Cécilia Tarpau

Javier Cebeiro

Mai K Nguyen

Genevieve Rollet

Marcela Morvidone

Geneviève Rollet

Analytic Inversion of a Radon Transform on Double Circular Arcs With Applications in Compton Scattering Tomography

come

Introduction

The seminal works of Radon [START_REF] Radon | Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten[END_REF] and Cormack [START_REF] Cormack | Representation of a function by its line integrals, with some radiological applications[END_REF] on the classical Radon transform on straight lines were followed by several studies attempting to generalize this integral transform to various manifolds in two and three dimensions. Many of these generalizations are formulations in either full or partial circular paths, hence they are called Circular Radon Transforms (CRTs) and Circular Arc Radon Transforms (CARTs) respectively. A major part of these works concerns inversion formulas of CRTs. One can refer for example to the work of Cormack on the family of circles passing through the origin [START_REF]The Radon transform on a family of curves in the plane[END_REF][START_REF]Radon's problem-old and new[END_REF], the works of Ambartsoumian [START_REF] Ambartsoumian | Inversion of the circular Radon transform on an annulus[END_REF] and Haltmeier [START_REF] Haltmeier | Thermoacoustic tomography and the circular Radon transform: exact inversion formula[END_REF] on circles centered on a circle for thermoacoustic and photoacoustic tomography or the work of Redding [START_REF] Redding | Inverting the circular Radon transform[END_REF] about a CRT with applications in synthetic aperture radar imaging. There is also the work of Palamodov [START_REF] Palamodov | Reconstruction from limited data of arc means[END_REF] on an inversion formula to reconstruct objects from their data over a family of half-circles with applications in seismic tomography and synthetic aperture radar. About Radon transform on circular arcs, Nguyen and Truong [START_REF] Nguyen | Inversion of a new circular-arc Radon transform for Compton scattering tomography[END_REF][START_REF] Truong | Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality[END_REF] proposed the inversion Radon transforms on different families of circular arcs modelling data acquisition and image reconstruction of new modalities of Compton Scattering Tomography (CST). Syed [START_REF] Syed | Numerical Inversion of Circular arc Radon Transform[END_REF] proposed a numerical inversion for circular arcs with a fixed angular span with applications in photoacoustic tomography. Other aspects of CRTs and CARTs are also studied such as injectivity and range conditions. The reader can refer respectively to [START_REF] Ambartsoumian | On the injectivity of the circular Radon transform[END_REF][START_REF] Agranovsky | Injectivity sets for the Radon transform over circles and complete systems of radial functions[END_REF] and [START_REF] Agranovsky | Range descriptions for the spherical mean Radon transform[END_REF][START_REF] Ambartsoumian | A range description for the planar circular Radon transform[END_REF][START_REF] Finch | The range of the spherical mean value operator for functions supported in a ball[END_REF] for these issues.

In this paper we introduce a new Radon transform on a family of double circular arcs. This study is motivated by the proposition of a new Compton scattering tomography. We will now the general functioning principle of CST modalities.

Compton Scattering Tomography (CST)

In Compton scattering tomography, the objective is to use radiation scattered according to Compton effect to reconstruct the electron density map of the object. This type of imaging has been early studied especially by Lale [START_REF] Lale | The examination of internal tissues, using gamma-ray scatter with a possible extension to megavoltage radiography[END_REF], Clarke [START_REF] Clarke | Compton-scattered gamma rays in diagnostic radiography[END_REF] and by Farmer [START_REF] Farmer | A new approach to the determination of anatomical cross-sections of the body by compton scattering of gamma-rays[END_REF]. The reader can refer to [START_REF] Truong | Recent developments on Compton scatter tomography: theory and numerical simulations[END_REF] for a more detailed history on CST systems.

arXiv:1912.10797v3 [math.NA] 16 May 2020

Conventional Computed Tomography (CT) uses only primary (transmitted) radiation and considers scattered radiation as noise. CST is a novel approach that takes advantage of scatter radiation to scan objects. Images generated in that way are claimed to offer better performance in medical imaging particularly to identify lung tumors [START_REF] Jones | Characterization of Comptonscatter imaging with an analytical simulation method[END_REF][START_REF] Redler | Compton scatter imaging: A promising modality for image guidance in lung stereotactic body radiation therapy[END_REF]. In fact, if a photon emitted by a source with energy E 0 finds an electron on its path inside matter, this photon is scattered subtending an angle ωwith its original direction. Its energy E(ω) after collision is given by the Compton formula

E(ω) = E 0 1 + E0 m c 2 (1 -cos(ω)) , (1.1)
with m the electron mass and c the speed of light. When a single scattered photon outgoing from the object with an energy E(ω) is collected by a detector, this one-to-one correspondence between energy and angle ensures that the photon was scattered on a circular arc. This circular arc passes through the source and the detector and is labeled by the scattering angle ω. Consequently, assuming Compton effect is the only source of energy attenuation for emitted radiation and considering only first order scattering, the modelling of data measurement with modalities of Compton scatter tomography leads to CARTs on different families of circular arcs according to the chosen geometry of the system. These assumptions are the basis for mathematical tractability [START_REF] Nguyen | Inversion of a new circular-arc Radon transform for Compton scattering tomography[END_REF][START_REF] Webber | Three dimensional Compton scattering tomography[END_REF][START_REF] Webber | Microlocal analysis of a Compton tomography problem[END_REF][START_REF] Cebeiro | New improved compton scatter tomography modality for investigative imaging of one-sided large objects[END_REF][START_REF] Truong | Compton scatter tomography in annular domains[END_REF][START_REF] Tarpau | A new concept of Compton Scattering tomography and the development of the corresponding circular Radon transform[END_REF][START_REF] Tarpau | Compton scattering imaging system with two scanning configurations[END_REF][START_REF] Cebeiro | An interior Compton Scatter Tomography[END_REF][START_REF] Cebeiro | New improved Compton scatter tomography modality for investigative imaging of one-sided large objects[END_REF] and some of their practical implications are discussed at the end of the paper. The inversion of these Radon transforms represents the theoretical challenge raised by CST modalities, required for image reconstruction. The first CST system whose data acquisition has been modelled by a Radon transform was proposed by Norton [START_REF] Norton | Compton scattering tomography[END_REF]. This modality is made of a fixed source and a line of detectors. Image acquisition is performed on half circular arcs having a fixed common end-point in the source and the other ends on the straight line of detectors. The reader can also refer to [START_REF] Rigaud | Novel numerical inversions of two circular-arc Radon transforms in Compton scattering tomography[END_REF] where Rigaud gave another numerical inversion method for the Radon transform associated to Norton's modality. The second CST modality, proposed by Nguyen and Truong [START_REF] Nguyen | Inversion of a new circular-arc Radon transform for Compton scattering tomography[END_REF][START_REF] Rigaud | Novel numerical inversions of two circular-arc Radon transforms in Compton scattering tomography[END_REF][START_REF] Rigaud | Combined modalities of Compton scattering tomography[END_REF], is composed of a pair source -detector diametrically opposed on a circle. This system has lead to a CART based on circular arcs having a chord of fixed length. A third modality has been proposed recently by us [START_REF] Tarpau | A new concept of Compton Scattering tomography and the development of the corresponding circular Radon transform[END_REF][START_REF] Tarpau | Compton scattering imaging system with two scanning configurations[END_REF][START_REF] Cebeiro | An interior Compton Scatter Tomography[END_REF]. This modality is composed of a fixed source and a detector array on a ring. This one has led to a new CART having a fixed common end-point in the source and the other ends on the detector ring. A system with a similar geometry has been studied by Rigaud [START_REF] Rigaud | Compton Scattering Tomography: Feature Reconstruction and Rotation-Free Modality[END_REF]. He proposed in this work a reconstruction algorithm of the singular support of the object under study.

Three dimensional CST systems have also been proposed. The reader can refer to the works of Webber on the extension of the Nguyen and Truong's system [START_REF] Webber | Three dimensional Compton scattering tomography[END_REF] and on the proposition of a system having a translational geometry [START_REF] Webber | Compton scattering tomography in translational geometries[END_REF] or to the work of Rigaud [START_REF] Rigaud | 3D Compton scattering imaging and contour reconstruction for a class of Radon transforms[END_REF] who proposed several potential 3D modalities associated to a general class of toric Radon transforms.

A new CST system with a collimation-free detector

The context of this study is the introduction of a new two-dimensional CST system. This modality is based on a fixed source S, assumed to be monochromatic and a detector D moving on a circle of radius R around the source (see Fig. 1), and localized by its angular position ϕ. Hence, D can be defined by its Cartesian coordinates as D(ϕ) = R (cos ϕ, sin ϕ). The object to scan is placed outside of this circle. In this two-dimensional setup, a plate collimator in the source restricts emitted photons to the plane (x, y). Thus, cross-sections of the object are scanned. In contrast with other two-dimensional CST designs [START_REF] Nguyen | Inversion of a new circular-arc Radon transform for Compton scattering tomography[END_REF][START_REF] Tarpau | A new concept of Compton Scattering tomography and the development of the corresponding circular Radon transform[END_REF][START_REF] Rigaud | Compton Scattering Tomography: Feature Reconstruction and Rotation-Free Modality[END_REF][START_REF] Webber | Compton scattering tomography in translational geometries[END_REF], no collimator is required at the detector. This notable feature enables an increase of the amount of acquired data for a given position of the detector and thus a possible reduction in the acquisition time. In this setting, for an angular position ϕ of the detector a scattering angle ω corresponds to two circular arcs. So, modelling of data acquisition with this collimation-free detector leads to a Radon transform on double circular arcs (DCART). Besides the uncollimated detector, this modality has other material advantages such as not requiring relative movement between the system and the object. In addition, the usage of a circular detector path enables reduction of the system size: with a linear detector, information corresponding to large scanning circles is recorded far from the source while with the proposed geometry photons are always collected at a fixed distance of it.

Objectives and outline of the paper

Image reconstruction from projections obtained with this modality requires inversion of the Radon transform on double circular arcs. In this paper, we derive the analytic inversion of this integral transform in order to reconstruct cross-sectional images of an object. This paper is outlined as follows. Section II introduces the measurement model of the CST system as well as the corresponding forward DCART. Section III presents the whole procedure for inverting the DCART. Section IV deals with the discrete formulations for the forward and inverse DCART used for simulations. Numerical simulations in section V illustrate the theoretical feasibility of the new system and the performance of the proposed reconstruction algorithm. Section VI opens a discussion about future works ranging from practical and physical considerations to the possibility of an extension in three dimensions. Concluding remarks of section VII summarize and end the paper. Technical details in demonstrations are presented in appendices A and B.

2 Measurement model of the proposed CST system Acquisitions using this system are made on a family of double circular arcs A C1 and A C2 of centre Ω 1 and Ω 2 whose union is denoted D in the rest of the paper: D(ρ, ϕ) = A C1 (ρ, ϕ) ∪ A C2 (ρ, ϕ). This family of double circular arcs is then defined relatively to ρ the diameter of scanning circles and ϕ the angular position of the detector (see Fig. 1). In fact, using ρ or ω in order to define the family of double circular arcs is equivalent, since they are related by ρ = R/ sin ω, where ω ∈ [π/2, π[. Given ϕ the angular position of D and ω the considered scattering angle, polar equations of A C1 and A C2 are defined as

A Cm (ρ, ϕ) : r = ρ cos (θ -(ϕ -(-1) m ψ)) , m ∈ {1, 2} (2.1)
where

θ ∈ [ϕ, ϕ + 2ω -π] for A C1 , θ ∈ [ϕ -2ω + π, ϕ] for A C2 and ψ = cos -1 (R/ρ).
Data measurement using this CST system is modelized by the generalized Radon transform R D on the family of double circular arcs D whose definition is:

Definition. Let f be an unknown function, non negative, continuous and compactly supported outside the disc of radius R centered at the origin. The Radon transform on double circular arcs R D maps f into the set of its integrals over the family of double circular arcs D as

R D f (ρ, ϕ) = D(ρ,ϕ) f (r, θ) ds, (2.2)
where ds is the elementary arc length measure on the considered double circular arc (s denotes the curvilinear abscissa).

In a physical sense, function f is a cross-sectional map of the electronic density of the object. Thus, when the object is scanned as explained in section 1.2, the flux of photons of energy E(ω) registered by the detector at site (R cos ϕ, R sin ϕ) is proportional to the integral of function f on the pair of circular arcs labelled by D(ρ, ϕ), i.e. R D f . This dependence is weighted by independent factors that model some physical effects. Thus, measurements may be corrected prior to reconstruction by applying the appropriate factors [START_REF] Rigaud | Novel numerical inversions of two circular-arc Radon transforms in Compton scattering tomography[END_REF]. For instance, the Klein-Nishina cross-sectional probability depends on ρ (i.e. ω) and accounts for different probabilities of scattering for different angles.

An analytic inversion formula for the DCART

In this section, we derive the main equations of the procedure for inverting the DCART. First, we use a circular harmonic expansion to establish a relation between the components of a function f and R D f . Then, with a similar approach as Cormack's one, we obtain (3.5), which is the inversion formula in circular harmonic expansion. Finally, a closed formulation (3.14), which will be used in numerical simulations, is obtained.

Circular harmonic expansion

Functions f (r, θ) and R D f (ρ, ϕ) are expanded in terms of Fourier series, where f n (r) and (R D f ) n (ρ) are respectively their circular harmonic expansion components

f (r, θ) = ∞ n=-∞ f n (r) e inθ (3.1) (R D f )(ρ, ϕ) = ∞ n=-∞ (R D f ) n (ρ) e inϕ (3.2)
where

f n (r) = 1 2π 2π 0 f (r, θ) e -inθ dθ (3.3) (R D f ) n (ρ) = 1 2π 2π 0 R D f (ρ, ϕ) e -inϕ dϕ (3.4)

Inversion formula in circular harmonics expansion

Proposition 1. Function f is completely recovered via its circular harmonic expansion f n from circular expansion of data measurement (R D f ) n with

f n (r) = 1 π d dr r R cosh n cosh -1 r ρ ρ r ρ 2 -1 (R D f ) n (ρ) 2 cos n cos -1 R ρ dρ (3.5)
Proof. Data projection on double circular arcs R D f (ρ, ϕ) can be decomposed on two Radon transforms on the families of circular arcs A C1 (ρ, ϕ) and A C2 (ρ, ϕ). Consequently, we denote R A C 1 (ρ, ϕ) and R A C 2 (ρ, ϕ) the Radon transforms on respectively A C1 (ρ, ϕ) and A C2 (ρ, ϕ), whose sum gives R D f . On a similar way, R A C 1 (ρ, ϕ) and R A C 2 (ρ, ϕ) can be decomposed in Fourier series to obtain respectively (R

A C 1 f ) n and (R A C 2 f ) n . By linearity, we have (R D f ) n (ρ) = (R A C 1 f ) n (ρ) + (R A C 2 f ) n (ρ)
. First step of this demonstration is to make explicit the circular expansions of R A Cm f , m ∈ {1, 2}. An interesting property of A C1 and A C2 is their symmetry about (ϕ + ψ) and (ϕ -ψ) respectively (see Fig. 1). This feature allows us to see each A Cm as a combination of two equal and symmetric elements of arcs. One of these elements of arc can be rewritten as an angular shift of the other one, hence the following relation for

R A Cm f R A Cm f (ρ, ϕ) = ∞ n=-∞ A + Cm (ρ,ϕ) f n (r) e inθ + e in[2(ϕ-(-1) m ψ)-θ] ds m (3.6)
where A + Cm denotes the half part of circular arc A Cm , θ ≥ ϕ -(-1) m ψ and ds m is the elementary arc length measure on A Cm .

Observing that

e inθ + e in[2+(ϕ-(-1) m ψ)-θ] = 2 e inϕ e inψ cos [n (θ -(ϕ -(-1) m ψ))] , (3.7)
and plugging (3.7) into (3.6), one obtains the circular harmonic expansions of

R A C 1 f and R A C 2 f (R A Cm f ) n (ρ) 2 e -(-1) m inψ = A + Cm (ρ,ϕ) f n (r) cos [n (θ -(ϕ -(-1) m ψ))] ds m . (3.8) Straightforward computations show that θ -ϕ -(-1) m ψ = cos -1 (r/ρ) and ds m = 1 -(r/ρ) 2 -1/2
dr.

Equation (3.8) becomes (R A Cm f ) n (ρ) 2 e -(-1) m inψ = ρ R f n (r) cos n cos -1 (r/ρ) 1 -(r/ρ) 2 dr. (3.9)
Hence, from the addition of the expressions in (R A C 1 f ) n and (R A C 2 f ) n in (3.9), the connection between circular components of f and R D f can be written

(R D f ) n (ρ) 4 cos (nψ) = ρ R f n (r) cos n cos -1 (r/ρ) 1 -(r/ρ) 2 dr. (3.10)
Then, denoting

G n (ρ) = (R D f ) n (ρ) 2 cos (nψ) (3.11)
and multipliying both sides of (3.10) by

t R cosh n cosh -1 (t/ρ) ρ (t/ρ) 2 -1 dρ, with t ∈ R, t > R, one gets 1 2 t R cosh n cosh -1 (t/ρ) ρ (t/ρ) 2 -1 G n (ρ)dρ = t R f n (r) t r cosh n cosh -1 (t/ρ) ρ (t/ρ) 2 -1 cos n cos -1 (r/ρ) 1 -(r/ρ) 2 dρdr, (3.12)
where the right ρ-integral is π/2 [START_REF]The Radon transform on a family of curves in the plane[END_REF]. Then, differentiating with respect to the variable t, one gets

f n (t) = 1 π d dt t R cosh n cosh -1 (t/ρ) ρ (t/ρ) 2 -1 G n (ρ)dρ (3.13)
Going back to coefficients (R D f ) n and substituting t by r, one finds (3.5).

Remark 1. Equation (3.5) demonstrates explicitly the Cormack's hole theorem: in order to determine f (r, θ) by its circular harmonic expansion f n (r), the knowledge of the coefficients (R D f ) n (ρ) in the annular domain R < ρ < r is sufficient.

A closed formulation of (3.5)

Proposition 2. Denoting G(ρ, ϕ) the function corresponding to the Fourier series expansion

∞ n=-∞ G n (ρ) e inϕ
with G n as defined in (3.11), f can be completely recovered from G as follows

f (r, θ) = 1 2π 2 r 2π 0 p.v. ∞ R ∂G(ρ, ϕ) ∂ρ ρ r -ρ cos (θ -ϕ) dρ dϕ (3.14)
where p.v. denotes the Cauchy principal value.

Proof. This result is achieved introducing consistency conditions [START_REF]Radon's problem-old and new[END_REF][START_REF] Truong | Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality[END_REF] in terms of Cormack sense. The complete demonstration is proposed in the Appendices.

Remark 2. Equation (3.14) can be rewritten using the Hilbert transform and implemented in a more efficient way using standard tools of discrete Fourier analysis, as we are going to show in the next section. A parametrisation in Cartesian coordinates (instead of equations in polar coordinates) is preferable to perform numerical simulations in order to have the same distance between adjacent running points on the considered scanning circle. Hence, a scanning arc A Cm , m ∈ {1, 2} can be seen as the shift of a circle centred at the origin of identical radius, to its center Ω m (x Ωm , y Ωm ) with a restriction of the domain of the variable γ (see Fig. 1) :

A Cm (ρ, ϕ) : (x m (γ), y m (γ)) = (x Ωm (ρ, ϕ), y Ωm (ρ, ϕ)) + ρ 2 (cos γ, sin γ), γ ∈ [γ mmin , γ mmax ] (4.1) Cartesian parametrisation of A Cm is A Cm (ρ, ϕ) : (x m (γ), y m (γ)) = ρ 2 (cos (ϕ -(-1) m ψ) + cos γ, sin (ϕ -(-1) m ψ) + sin γ) (4.2)
where γ ∈ [ϕ -ψ, ϕ + 3ψ] for A C1 and γ ∈ [ϕ -3ψ, ϕ + ψ] for A C2 . Then, the numerical computation of the forward DCART requires a discrete version of it. This process is achieved with a linear interpolation to make coincide the position of the object with the Cartesian parametrisation of the double family of circular arcs and an approximation of the integral with a sum according to the equation

R D f (ρ i , ϕ j ) = ρ i 2 ∆ γ γ k ∈[γ1 min ,γ1 max ] cup[γ2 min ,γ2 max ] f (x(γ k ), y(γ k )) (4.3)
where ∆ γ is the sampling angular distance of γ, ρ i and ϕ j are the discrete versions of ρ and ϕ respectively:

ρ i = i(ρ max -R)/N ρ , i = 1, ..., N ρ and ϕ j = j • (2π/N ϕ ), j = 1, ..., N ϕ . Hence, R D f (ρ i , ϕ j ) is a N ρ × N ϕ matrix.
ρ max refers to the largest diameter of the scanning double circular arcs. Fig. 2 shows an example of data measurement for the Shepp-Logan phantom.

Reconstruction strategy

For image reconstruction, we use the Hilbert transform, related to the Cauchy principal value as

H{u}(t) = 1 π p.v. ∞ -∞ u(τ ) t -τ dτ . (4.4)
The Hilbert transform can be computed in the Fourier domain, where we have :

H {u} (t) = F -1 (-i • sign(ν) • F(u)(ν))(t), (4.5)
where F denotes the one-dimensional Fourier transform. Equation (3.14) of image reconstruction becomes consequently

f (r, θ) = 1 2πr • 2π 0 1 cos(θ -ϕ) H ∂G(ρ, ϕ) ∂ρ • ρ r cos (θ -ϕ) dϕ. (4.6)
Finally, using the correspondence between polar coordinates (r, θ) and Cartesian coordinates (x, y) and r cos (θ -ϕ) = x cos ϕ + y sin ϕ, image reconstruction equation used for simulation is

f (x, y) = 1 2π 2π 0 1 x cos ϕ + y sin ϕ • F -1 -i • sign(ν)F ∂G(ρ, ϕ) ∂ρ ρ (ν) x 2 + y 2 x cos ϕ + y sin ϕ dϕ. (4.7)
The projections G(ρ, ϕ) are computed via the circular harmonic components of R D f (ρ, ϕ) with (3.11). However, zeros in the denominator may be source of instability and regularization may be required. According to [START_REF] Moon | Analytic inversion of a conical Radon transform arising in application of Compton cameras on the cylinder[END_REF], we add a regularization parameter in (3.11) (equal to 1 in the proposed simulation) in order to compute the circular harmonic components of G: 5 Experiments and study of the performance of the reconstruction algorithm

G n (ρ) = cos (nψ) 2 + cos (nψ) 2 (R D f ) n (ρ) 2 . ( 4

General parameter choices

We will perform simulations on three different phantoms, the Shepp-Logan, the Derenzo (also called Jaszczak) and bars phantoms. These phantoms will allow us to evaluate different criteria such as spatial resolution, contrast and the ability of the proposed algorithm to reconstruct singularities tangent to lines with arbitrary slopes. In all proposed simulations, the size of the object is 512 × 512 pixels. The detector is moving on a ring of radius R = 256 pixels with a constant step of arc length between two adjacent positions. This represents an amount of N ϕ = 2πR = 1609 different positions for the detector to collect data. We know the parameter choices should satisfy the condition N ρ × N ϕ ≥ N × N according to [START_REF] Bracewell | Numerical transforms[END_REF], where N ρ is the number of double circular arcs per detector position. To quantitatively asses the quality of reconstructions, we use NMSE = ||f -f 0 || 2 2 /N 2 and NMAE = ||f -f 0 || 1 /N 2 metrics, where f 0 and f are the respective original and reconstructed objects and ||.|| 1 and ||.|| 2 refer respectively to the 1 and 2-norm. These results are summed up in Table 1.

Study of the influence of some general parameters on reconstructions

Choice of ρ max

First step consists in choosing the maximum diameter for scanning circular arcs ρ max . This is equivalent to choosing the maximal scattering angle ω max since ω max = π -arcsin (R/ρ max ). We propose in Fig. 3 reconstructions of Derenzo and the bar phantoms for ρ max = 3000, 5000 and 7000 with a discretization step ∆ρ = 1 length unit. These choices correspond respectively to a maximal scattering angle ω max = 175, 177 and 178 degrees. A higher ρ max allows to reconstruct the upper right and lower left slopes of straight lines tangent to the ellipses of Shepp-Logan and to the circles of Derenzo and also, small structures of Shepp-Logan. However, one can observe a visual loss of contrast when ρ max increases and this may explain higher NMSE and NMAE values (see Table 1).

Choice of N ρ

For the rest of the simulations, ρ max is fixed and equal to 5000, which seems to be, a good trade off between contrast and good reconstruction of small structures. The objective is now to evaluate the ratio of required data N ρ × N ϕ relative to the number of pixels to reconstruct N 2 . We denote Q this ratio : for ρ are chosen uniformely between R and ρ max . Then, the other N ρ are the product of the chosen ratio Q and N ρ,min . Figure 4 shows the obtained reconstructions for Q = 1, 5 and 10. Although the reconstructions appears to be blurred for Q = 1, the algorithm already gives a first estimation of the objects to be reconstructed. The optimal choice of Q seems to be specific to each object and not directly linked to the proposed algorithm (see Table 1).

Q = (N ρ × N ϕ )/N 2 . Q = 1 represents the minimal case where N ρ = N ρ,min = N 2 /N ϕ = 163. This means that, for Q = 1, 163 values

Robustness against noise

Q is now fixed and equal to 10. We carry out simulations adding to projections a Gaussian noise of signal-tonoise ratios SNR= 10, 15 and 20 dB. Figure 5 shows the obtained results for the three phantoms that exhibit grain artifacts for the higher levels of noise.

Additional considerations and perspectives

Interesting issues arise when studying the new Compton scattering modality. We briefly detail some of them.

Practical considerations

Dominance of the Compton effect

Compton effect is dominant in matters with low atomic numbers in a wide range of middle-low energies. Particularly, these scenarios include biological tissues and medical applications. 3a), Derenzo (Fig. 3b) and Shepp-Logan phantoms (Fig. 3c) for Q = 1 (a,d,g), Q = 5 (b,e,h), Q = 10 (c,f,i)

Assumption of first order scattering

All models of CST based on Radon transforms [START_REF] Nguyen | Inversion of a new circular-arc Radon transform for Compton scattering tomography[END_REF][START_REF] Webber | Three dimensional Compton scattering tomography[END_REF][START_REF] Webber | Microlocal analysis of a Compton tomography problem[END_REF][START_REF] Cebeiro | New improved compton scatter tomography modality for investigative imaging of one-sided large objects[END_REF][START_REF] Truong | Compton scatter tomography in annular domains[END_REF][START_REF] Tarpau | A new concept of Compton Scattering tomography and the development of the corresponding circular Radon transform[END_REF][START_REF] Tarpau | Compton scattering imaging system with two scanning configurations[END_REF][START_REF] Cebeiro | An interior Compton Scatter Tomography[END_REF][START_REF] Cebeiro | New improved Compton scatter tomography modality for investigative imaging of one-sided large objects[END_REF] assume that photons undergo scattering only once before reaching detector, i.e. they consider only first order scattering. The consequences that multiple scattering (second order or higher) may have on image quality are still unclear and how to deal with it is an open question. Further work based on Monte Carlo simulations may help to understand this behaviour. In addition, the lower probability and the random nature of this event suggest that noise reduction strategies are a good starting point.

Inclusion of attenuation in the model

An analytic inversion formula for the non-attenuated Radon transform has been shown to play an important role in the reconstruction strategy for the attenuated case. In a practical setting, attenuation and scattering should be considered in the forward model. This leads to a major challenge from the mathematical point of view for which no exact solution is known at present. A number of ways to deal with the problem of attenuation in Table 1: NMSE / NMAE of reconstruction results phantoms ρmax (Fig. 3) Q (Fig. 4) SNR (Fig. 5 3a), Derenzo (Fig. 3b) and Shepp-Logan phantoms (Fig. 3c) with the addition of a Gaussian noise (SNR=10dB (a,d,g), SNR=15dB (b,e,h), SNR=20dB (c,f,i)

emission tomography have been studied like the generalized Chang Correction (GCC) [START_REF] Chang | A method for attenuation correction in radionuclide computed tomography[END_REF], an iterative algorithm that allows the a posteriori correction of the reconstruction or the Iterative Pre Correction algorithm [START_REF] Maze | Iterative reconstruction methods for non uniform attenuation distribution in SPECT[END_REF] which corrects data. This later method has been adapted for the previous CST Nguyen and Truong's modality in [START_REF] Guerrero | Attenuation correction in a new modality of Compton Scattering Tomography[END_REF]. This study requires having the analytic inverse Radon transform found in the non-attenuated case, i.e. the equivalent problem that we have efficiently solved here for our new CST modality. More recently, the GCC has been adapted by the authors [START_REF] Tarpau | A new bi-imaging NDT system for simultaneous recovery of attenuation and electronic density maps[END_REF] for a special bi-modality able to provide the attenuation map and electron density. Results shown in this paper encourage further work on that direction.

Realistic source and detector

In a practical scenario, source and detector are not punctual but have a finite dimension. Although no study explore the consequences of it, blur can be expected at reconstruction. Furthermore, non ideal energy resolution at detector may also produce some blur due to the fact that uncertainties in the energy of photons translate to uncertainties in the localization of scattering sites (electrons). Nevertheless, permanent progress in detector design has been achieved in recent years and is expected for future years. Regarding the source, angular dependence can be included in a correction factor inside the model as in [START_REF] Rigaud | Novel numerical inversions of two circular-arc Radon transforms in Compton scattering tomography[END_REF].

An extension of this modality in three dimensions

An extension of this modality to three dimensions can be also envisaged: a fixed source and a detector moving on a sphere centered at the source. This setup is even more appealing since no collimator is required neither at the source nor at the detector. This modality leads to a new toric Radon transform, resulting of the rotation of the circular arcs around the source-detector segment. The theoretical challenge raised by this extension consists in the inversion of this new toric Radon transform. Some work on this direction is on the way.

A Consistency conditions

We introduce consistency conditions [START_REF]Radon's problem-old and new[END_REF][START_REF] Truong | Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality[END_REF] in terms of Cormack sense in order to deduce a closed formulation of (3.5), more suitable for numerical computation. Equation (3.5) can be rewritten using the n-th order Tchebychev polynomial of the first kind T n as

f n (r) = 1 π d dr r R T |n| (r/ρ) ρ (r/ρ) 2 -1 (R D f ) n (ρ) 2 cos (n cos -1 (R/ρ)) dρ, (A.1)
Using the following relationship between Tchebychev polynomials of the first kind T n and the second kind

U n [4] T |n| (r/ρ) (r/ρ) 2 -1 = (r/ρ) -(r/ρ) 2 -1 |n| (r/ρ) 2 -1 + U |n|-1 (r/ρ) , (A.2)
and according to (3.11), (A.1) becomes

f n (r) = 1 π d dr      r R (r/ρ) -(r/ρ) 2 -1 |n| (r/ρ) 2 -1 G n (ρ) dρ ρ + r R U |n|-1 (r/ρ) G n (ρ) dρ ρ      . (A.3)
Furthermore, from (3.10), and changing the order of integration, one can obtain for n ∈ N and k

= n, n -2, n -4, ... > 0 ∞ R G n (ρ) dρ ρ k = 2 ∞ R f n (r) ∞ r cos n cos -1 (r/ρ) 1 -(r/ρ) 2 dρ ρ k dr. (A.4)
With the change of variables ρ = r/ cos ν, the previous ρ-integral becomes

∞ r cos n cos -1 (r/ρ) 1 -(r/ρ) 2 dρ ρ k = r -k+1 π 2 0 cos (nν) cos (ν) 2-k dν. (A.5)
We arrive to an ν-integral whose result is given in [START_REF] Magnus | Formulas and theorems for the special functions of mathematical physics[END_REF] π 2 0 cos (nν)

cos (ν) 2-k dν = π (k -1)2 k-1 Γ(k) Γ k+n 2 Γ k-n 2 , (A.6)
where Γ refers to the gamma function. For n > 0 and denoting h

(n) = Γ k+n 2 Γ k-n 2
, one can remark that k = n, n -2, n -4 > 0 are poles of h. Since h is even, then k = -n, -n -2, -n -4 < 0 for n < 0 are also poles of h. Consequently, one can conclude that this integral vanishes [START_REF] Truong | Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality[END_REF] Furthermore, since Tchebychev polynomials of second kind U |n|-1 (r/ρ) are a linear sum of polynomials 1/ρ k (see [START_REF] Magnus | Formulas and theorems for the special functions of mathematical physics[END_REF]), (A.7) leads to 2) is also an inversion formula based on circular harmonic expansion. However, the proposed formulation (3.14) is more practical for numerical reconstruction than (B.2). In fact, (B.2) can be numerically implemented using a method similar to Chapman and Cary numerical approach [START_REF] Chapman | The circular harmonic Radon transform[END_REF], see for instance [START_REF] Rigaud | Novel numerical inversions of two circular-arc Radon transforms in Compton scattering tomography[END_REF]. Nevertheless, the technique requires the evaluation of Tchebychev functions in (B.2) in terms of primitive integrals that are evaluated recursively. This implies either longer computational time or more memory.

Figure 1 :

 1 Figure 1: Setup and parametrisation of the new CST modality. Black dotted circle: detector path, Red continuous curves: scanning double arcs, Red dotted curves: portion of circles not used for acquisition

Figure 2 :

 2 Figure 2: Left: Object. Right: corresponding DCART for R = 256, ρmax = 7000, Nρ = 6744 and Nϕ = 1609.

Figure 3 :

 3 Figure 3: Reconstruction results of the bar (Fig. 3a), Derenzo (Fig. 3b) and Shepp-Logan phantoms (Fig. 3c) for Q = 1 (a,d,g), Q = 5 (b,e,h), Q = 10 (c,f,i)

Figure 4 :

 4 Figure 4: Reconstruction results of the bar (Fig.3a), Derenzo (Fig.3b) and Shepp-Logan phantoms (Fig.3c) for Q = 1 (a,d,g),

Figure 5 :

 5 Figure 5: Reconstruction results of the bar (Fig.3a), Derenzo (Fig.3b) and Shepp-Logan phantoms (Fig.3c) with the addition of

  , i.e., for n ∈ N and k = n, n -2, n -4, ... > 0, ∞ R (R D f ) n (ρ) 2 cos (nψ) dρ ρ k = 0.(A.7)

U 2 ∞ 1 nRemark 3 .

 213 f ) n (ρ) 2 cos (nψ) dρ = 0, n ∈ N . (A.8) B Derivation of formula (3.14) This section uses consistency conditions established in (A.8) to obtain the final closed-form inversion formula (3.14). From (A.8), we deduce ∀r ∈ [R, +∞[, (B.1) and the relation between Tchebychev polynomials of the first and the second kind (A.2), we obtain f n (rf and G the projections of circular harmonic components of G n and f n in (B.2), we obtain f (r, θ) |n|-1 (r/ρ) e in(θ-ϕ) dϕdρ . (B.3) We need the two following lemmas to simplify (B.3). Lemma B.1 ([10], Lemma 3.8). For 1 < ρ < r, we have 1 + n=1 r/ρ -(r/ρ) 2 -cos (n(θ -ϕ)) = -(r/ρ) 2 -1 cos (θ -ϕ) -(r/ρ) (B.4) Lemma B.2 ([10], Lemma 3.9). For 0 < r < ρ, we have n∈Z U |n|-1 (r/ρ) e in(θ-ϕ) = 1 cos (θ -ϕ) -(r/ρ) 14) is obtained first inverting the order of the r-derivative and the ρ-integral and then with the substitution u = ρ/r and the fact that G(ρ, ϕ) = 0 for ρ ∈ [0, R[. The intermediate result (B.

  Compute circular harmonic expansion of RDf (ρ, ϕ) to compute Gn(ρ) with (4.8) and recompose G(ρ, ϕ); 2 Compute discrete derivation of G(ρ, ϕ) relative to variable ρ and multiply the result by ρ; 3 Write the Hilbert transform as a filtering operation in Fourier domain using (4.5); 4 For each ϕ, interpolate the data on the considered scanning circles (x 2 + y 2 )/(x cos ϕ + y sin ϕ) of (4.7); 5 Weight the result using the factor 1/(x cos ϕ + y sin ϕ);

	.8)
	Algorithm 1 summarizes the different steps for reconstructing the object from (4.7).
	Algorithm 1: Reconstruction of object
	Data: RDf (ρ, ϕ), projections on double circular arcs of function f
	Result: f (x, y)

1 6

Sum the weighted interpolations on all directions ϕ; 7 Weight the result by 1 2π ;

  new modality of Compton Scattering Tomography has been studied in this paper. The design has attractive features such as compactness and simplicity. Furthermore, this CST modality uses uncollimated detectors. This may allow increasing data and simultaneously reducing acquisition time. This modality leads to a new Radon transform on double circular arcs, for which a closed analytic inversion formula is established in this paper. In addition, an efficient reconstruction algorithm based on the Hilbert transform has been developed. Simulation results illustrate the feasibility of this new system and the good performance of the reconstruction algorithm.

	7 Concluding remarks
	8 Funding and acknowledgements
	C. Tarpau's research work is supported by grants from Région Ile-de-France (in Mathematics and Innovation)
	2018-2021 and LabEx MME-DII (Modèles Mathématiques et Economiques de la Dynamique, de l'Incertitude
	et des Interactions) (No. ANR-11-LBX-0023-01). J. Cebeiro is supported a CONICET postdoctoral grant (#
	171800).

A


V Filipe Martins-Da-Rocha

Rafael Mouallem
email: srafaelmouallem@hotmail.com

Second-Order Beliefs and Second-Order Expected Utility

 by allowing for a general representation result without restricting the decision maker's attitude towards subjective uncertainty.

Introduction

In [START_REF] Anscombe | A definition of subjective probability[END_REF], a decision maker ranks objective (or roulette) lotteries p ∈ L(Z) with prizes in some set Z. 1 His preference ordering ca on L(Z) satisfies the axioms of expected utility theory and can then be numerically represented by the functional p -→ E p (u) := z∈Z p(z)u(z) 1 for some utility function u : Z → R. In order to accommodate subjective uncertainty, [START_REF] Anscombe | A definition of subjective probability[END_REF] introduce a finite set S of states of nature. An act (or horse lottery) is then defined as a function h : S → L(Z) from states of nature to objective lotteries on prizes in Z. The set of acts is denoted by H. [START_REF] Anscombe | A definition of subjective probability[END_REF] assumed that the decision maker can not only rank objective lotteries (elements of L(Z)), but can also rank lotteries on acts (elements of L(H)). His preference ordering on L(H) is assumed to extend the preference ordering ca on L(Z) by identifying constant acts with objective lotteries. The preference is also assumed to satisfy the axioms of expected utility theory and can then be numerically represented by a functional

P -→ f ∈H P (f )U (f )
for some utility function U : H → R. In additional to the assumptions of expected utility theory, the two preference orderings ca and are connected by two additional axioms: the monotonicity in the prizes and the reversal of order in compound lotteries. The first axiom requires that if two acts f and g are identical except for the lotteries f (s) and g(s) associated with one state s, then the ranking (according to ) between f and g is governed by the ranking (according to ca ) between the lotteries f (s) and g(s). The second axiom says that spinning a roulette wheel P ∈ L(H) on acts before the realization of the state of nature is equivalent to spinning the same roulette wheel after subjective uncertainty is resolved. Formally, the lottery P ∈ L(H) on acts is equivalent to the degenerate lottery on the act g with the prize g(s) defined as the compound lottery of P and f (s):

g(s) := f ∈H P (f )f (s).
Under monotonicity and the reversal of order in compound lotteries, [START_REF] Anscombe | A definition of subjective probability[END_REF] proved that there exists a unique probability measure µ over the set S of states of nature such that

U (f ) = s∈S µ(s)E f (s) (u).
This the so-called standard subjective expected utility model of [START_REF] Anscombe | A definition of subjective probability[END_REF] that has been challenged by the famous [START_REF] Ellsberg | Risk, ambiguity and the Savage axioms[END_REF] Paradox. To accommodate uncertainty aversion, numerous extensions of subjective expected utility have been proposed, including [START_REF] Schmeidler | Subjective probability and expected utility without addivity[END_REF]'s Choquet expected utility, [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF]'s maxmin expected utility, [START_REF] Segal | The Ellsberg paradox and risk aversion: An anticipated utility approach[END_REF]'s application of anticipated utility, [START_REF] Tversky | Advances in Prospect theory: Cumulative representation of uncertainty[END_REF]'s cumulative prospect theory (see also [START_REF] Wakker | An axiomatization of cumulative Prospect theory[END_REF] for an axiomatization) and a number of more recent models like [START_REF] Klibanoff | A smooth model of decision making under ambiguity[END_REF], [START_REF] Seo | Ambiguity and second-order belief[END_REF] and [START_REF] Nascimento | Second-order ambiguous beliefs[END_REF].

We follow this literature and propose a new axiomatization of the Second-Order Expected Utility (SOEU) preferences that were first axiomatized by [START_REF] Neilson | Ambiguity aversion: An axiomatic approach using second-order probabilities[END_REF][START_REF] Neilson | A simplified axiomatic approach to ambiguity aversion[END_REF]. Imposing the Savage axioms to the horse lotteries and the von Neumann-Morgenstern axioms to the roulette lotteries, [START_REF] Neilson | A simplified axiomatic approach to ambiguity aversion[END_REF] obtained the following SOEU representation

U (f ) = s∈S µ(s)v E f (s) (u)
yielding a subjective probability measure over states and two utility functions, the function u governing risk attitudes and the function v governing ambiguity attitudes. The SOEU model is able to accommodate the Ellsberg paradox and preferences for reductions in ambiguity. It has been used by [START_REF] Nau | Uncertainty aversion with second-order utilities and probabilities[END_REF], [START_REF] Ergin | A theory of subjective compound lotteries[END_REF], [START_REF] Chew | Small worlds: Modeling attitudes toward sources of uncertainty[END_REF], [START_REF] Grant | Second-order expected utility[END_REF], [START_REF] Strzalecki | Axiomatic foundations of multiplier preferences[END_REF] and [START_REF] Al-Najjar | Parametric representation of preferences[END_REF]. It is also very popular in applied work: see for instance [START_REF] Snow | Ambiguity and the value of information[END_REF][START_REF] Snow | Ambiguity aversion and the propensities for self-insurance and selfprotection[END_REF], [START_REF] Gollier | Portfolio choices and asset prices: The comparative statics of ambiguity aversion[END_REF], [START_REF] Alary | The effect of ambiguity aversion on insurance and self-protection[END_REF], [START_REF] Hoy | Take-up for genetic tests and ambiguity[END_REF], [START_REF] Huang | A mean-preserving increase in ambiguity and portfolio choices[END_REF].

To get an axiomatization of the SOEU model based on the axioms of [START_REF] Anscombe | A definition of subjective probability[END_REF], we drop reversal of order in compound lotteries and propose a natural extension of monotonicity to lotteries over acts. In our representation result, the probability µ is unique and the functions u and v • E u are cardinally unique. 2 We then show that the function v is concave if, and only if, we add the standard Uncertainty Aversion axiom of [START_REF] Schmeidler | Subjective probability and expected utility without addivity[END_REF]. This complements the analysis in [START_REF] Seo | Ambiguity and second-order belief[END_REF] who obtains an axiomatization of Second-Order Subjective Expected Utility (SOSEU) by considering a weaker axiom than our extended monotonicity.3 It also extends the contributions of [START_REF] Grant | Second-order expected utility[END_REF] by allowing for a general representation result without restricting the decision maker's attitude towards subjective uncertainty.

The Setup

Notations

Fix a non-empty space X. By a lottery (or a simple probability distribution) on X, we mean a map p ∈ R X + such that its support {x ∈ X : p(x) > 0} is finite and x∈X p(x) = 1. For any A ⊆ X, we slightly abuse notation and write p(A) := a∈A p(a) that can be interpreted as the probability of getting an element of A. The set of all lotteries on X is denoted by L(X). For any x ∈ X, we let δ x ∈ L(X) be the point mass at x, defined by δ x (y) = 0 if y = x and δ x (x) = 1. Lotteries of the form δ x are called degenerate lotteries. Observe that every lottery p ∈ L(X) can be written as a convex combination of degenerate lotteries

p = x∈X p(x)δ x
where the above sum is finite since the support of p is finite. For any lottery p ∈ L(X) and any real map u ∈ R X , the expected value of u with respect to p is defined as the real number

E p (u) := x∈X p(x)u(x).
If u is interpreted as a utility function on X, then E p (u) corresponds to the expected utility of the lottery p.

Anscombe-Aumann Setting

Let S be a finite set of states of nature to represent situations involving subjective uncertainty. Let Z denote a set of outcomes or prizes. A purely subjective act is a function f ∈ Z S . In order to provide a representation result of preferences on purely subjective acts, we follow [START_REF] Anscombe | A definition of subjective probability[END_REF] by considering an enrichment of items to which preference must apply. We denote by H the set of functions from S into L(Z) that are called acts (or horse race lotteries). 4 The decision maker is assumed to be able to rank lotteries on acts. Therefore, the decision maker's preference is a binary relation on L(H). Identifying an act f ∈ H with the degenerate lottery δ f ∈ L(H), we can view H as a subset of L(H). The ranking induced by on degenerate lotteries δ f for any f ∈ H is denoted by a .5 Similarly, identifying the lottery p ∈ L(Z) with the degenerate lottery δ p1 S on the constant act p1 S : s → p, we can view L(Z) as a subset of L(H) and denote by ca the induced ranking. 6We should be careful when considering compound lotteries in L(L(Z)). In this paper, a lottery P ∈ L(H) such that the support of P only contains constant acts is called a compound lottery and is identified as an element of L(L(Z)). Using the terminology of [START_REF] Anscombe | A definition of subjective probability[END_REF], the compound lottery P corresponds to a first spin of a roulette wheel before the horse race. The prizes of this first roulette are constant acts f : s → p f with known chance P (f ). Then, after the horse race, independently of the realized state s, a second roulette wheel is spun which prizes are elements of Z and chances are given by p f ∈ L(Z).

Typical elements of L(H) are denoted by P , Q, and R. We use f , g, and h for elements in H. Typical elements of L(L(Z)) are denoted by P , Q, and R. Finally, p, q, and r are typical elements of L(Z). Using this notation, an element P ∈ L(H) can be written

P = f ∈H P (f )δ f
where each f : S → L(Z) can be decomposed as follows

∀s ∈ S, f (s) = z∈Z f (s)(z)δ z .
A typical lottery on acts P ∈ L(H) is depicted in Figure 1. objective prob. subjective prob.

objective prob.

P α 1 -α f g s 1 x a y 1 -a s 2 z 1 s 1 x b y 1 -b s 2 x b y 1 -b act f ∈ H one stage lottery
Figure 1: P ∈ L(H) is a lottery on acts. The first and the last nodes are governed by the objectives probabilities (α, 1 -α), (a, 1 -a) and (b, 1 -b). The act g is constant and can be viewed as a one-stage lottery in L(Z).

We refer to an element in L(Z) as a one-stage lottery and refer to an element in L(L(Z)) as a two-stage (or compound) lottery. A constant act (taking the same value for every s ∈ S) is viewed as a one-stage lottery. If f, g ∈ H and E ⊂ S with E ∈ {∅, S}, then f E g denotes the act with f E g(s) = f (s) if s ∈ E and f E g(s) = g(s) if s ∈ E. When E = {s} is a singleton, we use the simpler notation f s g instead of f {s} g.

The spaces L(Z), H and L(H) are convex subsets of linear subspaces and are therefore endowed with their natural mixture. Formally, if p and q are one-stage lotteries in L(Z), then αp + (1 -α)q is also a one-stage lottery defined by [αp + (1 -α)q](z) = αp(z) + (1 -α)q(z) for each prize z ∈ Z. Similarly, if f and g are two acts in H and α ∈ [0, 1], then αf + (1 -α)g is also an act defined by the componentwise mixture

[αf + (1 -α)g](s) := αf (s) + (1 -α)g(s), for all s ∈ S where αf (s) + (1 -α)g(s)
is the mixture of the two one-stage lotteries f (s) and g(s). Finally, if P and Q in L(H) are two lotteries on acts, then αP + (1 -α)Q is also a lottery on acts defined by αP

+ (1 -α)Q := f ∈H [αP (f ) + (1 -α)Q(f )]δ f .
If p and q are one-stage lotteries, then αp + (1 -α)q is also a one-stage lottery.7 It should not be confused with the two stage lottery αδ p1 S + (1 -α)δ q1 S (also denoted by

αδ p + (1 -α)δ q )
that is a compound lottery where the first lottery occurs before the outcome of the subjective uncertainty and the second occurs after. See Figure 2 for an illustration.

Basic Axioms

The two following axioms are standard.

Order. The preference is complete and transitive.

Mixture Continuity.

If P, Q, R ∈ L(H), then the sets {α ∈ [0, 1] : αP + (1 -α)Q R} and {α ∈ [0, 1] : R αP + (1 -α)Q} are closed.
For any one-stage lottery p ∈ L(Z), we can consider the degenerate lottery δ p1 S ∈ L(H) which assigns with probability one the constant act p1 S : s → p. Denote by ca the induced preference on constant acts in L(Z) defined by

p ca q ⇐⇒ δ p1 S δ q1 S .
Since one-stage lotteries involve objective probabilities in the sense that everyone agrees on the likelihood of getting each alternative in Z, we impose the standard independence axiom.

Second-Stage Independence. For any α ∈ (0, 1] and one-stage lotteries p, q, r ∈ L(Z),

αp + (1 -α)r ca αq + (1 -α)r ⇐⇒ p ca q. p z 1 1/3 z 2 1/3 z 3 1/3 (a) One-stage lottery p ∈ L(Z) q z 1 1/2 z 4 1/2 (b) One-stage lottery q ∈ L(Z) αp + (1 -α)q z 1 α 3 + 1 -α 2 z 2 α/3 z 3 α/3 z 4 1 -α 2 (c)
One-stage lottery as a mixture of two one-stage lotteries

αδ p1 S + (1 -α)δ q1 S α 1 -α p1 S q1 S s 1 z 1 1/3 z 2 1/3 z 3 1/3 s 2 z 1 1/3 z 2 1/3 z 3 1/3 s 1 z 1 1/2 z 4 1/2 s 2 z 1 1/2 z 4 1/2 (d) Two-stage lottery Figure 2: αp + (1 -α)q is a mixture of one-stage lotteries but αδ p + (1 -α)δ q is a mixture of two-stage lotteries.
The interpretation of this assumption becomes transparent when the mixture one-stage lottery αp + (1 -α)r is viewed as a compound lottery compounded from two roulette wheels which spins occur after the realization of the subjective uncertainty (represented by the states in S). In a Lottery A, the first spin determines whether, with probability α, you get the lottery p, or, with probability 1 -α, you get r. Then, conditional on the result of the first spin, there is a second spin which chances to get a prize in Z are defined by p or r. In a Lottery B, the first spin determines whether, with probability α, you get the lottery q, or, with probability 1 -α, you get r. Then, conditional on the result of the first spin, there is a second spin which chances to get a prize in Z are defined by q or r. The Second-Stage Independence axiom requires that the decision maker's prefers Lottery A to Lottery B if, and only if, he prefers p to q. This axiom is related to dynamic consistency as explained in [START_REF] Gilboa | Theory of Decision under Uncertainty[END_REF].

In a second-stage lottery p ∈ L(Z), the chances (p(z)) z∈Z are objectively defined. Similarly, the chances (P (f )) f ∈H involved in a lottery on acts P ∈ L(H) are also objective. Therefore, replacing the set Z by H in the Second-Stage Independence axiom, we get the following natural axiom.

First-Stage Independence. For any α ∈ (0, 1] and lotteries P, Q, R ∈ L(H),

αP + (1 -α)R αQ + (1 -α)R ⇐⇒ P Q.
As before, the interpretation of the First-Stage Independence axiom becomes transparent when the mixture lottery αP + (1 -α)Q is viewed as a compound lottery compounded from two roulette wheels which spins occur before the realization of the subjective uncertainty (represented by the states in S). The prizes associated to the first spin are lotteries over acts, while the prize associated to the second spin are acts.

Remark 2.1. It is important to observe that first-stage independence is not related to the following property (Independence axiom on acts): for any α ∈ (0, 1] and acts f, g, h ∈ H,

αf + (1 -α)h αg + (1 -α)h ⇐⇒ f g.
This is because the mixture of lotteries αδ f +(1-α)δ g is different than the degenerate lottery δ αf +(1-α)g on the mixture act αf + (1 -α)g.

Extended Monotonicity and SOEU

For any act f ∈ H, we can consider the degenerate lottery δ f ∈ L(H) which assigns with probability one the act f . We can define a the induced preference on H as follows

f a g ⇐⇒ δ f δ g .
We start by recalling the standard Monotonicity axiom applied to acts.

Monotonicity. For any acts f, g ∈ H,

[f (s) ca g(s), ∀s ∈ S] =⇒ f a g.
Observe that the Monotonicity axiom deals only with acts in H. In order to extend it to lotteries over acts, we introduce the following notation.

Fix an act f ∈ H and a state s ∈ S. Recall that the property f (s) ca g(s) in the statement of the Monotonicity axiom is equivalent to f (s)1 S a g(s)1 S . We then consider the following notation: Ψ(f, s) := f (s)1 S is the constant act s → f (s) taking the value f (s) for every s ∈ S. Observe that monotonicity can be stated as follows: for any acts f, g ∈ H,

[Ψ(f, s) a Ψ(g, s), ∀s ∈ S] =⇒ δ f δ g , or, equivalently, δ Ψ(f,s) δ Ψ(g,s) , ∀s ∈ S =⇒ δ f δ g . (3.1)
Recall that a lottery P ∈ L(H) on acts can be canonically decomposed as

P = f ∈H P (f )δ f . (3.2)
We then let Ψ(P, s) ∈ L(H) be the lottery on (constant) acts defined by

Ψ(P, s) := f ∈H P (f )δ Ψ(f,s)
where each act f : s → f (s ) in the canonical decomposition (3.2) is replaced by the constant act Ψ(f, s) : s → f (s). See Figure 3 for an illustration. P P (f ) To interpret extended monotonicity, consider an agent who is not certain of the true state. If he believes that the state is s, then the two-stage lottery induced by P is Ψ(P, s). Extended dominance means that if the decision maker prefers the two-stage lottery induced by P to the one induced by Q independently of the true state s, then he must prefer P to Q. Choosing degenerate lotteries P = δ f and Q = δ g on single acts f, g ∈ H, we easily prove that extended monotonicity is stronger than monotonicity. It will follow from our characterization result that extended monotonicity is a much stronger axiom than monotonicity. [START_REF] Seo | Ambiguity and second-order belief[END_REF] used the terminology "extended AA dominance" for our axiom of extended monotonicity.

P (g) f f (s 1 ) s 1 f (s 2 ) s 2 g g(s 1 ) s 1 g(s 2 ) s 2 (a) P ∈ L(H) is an objective lottery on acts Ψ(P, s) P (f ) P (g) Ψ(f, s) f (s) s 1 f (s) s 2 Ψ(g, s) g(s) s 1 g(s)
Remark 3.1. As pointed out by [START_REF] Seo | Ambiguity and second-order belief[END_REF], extended monotonicity implies Krep's reversalof-order-style axiom (see (Kreps 1988, p. 107)). It states that if two lotteries P and Q over "purely subjective" (or Savage) acts map naturally to the same lottery over outcomes for each state s, then they must be indifferent.

We now state the formal definition of the Second Order Expected Utility (SOEU) representation.

Definition 3.1. A SOEU representation of the preference is a probability measure µ ∈ L(S), a function u : Z → R, and a strictly increasing function v : co(u(Z)) → R such that I represents on L(H), where

I(P ) = f ∈H P (f ) s∈S µ(s)v(E f (s) (u)).
(3.

3)

The main contribution of this paper is the following new representation result that obtains an axiomatization of SOEU without any restriction on the decision maker's attitude towards subjective uncertainty.8 Theorem 3.1. Preference on L(H) satisfies order, mixture continuity, first-stage independence, second-stage independence, and extended monotonicity if, and only if, it has an SOEU representation (µ, u, v).

Proof. We start by proving necessity of the axioms. Assume that is represented by the function I in (3.3). The proof that order, mixture continuity and first-stage independence are necessary is a straightforward exercise and is omitted. Observe that the preference ca on constant acts is represented by the function p → E p (u) and therefore second-stage independence is satisfied. To prove extended monotonicity, fix P ∈ L(H) and recall that Ψ(P, s) is the following lottery on constant acts

Ψ(P, s) := f ∈H P (f )1 f (s)1 S .
This implies that

I(Ψ(P, s)) = f ∈H P (f )v(E f (s) (u))
and we get that I(f ) = s∈S µ(s)I(Ψ(P, s)).

Extended monotonicity then follows immediately.

We now provide the proof of sufficiency. When P ∼ Q for all P, Q ∈ L(H), then the representation is trivial. It is sufficient to choose any arbitrary constant function u.9 Thus, assume that satisfies non-degeneracy, i.e., P Q for some P, Q ∈ L(H). Since the preference satisfies order, mixture continuity and first-stage independence, we can apply Theorem 5.15 in [START_REF] Kreps | Notes on the Theory of Choice[END_REF] to deduce that there exists a function U : H → R such that is represented by the function I : L(H) → R defined by

I(P ) = f ∈H P (f )U (f ).
Moreover, the function U is unique up to a positive affine transformation. That is, if I (P ) =

f ∈H P (f )U (f ) also represents for some function U : H → R, then there exist two real numbers a > 0 and b ∈ R such that U = aU + b. We normalize U such that

f ∈H U (f )R 0 (f ) = 0
for some lottery R 0 on constant acts. Recall that is R 0 can be identified with a compound lottery in L(L(Z)).

Let U ca be the restriction of U to constant acts in H. Since constant acts can be identified with lotteries on Z, we can assume that U ca is a function defined on L(Z). The function U ca is a representation of the preference ca on constant acts in L(Z) that satisfies order, mixture continuity and second-stage independence. Applying again Theorem 5.15 in [START_REF] Kreps | Notes on the Theory of Choice[END_REF], we deduce the existence of a function u : Z → R such that ca is represented by the function I ca : L(Z) → R defined by

I ca (p) = z∈Z p(z)u(z).
Moreover, the function u is unique up to a positive affine transformation. The functions U ca and I ca represent the same preference ca . Observe moreover that I ca (L(X)) = co(u(X)) is an interval of the real line.

Lemma 3.1. There exists a strictly increasing function v : co(u(X)) → R such that U ca = v • I ca .

Proof. We follow almost verbatim the arguments of Lemma B.9 in [START_REF] Seo | Ambiguity and second-order belief[END_REF]. Fix an arbitrary t ∈ co(u(X)). There exists a lottery p ∈ L(X) such that t = I ca (p). Observe that for any other lottery q ∈ L(X) satisfying t = I ca (q), we have q ∼ ca p. We then have U ca (p) = U ca (q) and we can define v(t) := U ca (p). To prove that v is strictly increasing, let t, t ∈ co(u(X)) such that t > t. There exist p, p ∈ L(X) such that t = I ca (p) and t = I ca (p ). In particular, we have p ca p. This implies that U ca (p ) > U ca (p) and we get that v(t ) > v(t).

We propose to show that extended monotonicity translates into the following no-arbitragetype property. Identify any act f ∈ H as an asset with state-dependent payoff given by the random variable (U ca (Ψ(f, s))) s∈S .

A portfolio is a simple function θ : H → R with finite support, that is the set {f ∈ H : θ(f ) = 0} is finite. We let Θ be the linear space of portfolios. The payoff of portfolio θ in state s is then Π(θ, s) :=

f ∈H θ(f )U ca (Ψ(f, s)).
The price of each asset f is U (f ) and the cost of portfolio θ is then

c(θ) := f ∈H θ(f )U (f ).
Extended monotonicity implies the following (weak) no-arbitrage property.

Lemma 3.2. Any portfolio with non-negative payoffs cannot have a negative price. Formally, for every portfolio

θ ∈ Θ [Π(θ, s) ≥ 0, ∀s ∈ S] =⇒ c(θ) ≥ 0.
Proof of Lemma 3.2. We can adapt the arguments in [START_REF] Seo | Ambiguity and second-order belief[END_REF]. We provide the details for the sake of completeness. Fix an arbitrary portfolio θ ∈ Θ such that Π(θ, s) ≥ 0, ∀s ∈ S.

(3.4)

We can always write θ = αP -βQ where P, Q ∈ L(H) and α, β ≥ 0. We provide the details of the proof when α ≥ β. The other case can be treated similarly. If α = 0 then β = 0 and θ = 0. The desired result is then obvious. Therefore, assume that α > 0. Note that (3.4) implies

f ∈H P (f )U ca (Ψ(f, s)) ≥ β α f ∈H Q(f )U ca (Ψ(f, s)). (3.5)
Given our normalization of U , we have for any state s

0 = f ∈H R 0 (f )U (f ) = f ∈H R 0 (f )U ca (Ψ(f, s))
since the support of the lottery R 0 only contains constant acts. Letting γ := β/α, we have

f ∈H P (f )U ca (Ψ(f, s)) ≥ γ f ∈H Q(f )U ca (Ψ(f, s)) + (1 -γ) f ∈H R 0 (f )U ca (Ψ(f, s)) ≥ f ∈H γQ(f ) + (1 -γ)R 0 (f ) U ca (Ψ(f, s)).
We have thus proved that I(Ψ(P, s)) ≥ I(Ψ(γQ + (1 -γ)R 0 , s)), for each s ∈ S, or, equivalently, Ψ(P, s)

Ψ(γQ + (1 -γ)R 0 , s), ∀s ∈ S.
Extended monotonicity then implies that

P γQ + (1 -γ)R 0 .
Using the representation function I, we deduce that

I(P ) ≥ I(γQ + (1 -γ)R 0 ) = γI(Q) + (1 -γ)I(R 0 ) = γI(Q)
since I(R 0 ) = 0 by the normalization choice of U . We then deduce that c(θ) = αc(P ) -βc(Q) = αI(P ) -βI(Q) ≥ 0 which is the desired result.

We can now apply a standard convex separation theorem to prove the following weak version of the Fundamental Theorem of Finance.

Lemma 3.3. There exists a probability measure µ ∈ L(S) such that

c(θ) = s∈S µ(s)Π(θ, s), ∀θ ∈ Θ.
Proof of Lemma 3.3. Assume, without any loss of generality, that 0 ∈ S and pose Σ := {0} ∪ S. Let A ⊆ R Σ be the set of financial transfers τ = (τ (σ)) σ∈Σ ∈ R Σ that can be implemented by a portfolio, i.e., there exists θ ∈ Θ such that τ (0) = -c(θ) and τ (s) = Π(θ, s)

for each s ∈ S. Since c is a weak no-arbitrage price (see Lemma 3.2), we have

A ∩ (0, ∞) × R S + =: B = ∅.
A is a linear subspace (and therefore convex) and B is convex. However, applying directly the standard Convex Separation Theorem does not allow to get the desired result. We follow an approximation argument. Fix ε ∈ (0, 1) and let C ε be the set of all vectors c = (c(σ)

) σ∈Σ such that c ∈ [ε, ∞) × R S + and σ∈Σ c(σ) = 1.
The set C ε is compact and convex. The set A is closed convex. Since A ∩ C ε = ∅, we can apply the Strict Convex Separation Theorem, to deduce the existence of two real numbers α, β ∈ R and some non-zero vector

ξ ε = (ξ ε (σ)) σ∈Σ ∈ R Σ such that ∀a ∈ A, ξ ε • a ≤ α < β ≤ ξ ε • c, ∀c ∈ C ε .
Since 0 ∈ A, we deduce that

ξ ε (0) > 0 and ξ ε (s) > - ε 1 -ε ξ ε (0).
Since A is a linear space, the inequality ξ ε • a ≤ α for each a ∈ A implies that ξ ε • a = 0 for each a ∈ A. We then deduce that

c(θ) = s∈S µ ε (s)Π(θ, s), ∀θ ∈ Θ (3.6)
where µ ε (s) := ξ ε (s)/ξ ε (0). Observe that µ ε (s) ≥ -ε/(1 -ε) for each s ∈ S. Fix now the portfolio θ that is the degenerate lottery 1 p1 S on the constant act s → p for some lottery p ∈ L(Z). Observe that

Π(θ , s) = U ca (p) = U (p1 S ) = c(θ ).
We then deduce from (3.6) that

s∈S µ ε (s) = 1.
Passing to a subnet if necessary, we can assume that we can pass to the limit in (3.6) and get the existence of some probability measure µ ∈ L(S) such that

c(θ) = s∈S µ(s)Π(θ, s), ∀θ ∈ Θ
Fix an arbitrary act f ∈ H. Choosing θ := δ f to be the degenerate lottery on f , we get that

U (f ) = s∈S µ(s)U ca (Ψ(f, s)).
Applying Lemma 3.1, we have

U (f ) = s∈S µ(s)U ca (Ψ(f, s)) = s∈S µ(s)v • I ca (Ψ(f, s)) = s∈S µ(s)v(E f (s) (u))
and we get the desired result.

Extended monotonicity plays a crucial role in the representation result of Theorem 3.1. It cannot be replaced by the weaker axiom of monotonicity. Indeed, for a given function u ∈ R Z , consider an arbitrary function

Θ : W → R where W := [co(u(X))] S
that is increasing on W and strictly increasing on the certainty line {t1 S : t ∈ co(u(X))}. If we pose J(P ) :=

f ∈H P (f )Θ((E f (s) (u)) s∈S ),
then the function J : L(H) → R defines a preference that satisfies order, mixture continuity, first-stage independence, second-stage independence, and monotonicity. However, we do not necessarily get a SOEU representation. Indeed, a possible choice for the function Θ is Θ gs (w) := min{w(s) : s ∈ S} where the subscript GS obviously stands for [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF].

Uniqueness

To analyze uniqueness of the representation in Theorem 3.1, we consider the following standard axiom.

Non-degeneracy. There exist P, Q ∈ L(H) such that P Q.

Before presenting the uniqueness properties, we introduce the following notation. Given a function u : Z → R, we can define the expected utility function E u from lotteries in L(Z) to R defined by

E u (p) := E p (u) = z∈Z p(z)u(z).
The function E u : L(Z) → R is affine. Moreover, any affine function from L(Z) to R is of the form E u for some utility function u ∈ R Z . Proposition 4.1. Consider a preference that admits an SOEU representation (µ, u, v). If also satisfies non-degeneracy, then (i) the probability measure µ is unique;

(ii) the function u is unique up to a positive affine transformation, 10 (iii) the function v • E u is unique up to a positive affine transformation.

Proof. Let I : L(H) → R be the representation of associated to (µ, u, v), i.e.,

I(P ) = f ∈H P (f )U (f ) where U (f ) := s∈S µ(s)v(E f (s) (u)).
The function U : H → R is the restriction of I to degenerate lotteries. Denote by I : L(H) → R be the representation of associated to (µ , u , v ), i.e.,

I (P ) = f ∈H P (f )U (f ) where U (f ) := s∈S µ (s)v (E f (s) (u )).
10 Equivalently, the function E u is unique up to a positive affine transformation.

Since satisfies order, mixture continuity and first-stage independence, it follows from Theorem 5.15 in [START_REF] Kreps | Notes on the Theory of Choice[END_REF] that there exists a > 0 and b ∈ R such that U = aU + b. That is, for every act f ∈ H,

s∈S µ (s)v (E u (f (s))) = b + a s∈S µ(s)v(E u (f (s))).
(4.1)

Fix an arbitrary lottery p ∈ L(Z). Letting f be the constant act p1 S , we deduce that

v (E u (p)) = av(E u (p)) + b. (4.2)
We have thus proved that v

• E u = a (v • E u ) + b.
The preference ca is represented by the functions p → E p (u) and p → E p (u ). Since ca satisfies order, mixture continuity and second-stage independence, it follows from Theorem 5.15 in [START_REF] Kreps | Notes on the Theory of Choice[END_REF] that there exists α > 0 and β ∈ R such that u = αu + β.

Combining (4.1) and (4.2), we deduce that for every act f ∈ H,

s∈S µ (s)v (E u (f (s))) = s∈S µ(s) [av(E u (f (s))) + b] = s∈S µ(s)v (E u (f (s))).
Letting U := co(u (Z)), we have that

s∈S µ (s)v (w(s)) = s∈S µ(s)v (w(s))
for any (w(s)) s∈S ∈ [U ] S . Since satisfies non-degeneracy, the interval U cannot be reduced to a singleton. Since v is strictly increasing, we deduce that the set

{(v (w(s))) s∈S : (w(s)) s∈S ∈ [U ] S }
admits a non-empty interior in R S . Since µ -µ is orthogonal to the above set, we deduce that µ = µ .

To characterize the support of the probability in the SOEU representation, we recall the standard definition of a null set. Definition 4.1. A set s ∈ S is null when for all acts f, g ∈ H and all lotteries p, q ∈ L(Z), f g ⇐⇒ p s f q s g.

In other words, the state s is null when it does not matter for the ranking of acts.

In a SOEU representation (µ, u, v) of a preference , the support of the probability µ corresponds to non-null states.

Proposition 4.2. Consider a preference that admits an SOEU representation (µ, u, v). If also satisfies non-degeneracy, then µ(s) = 0 if, and only if, s is null.

Proof. The "only if" part is obvious. To prove the "if" part, we assume that s is null. Fix an act f ∈ H and two lotteries p, q ∈ L(Z). Since s is null, we have p s f q s f if, and only if, p s f q s f . Using the SOEU representation (µ, u, v), we deduce that

µ(s)v(E p (u)) ≥ µ(s)v(E q (u)) ⇐⇒ µ(s)v(E q (u)) ≥ µ(s)v(E p (u)).
Since satisfies non-degeneracy, there exists p, q ∈ L(Z) such that p ca q, or, equivalently, E p (u) > E q (u). We then deduce that µ(s) = 0.

Continuity

In order to obtain continuity properties of the utility function, we consider a topology on Z. In this section, we assume that Z is a separable metric space. We endow the space L(Z) with the weak topology defined as the coarsest topology such that every linear mapping p → E p (u) is continuous for every bounded and continuous function u. This implies that a sequence (p n ) of lotteries in L(Z) weakly converges to another lottery p ∈ L(Z) when lim n→∞ z∈Z

p n (z)u(z) = z∈Z p(z)u(z)
for every bounded and continuous function u : Z → R. The weak topology on L(Z) is metrizable and separable. The space H of acts is identified with the product space L(Z) S and is therefore endowed with the product of weak topologies. This topology on H is also metrizable and separable. We then endow the choice space L(H) with the weak topology defined as the coarsest topology such that every linear mapping P → f ∈H P (f )U (f ) is continuous for every bounded and continuous function U : H → R.

We introduce the following strengthening of the mixture continuity axiom.

Continuity. The preference is continuous with respect to the weak topology.11

We can now characterize the continuity of the functions u and v in a SOEU representation.

Theorem 5.1. Preference on L(H) satisfies order, continuity, first-stage independence, second-stage independence, and extended monotonicity if, and only if, it has an SOEU representation (µ, u, v) where u is continuous and bounded and v is continuous.

Proof. We start by proving the "if" part. We shall prove that the function f

→ U (f ) defined on H by U (f ) = s∈S µ(s)v(E f (s) (u)) = s∈S µ(s)(v • E u )(f (s))
is bounded and continuous. Since u is bounded and continuous, the linear functional E u : L(Z) → R is weakly continuous. Since v is continuous, the mapping v • E u is weakly continuous and we get that U is continuous with respect to the product topology on H. Boundedness of U follows from boundedness of u and continuity of v.

The "only if" part corresponds to the proof of Theorem 3.1 replacing the use Theorem 5.15 in [START_REF] Kreps | Notes on the Theory of Choice[END_REF] by Theorem 5.21 in [START_REF] Kreps | Notes on the Theory of Choice[END_REF]. We then get that U : H → R is bounded and continuous with respect to the weak topology. Moreover, the weak topology on L(Z) coincides with the relative topology on L(Z) induced by L(H). This implies that the preference ca on constant acts is continuous with respect to the weak topology. We then get that u is bounded and continuous. Finally, to prove that v is continuous, recall that v is defined on the set co(u(X)) as follows

v(t) = U ca (p) for any p ∈ L(Z) such that t = E p (u),
where U ca (p) := U (p1 S ). Since L(Z) is connected and U ca is continuous, the set v(co(u(X))) = U ca (L(Z)) is also connected. Since v is strictly increasing, we necessarily have that v is continuous.

Uncertainty Aversion

Our general representation results (Theorem 3.1 and Theorem 5.1) do not require any specific assumption regarding the decision maker's attitude towards subjective uncertainty. We consider the following standard axiom introduced by [START_REF] Schmeidler | Subjective probability and expected utility without addivity[END_REF] to represent the decision maker's aversion towards subjective uncertainty.

Uncertainty Aversion. Let f, g ∈ H and α ∈ (0, 1).

f ∼ a g =⇒ αf + (1 -α)g a f.
This axiom captures the plausible assumption that "smooting out" acts should provide a hedge against subjective uncertainty and consequently should be desirable. That is, we assume that a decision maker who finds two acts equally attractive, will be embarrassed to state that their mixture is worse than both.

Under the basic axioms and extended monotonicity, the axiom of uncertainty aversion translates into the concavity of the function v. Theorem 6.1. Preference on L(H) satisfies order, continuity, first-stage independence, second-stage independence, extended monotonicity and uncertainty aversion if, and only if, it has an SOEU representation (µ, u, v) where u is bounded continuous and v is concave continuous.

Proof. We omit the straightforward arguments of the "if" part and focus on the proof of the "only if" part. Let be a preference on L(H) satisfying order, continuity, first-stage independence, second-stage independence, extended monotonicity and uncertainty aversion. It follows from Theorem 5.1 that has an SOEU representation (µ, u, v) where u is bounded continuous and v is continuous. We only have to prove that v is also concave. If u is constant, then the domain of v is a singleton and v is obviously concave. We now assume that u is not constant. This implies that co(u(Z)) is an interval of R with a non-empty interior denoted by int(co(u(X))).

Let W := [co(u(Z))] S and define Ψ : W → R defined as follows: for every w = (w(s)) s∈S ∈ W , Ψ(w) := s∈S µ(s)v(w(s)).

Replacing S by the support of µ is necessary, we can assume without any loss of generality that µ(s) > 0 for each s ∈ S.

Lemma 6.1. The function Ψ is quasi-concave.

Proof of Lemma 6.1. Fix w, w ∈ W and assume, without any loss of generality, that Ψ(w) ≤ Ψ( w). We analyze two cases. If w(s) = inf co(u(Z)) for each s, then Ψ(w) = inf Ψ(W ), and for any α ∈ [0, 1], we have Ψ(αw + (1 -α) w) ≥ Ψ(w).

Assume now that there exists u ∈ co(u(Z)) such that w := u1 S < w.12 Replacing u by min{min{u, ŵ(s)} : s ∈ S} if necessary, we can assume that w ≤ w. Since v is strictly increasing, we deduce that Ψ(w) < Ψ(w). Since Ψ is continuous, we can apply the Intermediate Value Theorem to get the existence of some γ ∈ [0, 1] close enough to 1 such that Ψ( ŵ) = Ψ(w) where ŵ := γw + (1 -γ) w.

Since w(s) ∈ co(u(Z)), there exists f (s) ∈ L(Z) such that w(s) = E f (s) (u). Similarly, there exists an act g ∈ H such that ŵ(s) = E g(s) (u). Observe that f ∼ a g. Uncertainty aversion implies that αf + (1 -α)g a f . In particular, we get that Ψ(αw + (1 -α) ŵ) ≥ Ψ(w).

Since w ≤ w, we deduce that αw + (1 -α) ŵ ≤ αw + (1 -α) w. Monotonicity of Ψ implies the desired result: Ψ(αw + (1 -α) w) ≥ Ψ(w).

The restriction of the function Ψ to the open set [int(co(u(X)))] S satisfies the assumptions of Theorem 2 in [START_REF] Debreu | Additively decomposed quasi-convex functions[END_REF] and deduce that with at most one exception, every function µ(s)v is concave on int(co(u(X))). This implies that v is concave on the interior of the interval co(u(X)). Since we already proved that v is concave on the whole interval co(u(X)), we can follow a standard limiting argument to deduce that v is concave on the whole interval co(u(X)). [START_REF] Grant | Second-order expected utility[END_REF] also provide an axiomatization of SOEU preferences with a concave function v. In Theorem 6.1, we obtain concavity of v under the mere assumption of uncertainty aversion, while [START_REF] Grant | Second-order expected utility[END_REF] impose an additional axiom, called weak translation invariance at certainty, which requires that the supporting hyperplanes are equal along the certainty line whenever they are unique.

  s 2 (b) Ψ(P, s) is an objective lottery on constant acts

Figure 3 :

 3 Figure 3: If the decision maker believes that s occurs with probability one, then P and Ψ(P, s) are equivalent.

E u is mapping p → E p (u) from L(Z) to R.

Our representation result cannot be directly obtained from the representation result in[START_REF] Seo | Ambiguity and second-order belief[END_REF]. This is because[START_REF] Seo | Ambiguity and second-order belief[END_REF] does not provide any information about the support of the second order belief.

By identifying a prize z with the corresponding degenerate lottery δ z , the set Z S of purely subjective acts can be embedded in H.

The upper-script "a" stands for act.

The upper-script "ca" stands for constant act.

We can interpret αp + (1 -α)q as the successive spin of two different roulette wheels, but both spins occurring after the realization of the subjective uncertainty.

This corresponds to one of the open problems suggested by[START_REF] Grant | Second-order expected utility[END_REF].

Observe that in this case, the probability µ is indeterminate.

That is, for every P ∈ L(H), the set {Q ∈ L(H) : Q P } and {Q ∈ L(H) : P Q} are closed for the weak topology.

That is, u ≤ w(s) for each s with a strict inequality for at least one s.