority (EBA). Stress tests are among the post-crisis regulatory framework and are used to both microprudential supervision of individual firms and macroprudential assessments of financial imbalances. The Banks' balance sheets were subjected to macro-economic scenarios, one considered as "baseline" and based on the European Commission's forecasts from early 2014, and the other as "stressed", representing worsening European economic conditions.

The exercise revealed a capital shortfall of €25 billion for 25 of the 130 assessed banks, on the basis their balance sheets at the end of 2013. Given the recapitalisation already implemented for most of these banks in 2014, in the end only 8 banks5 were required to raise €6 billion of capital over a period of 6 to 9 months. Failed banks had to submit recapitalisation plans and were mainly concentrated in Italy with 4 banks over 8. None of the major European banks failed the test and especially no banks from the "core" of the Eurozone (Germany, Belgium, Netherlands and France) failed either, resulting in high praise for the European universal banking model. Overall, no surprise or worrying situation was announced. [START_REF] Sahin | Market reactions to the ECB's Comprehensive Assessment[END_REF] highlight that banks' stock market prices and CDS spreads did not react to the results. One should recognize however that it is a time consuming and highly complex exercise since the ECB faces a dilemma of requiring severity in order to be credible, but not to excess, in order to avoid potential destabilisation of financial markets.

Nevertheless, concluding to the soundness of European banks would be a mistake.

First, the assumptions on which the stress tests were based resulted in mild scenarios and requirements. Unlike the Fed and BoE stress tests, ECB stress test and communication about its results was restricted to the risk-weighted capital ratio, whereas the leverage ratio ("Tier 1" capital -T1, to non-weighted assets) paints a more varied picture. Our calculations of a stressed leverage ratio of 3% show a capital shortfall of €13 billion Tier1 capital, in 2015. However, if consider a 4% ratio, which will be the case in many countries (see section 3 on leverage), we find an important shortfall estimate at €88 billion Tier1 capital, in 2015.

Second, as mentioned by the ECB this exercise suffers from a problem of comparability between banks (different risk models) but also between countries (more than 150 different supervisory options and discretions applied by national regulators, such for example concerning provisions and non-performing loans). Additionally, the risk-weighted capital ratio cannot be taken literally, because its scope is evolving over the next four years during the transition to fully loaded Basel III 3 requirements since many elements considered until now as equity will disappear (intangible assets and deferred tax assets for example). Furthermore all countries don't keep the same pace of implementation. If we consider now the final deductions of equity, banks need €130 billion to keep the same level of capital CET1.

Third, when transposing the Basel committee recommendations into European law, the EU selected a zero risk weight coefficient for all sovereign debt securities of European Union countries while Basel regulation offered this possibility only for those considered as risk-free. An application of Basel risk weights for sovereign debt leads to a capital shortfall of €34 billion.

Finally, banks succeed generally to meet the risk weighted capital requirement by rising capital, deleveraging or rebalancing portfolios towards sovereign debt. However, if raising capital is the optimal solution for the regulator, it not the case for the bank. Furthermore, the changing environment illustrated by weak profitability, tightening regulation and even, for some banks, costly lawsuits due to misconducts, makes raising capital quite challenging. This article continues as follows: section 2 evaluates soundness of the results of the AQR and stress tests; section 3 assess eurozone banks' capital requirements according to the regulatory evolution and section 4 discusses the eurozone banks' ability to meet those capital requirements. Section 5 concludes.

Section 2. Assessing banking system soundness through AQR and stress-tests

Size of assets as primary selection criterion

The 1306 banks that participated to the comprehensive assessment undertaken by the ECB were selected according to the same criteria as for the SSM (ECB, 2014), namely:

 An asset value exceeding €30 billion.  A size exceeding 20% of national GDP, except for banks smaller than €5 billion.  Being part of the 3 largest credit institutions in a country participating in the SSM.  Having received European financial subsidies.

The size of the institutions is clearly the main criterion for this sample, and as emphasised by [START_REF] Drehmann | Measuring the Systemic Importance of Interconnected Banks[END_REF], a relatively large size goes hand in hand with increased complexity, denser financial links and a sufficient market share to trigger asset fire sales (AFS) in the event of financial difficulties [START_REF] Arnould | Risque systémique et politique macroprudentielle[END_REF]. Since 2011, the BIS has developed a scoring system for identifying systemically important banks 7 (G-SIB). This combines 5 criteria (BCBS, 2014a): size, interconnections, substitutability, international activities and complexity. Moreover, the Federal Reserve has recently added a sixth criterion by including the reliance on short-term 4 funding (Fed 2014a). Indeed, size is not sufficient to address the multiple systemic risk dimensions of financial institutions -as recently demonstrated by the problems of relatively small Spanish and German regional banks (respectively CajaSur in 2010 andSachsen LB in 2007). Moreover, Switzerland has recently decided to classify the Raiffeisenbank Schweiz Genossenschaft as systemic at domestic level, to the extent that its activities are considered to be difficult to substitute in the Swiss banking system in case of bankruptcy, yet the size of its balance sheet is just €28 billion (compared for example with the €880 billion assets of UBS).

Unambitious methodological choices

As reminded by the ECB, stress tests are not used to predict future events, but to evaluate the banks' capacity to withstand deteriorating economic and financial conditions. Recent stress tests are relatively more rigorous than those performed by the Committee of European Banking Supervisors (CEBS) in 2010 and by the EBA in 2011 (a larger number of banks, more rigorous scenarios and a longer time-scale). In addition, ECB capital requirements8 (8% for the baseline and 5.5% for the severe case) are tougher than those of the Fed (5%) and BoE9 (7% for the baseline and 4.5% for the severe case). Nevertheless, they are less ambitious than those conducted in the United States since 2009 and in UK in many aspects.

The scenarios used in US stress tests are more severe (Table 1), and refer to dynamic bank balance sheets which may be adjusted during the tests in accordance with changes in the risk profiles. Systemic institutions referred to as "too big to fail", "too costly to save" or "too complex to resolve" are distinguished from other banks and must satisfy stricter capital requirements. Finally, the assessment is not limited to the risk-weighted capital ratio but includes the stressed leverage ratio and qualitative aspects such as the pertinence of internal risk models, governance and recapitalisation plans [START_REF] Fed | CCAR 2014-Assessment Framework and Results[END_REF]. If banks fail the tests they may be denied permission to distribute dividends or to repurchase shares until they are recapitalised [START_REF] Hirtle | Supervisory Stress Tests[END_REF]. Finally, the UK stress test, conducted at the end of 2014, incorporated a monetary policy exit scenario by increasing the main interest rate by 4% over two years, to evaluate how banks would withstand risks stemming from increased defaulting on real estate loans 10 (BoE, 2014). The main critics addressed to the ECB exercise were about the severity of the scenarios, such as the absence of deflation or of a major shock on the sovereign debt market. It should also be noted that the transposition of macro-economic scenarios into losses for each bank was calculated through a risk assessment model specific to each institution, ultimately limiting the comparability of the level of losses between banks and the reliability of results -as the error of the British bank RBS, revealed in November 2014, reminded it. In fact, a few weeks after the announcement of the EBA results, RBS declared it had been mistaken when calculating the data communicated to the EBA: its final risk-weighted capital ratio fell consequently from 6.7% to 5.7%, just 0.2 percent above the floor of 5.5%. It is also and above all, the purely micro-prudential nature of this test which poses a problem. By definition, it excludes the systemic aspect of a shock. Second-round effects are disregarded since the banks' balance sheets are static over the exercise. Furthermore, these scenarios are non-systemic as they applied to a bank as if it was isolated from the rest of the financial sector. As such, the scenarios fail to inform macro-prudential policymakers. The 2008 financial crisis clearly illustrated the extent of contagion effects on market liquidity and during the assets fire sales.

Moreover, these stress tests barely address liability-related vulnerability, notably regarding the financing structure of banks and the liquidity of its resources (cf. the freezing of the inter-bank market following the collapse of Lehman Brothers, or the one of Northern Rock and Dexia and their difficulties in refinancing themselves on short-term markets). An assessment of liquidity ratios such as those planned in the Basel III framework (LCR12 short-term ratios and NSFR13 long-term ratios), incentivising banks to make use of more stable resources would have been welcomed. Since 2012, the Federal Reserve has been conducting a confidential liquidity analysis "CLAR -Comprehensive Liquidity Analysis and Review", testing the capacity of the major banks to withstand liquidity disruption [START_REF] Elliott | Bank Liquidity Requirements: An Introduction and Overview[END_REF]. More generally, a review of the stability of bank funding appears necessary. The dependency on wholesale funding, (market short-term funding) in particular, has become extremely significant [START_REF] Imf | Changes in bank funding patterns and financial stability risks[END_REF]. This causes banks to be highly sensitive to a liquidity shock. In the UK, the Prudential Regulation Authority's (PRA's) has the supervisory role to examine liquidity and funding risks [START_REF] Pra | The PRA's approach to supervising liquidity and funding risks[END_REF]. In their stress testing banks should consider the impact of a range of severe but plausible stress scenarios on their cash flows, liquidity resources, profitability, solvency, asset encumbrance and survival horizon. An ECB's Liability Quality Review (LQR) would seem essential.

Yet the stress test conducted by the ECB is by far the most comprehensive up to now because it expands the number of banks involved. And although it is difficult to compare the capital requirements for the particularly severe US stress test, conducted in 2009 just after the crisis, with the one performed by the ECB five years later, the amounts are highly comparable with the EBA stress test of 2011, despite an increase of almost 50% in the number of banks (Table 2). Source: [START_REF] Fed | The Supervisory Capital Assessment Program: Overview of Results[END_REF], [START_REF] Cebs | Aggregate outcome of the 2010 EU wide stress test exercise coordinated by CEBS in cooperation with the ECB[END_REF], EBA ( 2011), [START_REF] Fed | CCAR 2014-Assessment Framework and Results[END_REF], [START_REF] Eba | European banking authority 2011 Eu-wide stress test aggregate report[END_REF] and BoE (2014a). (i) EU banks, (ii) banks of Eurozone and of the European Union that want to join the banking union.

Comparability issues

Non-performing loans (NPL)

The AQR conducted by the ECB in 2014 pinpointed non-performing loans (NPL henceforth) that was not accurately assessed by banks. Indeed, €136 billion of NPL must be added to the €740 billion already declared by the banks (summing to €876 billion), denoting a significant increase of almost 20%. The most affected countries are Slovenia with 32%, followed by Greece, Malta and Estonia each with a 20% reclassification of their loans [START_REF] Eba | European banking authority 2011 Eu-wide stress test aggregate report[END_REF].

The October 2014 AQR study marked an upward revision of NPL indicates chronic overvaluation of the quality of assets by banks. The ECB report following the stress tests (ECB, 2014) emphasises that almost a third of audited banks were more lax than the ECB in the recognition of NPL. This assessment discrepancy was found even within so-called "level 1" assets, where 13% of banks disagreed with the ECB classification. Other differences were also brought to light by the AQR related to the number of days used to assess whether a debt is considered non-performing and to the date for constituting provisions for losses. The implication of the AQR results was that the lack of European harmonisation and possible indulgence of national supervisory bodies has hindered a rigorous assessment of risks.

Internal risk models

The statistics highlighted by the ECB during its presentation of stress tests results are related to risk-weighted capital ratio ("Common Equity Tier1" capital -CET1 to "Risk Weighted Assets" -RWA) outcomes. Contrary to the leverage ratio which does not discriminate between assets according to their degree of risk, the risk-weighted capital ratio allows, in theory, for the capital requirement to be adjusted according to the assumed risk of assets. However, the amount of weighted assets may vary significantly from the non-weighted amount (Figure 1). In the case of Deutsche Bank, for example, weighted assets account for only 18% of total assets, which substantially reduces the capital requirement. This discrepancy may be explained either by a more advantageous risk assessment methodology or by the holding of low capital-consuming assets. The risk assessment methodology submitted to the ECB may be developed either by the bank itself, with models estimated internally (so-called "Internal Rating Based" approach) or provided by the regulator (standard approach). In theory, the internal models allow a more accurate estimation of the risks and avoid strategies seeking investment yields from the highest risk assets for a given capital requirement. In practice, the difference between the risk-weighted capital ratio and the non-weighted ratio is significantly higher in the case of the internal rating based model (Figure 2), reflecting, inter alia, greater flexibility, or even gaming, when assessing risks (Le Leslé and Avramova, 2012). Capital shortfalls report by banks were all computed through their own risk model, meaning that with the same inputs of the economic scenarios, results, as of capital shortfall, that come out this black box are barely comparable.

Section 3. Eurozone banks capital requirements according to regulatory timeline

The most frequently drawn conclusion from the stress test results is that the European banking sector is once again in good health. Almost all French and German banks passed the tests with flying colours: they maintained their risk-weighted ratios above 8% for the baseline scenario and 5.5% for the stressed one. Using data provided by the EBA, we can draw our own assessment of eurozone banks' fragility and capital requirements on a 5 year horizon (2019). We mainly rely on 101 common banks of the ECB and EBA database from 15 countries that represent the majority of banks from the eurozone 17 (see table C in the appendix) and build a recapitalization trajectory from 2014 to 2019 based on regulation implementation plans and potential new evolutions.

Basel III phase-in and capital constraints

The Basel III agreements, signed in 2010, plan a transitional phase prior to the full and total application of the new requirements comprising an increase of capital ratio and a narrowed definition of capital (see table A in the appendix) 18 . Scheduled between 2014 and 2019 this timeline is also incorporated in the CRR/CRD4 legislation.

The transition from Basel II to Basel III is notably characterised by an improved quality of equity (CET1) and an increase in related deductions. More specifically, intangible assets (including "goodwill") are now 100% deducted from CET1 19 . Equity participations in other financial entities are now consolidated to avoid any double counting and, if they exceed 10%, are deducted from CET1. Finally, deferred taxes have been withdrawn from CET1 scope, whereas unrealised profits in the portfolio of assets available for sale have been added [START_REF] Bcbs | Basel III: A global regulatory framework for more resilient banks and banking systems[END_REF]. The timetable for 17 This induces the absence of banks from Estonia, Denmark, Lithuania, Poland, Sweden, Slovakia, Norway, Hungary and United-Kingdom. But the total assets of those 101 banks represent €20.4 trillion compare to the €22 trillion of the banks of the CA.

18 [START_REF] Cecchetti | The Road to Financial Stability: Capital Regulation, Liquidity Regulation, and Resolution[END_REF] declared "Putting this all together, we find that if we were to use the Basel III capital and asset definitions, the Basel II requirement of 4 percent would in fact be something between 1/2 percent and 3/4 percent. As the authorities were fond of saying, Basel III increased capital requirements by a factor of 10" 19 Under Basel II, intangible assets, including goodwill, were deducted at 50% from Tier1 and 50% from Tier2. National supervisory authorities are not all at the same stage [START_REF] Eba | European banking authority 2011 Eu-wide stress test aggregate report[END_REF]. Whereas the majority of States apply floor thresholds (for example 20% in 2014), others have acted in advance, such as Luxembourg, for instance, which already applies a rate close to 100%. Ultimately, banks in various EU countries do not have the same constraints in terms of capital, and the stress test was carried out without seeking to correct these differences. The only harmonisation measure adopted was to refrain from applying facilities granted by national supervisory authorities to their banks, which allow them to, for example, include in their accounts subsidies granted but not yet paid. Although these differences are due to be resolved by 2018, in the interim period, all eurozone banks are not singing from the same hymnal. The lack of harmonisation gives them margins for regulatory arbitrage. The effects of the transitional phase also depend on the composition of the capital held by each bank. The two main items which must be deducted from CET1, as part of the changeover to Basel III, are intangible assets and deferred taxes which can represent a significant proportion of CET1. Transitional arrangement figures are directly available from EBA data. Greece is in the lead with almost 80% of the CET1 of its banks (Figure 3), which will disappear by 2018 due to the end of the phase-in period for transitional arrangements (see table A in the appendix). In addition, German and Spanish banks whose solidity was much boasted about after the stress tests, have the greatest capital requirement, estimated at €30 billion each by 2018. In total, by 2018 the banks must raise more than €130 billion of CET1 (approximately 10% of their CET1 in 2013).

Sovereign debt treatment

Contrary to the spirit of the Basel regulations (since Basel II), at least for the standard approach 20 (BCBS, 2014b) which does not tolerate a nil weighting coefficient except for securities considered risk-free (AAA to AA-), when transposing the Basel committee recommendations into European law 21 , the EU selected a nil coefficient for all sovereign debt securities of European Union countries [START_REF] Nouy | Is sovereign risk properly addressed by financial regulation?[END_REF]. Furthermore, the risk-weighting rarely evolves in relation to risk (cf. the recent tensions on sovereign debt). The banks' pattern of holding of assets which consume less capital also allows the reduction of RWAs in relation to total assets. Banks can therefore perform regulatory arbitrage (Figure 4) in favour of highly lucrative sovereign debt securities, notably for peripheral countries, which are not expensive in terms of capital but nonetheless reflect a range of risks. Overall, our calculations show that the exposure of European banks to sovereign debt securities represents 280% of CET1 22 . We estimated that the application of the Basel III recommended weights would constrain European Union banks to raise €34 billion of capital in 2014 23 (Figure 4).

Figure 4 Eurozone banks' sovereign exposition to other eurozone countries (% of CET1) by risk weights and capital needs following Basel III standards in 2014

Source: Authors, data from SNL, ECB-EBA. Data available for 123 banks from EBA ; countries with sovereign rating, long term in local money, end 2013, between AAA et AA-are weighted at 0%, those between A+ and A-at 20%, those between BBB+ and BBBat 50% and finally those between BB+ and B-at 100% [START_REF] Nouy | Is sovereign risk properly addressed by financial regulation?[END_REF]. 20 The IRB approach allows banks to carry out an internal risk assessment of their exposure or to rely on rating agency scoring. It has been established that the banks use very low coefficients, around 1.4% [START_REF] Korte | Revisiting risk-weighted assets: why do RWAs differ across countries and what can be done about it?[END_REF], and all the more so as there is no floor on the probability of default on sovereign debt securities contrary to other forms of exposure and they are allowed to switch to the standard approach when assessing sovereign debt securities (partial use). This arbitrage is manifested as a financing facility for States but can reduce the resources available for the private sector, notably SMEs, which find it difficult to obtain finance on the markets (crowding out effect, [START_REF] Brone | Sovereign debt markets in turbulent times: Creditor discrimination and crowding-out effects[END_REF]. In addition, it favours the significant exposure of banks to domestic or foreign sovereign debt securities (Figure 5). Especially banks of the southern countries (Italy, Spain, Portugal, Greece, and Cyprus) are greatly exposed to the domestic debt, making them vulnerable to losses in the event of a debt crisis in those States and potentially compelling their governments to support them, fuelling the vicious circle between banks' balance sheet and domestic bonds' rates. An additional risk is that increased financial integration in Europe will concurrently increase the risk of contagion. Banks from Belgium, Austria or Luxembourg hold a great deal of foreign sovereign bonds.

Although Basel III introduces an incremental risk charge on the trading book which is assumed to take into consideration the risk of default and a change in the rating of trading positions (including sovereign debt), it rather favours the retention of sovereign debt. In recent years, sovereign debt falls within the category of the most liquid "level 1" 24 assets, through the liquidity ratio. Finally, sovereign debts are included in assets which may be held without any cap in the framework of Basel III (BCBS, 2014c). If a limit on holding sovereign debt was set at 25% of capital, more than €1.1 trillion of debt held by European banks would have to change hands [START_REF] Fitch | Protecting Eurozone Banks from the Sovereigns[END_REF].

The leverage ratio

The risk-weighted capital ratio indicated in 2007-2009 that the major American and European banks were sufficiently capitalised, whereas, several of them failed (Lehman Brothers in 2008) or had to be rescued (Northern Rock in 2007). Studies such as Haldane and Madouros (2012) indicate 24 The assets are ranked into three categories according to the ease of allocating a price, the first level being the most liquid assets, i.e. those which can most easily be priced directly on the market. that the risk-weighted capital ratio discriminated only marginally between banks in difficulty and more healthy ones. [START_REF] Pagano | Is Europe Overbanked?[END_REF] indicates that using the leverage ratio, banks had actually very little capital compare to their total of assets. In 2011, for example, Dexia had a risk-weighted capital ratio of 10.4% during the EBA stress test (for a requirement of 5%) whereas its leverage ratio was 1.34%. On the whole, the risk-weighted ratio tends to underestimate risks during the ascending phase of the cycle leading to an artificial increase of the risk-weighted capital ratio.

Although the leverage ratio was not disclosed in the results of the stress tests, it can be calculated on the basis of public data made available by ECB and EBA, using T1 capital and total exposures. The level of capital (T1) is given by EBA for each date of the whole stress-test period, 2014-2016 and for both the baseline and the adverse scenario [START_REF] Eba | European banking authority 2011 Eu-wide stress test aggregate report[END_REF]. In addition, we take in account the capital raised by banks during 2014 (see table B in the appendix), using data made available by the ECB (ECB, 2014) and the transitional arrangements as given by the EBA (EBA, 2014). The total assets, as defined by article 429 of CRR for « Leverage Exposure », are provided by the ECB for the end of 2013 (ante-AQR). Since the stress tests are static (see Section 2), we can make the assumption of fixed total assets for the whole test period. The leverage ratio can thus be measured throughout the entire test period for the 101 banks common to the ECB and EBA samples.

We calculate capital shortfall using stressed leverage ratio (Table 3), and show that for the 3% threshold of leverage, if capital shortfall are low in 2015 25 , they are not to be neglected in 2018. This increase is mainly due to the end of transitional arrangements (see Section 3.1) and leads to a shortfall of €67 billion. It is also worth noting that recapitalisation requirements literally explode when the ratio increases from 3% to 4% 26 . A potential increase of leverage ratio is not an unrealistic hypothesis since the United Kingdom 27 and the United States 28 have both planned to increase this ratio in comparable proportions during next years, along with Switzerland and the Netherlands, rendering an amendment of this ratio by the Basel Committee quite probable. A study by Steffen and Acharya (2014), using data from the end of 2013 and imposing a leverage requirement of 4%, consider that more than €85 billion would have to be raised by 109 European banks and €373 billion using the systemic risk measure known as the SRISK29 [START_REF] Brownlees | SRISK: A Conditional Capital Shortfall Index for Systemic Risk Measurement[END_REF]. 25 Notice that the leverage ratio came into force only in 2015, see table A in the appendix 26 [START_REF] Haldane | The dog and the frisbee" Speech given at the Federal Reserve Bank of Kansas City's 36th economic policy symposium[END_REF] considers that a ratio of at least 7% for systemic entities is required for financial stability. On this issue, 20 renowned economists, including Nobel prize-winners, have expressed a wish to increase the leverage ratio to 15% in an article published in the Financial Times ("Healthy Banking System is the Goal, not Profitable Banks", November 9, 2010). 27 The basic leverage remains at 3% as provided by Basel III. But, it is supplemented by capital buffers and may increase up to 4.95% in the ascending phase of the economic cycle for systemic banks by 2018 (BoE, 2014b). 28 Leverage ratio increased to 5% for major American banks, and even 6% for those participating in the deposit insurance system (BoE, 2014b). Nevertheless, the European accounting standards (IFRS) are more rigorous than those in the United States (GAAP) [START_REF] Brei | The leverage ratio over the cycle[END_REF] and the American ratio of 6% is equivalent to approximately 4% of the European ratio.

Austria 0 0 -0,3 -0,1 -3,4 -7,4 Belgium 0 -1,0 -3,4 -0,7 -3,4 -8,4 Cyprus 0 0 0 0 0 0 Finland 0 0 0 0 0 -0,7 France -2,0 -2,1 -32,4 -2,1 -5,6 -56,0 Germany -0,5 -2,6 -21,6 -2,3 -11,0 -49,2 Greece 0 -0,1 -1,8 -4,1 -15,7 -19,3 Ireland 0 -0,5 -1,0 -3,1 -10,8 -14,0 Italy 0 -3,1 -7,2 0 -8,7 -18,3 Luxembourg 0 0 0 0 0 -0,1 Malta 0 0 0 0 0 0 Netherland -1,7 -3,1 -17,6 -1,6 -5,1 -24,1 Portugal 0 -0,2 -1,1 0 -2,7 -5,1 Slovenia 0 0 -0,1 0 -0,2 -0,4 Spain 0 -0,4 -1,2 0 -0,6 -10,5
Total -4,3 -13,1 -87,7 -13,9 -67,2 -213,6 Source: Authors, SNL, data from ECB-EBA. Leverage ratio is based on 101 banks in common on both ECB and EBA samples. It is calculated by dividing the sum of CET1 and additional Tier1 (from the baseline or the adverse scenario), all provided by the EBA, by the total assets as defined by article 429 of CRR for « Leverage Exposure » communicated by ECB, end 2013. We also take in account the additional CET1 and T1 that have been raised in 2014. (i) We use data from 2016 scenarios and we apply on CET1 a complete deduction of transitional arrangements following Basel III phase-in (see table A in the appendix).

Predominantly, some banks of the "core" of Europe (France, Germany and Netherlands) record the greatest discrepancies between the 3% leverage threshold and the 4%, meaning that they are contenting themselves with the legal minimum regulatory capital and their risk-weighted capital ratios. Italian banks, although branded as the black sheep of the Eurozone, are far behind the leading duo of French and German banks. By over-emphasising the risk-weighted capital ratio, the ECB has drawn overly-optimistic conclusions while dismissing a significant proportion of the underlying risks and leaving major banks free to assess risks on their own with wide latitude.

Section 4. Eurozone banks' ability to meet capital requirements

Raising capital ratios by deleveraging

In theory, banks can adjust their capital ratio in three ways: 1-they can increase their capital by issuing new securities (shares or hybrid securities) or retaining a larger proportion of their profits; 2-they can adjust the size of their asset (financial securities or loans) or 3-they can reallocate their portfolio in favour of assets that consume less capital (e.g. sovereign debt).

In general, banks prefer the second option so as to avoid issuing new shares or retaining profits, both of which could be detrimental to their shareholders. They favour debt to the detriment of Documents de travail du Centre d'Economie de la Sorbonne -2015.61 equity, given the more advantageous tax regime and the implicit government guarantee [START_REF] Admati | The Bankers' New Clothes. What's Wrong with Banking and What to Do about It?[END_REF][START_REF] Imf | Moving From Liquidity-To Growth-Driven Markets[END_REF]. The regulator would prefer the first option in order to avoid a contraction in the financing of the economy. The regulator would also prefer to prevent widespread sales of assets which would -should the banks reduce their debts simultaneously30 -result in a deflationary spiral in the price of assets [START_REF] Greenlaw | Stressed Out: Macroprudential Principles for Stress Testing[END_REF]. Still, if banks adjust their assets, an economy financed mainly through banking intermediation, as is the case in the eurozone with 74% of credit intermediation provided by the banking system [START_REF] Goldstein | Too Big to Fail: The Transatlantic Debate[END_REF], would suffer less from deleveraging through the sale of financial assets than from restricting credit distribution. Another possible solution for fragile institutions would be a merger/acquisition operation. The results of the stress tests will probably bring about a consolidation of the banking system in certain countries such as Italy or Austria, comparable to the one embarked on by Spain since 2010.

In practice, Bologna and co-authors (2014) consider that between the end of 2011 and mid-2013 in Europe, the capital ratio was adjusted by a 9.6% increase in equity capital and a reduction in assets of 4.3% (including a reduction in financial assets other than sovereign debt and derivatives of 18.7% and of credit of 3.8%). However, the authors demonstrate that exposure to sovereign debt securities simultaneously increased by 68%. In the United States, by contrast, the capital ratio increases were achieved almost exclusively by a de facto capital increase [START_REF] Schoenmaker | The State of the Banking Sector in Europe[END_REF]. The Fed in fact demanded that recapitalisation requirements be implemented exclusively via this channel, something that was not stipulated by the EBA during its initial stress tests. The ECB does not have the same powers, but it has recently recommended that failed banks respect regulatory constraints of full Basel III should limit distribution of dividends and bonuses or even refrain from those distributions if they failed the stress-tests.

The scarcity of equity

Low profitability of eurozone banks

Combining structural weaknesses (e.g. high level of NPL as well as low interest rate environment which puts pressure on net interest margins (BIS, 2015)), evolution of competition (due to further financial integration and the development of e-banking), regulatory reforms and a grim macroeconomic environment raise many uncertainties on the profitability of the European banking sector31 and its capacity to build sufficient capital necessary to withstand adverse shocks. The IMF (2012) highlighted that many European banks are modifying their business plans to address several weaknesses exposed by the financial crisis and reduce some activities becoming less profitable and requiring more regulatory buffers, as the selling by Deutsche Bank of its subsidiary Postbank illustrates it. A study by Morgan Stanley and Oliver Wyman32 shows this lower profitability with a durable decline of return on equity (ROE) of investment bank over the last decade from 20% in early 2000's to less than 10% recently. Similarly, an ECB's study (2010) emphasises similar but less marked trend for universal banks.

Dividend's distribution

Unlike the Federal Reserve where failed banks are prohibited from distributing profits or buyback shares, the ECB does not have the same powers of coercion. Instead the ECB issued a recommendation that its banks limit dividend distributions. [START_REF] Shin | On book equity: why it matters for monetary policy" Speech at the joint workshop by the Basel Committee on Banking Supervision[END_REF] considers distributing dividend or repurchasing shares as mechanisms whereby banks routinely dissipate their equity. He shows that, from 2007 to 2012, the six-year cumulated dividend distributions of €150 billion, accounted for fully 50% of the retained earnings of a euro area group of banks (Figure 6).

Fines

In addition to weak profitability, many banks are facing costly lawsuits linked to misconducts. We can distinguish between direct costs from fines and indirect costs from market valuation since banking stocks with legacy litigation risks have underperformed those without litigation33 (reputation and risk of hidden misconduct). Banks are facing many legacy litigation liabilities and expenses34 . U.S. and European banks have paid out $190 billion and provisioned about $60 billion in fines since 2009 for misconduct and further $72 billion of to be provisioned by 2017 simply from ongoing litigation 35 . EBA database highlights that, from January to September 2014, eurozone banks incurred fines/litigation costs (net of provisions) of $7.3 billion (see Table B in the appendix for details per country). The equity banks can raise for incoming capital requirements is getting scarce, jeopardizing eurozone banks' ability to meet capital requirement of the incoming years.

Section 5. Conclusion

The transfer of national diverse euro-area prerogatives to a supra-national institution at the ECB will result in improved comparability, enhanced control, and a level playing field. The work of the ECB on the AQR is a first step in this direction, but much work remains. Certainly, the deadlines imposed, the required human resources, and the absence on the European stage of a credible resolution mechanism sufficiently funded to overcome prospective vulnerabilities may continue to limit the ECB's ambitions in the conduct of the "Comprehensive Assessment". Fortunately, as this only the first stress test of the new single supervisory body, and hopefully that future ones36 will be more complete. The US experience is instructive and it is regrettable that the ECB did not draw more inspiration therefrom. This exercise is all the more important for the eurozone since its financing structure is based primarily on banking intermediation.

As mentioned by the ECB, this exercise suffers not only from a problem of comparability between banks (different risk models) but also between countries (many discretions applied by national regulators, such for example the definition of non-performing loans). With regard to the riskweighted capital ratio, the use of the finalised Basel III ratio is to be preferred in order to remove transitional arrangements which renders any comparison between two banks in two different countries almost impossible.

We found that, considering the transitional requirements as of the Basel III transitional phase, eurozone banks still have to raise €130 billion by 2018, in order to keep the same amount of CET1. We further estimated that if sovereign debt capital requirements of Basel III recommendations were applied to eurozone countries, and if eurozone banks wanted to keep the same amount of sovereign debt, they had to raise €45 billion of capital (of which €29 billion of CET1) by 2018. Finally if the 3% threshold for the leverage is quite well observed in the baseline scenario, it leads to capital shortfall in the case of the adverse scenario, especially in the hypothesis of an increase of the threshold to 4%.

Our calculations can be summarised into two scenarios: one with absolute figures of capital shortfall, using Basel III phase-in and corresponding transitional arrangements, and a second with probable scenarios of the evolution of bank regulation in Europe. The latter scenarios include implementation of Basel III capital requirements for eurozone countries sovereign debt (see Section 2.2), increase of the leverage threshold to 4%37 and a macro outcome of adverse scenario for ECB's stress tests. For the first scenario we measure a CET1 shortfall of €130 billion by 2018, and for the second we reach €214 billion of T1 capital (of which €68 billion of CET1). This second scenario figure is not the sum of the 3 sources of recapitalisation we introduced: since we increase the level of capital required (4% for the leverage), the requirements according to portfolio composition (sovereign debt) and the type of capital (transitional arrangements), some requirements offset one another.

One thing is certain; reporting of stress tests' result must not gloss over the limitations of this type of exercise. It is dangerous to harbour the illusion of financial security [START_REF] Borio | Stress-testing macro stress testing: does it live up to expectations?[END_REF], and any result slack in efforts to restore banking sector soundness only increase the risk of a future crisis.

Finally, banks succeed generally in meeting risk weighted capital requirement by rising capital, deleveraging, or by rebalancing portfolios towards sovereign debt. However, the low profitability of eurozone banks, their massive dividend distributions (between 30%-50%), their substantial current and prospective fines all give rise to concerns about their ability of eurozone banks to meet the incoming regulatory capital requirements. Documents de travail du Centre d'Economie de la Sorbonne -2015.61

Figure 3

 3 Figure 3 Transitional arrangements (deferred taxes and intangible assets) to disappear by 2018, end 2013

  Documents de travail du Centre d'Economie de la Sorbonne -2015.61

Figure 5

 5 Figure 5 Eurozone banks' sovereign exposure to domestic and foreign sovereign debt (% of CET1)

  du Centre d'Economie de la Sorbonne -2015.61

Figure 6

 6 Figure 6 Dividend distribution of a group of 28 euro area banks

Table 1

 1 Comparison of the severity of stress tests'

scenarios Scenario USA, CCAR 11 2014 EU, EBA-ECB 2014 (i) UK, BoE 2014

	Real GDP	(2015), +1.5% (2016) -4.75% (2014), -1%	(2016) -0.7% (2014), -1.4% (2015), 0%	-3.5% (2015), -2.5% (2016)
	Unemployment	mid-2015 From +4% to 11.25%	+1.5% (2016) +0.3% (2014), +0.9% (2015),	Up to 12% (2016)
	Inflation	(2016) <1% (2015), 1.5%	+0.3% (2016) +1% (2014), +0.6% (2015),	+6.5% (2015)
	Stocks	-50%	-16% à -18% (ii)	-30%
	Housing market	-25%	-7% à -11% (ii)	-35%
	Source: Fed (2013), ESRB (2014) and BoE (2014a). (i): average for the Eurozone (ii): average of scenarios for 2014 and 2016.

Table 2

 2 Comparison of stress tests' results

	Year	2009	2010	2011	2014	2014	2014
	Stress test	USA-SCAP 14	CEBS 15 (EBA) (i)	EBA (i)	USA-CCAR	ECB-EBA (ii)	RU-BoE
	Banks number	19	91	90	30	130	8
	Failed banks	10 (53%)	7 (8%)	20 (22%)	3 (10%)	25 (19%)	3 (38%)
	Capital shortfall in billions	$185	€3.5	€27	n.a. (iii)	€25	n.a. (iii)

  Documents de travail du Centre d'Economie de la Sorbonne -2015.61 application of Basel III in Europe provides for a gradual increase in deductions and the inclusion of the new elements of CET1 (see table A in the appendix). Only the floor level is binding and if States so wish, they can apply final requirements in advance.

	Standard
	Internal
	0,0%	2,0%	4,0%	6,0%	8,0%	10,0%	12,0%	14,0%
			Leverage ratio (CET1/Assets)	Capital ratio (CET1/ RWA)

Table 3

 3 Capital shortfall in 2015 and 2018 under several stressed leverage thresholds (billions €)

		2015 (as of 1st January)	2018 (as of 1st January) (i)
	Pays	Baseline	Adverse	Adverse	Baseline	Adverse	Adverse
		-3%	-3%	-4%	-3%	-3%	-4%

  Table B in the appendix illustrates the repayment of CET1 and buyback by eurozone banks, per country, estimated at €14.3 billion in 2014 or almost 30% of the CET1 raised in 2014.

The banking union is based on three founding pillars: a single supervision, a resolution mechanism and deposit guarantee schemes. See[START_REF] Véron | From supervision to resolution Next steps on the road to European banking union[END_REF].

The comprehensive assessment was conducted not only by the ECB but also by the consultancy firm Oliver Wyman.

Of the 25 banks listed by the ECB, 12 already raised sufficient capital during 2014, and of the 13 remaining, 5 are in the process of restructuring or have implemented adequate capitalisation plans (ECB, 2014). Documents de travail du Centre d'Economie de la Sorbonne -2015.61

The ECB comprehensive assessment included more banks than for the SSM (120), since several banks close to but below the threshold criteria and not included in the SSM were reviewed nevertheless. See part 3 and annexe 9.1 (ECB, 2014).

The Financial Stability Board (FSB) adopts the BIS methodology to identify G-SIBs, numbering 30 in 2014, of which the majority were European banks. Documents de travail du Centre d'Economie de la Sorbonne -2015.61

CET1/RWA. See section 3 for details.

However, Fed and BoE stress tests are based also on a leverage ratio (3%).

Since variable rate mortgage debt is estimated at 68% of the real estate market, the latter is extremely sensitive to any rise in interest rates. Documents de travail du Centre d'Economie de laSorbonne -2015.61

Comprehensive Capital Analysis and Review.

The "Liquidity Coverage Ratio" enters into force in 2015, and requires that over a period of 30 days bank's liquidity reserves must be higher than the leakage caused by the loss of access to short term market and the flight of deposits. See Appendix.

Work on establishing the "Net Stable Funding Ratio" will start in 2018. It stipulates that over a period of one year, the stable funding must be less than the amount of available resources. See Appendix. Documents de travail du Centre d'Economie de laSorbonne -2015.61

Supervisory Capital Assessment Program.

Committee of European Banking Supervisors. Documents de travail du Centre d'Economie de la Sorbonne -2015.61

This measurement of risk is based on the evolution of the bank's capital, at the market price, during a period of systemic risk (see OFR, 2012). Documents de travail du Centre d'Economie de laSorbonne -2015.61

Although a reduction in banking debt may be viable on an individual basis, the global impact could prove disastrous (see the balance sheet recession described by[START_REF] Koo | The world in balance sheet recession: causes, cure, and politics[END_REF], hence the advantages of a macro-prudential approach to supervision[START_REF] Borio | Stress-testing macro stress testing: does it live up to expectations?[END_REF].

Speech by Danièle Nouy, Chair of the Supervisory Board of the Single Supervisory Mechanism, at the Economic Council in Berlin, 9 June 2015.

Morgan Stanley and Oliver Wyman, March 19, 2015, "Wholesale & Investment Banking Outlook. Liquidity Conundrum: Shifting risks, what it means", Blue paper. Documents de travail du Centre d'Economie de la Sorbonne -2015.61

Credit Suisse, Equity Research Europe, 4 June 2014, "Litigation-more risk, less return".

Manipulation and violation of rules range from: Libor, Forex, mortgages, insurance mis-selling, money-laundering rules, violating U.S. sanctions and tax evasion.

Source: WSJ, 3 June 2015, "How Regulators Can Keep Bankers Honest" and Le Monde, 23 June 2015, "Depuis le début de la crise, les banques ont payé 190 milliards de dollars d'amende" quoting a Morgan Stanley study. Documents de travail du Centre d'Economie de la Sorbonne -2015.61

The next stress test is planned in 2016.

It can be achieved through a real increase of the threshold, which we don't consider so credible, or a redenomination of the denominator (Leverage Exposure), which seems more likely. Documents de travail du Centre d'Economie de laSorbonne -2015.61

Documents de travail du Centre d'Economie de la Sorbonne -2015.61

This work was achieved through the Laboratory of Excellence on Financial Regulation (Labex ReFi) supported by PRES heSam under the reference ANR-10-LABX-0095. It benefitted from a French government support managed by the National Research Agency (ANR) within the project Investissements d'Avenir Paris Nouveaux Mondes (investments for the future Paris-New Worlds) under the reference ANR-11-IDEX-0006-02. Documents de travail du Centre d'Economie de la Sorbonne -2015.61

We remain responsible for any errors and omissions. All data are available upon request.

Appendix


Christophe Chorro
email: cchorro@univ-paris1.fr.

A Simple Probabilistic Approach of the Yard-Sale Model

Keywords: Wealth distribution, Yard-Sale model, Diffusion approximations, Impact of taxation

 concerning both wealth condensation in the absence of redistribution mechanisms and steady state distributions when a uniform capital taxation is imposed.

Introduction

Several recent studies have questioned the impact of randomness on wealth concentration and proved that luck alone may generate extreme disparities in wealth dynamics even if economical agents are identical in terms of their patience and their abilities (see for example, [START_REF] Fernholz | Instability and concentration in the distribution of wealth[END_REF], [START_REF] Boghosian | An H theorem for Boltzmann's equation for the Yard-Sale Model of asset exchange (The Gini coefficient as an H functional)[END_REF] and [START_REF] Bouleau | The impact of randomness on the distribution of wealth: Some economic aspects of the Wright-Fisher diffusion process[END_REF]). In these models, when there are no redistributive mechanisms, situations of strong concentration appear in which all the wealth ends up in the hands of a single oligarch and taxation is necessary to ensure the existence of a steady state distribution of wealth. Among these approaches, Multiplicative Random Asset Exchange Models (MRAEM), introduced in the framework of Econophysics by Ipsolatov et al. (1998), have recently emerged as interesting alternatives to proportional random growth models (see champernowne (1953) or [START_REF] Levy | Market efficiency, the Pareto wealth distribution, and the Levy distribution of stock returns[END_REF]) to describe the evolution of wealth distribution in a population interacting economically. In MRAEM, economy is considered in its simplest form as an interchange of wealth between pairs of agents.

This simplicity makes it possible to study social and collective phenomena. In particular, for the so-called [START_REF] Hayes | Follow the Money[END_REF] Yard-Sale model, where a player can win with probability 0.5 a fraction of wealth of the poorer agent, [START_REF] Boghosian | An H theorem for Boltzmann's equation for the Yard-Sale Model of asset exchange (The Gini coefficient as an H functional)[END_REF] show wealth concentration proving that the associated [START_REF] Gini | Measurement of inequality of income[END_REF] coefficient, a classical measure of wealth inequality, is an H function of the Boltzmann equation derived in [START_REF] Boghosian | Kinetics of Wealth and the Pareto Law[END_REF]. In the latter paper, the author also studies the impact of a uniform taxation of capital on wealth dynamics that leads to a steady state distribution for this economic system. The aim of this paper is to recover these results, at the very least for small and high frequency transactions, using classical probabilistic diffusion approximations in the spirit of Chorro and [START_REF] Bouleau | The impact of randomness on the distribution of wealth: Some economic aspects of the Wright-Fisher diffusion process[END_REF]. The main idea is to prove the convergence of the infinitesimal generator associated to the Yard-Sale repeated Markov chain game by transforming the time scales and state spaces appropriately and to study the limiting diffusion process.

For simplicity, we start, in Section 2, with the case of two players where the computations are explicit. The extension to the N players game is discussed in Section 3. Section 4 provides some conclusions.. In the so-called Yard-Sale model, at each round, each player may win or lose, with the same probability, a fraction a of the wealth owned by the poorest agent1 . This model has been introduced in [START_REF] Hayes | Follow the Money[END_REF] to overcome some economic bias of the Theft-and-Fraud model of Ipsolatov et al. (1998) where the exchanged amount is a fraction of the losing agent's wealth.

In fact, the last model drastically favors the poorest player while the Yard-Sale one is fair in expectation. More precisely, in the Yard-Sale model, if we denote by X i n the wealth of player i ∈ {1, 2} after n transactions, supposing that2 X 1 0 + X 2 0 = 1, we have X 1 n + X 2 n = 1 (zero-sum game) and

X i n+1 = X i n + a min(X i n , 1 -X i n )1 U n+1 1 2 -a min(X i n , 1 -X i n )1 U n+1 > 1 2
where

(U k ) k∈N * is a sample of the uniform distribution on [0, 1]. The sequence (X i n ) n∈N is a Markov chain with E[X i n+1 | X i n ] = X i n . Thus (X i n )
n∈N is a non-negative and bounded martingale that converges almost-surely and in L p (1 p < ∞) toward a random variable

X i ∞ . From E[(X i n+1 -X i n ) 2 | X i n ] = a 2 min(X i n , 1 -X i n ) 2 ,
we deduce that X i ∞ follows a Bernoulli distribution of parameter X i 0 . Thus, even if it is not possible for one player to be ruined after a finite number of rounds, all the wealth is concentrated at the limit in the hands of a single player.

By transforming the time scales and state spaces appropriately it is possible to study the continuous time limit of the preceding Markov chain game. This methodology was used in [START_REF] Bouleau | The impact of randomness on the distribution of wealth: Some economic aspects of the Wright-Fisher diffusion process[END_REF] to obtain theoretical approximations of almost-bankruptcy times in elementary market games with proportional bets: Let f : R → R be a measurable and bounded mapping. For all a ∈ R + and x ∈]0, 1[ we define the generator A a of the elementary market game with parameter a:

A a [f ](x) = E[f (X 1 1 )-f (X 1 0 ) | X 1 0 = x] = 1 2 f (x+a min(x, 1-x))+ 1 2 f (x-a min(x, 1-x))-f (x).
In particular, when f is of class C ∞ with a compact support in the interval ]0, 1[, we obtain from Taylor expansion that 1 a 2 A a f uniformly converges toward 1 2 min(x, 1 -x) 2 f "(x) when a goes to 0.

Considering the process (Z a t ) t∈R + that is the rescaled (at frequency a 2 ) continuous time

linear interpolation of the sequence (X 1 n ) n∈N with X 1 0 = x: Z a na 2 = X 1 n ∀n 0 Z a (n+θ)a 2 = Z a na 2 + θ(Z a (n+1)a 2 -Z a na 2 ) θ ∈ [0, 1] ∀n 0,
we obtain from classical arguments (see for example [START_REF] Stroock | Multidimensional diffusion processes[END_REF], Chap.

11) the uniform weak convergence of (Z a t ) t∈R + when a goes to 0 toward the diffusion process (X t ) t∈R + , associated to the infinitesimal generator

A[f ](x) = 1 2 min(x, 1 -x) 2 f "(x), (1)
that is the unique strong solution of the Stochastic differential equation

dX t = min(X t , 1 -X t )dB t 0 < X 0 < 1 (2)
where B t is a standard Brownian motion 3 . The points 0 and 1 are absorbing since the constant processes 0 and 1 are solutions of (2). The process (X t ) t∈R + is then a continuous and uniformly integrable martingale that converges almost-surely toward 1 with probability X 0 and toward 0 with probability (1 -X 0 ). Nevertheless, since the Green's function (see Etheridge (2012), p.44) associated to the diffusion is given by ∀x ∈ [0, 1],

G(x, ξ) = 2x(1-ξ) min(ξ,1-ξ) if x < ξ < 1 = 2ξ(1-x) min(ξ,1-ξ) if 0 < ξ < x
we obtain from the Dynkin's formula that

E[T | X 0 = x] = +∞
where T is the hitting time of the boundary {0, 1} that is inaccessible in finite time. The economy never collapses in finite time in this case (contrary to what happens in [START_REF] Bouleau | The impact of randomness on the distribution of wealth: Some economic aspects of the Wright-Fisher diffusion process[END_REF] for elementary market games with proportional bets), even if the concentration of wealth becomes stronger and stronger. In this case, this is not absorption because the absorption time is infinite with probability one, rather it is quasi-absorption 4 . Now we study the impact of a small tax rate on the preceding dynamics to show that it is possible to affect the dispersion of the wealth previously observed. One of the simplest redistribution mechanism is to consider a proportional capital tax rate that is collected at any stage and uniformly reallocated to the players 5 . With a tax rate b (b fulfilling 0 < a + b < 1), the transition of the Markov chain becomes

X i n+1 = (1 -b)X i n + a min(X i n , 1 -X i n )1 U n+1 1 2 -a min(X i n , 1 -X i n )1 U n+1 > 1 2 + b 2
where (U k ) k∈N * is a sample of the uniform distribution on the unit interval. In this case, the Markov chain is not a martingale anymore because

E[X 1 n+1 -X 1 n ) | X 1 n ] = b( 1 2 -X 1 n ).
The Yard-Sale model with capital taxation favors the poorest player during each transaction to mitigate the negative impact of randomness on wealth concentration. However, in spite of its simplicity, it is a priori difficult to obtain explicitly one invariant measure of such a Markov chain when b = 0. Therefore, we study directly the diffusion limit of the model for small and high-frequency transactions. Using the preceding argument the generator A a of the elementary taxed market game with parameters a and b becomes

A a [f ](x) = E[f (X 1 1 ) -f (X 1 0 ) | X 1 0 = x] = 1 2 f ((1 -b)x + a min(x, 1 -x) + b 2 ) + 1 2 f ((1 -b)x -a min(x, 1 -x) + b 2 ) -f (x) in particular, when f is of class C ∞ with a compact support in the interval ]0, 1[, 1 a 2 A a f uniformly converges toward A[f ](x) = 1 2 min(x, 1 -x) 2 f + λ 2 (1 -2x)f (3)
when a goes to 0 and b = λa 2 . 6 The generator A is associated to the diffusion

dX t = min(X t , 1 -X t )dB t + λ 2 (1 -2X t )dt 0 < X 0 < 1. (4)
In this case, the scale function of the generator (3) is given by

S(x) = x 1/2 exp[- y 1/2 λ(1 -2z) min(z, 1 -z) 2 dz]dy, thus, S (x) = K e λ/x x 2λ 1 x 1 2 + e λ/(1-x) (1 -x) 2λ 1 x> 1 2 .
In particular, S(0+) = -∞ and S(1-) = +∞ so the process is recurrent (see [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] Ex. 3.21 p. 298) and the associated invariant measure is given by the density (symmetric about 0.5)

7 m 2 (x) = C e -λ/x x -2(λ+1) 1 x 1 2 + e -λ/(1-x) (1 -x) -2(λ+1) 1 x> 1 2
where C is a constant of integration. This density function is represented in Figure 1 for different values of λ. We remark that the mode of the invariant density on [0, 0.5] that is equal to λ 2λ+2 converges toward 0.5 (egalitarian case) as λ goes to infinity. For small x, we interestingly recover the expression provided, using the techniques of mathematical physics, in Boghosian (2014), Eq. 91 (with W = 1 and N = 2) with a different probabilistic approach based on diffusion approximations of Markov chains. In Figure 1 we also observe the impact of λ on the wealth concentration process and we see that the proportion of wealth held by the wealthiest player falls drastically even for small values of the capital tax rate 8 .

H H H ¡ H H ¢ H H £ H H ¤ H

Remarks on the N players case

In a population of N players, we select, during each round, two agents randomly and without replacement for a transaction according to the 2-player Yard-Sale mechanism described in the preceding section. We also consider in this framework a proportional capital taxation (associated to the rate b) uniformly distributed among all the players. The aim of this section is to present the main lines of the extension of the preceding results.

If we denote by X n = (X 1 n , ..., X N n ) the vector of wealth after n rounds and by S N the set

{k = (k 1 , k 2 ) ∈ N 2 | N k 1 > k 2 1} , we have, ∀n ∈ N, X n+1 = (1 -b)X n + b N + k∈S N Y k 1 ,k 2 n+1 1 V n+1 =(k 1 ,k 2 ) with Y k 1 ,k 2 n+1 = a min(X k 1 n , X k 2 n )1 U (k 1 ,k 2 ) n+1 1 2 (e k 1 -e k 2 ) + a min(X k 1 n , X k 2 n )1 U (k 1 ,k 2 ) n+1 > 1 2 (e k 2 -e k 1 )
where (V n , U k n ) (n,k)∈N * ×S N are independent random variables such that V n (resp. U k n ) follows a uniform distribution on S N (resp. [0, 1]) and where e i denotes the vector of R N with a 1 in the i-th coordinate and 0's elsewhere. Starting from a point X 0 in the sim-

plex x = (x 1 , ..., x N ) ∈ R N | N i=1
x i = 1, x i 0 , (X n ) n∈N stays in the simplex and for b = 0, (X n ) n∈N is once again a bounded martingale that converges almost surely and in L p 1 p < +∞, toward X ∞ that is invariant with respect to the transition of the chain. From

E[| X n+1 -X n | 2 | X n ] = k=(k 1 ,k 2 )∈S N N (N -1)a 2 min(X k 1 n , X k 2 n ) 2
we deduce that X ∞ almost surely belongs to the vertices of the simplex and that X ∞ = e i with probability X i 0 . This strong tendency for the Yard-Sale model to drive increasing fractions of wealth into the hands of a single agent has been already observed numerically in [START_REF] Sinha | Stochastic Maps, Wealth Distribution in Random Asset Exchange Models and the Marginal Utility of Relative Wealth[END_REF] and [START_REF] Boghosian | Kinetics of Wealth and the Pareto Law[END_REF] and we provide in this paper a very simple probabilistic proof of this phenomenon9 . Now, supposing that b = λa 2 > 0, we can prove, along the same lines as the preceding section, that the generator associated to the N players game with proportional capital taxation is equal, for F : R N → R of class C ∞ with compact support, to

A a [F ](x) = 1 N (N -1) k=(k 1 ,k 2 )∈S N [F (x + a min(x k 1 , x k 2 )(e k 1 -e k 2 )) -F (x) F (x + a min(x k 1 , x k 2 )(e k 2 -e k 1 )) -F (x)]
and that the rescaled continuous time linear interpolation at frequency a 2 of the sequence (X n ) n∈N converges in distribution towards the diffusion associated to the infinitesimal generator

A N λ [F ](x) = 1 N (N -1) k=(k 1 ,k 2 )∈S N min(x k 1 , x k 2 ) 2 (F k 1 k 1 (x) + F k 2 k 2 (x) -2F k 1 k 2 (x)) + N i=1 λ N (1 -N x i )F i (x).
. Of course we recover for N = 2 the expression of Equation ( 3), however, up to our knowledge, there exists no classical expression for the steady state distribution associated to this generator for N > 2. Nevertheless, it can be proved easily that the steady state one-dimensional marginal distributions of this model are obtained, for small x, from the one dimensional generator

x 2 N f + λ N (1 -N x)f .
Thus, for small x, the steady state one-dimensional marginal distributions take the form

m N (x) = C N e -λ/x x -N λ-2
(where C N is a constant of integration) with a mode that is equal to λ N λ+2 . This expression is similar10 to Boghosian (2014), Eq. 91 obtained by the author using a random-agent approximation that is classical in Econophysics but different from our approach where diffusion approximations of Markov chains are simply used. In Figure 2 Remark: Another possible extension of the two players game is to consider that all the agents play simultaneously and that during each round one player can win a proportion a of min(X i n ) (divided equally between the other players) with probability 1 N . In its taxed version, this Markov chain game may be written in the following way:

X n+1 = (1 -b)X n - a N -1 min(X i n ) + aN N -1 min(X i n )e i + b N with probability 1 N .
Supposing once again that b = λa 2 > 0, we can prove that the associated generator converges toward

ÃN λ [F ](x) = min(x i ) 2 2(N -1) 2 N i,j=1 (N δ ij -1)F ij (x) + N i=1 λ N (1 -N x i )F i (x)
where δ ij = 1 if i = j and 0 otherwise. For N = 2 we recover the Equation (3) and for N > 2 this generator is different from A N λ .

Conclusion

This paper provides a simple alternative probabilistic study of the Yard-Sale model. First, we give an elementary proof of the concentration of wealth, when there is no redistribution, with a different approach than [START_REF] Boghosian | An H theorem for Boltzmann's equation for the Yard-Sale Model of asset exchange (The Gini coefficient as an H functional)[END_REF]. Then, using classical approximations of Markov chains by continuous diffusions, we study the steady state probability of the system when proportional capital taxation is added to the initial dynamics. In particular, we recover for small and high frequency transactions, the approximation, in the vicinity of 0, of the one-dimensional steady state marginals obtained in [START_REF] Boghosian | Kinetics of Wealth and the Pareto Law[END_REF] with an Econophysics view.

Figure 1 :

 1 Figure 1: Steady state density (left) of the two players Yard-Sale model for different values of the capital tax rate λ and proportion of the wealth (right) owned by the wealthiest player without (bold) or with (dashed) a 1% capital taxation in the two players game with n = 1000 transactions. The invariant densities are represented on the interval [0, 0.5] because they are symmetric about 0.5.

  , we have represented these theoretical approximations for the games with N = 10 and N = 50 players (with λ = 1%) to compare them with the one-dimensional empirical marginals obtained from the components of the Markov chain dynamics by Monte Carlo simulations: the results are close together for small values of x.

Figure 2 :

 2 Figure 2: Invariant density function of the Yard-Sale model with proportional capital taxation (λ = 1%) for N = 10 (left part) and N = 50 (right part). We represent, for small values of x, the empirical density function (bold) obtained using 1000 independent Monte Carlo simulations of X 1 100000 in the Markov chain market game with parameters a = 0.1 and b = a 2 and the density obtained from the diffusion approximation.

The parameter 1 -a a may be seen as a measure of the saving propensity supposed to be constant among all the participants to ensure an identical involvement.

The zero-sum game hypothesis implies that no wealth is imported, exported, generated, or consumed, wealth can only change hands. Working in a closed economic system, we use in this paper proportions of wealth instead of absolute values.

Documents de travail du Centre d'Economie de laSorbonne -2015.62

This result of convergence requires the existence and the unicity of the martingale problem associated to the generator (1) that is equivalent to the weak existence and the unicity in distribution of the solution of the associated stochastic differential equation. Here, (2) having Lipschitz we classically deduce the strong existence and unicity of the solution.

Documents de travail du Centre d'Economie de laSorbonne -2015.62

If we denote by T ε the first exit time of the interval [ε, 1 -ε] we can prove using[START_REF] Etheridge | Some Mathematical Models from Population Genetics[END_REF], p.44, that E[T ε | X 0 = x] ∼ 0 + -2log(ε)quantifying how inaccessible the boundaries of this model are. This quantity is greater for small ε than -2[(1 -x)log(1 -x) + xlogx], the limit that was obtained in[START_REF] Bouleau | The impact of randomness on the distribution of wealth: Some economic aspects of the Wright-Fisher diffusion process[END_REF] for the Wright-Fisher diffusion without mutations associated to elementary market games with proportional bets.

Is is easy to prove that if we consider instead a tax on the income a completely different asymptotic behavior is obtained. In this case, we recover (2) up to a constant: a proportional income tax is not sufficient to prevent agents from being ruined. 5 Documents de travail du Centre d'Economie de la Sorbonne -2015.62

Let us remark that in[START_REF] Boghosian | Kinetics of Wealth and the Pareto Law[END_REF], the same hypothesis is made on the scale of the tax rate with respect to a in order to obtain a Fokker-Planck equation for the steady state density of wealth when a is small.6 Documents de travail du Centre d'Economie de laSorbonne -2015.62

In order to introduce asymmetry in the steady state density, we may consider a different tax rate for the two players.

In simulations that are not reported here, we also observe that richest individuals tend not to remain the richest ones over time when the capital tax rate increases. In the Yard-Sale model social mobility and wealth concentration are highly correlated as already observed in[START_REF] Bouleau | The impact of randomness on the distribution of wealth: Some economic aspects of the Wright-Fisher diffusion process[END_REF] and[START_REF] Fernholz | A model of economic mobility and the distribution of wealth[END_REF] for different models. 7 Documents de travail du Centre d'Economie de la Sorbonne -2015.62

In[START_REF] Boghosian | An H theorem for Boltzmann's equation for the Yard-Sale Model of asset exchange (The Gini coefficient as an H functional)[END_REF], the authors prove this result demonstrating that the[START_REF] Gini | Measurement of inequality of income[END_REF] coefficient is a Lyapunov function of the Boltzmann equation for the Yard-Sale model. Thus this coefficient increases in time and asymptotically approaches the state of perfect inequality. 9 Documents de travail du Centre d'Economie de laSorbonne -2015.62

Such a form for the steady state distribution is also obtained for wealth distribution problems in a slight different model by[START_REF] Bouchaud | Wealth Condensation in a Simple Model of Economy[END_REF]. 10 Documents de travail du Centre d'Economie de la Sorbonne -2015.62

Documents de travail du Centre d'Economie de laSorbonne -2015.62