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Mathematical Finance, Vol. 15, No. 4 (October 2005), 613–634
A REPRESENTATION RESULT FOR CONCAVE
SCHUR CONCAVE FUNCTIONS
ROSE-ANNE DANA
Université Paris Dauphine, CEREMADE, UMR CNRS 7534, France
A representation result is provided for concave Schur concave functions on L∞ ().
In particular, it is proven that any monotone concave Schur concave weakly upper semicontinuous function is the infinimum of a family of nonnegative affine combinations
of Choquet integrals with respect to a convex continuous distortion of the underlying
probability. The method of proof is based on the concave Fenchel transform and on
Hardy and Littlewood’s inequality. Under the assumption that the probability space is
nonatomic, concave, weakly upper semicontinuous, law-invariant functions are shown
to coincide with weakly upper semicontinuous concave Schur concave functions. A
representation result is, thus, obtained for weakly upper semicontinuous concave lawinvariant functions.
KEY WORDS: concave order, second-order stochastic dominance, concave Schur concave functions,
law-invariant concave functions, law-invariant risk measures, concave rearrangement-invariant functions
1. INTRODUCTION
The concave and the convex orders, the increasing concave and the increasing convex
orders were first studied by Hardy, Littlewood and Polya (1988). They have been generalized and used in many areas of mathematics (graph theory, matrix theory, numerical
analysis, geometry, statistics (rank-order tests), rearrangement theory, calculus of variations, and probability theory (Blackwell 1953; Strassen 1965). Marshall and Olkin (1979)
provide a survey of fields of applications. These orders have so often been used that they
are also known under various other names: for example, the convex order is known as
the “majorization order” when the state space is finite and the probability uniform, the
increasing convex order is also called the stop-loss order while the increasing concave
order is called second order stochastic dominance (S.S.D. from now on) and uniform
order by Föllmer and Schied (2002). Introduced in Economics by Rothschild and Stiglitz
(1970) as measures of risk, the concave and increasing concave orders have been used
in various fields of economics of uncertainty such as efficiency pricing (Peleg and Yaari
1975; Chew and Zilcha 1990), finance (Dybvig and Ross 1982; Dybvig 1988; Jouini and
Kallal 2000), equilibrium theory (Landsberger and Meilijson 1994, Dana 2004, Dana
and Meilijson 2003), and also in others such as measurement of inequalities (Atkinson
1970).
I am indebted to G. Carlier for many helpful conversations and for our joint work on which this paper
heavily builds.
Manuscript received May 2004, final revision received January 2005.
Q1
Address correspondence to the author at Université Paris Dauphine, CEREMADE, UMR CNRS 7534,
France; e-mail: [email protected].
C
2005 Blackwell Publishing Inc., 350 Main St., Malden, MA 02148, USA, and 9600 Garsington Road, Oxford
OX4 2DQ, UK.
613
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R. A. DANA
Convex order preserving functions have first been introduced and characterized by
Schur (1923) in the finite number of states, uniform probability case under the name of
“majorization order” preserving functions. Convex order preserving functions were later
called “Schur convex” by Ostrowski (1952). Many examples of “Schur convex” functions
may be found in Marshall and Olkin (1979). We emphasize that these functions are not
necessarily convex. A somewhat different concept of “Schur convexity” was considered
in the rearrangement literature of the sixties by Luxemburg (1967) and Chong and Rice
(1971) who defined a Schur convex function to be a convex “rearrangement invariant”
(in other words law-invariant) function. Under assumptions on the underlying measure
space, they provided representations theorems for Schur convex functions and showed
that they were “convex order-preserving”. More recently, the concepts of “law-invariant
convex risk measure” and “law-invariant coherent measure” have been introduced in
the mathematical finance literature (see Delbaen 2001 and Föllmer and Schied 2002 for
an overview). They are law-invariant convex functionals with further restrictions. Lawinvariant and order preserving convex functions, thus have a long history in mathematics.
“Schur concave” (or concave order preserving) and “Schur convex” functions also have a
long history in economics. They have been used in many economic fields, such as insurance
(Gollier and Schlesinger 1996; Carlier and Dana 2003), finance (Kim 1998; Dana 2004,
Dana and Meilijson 2003), measurement of inequalities (many examples may be found
in Marshall and Olkin 1979).
In this paper, we focus on “Schur concave” functions. As already mentioned, these
functions are, in general, neither concave nor quasi-concave (see below and Chew and Mao
1995 and Marshall and Olkins 1979 for counterexamples). However, as mentioned above,
“convex Schur convex” functions have played an important role in mathematics, in the
rearrangement literature, and in economics (in the theory of measurement of inequalities
for example). Although existence and qualitative results can often be obtained without
assuming concavity of the criterion (an example is provided in the paper but others may
be found in Carlier and Dana 2005 and Dana and Meilijson 2003), concavity is almost
necessary for describing the solution when it exists. Convex analysis tools may then be
used (see Carlier and Dana 2003a, 2003b and Schied 2004).
There are three purposes of the paper. It first provides a representation result for
weakly upper semicontinuous concave “Schur concave” functions “without a nonatomic
hypothesis”. In particular, it shows that weakly upper semicontinuous, monotone concave “Schur concave” functions, may be represented as infinimum of nonnegative affine
combinations of Choquet integrals with respect to a convex, continuous distortion of the
underlying probability measure. It next shows that, in the nonatomic case, weakly upper
semicontinuous, concave “Schur concave” functions coincide with weakly upper semicontinuous, concave law-invariant functions. It lastly links the decision theory literature
that deals with order preserving maps and their use in economics, the rearrangement
literature of the sixties, and the more recent mathematical finance literature that deals
with law-invariant convex or coherent risk measures. More precisely, it links a representation result obtained in the rearrangement literature (see Chong and Rice 1971) and
a version of representation results more recently obtained for law-invariant convex or
coherent risk measures in the mathematical finance literature (see Kusuoka 2001; Fritelli
and Rosazza Gianin 2005; Rosazza Gianin 2002; Kunze 2003; Leitner 2004). The main
tools of the paper are the concave Fenchel transform and Hardy and Littlewood’s inequality. The use of the concave Fenchel transform seems to be adequate here since it defines a bijection between σ (L∞ (), L1 ()) upper semicontinuous concave Schur concave
functions and σ (L1 (), L∞ ()) upper semicontinuous concave Schur concave functions
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(and therefore between σ (L∞ (), L1 ()) upper semicontinuous concave law-invariant
functions and σ (L1 (), L∞ ()) upper semicontinuous concave law-invariant functions
in the nonatomic case).
The paper is organized as follows: In Section 2, we give an overview of the definitions of the various orders quoted in the Introduction and of their connections. We next
prove an application of Hardy-Littlewood’s theorem, which will turn out to be the basic tool of the paper and provides an interpretation of the result in terms of a Choquet
integral with respect to a convex, continuous distortion of the underlying probability
measure. We then define Schur concave functions. We end the section with examples of
Schur concave functions and an example of the use of Schur concave functions which
are nonconcave. Section 3 contains the main results of the paper. Using the concave
Fenchel transform, we provide a representation result for weakly upper semicontinuous concave Schur concave functions, “without a nonatomic hypothesis”. In particular,
we show that weakly upper semicontinuous, monotone concave “Schur concave” functions, may be represented as infinimum of nonnegative affine combinations of Choquet
integrals with respect to a convex, continuous distortion of the underlying probability
measure, “without a nonatomic hypothesis”. Section 4 is devoted to the nonatomic case.
Using a geometric characterization of preferred sets for the concave order, we show that
weakly upper semicontinuous, concave Schur concave functions coincide with weakly
upper semicontinuous, concave, law-invariant functions. As a corollary, we give, under
the nonatomic hypothesis, a version of representation results more recently obtained for
law-invariant convex or coherent risk measures in the mathematical finance literature.
We end the paper by a short and incomplete bibliography of representation theorems.
2. A FEW BASIC DEFINITIONS AND PROPERTIES
Two random variables X on (, B, P) and Y on ( , B , P ) with same distribution will
d
be denoted X ∼ Y. Given as primitive is a probability space (, B, P).
2.1. Second Order Stochastic Dominance and the Concave Order
In this subsection, we briefly review some well-known definitions on stochastic orders.
−1
Let X ∈ L1 (). Let FX denote its distribution function and FX a version of its generalized
inverse
FX−1 (t) = inf{z ∈ IR | FX (z) ≥ t}
and
FX−1 (0) = essinf X,
d
where essinf X = sup{t ∈ IR : P({X ≥ t}) = 1}. We recall that FX−1 ∼ X.
DEFINITION 2.1. The random variable X (strictly) dominates Y in the sense of S.S.D.
or in the increasing concave order denoted X 2 Y (resp X 2 Y ) if any of the following
equivalent
t conditions are
t fulfilled:
1. −∞ FY (s) ds ≥ −∞ FX (s) ds, ∀ t ∈ IR (resp with a strict inequality for some t),
t
t
2. 0 FX−1 (s) ds ≥ 0 FY−1 (s) ds, ∀ t ∈ ]0, 1], (resp with a strict inequality for some
t),
3. E[u(X)] ≥E[u(Y )], ∀ u: IR →IR concave increasing (resp with a strict inequality
for some u concave increasing),
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R. A. DANA
d
4. Y ∼ X + ε for some ε such that E[ε | X] ≤ 0 a.s. (resp where in addition P(ε =
0) > 0). X and Y are S.S.D. equivalent, denoted by X ∼2 Y , if 1, 2, or 3 holds
d
with equality throughout. In other words, X ∼2 Y iff X ∼ Y. It follows from
condition 4 and Jensen’s inequality that X 2 Y if and only if E[u(X)] > E[u(Y )]
for every increasing and strictly concave u: IR → IR.
DEFINITION 2.2. The random variable X (strictly) dominates Y in the concave order
denoted X c Y (resp X c Y ) if any of the following equivalent conditions are fulfilled
1. E[u(X)] ≥ E[u(Y )] for all u: IR →IR concave (resp with a strict inequality for
some u concave).
d
2. Y ∼ X + ε for some ε such that E[ε | X] = 0 a.s. (resp where in addition P(ε =
0) > 0)
X and Y are concave-order equivalent, denoted by X ∼c Y , if E[u(X)] = E[u(Y )] for all
d
u: IR →IR concave. In other words, X ∼c Y iff X ∼ Y. It follows from condition 2 and
Jensen’s inequality that X c Y if and only if E[u(X)] > E[u(Y )] for every strictly concave
u: IR →IR.
The concave order and S.S.D. are related as follows (see Müller and Stoyan (2002),
theorem 1.5.3.) for a proof:
LEMMA 2.1.
X c Y (resp X c Y ) iff X 2 Y (resp X 2 Y ) and E(X) = E(Y ).
For further use, let us quote an important result. Let r ∈ {1, ∞} and r be its
conjugate.
LEMMA 2.2. 1. X ∈ Lr () dominates Y ∈ Lr () in the concave order iff
1
1
g(s)FX−1 (s) ds ≥
g(s)FY−1 (s) ds for any g ∈ Lr [0, 1] nonincreasing.
0
0
2. X ∈ L () dominates Y ∈ Lr () in the sense of S.S.D., iff
1
1
g(s)FX−1 (s) ds ≥
g(s)FY−1 (s) ds for any g ∈ Lr+ [0, 1] nonincreasing.
r
0
0
The proof of Lemma 2.2 may be found in the Appendix. We shall see in Section 2.3 that
the second assertion of Lemma 2.2 amounts to say that X ∈ Lr () dominates Y ∈ Lr ()
in the sense of S.S.D., iff for any continuous convex distortion f (see Section 2.3 for a
definition), the Choquet integral with respect to f (P) of X is greater than the Choquet
integral of Y .
For sake of completeness, let us finally define the convex and increasing convex orders
and show how they are related to the concave order and S.S.D.
DEFINITION 2.3. The random variable X dominates Y in the convex order if
E[ f (X)] ≥ E[ f (Y )] for all f : IR →IR convex. Equivalently X dominates Y in the convex
order, if −Y c −X. The random variable X dominates Y in the increasing convex order
if E[ f (X)] ≥ E[ f (Y )] for all f : IR →IR convex increasing. Equivalently X dominates Y
in the increasing convex order, if −Y 2 −X.
2.2. Hardy and Littlewood’s Inequality and a Useful Result
We shall make extensive use of the following inequality (see Hardy, Littlewood, and
Pòlya 1988):
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2.2.1. Hardy and Littlewood’s Inequality.
1
FX−1 (1 − t)FY−1 (t) dt ≤ E(XY) ≤
0
1
0
617
FX−1 (t)FY−1 (t) dt.
The integrals may possibly take infinite values.
Let X ∈ L1 () and X ∈ L∞ () be given. Consider the following problem


 min E(X C)
.
(E) Cc X


∞
C ∈ L ()
When X ∈ L+ (), then X can be interpreted as a pricing density and the problem is to
find a contingent claim C that minimizes expenditure according to the pricing density X among those that dominate X for the concave order. A S.S.D. version of the problem


 min E(X C)
,
(D) C2 X


C ∈ L∞ ()
1
was first introduced by Dybvig (1988) in the finite states, uniform probability case, to
provide a measure of the risk of a contingent claim X that retains some features of the
mean-variance measure (that is increasing in expectation and decreasing in dispersion).
The problem (E) was generalized to incomplete markets, in the finite states, uniform
probability case, by Jouini and Kallal (2000). The problem (E) was finally solved in the
infinite dimensional case (see for example Dana and Meilijson 2003, Föllmer and Schied
2002, and Kunze 2003). It may easily be seen that (D) does not have a solution when
∞
/ L1+ (). Indeed if X ∈
/ L1+ (), there exists Z ∈ L+ () such that E(X Z) < 0. Hence
X ∈
if C 2 X, C + λZ 2 X for λ ≥ 0 and infλ≥0 E(X (C + λZ)) = −∞.
Let e(X, X ) denote the value function of (E). When X is a pricing density, e(X, X )
is called the utility price of X by Jouini and Kallal (2000). The following result will be the
key tool of the note:
THEOREM 2.1.
and
For every X ∈ L∞ (), X ∈ L1 (), the two problems


 min E(X C)
(E) Cc X


C ∈ L∞ ()


 min E(XC )
(Ẽ) C c X 
 C ∈ L1 ()
have a solution and their value function is
1
FX−1 (1 − t)FX−1 (t) dt.
e(X, X ) =
0
The proof of Theorem 2.1. is given in the Appendix. It is an adaptation of the proof in
Dana and Meilijson (2003). If follows from Lemma 2.2 that if X c Y , then e(X, X ) ≥
e(Y , X ) for all X ∈ L1 () and if X c Y , then e(X, X ) ≥ e(X, Y ) for all X ∈ L∞ ().
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Let h ∈ L1 [0, 1] be nondecreasing
and C ∈ L1 (). Define C c h by the
s
s −1
1
conditions 0 FC (t) dt ≥ 0 h(t) dt for all s ∈ [0, 1[ and E(C) = 0 h(t) dt. Let X ∈
∞
L (). By the same proof as that of Theorem 2.1, one can show that the problem


 min E(XC )
C c h

 C ∈ L1 ()
1
has a solution and that its value function is v h (X) = 0 FX−1 (1 − t)h(t) dt. As vh is the
minimum of a family of linear continuous functions, it is concave and σ (L∞ (), L1 ())
1
upper semicontinuous and vh is monotone if h ∈ L+ [0, 1]. From Lemma 2.2., if X c Y ,
then vh (X) ≥ vh (Y ). For further use, let us gather these remarks:
COROLLARY
Let h ∈ L1 [0, 1] be nondecreasing and vh : L∞ () → IR be defined
1 2.1.
−1
by v h (X) = 0 FX (1 − t)h(t) dt. Then vh is concave and σ (L∞ (), L1 ()) upper semicon1
tinuous and if X c Y , then vh (X) ≥ vh (Y ). Moreover, if h ∈ L+ [0, 1], then X ≥ Y implies
vh (X) ≥ vh (Y ).
2.3. Choquet Integral with Respect to a Convex Distortion
We recall that a capacity on a measurable space (, B) is a set function ν : B → [0, 1]
such that ν(∅) = 0, ν() = 1 and for all A, B ∈ B, A ⊂ B implies ν(A) ≤ ν(B). A capacity
ν is convex if for all A, B ∈ B, ν(A ∪ B) + ν(A ∩ B) ≥ ν(A) + ν(B).
A convex distortion is a convex increasing map f : [0, 1] → [0, 1] such that f (0) =
0, f (1) = 1. Since f is convex and f (0) = 0, f is continuous on [0, 1]. It may easily be
verified that ν := f (P) is a convex capacity. Let X ∈ L∞ (). The Choquet integral of X
with respect to the capacity f (P), denoted Ef (X) is defined by
E f (X) =
0
−∞
( f (P({X > t})) − 1) dt +
∞
f (P({X > t})) dt.
0
Assume further that f is continuous at one. Since f is nondecreasing and convex,
1
it is differentiable a.e. and f ∈ L+ [0, 1]. This implies from Lemma A.1 in Appendix
that,
1
(2.1)
f (1 − t)FX−1 (t) dt.
E f (X) =
0
From Corollary 2.1, Ef is concave monotone, σ (L∞ (), L1 ()) upper semicontinu1
ous and if X 2 Y , then Ef (X) ≥ Ef (Y ). Conversely, let g ∈ L+ [0, 1] be nondecreas1
1
to the Choquet
ing with 0 g(t) dt = 1. Then 0 g(t)FX−1 (1 − t) dt may be identified
x
integral with respect to φg : [0, 1] → [0, 1] defined by φg (x) = 0 g(t) dt. Clearly φg is
a convex
1 continuous distortion with derivative a.e. equal to g, hence from equation
(2.1), 0 g(t)FX−1 (1 − t) dt = Eφg (X).
COROLLARY 2.2. Let X ∈ L∞ () and f be a convex continuous distortion.
1
Then E f (X) = 0 f (1 − t)FX−1 (t) dt. Hence Ef : L∞ () →IR is concave, monotone and
∞
1
σ (L (), L ()) upper semicontinuous and if X 2 Y , then Ef (X) ≥ Ef (Y ).
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2.4. Schur Concave and S.S.D. Preserving Maps
Let v: Lp () →IR ∪ {−∞}, p ∈ {1, ∞}. Throughout the paper, we assume that v ≡
−∞. Let dom v = {X ∈ Lp () | v(X) > −∞}. We first recall a definition:
DEFINITION 2.4.
1. A map v: Lp () →IR ∪ {−∞}, p ∈ {1, ∞} is (strictly) monotone if X ≥ Y a.e.
implies v(X) ≥ v(Y ) (resp v(X) > v(Y ) whenever X ≥ Y a.e. and X = Y with
positive probability and Y ∈ dom v).
2. A map v: Lp () →IR ∪ {−∞}, p ∈ {1, ∞} is (strictly) Schur concave if X c Y
implies v(X) ≥ v(Y ) (resp v(X) > v(Y ) whenever X c Y and Y ∈ dom v).
3. A map v: Lp () →IR ∪ {−∞}, p ∈ {1, ∞} (strictly) preserves S.S.D. if X 2 Y
implies v(X) ≥v(Y )(resp v(X) > v(Y ) whenever X 2 Y and Y ∈ dom v).
We recall that a Schur concave map is also called a concave-order preserving map and
that S.S.D. preserving maps are also called increasing concave order preserving maps.
Let us now relate S.S.D. preserving maps and Schur concave map. If v preserves S.S.D.,
then from Lemma 2.1., v is Schur concave and monotone. To show the converse, let us
first provide a geometric characterization of preferred sets:
LEMMA 2.3. Let Y ∈ Lp (), p ∈ {1, ∞}. Then {X 2 Y } ∩ Lp () = {X c Y } ∩
L () + Lp + ()
p
The proof of Lemma 2.3. may be found in the Appendix. We now obtain
PROPOSITION 2.1. v : Lp () → IR∪ {−∞}, p ∈ {1, ∞} is monotone and Schur concave
iff v preserves S.S.D.
Proof. It remains to show one direction. Let Y , Y ∈ Lp (), p ∈ {1, ∞} be such that
Y 2 Y . From Lemma 2.3, there exists Y1 c Y , Y1 ∈ Lp () and Y2 ∈ Lp + () such that
Y = Y1 + Y2 . Since v is monotone and Schur concave, v(Y ) ≥ v(Y1 ) ≥ v(Y ) as was to
be proven.
Schur concave functions fulfill Jensen’s inequality. This property turns out to be very
important in applications.
PROPOSITION 2.2 (Jensen’s inequality). Let v : Lp () → IR ∪{−∞}, p ∈ {1, ∞} be
Schur concave, then
v(E(X | A)) ≥ v(X) for every σ − field A ⊂ B and X ∈ L p (), p ∈ {1, ∞}
If v is strictly Schur concave, then the inequality is strict for any X∈ dom v unless X
is B-measurable.
Proof. The result follows from E(X | B) c X with strict inequality unless X is
B-measurable.
2.5. Examples
2.1 (A bi-Schur concave function). Let X ∈ L1 () be fixed and let
: L () → IR be defined by v X = e(·, X ). From Corollary 2.1, v X is concave,
EXAMPLE
vX ∞
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R. A. DANA
σ (L∞ (), L1 ()) upper semicontinuous and Schur concave. If X ∈ L+ (), then v X is
1
monotone. When X ∈ L+ () and E(X ) = 1, v X is a Choquet integral with respect to
a convex continuous distortion and is called a Yaari utility. Symmetrically, for a fixed
X ∈ L∞ (), let ṽ X : L1 () → IR be defined by ṽ X = e(X, ·). Then ṽ X is concave monotone, σ (L1 (), L∞ ()) upper semicontinuous and Schur concave.
1
In order to present our next example, let us first introduce a definition.
DEFINITION 2.5. Let X ∈ Lp (), p ∈ {1, ∞}. The orbit O(X) of X with respect to
the concave order is the set {C ∈ Lp () | C c X}. A subset H ⊆ Lp (), p ∈ {1, ∞} is
concave-order invariant if H = ∪X∈H O(X).
2.2 (A Schur concave Indicator function). Let H ⊆ L∞ () and χH :
/ H.
L () → IR ∪ {−∞} be defined by χH (x) = 0 if x ∈ H and χH (x) = −∞ if x ∈
Then χH is Schur concave iff H is concave-order invariant, concave iff H is convex
and σ (L∞ (), L1 ()) upper semicontinuous iff H is σ (L∞ (), L1 ()) closed. If H is
concave-order invariant but not convex, v is Schur concave without being concave.
EXAMPLE
∞
EXAMPLE 2.3 (The quadratic utility). We give a more elaborate example of a function
that is Schur concave but not concave. We refer to Chew, Epstein, and Segal (1991)
for decision theoretic foundations of the example. Let ψ : IR × IR → IR be continuous.
Assume that ψ satisfies ψ(., y) is concave for all y ∈ IR and ψ(x, .) is concave for all x ∈ IR
and that ψ is not concave on the diagonal. In other words, there exist a, b ∈ IR, a = b,
and λ ∈]0, 1[ such that ψ(λa + (1 − λ)b, λa + (1 − λ)b) < λψ(a, a) + (1 − λ)ψ(b, b). For
example, the function ψ(x, y) = xy fulfills the above properties. Let X ∈ L∞ () and let
v : L∞ () → IR be defined by
1 1
v(X) =
ψ FX−1 (t), FX−1 (t ) dt dt .
0
0
Let us show that v is Schur concave. Let X c Y . Then for every fixed y ∈ IR, since ψ(., y)
is concave for all y ∈ IR,
1
1
ψ FX−1 (t), y dt ≥
ψ FY−1 (t), y dt.
0
0
1
Since ψ(x, .) is concave for all x ∈ IR, 0 ψ(FX−1 (t), .) dt and 0 ψ(FY−1 (t), .) dt are concave, therefore
1 1
1 1
V(X) =
ψ FX−1 (t), FX−1 (t ) dt dt ≥
ψ FY−1 (t), FX−1 (t ) dt dt 1
0
0
0
1
≥
0
0
1
0
ψ FY−1 (t), FY−1 (t ) dt dt = V(Y).
To show that V is not concave, let a, b ∈ IR be such that ψ(λa + (1 − λ)b,
λa + (1 − λ)b) < λψ(a, a) + (1 − λ)ψ(b, b) for some λ ∈]0, 1[. Let X = a and Y =
b. Then V (X) = ψ(a, a), V (Y ) = ψ(b, b), V (λX + (1 − λ)Y ) = ψ(λa + (1 − λ)b, λa +
(1 − λ)b) < λV (X) + (1 − λ)V (Y ). Furthermore, one easily verifies that if ψ(., y) is
(strictly) concave and increasing for all y ∈ IR and ψ(x, .) is (strictly) concave and increasing for all x ∈ IR, then v is (strictly) S.S.D. preserving.
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We end this section by giving an example of the use of nonconcave Schur concave
functions.
2.5.1. Application: A Demand Problem. Let v : L∞ () → IR ∪ {−∞} be Schur concave (but not necessarely concave) and satisfy
H1 For Xn , X ∈ L∞ (), if Xn → X pointwise, then limsup v(Xn ) ≤ v(X).
∞
Let ψ ∈ L+ () be a pricing density with support ⊆ [0, a]. Let w ∈ IR+ . Consider the
demand problem


 max v(C)
(P) E(ψC) ≤ w .


0≤C≤M
(P) may also be interpreted as a Neyman–Pearson problem (see Schied 2004). W.l.o.g.
we may assume that C ∈ dom v. Furthermore if C satisfies E(ψC) ≤ w and 0 ≤ C ≤
M, then E(C | ψ) also satisfies these constraints and v(E(C | ψ)) ≥ v(C) (with a strict
inequality if v is strictly Schur concave). Hence to prove existence, we may assume that
C = f (ψ). If ψ is nonatomic (there exists no c such that (P(ψ = c) > 0), then there
d
exists a nonincreasing function f̃ , unique dF ψ a.e., such that f (ψ) ∼ f̃ (ψ)) and such
that E(ψ f̃ (ψ)) ≤ E(ψ f (ψ)) with a strict inequality if f is not nonincreasing (see Carlier
and Dana 2005). Hence f̃ (ψ) also satisfies the constraints. To show the existence of a
solution, we have to solve the problem


 max v( f (ψ))
E(ψ f (ψ)) ≤ w
.


f : [0, a] → [0, M] nonincreasing
Let fn be a maximizing sequence. From Helly’s theorem, fn has a subsequence that converges pointwise to some f nonincreasing. Hence fn (ψ) → f (ψ)dF ψ a.e. and from H1,
v(f (ψ)) ≥ limsup v(fn (ψ)). Hence f (ψ) is a solution to (P). To summarize, we have thus
proved that:
Assuming H1, (P) has a solution of the form f (ψ) with f nonincreasing. If v is strictly
Schur concave, then from Jensen’s inequality, any solution is a function of ψ. If further
v is strictly monotone, then any solution is a nonincreasing function of ψ.
We would like to emphasize that v is not assumed to be concave. In the concave case,
to show existence, one can also use standard σ (L∞ (), L1 ()) upper semicontinuity
arguments (see Schied 2004). However, the nonatomicity assumption may not be avoided
in the previous reasoning. If ψ has atoms, for some f : [0, a] → IR, there exist no f˜ :
d
[0, a] → IR nonincreasing such that f (ψ) ∼ f˜ (ψ) (see Carlier and Dana 2005).
3. A CHARACTERIZATION OF SCHUR CONCAVE CONCAVE MAPS
3.1. Concave Transform
From now on, we assume that v : L∞ () → IR ∪ {−∞} is concave Schur concave. In
order to deal with the duality (L∞ (), L1 ()), we assume that v is σ (L∞ (), L1 ()) upper
semicontinuous. We shall show that the concave Fenchel transform defines a bijection
between σ (L∞ (), L1 ()) upper semicontinuous concave Schur concave functions and
σ (L1 (), L∞ ()) upper semicontinuous concave Schur concave functions. Let us therefore first recall the definitions of the concave Fenchel conjugate and biconjugate and their
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properties (see Ekeland and Temam 1974, chap 1 for a review of the mathematics and
Frittelli and Rosazza Gianin 2002 for an application to finance).
The concave transform and the concave biconjugate of the concave σ (L∞ (), L1 ())
upper semicontinuous function v : L∞ () → IR ∪ {−∞} are respectively defined by
(3.1)
(3.2)
v ∗ (X ) =
v ∗∗ (X) =
inf [E(XX ) − v(X)]
X∈L∞ ()
inf [E(XX ) − v ∗ (X )]
X ∈L1 ()
for all X ∈ L1 (),
for all X ∈ L∞ ().
Clearly
(3.3)
(3.4)
v ∗ (X ) = inf [E(XX ) − v(X)]
dom v
v ∗∗ (X) =
inf [E(XX ) − v ∗ (X )]
dom v ∗
for all X ∈ L1 (),
for all X ∈ L∞ ().
We recall that v ∗ : L1 () → IR ∪ {−∞} and that it is concave and σ (L1 (), L∞ ())
upper semicontinuous (see Ekeland and Temam 1974, p 16). The next proposition is a
minor extension of analogous results obtained in Frittelli and Rosazza Gianin (2002) for
v finite valued.
Let v : L∞ () → IR ∪ {−∞} be concave, σ (L∞ (), L1 ()) upper
PROPOSITION 3.1.
semicontinuous.
1. v = v ∗∗ or equivalently
v(X) =
(3.5)
=
inf [E(XX ) − v ∗ (X )]
X ∈L1 ()
inf [E(XX ) − v ∗ (X )]
dom v ∗
for all X ∈ L∞ ().
2. v is monotone iff
(3.6)
v(X) =
inf
X ∈L1+ ()
[E(XX ) − v ∗ (X )],
for all X ∈ L∞ ().
Proof. The first assertion follows from Ekeland and Temam (1974), proposition 4.1
chapter 1, applied to L∞ () and the duality (L∞ (), L1 ()). To prove the second, clearly
if v(X) = inf X ∈L1+ () [E(XX ) − v ∗ (X )], then v is monotone as infinimum of monotone
/ L1+ (). Indeed
functions and conversely if v is monotone, then v ∗ (X ) = −∞ if X ∈
∗
there exists X ≥ 0, X = 0 such that E(XX ) < 0 and v (X ) ≤ infn E(nXX ) − v(nX) ≤
infn [nE(XX ) − v(0)] = −∞, hence using equations (3.4) and (3.5), one obtains (7). DEFINITION 3.1.
A map v : L∞ () → IR ∪ {−∞},
1. is translation invariant if v(X + c) = v(X) + c for all c ∈ IR, X ∈ dom v,
2. superlinear if v(λX) = λv(X) for all λ > 0 and v(X1 + X2 ) ≥ v(X1 ) + v(X2 ).
In order to study the relation with risk measures, we recall that a convex measure of
risk ρ : L∞ () → IR is a map such that −ρ is u.s.c. concave, monotone, and translation
invariant and a coherent measure of risk ρ : L∞ () → IR is a map such that −ρ is an
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u.s.c., superlinear, monotone, and translation invariant. We extend the definition of a risk
measure by allowing the value +∞ for ρ.
Let D = {X ∈ L1+ (), E(X ) = 1}.
PROPOSITION 3.2.
Let v : L∞ () → IR ∪ {−∞} be concave upper semicontinuous.
1. v is translation invariant and monotone iff ,
(3.7)
v(X) = inf
[E(XX ) − v ∗ (X )]
X ∈D
for all X ∈ L∞ ().
2. v is superlinear iff v fulfills one of the following equivalent conditions.
r There exists a closed convex subset H ⊆ L1 () such that v(X) = infX ∈H
[E(XX )], for all X ∈ L∞ ().
r There exists a closed convex set H ⊆ L1 () such that v ∗ is the indicator function
of H.
3. v is superlinear, monotone, and translation invariant iff v fulfills one of the equivalent
conditions.
r There exists a closed convex subset H ⊆ D such that v(X) = infX ∈H
[E(XX )], for all X ∈ L∞ ().
r There exists a closed convex set H ⊆ D such that v ∗ = χH .
Proof. To prove the first assertion, if v(X) = inf X ∈D [E(XX ) − v ∗ (X )] for all X ∈
L (), then v is monotone as infinimum of monotone functions and v(X + c) =
inf X ∈D [E(XX ) + cE(X ) − v ∗ (X )] = v(X) + c for all X ∈ dom v. Conversely, let
X ∈ dom v and v(X + c) = v(X) + c and v be monotone. From Proposition 3.1,
v ∗ (X ) = −∞ if X ∈
/ L1+ (). Let m ∈ IR. Then, v ∗ (X ) ≤ infm [E[X (X + m)] − v(X +
m)] = infm [E(X X) − v(X) + m(E(X ) − 1)] = −∞ if E(X ) = 1. Hence v ∗ (X ) = −∞ if
/ D. The second assertion is a standard result in convex analysis and may be found
X ∈
in Ekeland and Temam (1974), (example 4.3, chap 1). The third assertion follows from
the first and the second assertions.
∞
3.2. A Representation Result
We now prove the main result of the paper.
THEOREM 3.1. Let v : L∞ () → IR ∪ {−∞} be concave, σ (L∞ (), L1 ()) upper
semicontinuous. The following assertions are equivalent:
1. v is Schur concave,
2. v ∗ has the following representation
1
−1
−1
(3.8) v ∗ (X ) = inf
F
(t)F
(1
−
t)
dt
−
v(X)
, for all X ∈ L1 (),
X
X
∞
X∈L ()
0
3. v ∗ is Schur concave,
4. v has the following representation
1
(3.9) v(X) = inf
FX−1 (t)FX−1 (1 − t) dt − v ∗ (X ) , for all X ∈ L∞ ().
X ∈L1 ()
0
Assume further v monotone. Then 1–4 are also equivalent to
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5. v has the following representation
1
(3.10) v(X) = inf
FX−1 (t)FX−1 (1 − t) dt − v ∗ (X ) , for all X ∈ L∞ ().
X ∈L1+ ()
0
Proof. Let us show that assertion 1 implies assertion 2. From the definition of v ∗ and
from Hardy & Littlewood’s inequality, we first have
1
−1
−1
∗
(3.11) v (X ) ≥ inf
FX (t)FX (1 − t) dt − v(X) , for all X ∈ L1 ().
∞
X∈L ()
0
To obtain the reverse inequality, let X ∈ L1 () and X ∈ L∞ () be fixed. We have
inf
inf
E(C X ) − v(C) ≤
E(C X ) − v(X)
v ∗ (X ) ≤
C c X, C∈L∞ ()
C c X, C∈L∞ ()
1
FX−1 (t)FX−1 (1 − t) dt − v(X) .
=
0
The second inequality follows from v being Schur concave and the last equality follows
from Theorem 2.1. Taking the infinimum over X, we get
1
−1
−1
(3.12) v ∗ (X ) ≤ inf
F
(t)F
(1
−
t)
dt
−
v(X)
, for all X ∈ L1 ().
X
X
∞
X∈L ()
0
proving (9). To show that assertion 2 implies assertion 3, let Y c X . From Lemma 2.2.,
we have,
1
−1
−1
FX (t)FX (1 − t) dt − v(X)
0
1
≤
0
−1
FY−1
(t)FX (1 − t) dt − v(X) ,
for all X ∈ L∞ ()
hence, taking the infinimum with respect to X and using equation (3.8), we obtain v ∗ (Y ) ≤
v ∗ (Y ). Since v ∗ is concave, σ (L1 (), L∞ ()) upper semicontinuous, the implications from
3 to 4 (4 to 1) goes like the implication from 1 to 2 (2 to 3) except that X , X, and L1 (),
1
and L∞ () are exchanged. From Proposition 3.2, v is monotone iff dom v ∗ ⊆ L+ ().
The equivalence between 5 and 6 follows from the equivalence between 1 and 4 and (5)
and v = v ∗∗ .
1
For further use, let
us introduce a notation. For Y ∈ L+ (), let φY : [0, 1] → [0, 1] be
defined by φY (x) =
x
01
0
FY−1 (t) dt
FY−1 (t) dt
. Clearly φY is a convex continuous distortion. From Section
2.3, we may reformulate the previous theorem as follows
COROLLARY 3.1. Let v : L∞ () → IR ∪ {−∞} be concave σ (L∞ (), L1 ()) upper
semicontinuous. The following are equivalent.
1. v is monotone and Schur concave,
2. v has the following representation:
v(X) = inf [E(Y)EφY (X) − v ∗ (Y)].
Y∈L1+
In other words, v is the infinimum of a family of nonnegative affine combinations of
Choquet integrals with respect to a convex, continuous distortion of P.
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Proof. Since v is monotone, from equation (3.10) in Theorem 3.1. and Section 2.3,
v(X) = inf [E(Y)EφY (X) − v ∗ (Y)].
Y∈L1+
Hence assertion 1 implies assertion 2. From Corollary 2.2, as the infinimum of a family of nonnegative affine combinations of Choquet integrals with respect to a convex,
continuous distortion of P is monotone and Schur concave, assertion 2 implies 1.
In order to study the relation with risk measures, assume that − v is a convex measure
of risk that is Schur concave.
COROLLARY 3.2. Let v : L∞ () → IR ∪ {−∞} be concave σ (L∞ (), L1 ()) upper
semicontinuous. The following are equivalent.
1. v is monotone, Schur concave and translation invariant,
2. v has the following representation
v(X) = inf [EφY (X) − v ∗ (Y)], for all X ∈ L∞ ().
Y∈D
Proof. Assertion 1 implies 2 follows from Proposition 3.2. and Corollary 3.1. Assertion 2 implies 1 since the infinimum of monotone, Schur concave and translation invariant
maps is monotone, Schur concave and translation invariant.
Let us now assume that −v is a coherent measure of risk that is Schur concave.
COROLLARY 3.3. Let v : L∞ () → IR ∪ {−∞} be concave σ (L∞ (), L1 ()) upper
semicontinuous. The following are equivalent
1. v is superlinear, monotone, Schur concave and translation invariant,
2. there exists a closed convex concave order invariant set H ⊆ D such that v ∗ = χH ,
3. v is the infinimum of a family of Choquet integrals with respect to continuous
distortions of P.
Proof. Assertion 1 implies 2 follows from Proposition 3.2. and Theorem 3.1. and Example 2.2. To show that Assertion 2 implies 3, from Theorem 3.1., Section 2.3, Proposition
3.2., and Corollary 3.2., there exists a closed convex concave-order invariant set H ⊆ D
such that v(X) = infY ∈H EφY (X) for all X ∈ L∞ (). To show that assertion 3 implies 1, let
v be the infinimum of a family of Choquet integrals with respect to continuous distortions of P. Then v is superlinear, monotone, Schur concave and translation invariant as
infinimum of monotone, Schur concave, translation invariant, superlinear maps.
REMARKS
3.1.
1. If v : L∞ () → IR ∪ {−∞} is concave, upper semicontinuous and Schur concave,
then it follows from Theorem 3.1. that there exists : L∞ [0, 1] → IR ∪ {−∞}
concave σ (L∞ [0, 1], L1 [0, 1]) upper semicontinuous and Schur concave such that
−1
v(X) = (FX ). Moreover,
r if v is monotone, then is monotone,
r if v is translation invariant and monotone, then is translation invariant and
monotone,
r if v is superlinear and monotone, then is superlinear and monotone.
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2. Let v : L∞ () → IR ∪ {−∞} be concave σ (L∞ (), L1 ()) upper semicontinuous, superlinear, monotone, Schur concave and translation invariant. In Corollary 3.3., H consists of a unique orbit iff v is a Choquet integral with respect
to a convex, continuous distortion of P (see also Carlier and Dana 2003a and
Wasserman and Kadane 1992).
4. LAW-INVARIANT CONCAVE UTILITIES
4.1. Definition and First Properties
DEFINITION 4.1. A map v : Lp () → IR ∪ {−∞}, p ∈ {1, ∞} is law-invariant if
d
v(X) = v(Y ) whenever X ∼ Y and X∈ dom v.
If v is Schur concave, then v is law-invariant. The following simple example shows that
the converse is not true.
4.1.1. A Law-invariant Map that is not Schur Concave. Suppose that there are two
states of the world (1, 2) and that state 1 has probability 14 . The space of random variables
is identified to IR2 (the random variable X, which equals x1 in state 1 and x2 in state 2,
is identified with the vector X = (x1 , x2 ) ∈ IR2 ). Let X be given. There is no Y = X such
d
that Y ∼ X, hence all maps on IR2 are law-invariant.
• If x1 ≤ x2 ,
{Yc X} = {Y | x1 ≤ y1 ≤ y2 , y1 + 3y2 = x1 + 3x2 }
∪ {Y | x1 ≤ y2 ≤ y1 ≤ x2 , y1 + 3y2 = x1 + 3x2 }
{Y c X} is a segment with extreme points X and YX = (x2 ,
• If x2 ≤ x1 ,
x1 +2x2
).
3
{Yc X} = {Y | x2 ≤ y1 ≤ y2 ≤ x1 , y1 + 3y2 = x1 + 3x2 }
∪ {Y | x2 ≤ y2 ≤ y1 , y1 + 3y2 = x1 + 3x2 }
is a segment with extreme points X and YX = (−2x1 + 3x2 , x1 ).
If v is assumed to be concave, v is Schur concave iff v(YX ) ≥ v(X) for every X. This
provides a restriction on v.
4.2. The Nonatomic Case
We recall that a probability space (, B, P) is nonatomic if it has no atoms.
We will make extensive use of a result due to Ryff (1970) under the assumption
that (, B, P) is nonatomic. For a proof of this result, we refer to Chong and Rice
(1971).
LEMMA 4.1. Let Y be a random variable on the nonatomic space (, B, P). Then there
−1
exists a random variable U : → [0, 1] with a uniform law such that Y = FY (U) a.e.
For sake of completeness, we recall a definition:
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DEFINITION 4.2. A pair (X, Y ) of random variables is comonotone (resp anticomonotone) if there exists a subset B ⊆ of P-measure comonotone such that
X(s) − X(s ) Y(s) − Y(s ) ≥ 0, ∀s, s ∈ B (resp ≤).
LEMMA 4.2. Let X and Y be a pair of random variables on the nonatomic
d
space (, B, P). Then there exists a random variable X̃ on (, B, P), X̃ ∼ X, comonotone (resp anticomonotone) with Y .
Proof. By Lemma 4.1., let U be a random variable on (, B, P) with a uniform
−1
−1
−1
law be such that Y = FY (U) a.e. Then FX (U) (resp FX (1 − U)) is comonotone (resp
anticomonotone) with Y and is distributed like X.
Lemma 4.2. implies that
COROLLARY 4.1. Let r ∈ {1, ∞} and r be its conjugate. Let (, B, P) be nonatomic
and let X ∈ Lr () and X ∈ Lr (). Then
1
min E(X C) = min E(X C) =
FX−1 (1 − t)FX−1 (t) dt
Cc X
C ∈ Lr ()
d
0
C∼X
C ∈ Lr ().
d
The minimum is attained for C ∼ X, C anticomonotone with X .
d
Proof. Since {C ∼ X} ⊂ {Cc X}, we have from Theorem 2.1.
1
FX−1 (1 − t)FX−1 (t) dt = min E(X C) ≤ min E(X C)
0
Cc X
C ∈ Lr ()
d
C∼X
C ∈ Lr ().
−1
The converse inequality follows from Lemma 4.2. by taking C = FX (1 − U) for U such
1
−1
that X = FX (U) and using E([FX−1 (1 − U)FX−1 (U)] = 0 FX−1 (1 − t)FX−1 (t) dt.
Let r ∈ {1, ∞} and r be its conjugate. Let A ⊆ Lr () and coAdenote the convex closure
of A for the σ (Lr (), Lr ()) topology. We have the following result due to Ryff (1970)
which may also be proven by the method of Lemma 2.3.
PROPOSITION 4.1. Let r ∈ {1, ∞} and let (, B, P) be nonatomic. Let Y ∈ Lr ().
d
Then {Ỹ ∼ Y} is the set of extreme points of {X c Y } ∩ Lr () and {X c Y} ∩ Lr () =
d
co{Ỹ ∼ Y}.
THEOREM 4.1. Let r ∈ {1, ∞}. Let (, B, P) be nonatomic and let v : Lr () → IR ∪
{−∞} be concave, σ (Lr (), Lr ()) upper semicontinuous, the following statements are
equivalent
1. v is Schur concave,
2. v is law-invariant.
The following statements are also equivalent to each other
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3. v is S.S.D. preserving,
4. v is law-invariant and monotone.
Proof. To prove that statement 1 is equivalent to statement 2, it suffices to show that 2
implies 1. Let X c Y be in Lr (). We may w.l.o.g. assume that Y ∈ dom v. From Proposid
tion 4.1., X ∈ co{Ỹ ∼ Y}. Since v is σ (Lr (), Lr ()) upper semicontinuous and concave,
d
{Z ∈ Lr () | v(Z) ≥ v(Y )} is σ (Lr (), Lr ()) closed and convex and contains {Ỹ ∼ Y}
d
since v is law-invariant, hence it contains co{Ỹ ∼ Y}. Therefore v(X) ≥ v(Y ). The equivalence between statement 3 and 4 follows from the equivalence between statement 1 and
2 and Proposition 2.1.
Hence, under the nonatomic hypothesis, concave σ (Lr (), Lr ()) upper semicontinuous, Schur concave maps v : Lr () → IR ∪ {−∞} coincide with concave σ (Lr (), Lr ())
upper semicontinuous law-invariant maps. A version of this remark goes back to
Grothendieck (1955).
Let us now state a law-invariant version of Theorem 3.1. An early version is given by
Grothendieck (1955). It may be found in Luxemburg (1967) and Chong and Rice (1971).
THEOREM 4.2. Let (, A, P) be nonatomic and let v : L∞ () → IR ∪ {−∞} be concave and σ (L∞ (), L1 ()) upper semicontinuous. The following conditions are equivalent
1. v is law-invariant,
2. v ∗ is law-invariant,
3. v has the following representation:
1
(4.1) v(X) = inf
FX−1 (t)FX−1 (1 − t) dt − v ∗ (X )
X ∈L1 ()
for all X ∈ L∞ ().
0
If furthermore, v is monotone, then the following conditions are also equivalent to
1–3,
4. v has the following representation
1
(4.2) v(X) = inf
FX−1 (t)FX−1 (1 − t) dt − v ∗ (X )
for all X ∈ L∞ (,
X ∈L1+ ()
0
5. v has the following representation
v(X) = inf [E(Y)EφY (X) − v ∗ (Y)]
Y∈L1+
for all X ∈ L∞ ().
In other words v is the infinimum of a family of nonnegative affine combinations of
Choquet integrals with respect to a convex, continuous distortion of P.
Let us finally state a representation result for law-invariant convex measures of risk measures and coherent measures. Representations theorems have been proven by Kusuoka
(2001), Frittelli and Rosazza Gianin (2005), Rosazza Gianin (2002), Kunze (2003) and
by others.
COROLLARY 4.2. Let (, B, P) be nonatomic. Let v : L∞ () → IR ∪ {−∞} be concave, σ (Lr (), Lr ()) upper semicontinuous and translation invariant. The following are
equivalent
1. v is monotone and law-invariant
2. v(X) = infY∈D [EφY (X) − v ∗ (Y)]
for all X ∈ L∞ ().
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DEFINITION 4.3.
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A subset H ⊂ Lr (), r ∈ {1, ∞} is law-invariant if for all X ∈
d
H, {Y ∼ X} ⊆ H.
COROLLARY 4.3. Let (, B, P) be nonatomic. Let v : L∞ () → IR ∪ {−∞} be concave, σ (Lr (), Lr ()) upper semicontinuous and translation invariant. Then the following
are equivalent
1. v is monotone, law-invariant, and superlinear,
2. there exists a closed convex law-invariant subset H ⊆ D such that v ∗ = χH ,
3. v is the infinimum of a family of Choquet integrals with respect to continuous
distortions of P.
REMARK 4.1.
Theorem 4.2. obviously follows from Theorems 3.1. and 4.1. It may be
proven directly using Corollary 4.1. Indeed, let X ∈ L∞ () be fixed, X ∈ domv. We have
inf
inf
E(C X ) − v(C) ≤
E(C X ) − v(X)
v ∗ (X ) ≤
d
d
C ∼X, C∈L∞ ()
C ∼X, C∈L∞ ()
=
0
1
FX−1 (t)FX−1 (1 − t) dt − v(X) .
The second inequality follows from v being law-invariant and the last equality follows from Corollary 4.1. Taking the infinimum over X ∈ domv, we get v ∗ (X ) ≤
1
inf X∈dom v [ 0 FX−1 (t)FX−1 (1 − t) dt − v(X)] for all X ∈ L1 (). As the other direction
follows from Hardy and Littlewood’s inequality, we get the desired assertion.
The following counterexample shows that without the nonatomic hypothesis, one cannot expect that v law-invariant implies v ∗ is law-invariant.
4.2.1. A Concave Law-invariant Map v Such that v ∗ not Law-invariant. Assume that
there are three states of the world (1, 2, 3) with probability p1 = 18 , p2 = 38 , p3 = 12 . The
only random variables with same distribution are of the form (x, x, y), (y, y, x). Let X =
= v(y, y, x),
(x1 , x2 , x3 ) and v(X) = min( x1 +x42 +2x3 , 3x1 +x82 +4x3 ). Then v(x, x, y) = x+y
2
hence v is law-invariant and superlinear. Since v is superlinear, v ∗ = χH with H = {Y ∈
R3 , | Y · X ≥ v(X)} Hence H is the segment with extreme points ( 14 , 14 , 12 ) and ( 38 , 18 , 12 ).
H is not law-invariant since it does not contain ( 12 , 12 , 14 ).
5. A BIBLIOGRAPHICAL REVIEW
Grothendieck (1955) introduces “Polya functions” as convex, lower semicontinuous “rearrangement invariant” functions on L∞ (M, m) where (M, m) is an open interval of IR
and provides a representation result for these functions. Luxemburg (1967) and Chong
and Rice (1971) define Schur convex function as convex, lower semicontinuous “rearrangement invariant” functions. They provide a representation theorem based on Hardy
and Littlewood’s inequality and on the assumption that Corollary 4.1. is fulfilled. In the
statistics literature, Wasserman and Kadane (1992) give a representation theorem for
symmetric upper probabilities on a nonatomic space. Symmetric upper probabilities are
concave functionals that assign same value to the indicators of sets of same probability.
Finally, the concepts of law-invariant coherent measure of risk and of law-invariant convex measure of risk are introduced in mathematical finance (see Delbaen 2001, 2002 and
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Föllmer and Schied 2002 for a review). A number of representation theorems for lawinvariant risk measures on a nonatomic space have been given (with more or less long
proofs) by Kusuoka (2001), Frittelli and Rosazza Gianin (2002), (2005), Kunze (2003).
Jouini and Schachermayer and Touzi (2005) have recently shown that, on a nonatomic
space, a concave law-invariant finite valued map is σ (L∞ (), L1 ()) upper semicontinuous. Hence, in Corollaries 4.2. and 4.3., the hypothesis that v is σ (L∞ (), L1 ()) upper
semicontinuous is not needed. A representation theorem for S.S.D. preserving, coherent
measures is given by Leitner (2004) without the nonatomic assumption.
APPENDIX
The following lemma is recalled for sake of completeness.
LEMMA A.1. Let (, B, P) be a probability space and let X ∈ L∞ (). Let f : [0, 1] →
[0, 1] be non decreasing, absolutely continuous and such that f (0) = 0, f (1) = 1. Then
1
f (1 − t)FX−1 (t) dt.
E f (X) =
0
1
Proof. Since Ef and X → 0 f (1 − t)FX−1 (t) dt are translation invariant and since
∞
X+ X∞ ≥ 0, we may w.l.o.g. assume that X ≥ 0, hence E f (X) = 0 f (P({X > t})) dt.
Thus
∞
∞ P(X>t)
f (P(X > t)) dt =
f (u) du dt =
f (u) du dt
0
0
0
=
{t≥0,u≥0,u<1−FX (t)}
{t≥0,u≥0,t<FX−1 (1−u)}
1
=
f (u)
0
=
f (u) du dt
FX−1 (1−u)
dt du
0
1
f (u)FX−1 (1 − u) du.
0
The first equality follows from f being absolutely continuous, it is the indefinite integral
of its derivative, the fourth equality follows from Fubini’s theorem.
Proof of Lemma 2.2. We first consider the case r = 1, r = ∞. To prove assertion 1,
1
let
kX c Y , with X, Y ∈ L (). For any nonincreasing, nonnegative step function g(t) =
i =1 ai 1[0,bi ] (t) with 0 < b1 < · · · < bk = 1 and ai > 0 for all i, we have from Lemma ?? Q2
and the definition of S.S.D.
bi
bi
1
1
k
k
−1
−1
−1
(A.1)
g(t)FX (t) dt =
ai
FX (t) dt ≥
ai
FY (t) dt =
g(t)FY−1 (t) dt.
0
i =1
0
i =1
0
0
It follows from the Lebesgue-dominated convergence theorem that Equation (A.1)
holds true for any nonnegative, nonincreasing, bounded g. If g is any nonincreas1
1
ing bounded function, then g+ g∞ ≥ 0, hence 0 (g(t) + g∞ )FX−1 (t) dt ≥ 0 (g(t) +
1
1
g∞ )F −1 (t) dt which implies that 0 g(t)FX−1 (t) dt ≥ 0 g(t)FY−1 (t) dt since E(X) =
1 −1 Y
1 −1
0 FX (t) dt = E(Y) = 0 FY (t) dt. Conversely if assertion 1 holds true, by taking
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g(s) = 1[0,t] (s), we obtain that X 2 Y . By taking g = 1 and g = −1, we obtain that
E(X) = E(Y ) and that X c Y .
The proof of the second assertion follows easily from the proof of the first assertion. To
prove the case r = ∞, r = 1, one first proves the property for g ∈ L∞ [0, 1] nonincreasing
as in the proof above. The general case follows from Lebesgue-dominated convergence
theorem and the fact that g ∈ L1 [0, 1] nonincreasing can be approximated in the L1 norm
by a sequence of nonincreasing L∞ [0, 1] functions.
Proof of Lemma 2.3.
r Let us first remark that {X 2 Y } ∩ L∞ () is convex σ (L∞ (), L1 ()) closed. To
+
r
prove this last assertion, it suffices to prove that {X 2 Y } ∩ {X∞ ≤ n} is closed
with respect to convergence in probability (see Delbaen 2001). But this follows easily from the characterisation X 2 Y iff E(u(X)) ≥ E(u(Y )) for every u continuous,
∞
increasing concave. By a similar type of argument, {Y c Y } ∩ L∞
+ () + L+ () is
convex σ (∞ (), L1 ()) closed.
∞
Let B = {X c Y } ∩ L∞
+ () + L+ (). Clearly B ⊂ {X 2 Y }. Assume that there
exists X0 ∈ ({X 2 Y } ∩ L∞
()
\
B). By the Hahn–Banach theorem, there exist
+
ψ ∈ L1 () and ε > 0 such that
(A.2)
ψ Xd Q.
ψ X0 d Q ≤ −ε + inf
X∈B
Since X + Z ∈ B, for any X ∈ B and Z ∈ L∞
+ (), we must have ψ ≥ 0, ψ = 0,.
Thus Fψ−1 (1 − Id) is nonincreasing and positive. Furthermore inf X∈B ψ Xd Q =
1
inf(Xc Y)∩L∞
ψ Xd Q = 0 Fψ−1 (1 − t)FY−1 (t) dt (the last equality follows from
+ ()
Theorem 2.1.). From Hardy and Littlewood’s inequality
1
1
(A.3)
Fψ−1 (1 − t)FX−1
(t)
dt
≤
ψ
X
d
Q
≤
−ε
+
Fψ−1 (1 − t)FY−1 (t) dt.
0
0
0
0
r Since X0 2 Y , we have from Lemma 2.2.,
1
(A.4)
0
g(t)FX−1
(t) dt ≥
0
1
0
g(t)FY−1 (t) dt.
for any g ∈ L1+ [0, 1] nonincreasing, in particular, it holds true for Fψ−1 (1 − Id)
contradicting equation (A.3). Hence B = {X 2 Y } ∩ L∞
+ ().
Proof of Theorem 2.1. We prove that problem (E) has a solution and that its value
function is e(X, X ). Let X ∈ L1 () and X ∈ L∞ (). From Hardy and Littlewood’s
inequality and Lemma 2.2., we first have, for C ∈ L∞ ()
1
1
−1
−1
FX (1 − t)FC−1 (t) dt ≥
FX (1 − t)FX−1 (t) dt.
e(X, X ) = inf E(X C) ≥ inf
C c X
C c X 0
0
To prove the reverse inequality, it suffices to prove that there exists C c X, C ∈ L∞ ()
1
such that E(X C) = 0 FX−1 (1 − t)FX−1 (t) dt. If C c X, then E(C | X ) c X. Hence, we
may assume w.l.o.g. that C = f (X ). We need to find f such that
1
1
−1
−1
E(X f (X )) =
FX (t) f (FX−1 (t)) dt =
FX (1 − t)FX−1 (t) dt,
0
0
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Let f be defined by
f FX−1 = Eλ FX−1 (1 − Id) | FX−1 ,
(A.5)
with λ the Lebesgue measure. Since at continuity points x of FX , FX−1 is uniquely determined,
f (x) = FX−1 (1 − FX (x)).
At discontinuity points x of FX ,
f (x) =
1
FX (x) − FX (x− )
FX (x)
FX (x− )
FX−1 (1 − t) dt.
Let us now show that f (X ) ∈ L∞ () and that f (X ) c X. Since FX−1 ∈ L∞ [0, 1], C =
d
f (X ) ∈ L∞ (). Since X ∼ FX−1 , for every u concave, we have,
1
1
u f FX−1 (t) dt ≥
u FX−1 (1 − t) dt
E(u( f (X )) =
0
(A.6)
0
1
=
0
u FX−1 (t) dt = E(u(X) .
the inequality in equation (A.6) follows from the definition of f (FX−1 ) and Jensen’s inequality. By taking u = Id and u = − Id, we further obtain that E(f (X )) = E(X) proving
that f (X ) c X.
The proof for problem (Ẽ) is similar. It is left to the reader to verify that “C = f (X)”
with f defined by equation (A.5) is in L1 ().
REFERENCES
ATKINSON, A. B. (1970): On the Measurement of Inequality, J. Econ. Theo., 2, 244–263.
BLACKWELL, D. (1953): Equivalent Comparaisons of Experiments, Ann. Math. Stat. 24, 265–
272.
CARLIER, G., and R. A. DANA (2003a): Core of a Convex Distortion of a Probability on a Non
Atomic Space, J. Econ. Theo. 113, 199–222.
CARLIER, G., and R. A. DANA (2003): Modelling the Effect of Aversion to Catastrophic Risk on
Q3
Insurance Contracts, Cahier du Ceremade.
CARLIER, G., and R. A. DANA (2005): Rearrangement Inequalities in Non Convex Insurance
Models, Cahier du Ceremade 0238 (2002). To appear in J. Math. Econ.
CARLIER, G., and R. A. DANA (2003): Pareto Efficient Insurance Contracts When the Insurer’s
Cost Function is Discontinuous, Econ. Theo. 21, 871–893.
CHEW, S. H., L. G. EPSTEIN, and U. SEGAL (1991): Mixture Symmetry and Quadratic Utility,
Econometrica 59, 165–187.
CHEW, S. H., and M. H. MAO (1995): A Schur Concave Characterization of Risk Aversion for
Non Expected Utility Preferences, J. Econ. Theo. 67, 402–435.
CHEW, S. H., and I. ZILCHA (1990): Invariance of the Efficient Sets When the Expected Utility
Hypothesis is Relaxed, J. Econ. Beha. Org. 13, 125–131.
CHONG, K. M., and N. M. RICE (1971): Equimeasurable Rearrangements of Functions, Queen’s
Q4
Papers in Pure and Applied Mathematics 28.
mafi˙253
MAFI.cls
June 24, 2005
23:25
RESULT FOR CONCAVE SCHUR CONCAVE FUNCTIONS
633
DANA, R. A. (2004): Market Behavior when Preferences are Generated by Second Order Stochastic Dominance, J. Math. Econ. 40, 619–639.
DANA, R. A., and I. MEILIJSON (2003): Modelling Agents’ Preferences in Complete Markets by
Q5
Second Order Stochastic Dominance, Cahier du Ceremade 0333.
DELBAEN, F. (2001): Coherent Risk Measures, Lectures notes, Pisa.
DELBAEN, F. (2002): Coherent Measures of Risk on General Probability Space In: Advances
in Finance and Stochastics. Essays in Honour of Dieter Sondernann, Sandmann and J. P.
Schönbucker, eds, Springer-Verlag Berlin, 1–37.
DYBVIG, P. (1988): Distributional Analysis of Portfolio Choice, J. Business 61, 369–393.
DYBVIG, P., and S. ROSS (1982): Portfolio Efficient Sets, Econometrica 50, 1525–154.
Q6
EKELAND, I., and R. TEMAM (1974): Analyse convexe et problèmes variationnels, Dunod Gauthier, Villars editors.
FÖLLMER, H., and A. SCHIED (2002): Stochastic Finance. An introduction in discrete time, De
Gruyter editor, Berlin.
FRITTELLI, M., and E. R. ROSAZZA GIANIN (2002): Putting Order in Risk Measures, J. Banking
and Fin. 26-7, 1473–1486 (2002).
FRITTELLI, M., and E. R. ROSAZZA GIANIN (2005): Law Invariant Convex Risk Measures, Adv.
Math. Eco. 7, 33–44.
GOLLIER, C., and H. SCHLESINGER (1996): Arrow’s Theorem on the Optimality of Deductibles:
A Stochastic Dominance Approach, Econ. Theo. 22, 107–110.
GROTHENDIECK, A., MARS (1955): Réarrangements de Fonctions et Inégalités de convéxité dans
les Algèbres de von-Neumann munies d’une trace, Séminaire Bourbaki.
HARDY, G. H., J. E. LITTLEWOOD, and G. PÒLYA (1988): Inequalities, Reprint of the 1952 edition,
Cambridge: Cambridge University Press.
JOUINI, E., and H. KALLAL (2000): Efficient Trading Strategies in the Presence of Market Frictions, Rev. Finan. Stud. 14, 343–369.
JOUINI, E., W. SCHACHERMAYER, and N. TOUZI (2005): Law invariant risk measures have the
Fatou Property, preprint.
KIM, C. (1998): Stochastic Dominance, Pareto Optimality and Equilibrium Pricing, Rev. Econ.
Stud. 65, 341–356.
KUNZE, M. (2003): Verteilungesinvariante konvexe Risikomabe, Diplomarbeit, HumboltUniversität zu Berlin.
KUSUOKA, S. (2001): On Law Invariant Coherent Measures, Adv. Math. Eco 3, 83–95.
LANDSBERGER, M., and I. MEILIJSON (1994): Comonotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt Measure of Risk Aversion, Ann. Oper. Res., 52, 97–106.
LEITNER, J. (2004): A Short Note on Second Order Preserving Coherent Risk Measures. To
appear in, Mathematical Finance.
LUXEMBURG, W. A. J. (1967): Rearrangement Invariant Banach Function Spaces, Queen’s Papers
Q7
in Pure and Applied Mathematics 10.
MARSHALL, A. W., and I. OLKIN (1979): Inequalities: Theory of Majorization and its Applications.
Academic Press.
MÜLLER, A., and D. STOYAN (2002): Comparaisons Methods for Stochastic Models and Risks.
New York: Wiley.
OSTROWSKI, A. M. (1952): Sur Quelques Applications des fonctions convexes et concaves au sens
de Schur, Journal of Math Pures Appli 31, 253–292.
PELEG, B., and M. E. YAARI (1975): A Price Characterisation of Efficient Random Variables,
Econometrica 43, 283–292.
mafi˙253
MAFI.cls
634
June 24, 2005
23:25
R. A. DANA
ROSAZZA GIANIN, E. (2002): Convexity and Law Invariance of Risk Measures, PhD thesis,
Università di Bergamo.
ROTHSCHILD, M., and J. E. STIGLITZ (1970): Increasing risk, I. A definition, J. Econ. Theo. 2,
225–243.
RYFF, J. V. (1970): Measure Preserving Transformations and Rearrangements, J. Math. Anal.
and Appl. 31, 449–458.
SCHIED, A. (2004): On the Neyman-Pearson problem for law-Invariant Risk Measures and
Robust Utility Functionals, Ann. Appl. Probab. 14, 1398–1423.
SCHUR, I. (1923): Uber eine Klasse von Mittelbindungen mit Anwendungen in der Determinantentheorie, Sitzungsber Math Gesellschaft, 22, 9–20.
STRASSEN, V. (1965): The Existence of Probability Measures with Given Marginals, Ann. Math.
Stat. 36, 423–439.
WASSERMAN, L., and J. B. KADANE (1992): Symmetric Upper Probabilities, Ann. Stat. 20-4,
1720–1736.
mafi˙253
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QUERIES
Q1 PE: Please provide history dates for the article
Q2 Au: Please provide Lemma No.
Q3 Au: Please provide details of the reference Carlier and Dana (2003).
Q4
Q5
Q6
Q7
Au: Please provide page range for Chong and Rice (1971).
Au: Please provide page range for Dana and Meilijson (2003).
Au: Please check page range of Dybvig and Ross (1982).
Au: Please provide page ragnge for Luxemburg (1967).
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