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Law invariant risk measures have the Fatou property

TL;DR: In this paper, a dual characterization of law invariant coherent risk measures, satisfying the Fatou property, was given, and it was shown that the hypothesis of Fatou properties may actually be dropped as it is automatically implied by the hypothesis for law invariance.
Abstract: S. Kusuoka [K01, Theorem 4] gave an interesting dual characterization of law invariant coherent risk measures, satisfying the Fatou property. The latter property was introduced by F. Delbaen [D 02]. In the present note we extend Kusuoka’s characterization in two directions, the first one being rather standard, while the second one is somewhat surprising. Firstly we generalize — similarly as M. Fritelli and E. Rossaza Gianin [FG 05] — from the notion of coherent risk measures to the more general notion of convex risk measures as introduced by H. Follmer and A. Schied [FS 04]. Secondly — and more importantly — we show that the hypothesis of Fatou property may actually be dropped as it is automatically implied by the hypothesis of law invariance.

Summary (2 min read)

1 Introduction

  • This paper is a twin to [JST 05] and the authors shall use similar notation.
  • The precise statement of this result is the following.
  • The first main contribution of this paper is to drop the Fatou property in condition (a) of the above Theorem 1.1 by proving that it is automatically satisfied by law-invariant monetary utility functions.
  • In fact, the authors prove more generally that the Fatou property is implied by the concavity, the L ∞ -u.s.c. and the law-invariance properties.

2.1 Definitions

  • The assumption of F being free of atoms is crucial (otherwise one is led to combinatorial problems which are irrelevant from the economic point of view).
  • ∞ -continuity property of monetary utility functions.

2.2 Strong and weak upper semi-continuity of law invariant maps

  • The authors now have assembled all the concepts that are needed to formulate their first main result.
  • In particular the AV@R representation of such utility functions holds without any further condition.
  • The authors novel contribution is that the Fatou property is automatically implied by the law invariance and the strong upper semi-continuity; recall that monetary utility functions are L ∞ -Lipschitz continuous.
  • The reader only interested in Theorem 2.2 may directly proceed to this section.
  • For the sake of completeness the authors formulate this result.

3 Reduction of the probability space by law invariance

  • In this section the authors shall show the equivalence of (ii) and (iii) in Theorem 2.1.
  • The authors shall see that a rather straight-forward application of the formula of integration by parts translates (ii) into (iii) and vice versa.
  • D it suffices to interpret the above partial integration formula in a generalized sense, using Stieltjes integration.

Proof of Theorem 2.1 (ii) ⇔ (iii)

  • Conversely, given a law invariant convex function EQUATION where −q −Y runs through D when Y ranges through P(Ω, F, P).
  • This establishes a bijective correspondence between the functions V and v as appearing in items (ii) and (iii) of Theorem 2.4.
  • The authors then have EQUATION ) Note that the latter integral is just the term appearing in (5).
  • Hence the authors have found a (lower semi-continuous, convex) function v, defined on P([0, 1]), such that the above infima coincide.
  • The authors also remark that the above proof also shows that in item (ii) above one may equivalently drop the word "convex" and/or the word "lower semicontinuous".

4 The Fatou property for law invariant utility functions

  • In this section the authors shall prove Theorem 2.2 which will follow from the subsequent result whose proof will be reported at the end of this section.
  • In Lemma A.5 below the authors justify that in their setting they may use the notions of law invariance and transformation invariance synonymously.
  • Admitting the above Proposition 4.1 the proof of Theorem 2.2 is straightforward.
  • To prepare the proof of Proposition 4.1 the authors need some auxiliary results.

5 The Lebesgue property for law invariant utility functions

  • The authors first state (without proof) an easy result from measure theory which will be used in the implication (ii) =⇒ (i) below.
  • Hence the authors only have to show the reverse inequality of (25).
  • In the above proof of the equivalence of (i) and (ii) of Theorem 2.4 the authors have not used the law invariance of U and V respectively, which is irrelevant for this equivalence (while for the formulation of (iii) it is, of course, indispensable).
  • As these results are somewhat scattered in the previous literature [D 03], [FS 04], [K 05] the authors resume them in the subsequent theorem and give proofs.
  • Assume that for every EQUATION is attained and let us work towards a contradiction.

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Law Invariant Risk Measures have
the Fatou Property
E. Jouini
W. Schachermayer
N. Touzi
§
Abstract
S. Kusuoka [K 01, Theorem 4] gave an interesting dual characteriza-
tion of law invariant coherent risk measures, satisfying the Fatou prop-
erty. The latter property was introduced by F. Delbaen [D 02]. In the
present note we extend Kusuoka’s characterization in two directions, the
first one being rather standard, while the second one is somewhat sur-
prising. Firstly we generalize similarly as M. Fritelli and E. Rossaza
Gianin [FG 05] from the notion of coherent risk measures to the more
general notion of convex risk measures as introduced by H. ollmer and
A. Schied [FS 04]. Secondly and more importantly we show that
the hypothesis of Fatou property may actually b e dropped as it is au-
tomatically implied by the hyp othesis of law invariance.
We also introduce the notion of the Lebesgue property of a convex
risk measure, where the inequality in the definition of the Fatou property
is replaced by an equality, and give some dual characterizations of this
property.
1 Introduction
This paper is a twin to [JST 05] and we shall use similar notation. In particular
we rather use the language of “monetary utility functions” which up to the
We thank S. Kusuoka, P. Orihuela and A. Schied for their advise and help in preparing
this paper.
Universit´e Paris Dauphine and CEREMADE, Place du Mar´echal de Lattre de Tassigny,
F-75775 Paris Cedex 16, France.
Vienna University of Technology, Wiedner Hauptstrasse 8-10/105, A-1040 Wien, Austria
and Universit´e Paris Dauphine, Place du Mar´echal de Lattre de Tassigny, F-75775 Paris
Cedex 16, France. Financial support from the Austrian Science Fund (FWF) under the
grant P15889 and from Vienna Science and Technology Fund (WWTF) under Grant MA13
is gratefully acknowledged.
§
CREST, Laboratoire de Finance et Assurance, 15 Bd Gabriel eri, F-92245 Malakoff
Cedex, France and Universit´e Paris Dauphine, Place du Mar´echal de Lattre de Tassigny,
F-75775 Paris Cedex 16, France.
1

sign is identical to the notion of convex risk measures [FS 04]. We do so in
order to point out more directly how the present theory is embedded into the
framework of classical utility theory.
Throughout the paper we work on a standard probability space (Ω, F, P),
i.e., we suppose that (Ω, F, P) does not have atoms and that L
2
(Ω, F, P) is
separable.
A monetary utility function is a concave non-decreasing map U :
L
(Ω, F, P) [−∞, [ with dom(U) = {X | U(X) R} 6= , and
U(X + c) = U(X) + c, for X L
, c R.
Note that a monetary utility function is Lipschitz with respect to k . k
, and
that dom(U) = L
. By adding a constant to U if necessary, we may and shall
always assume that U(0) = 0.
Defining ρ(X) = U(X) the above definition of a monetary utility function
yields the definition of a convex risk measure [FS 04]. Convex risk measures are
in turn a generalization of the concept of coherent risk measures [ADEH 97],
which are particularly relevant in applications, and where one imposes the
additional requirement of positive homogeneity ρ(λX) = λρ(X), for X L
and λ 0. A characterization of coherent (resp. convex) risk measures ρ :
L
(Ω, F, P) R in terms of their Fenchel transform, defined on L
1
(Ω, F, P),
was obtained in [D 02] under the condition that ρ satisfies the Fatou property,
i.e.,
ρ(X) lim inf
n→∞
ρ(X
n
) whenever sup
n
kX
n
k
< and X
n
P
X, (1)
where
P
denotes convergence in probability. In the present context, this
condition is equivalent to the upper semi-continuity condition with respect to
the σ(L
, L
1
)-topology.
For fixed X L
1
(Ω, F, P), we introduce the function
U
α
(X) := α
1
Z
α
0
q
X
(β) , α ]0, 1[ , (2)
U
0
(X) = ess inf(X), and U
1
(X) = E[X], where q
X
denotes the quantile func-
tion of the random variable X, i.e. the generalized inverse of its cumulative
distribution function (see (3) below). For every α [0, 1], U
α
is a positively
homogeneous monetary utility function, which is in addition law invariant.
The corresponding coherent risk measure ρ
α
= U
α
is the so-called average
value at risk at level α, sometimes denoted by AV@R
α
(see [FS 04]). The fam-
ily {U
α
, 0 α 1} plays an important role as any law invariant monetary
utility function U may be represented in terms of the utility functions U
α
,
α [0, 1]. This result was obtained by [K 01] in the context of coherent risk
measures, and later extended by [FG 05] to the context of convex risk mea-
sures, see also [FS 04], Theorem 4.54 and 4.57 as well as Corollary 4.72. The
precise statement of this result is the following.
2

Theorem 1.1 Suppose that (Ω, F, P) is a standard probability space. For a
function U : L
(Ω, F, P) R the following are equivalent:
(a) U is a law invariant monetary utility function satisfying the Fatou property.
(b) There is a convex function v : P([0, 1]) [0, ] such that
U(X) = inf
m∈P([0,1])
Z
1
0
U
α
(X)dm(α) + v(m)
for every X L
.
Here, P([0, 1]) denotes the set of all Borel probability measures on the
compact space [0, 1]. The crucial observation of Kusuoka [K 01] is that, for law
invariant monetary utility functions, condition (b) is equivalent to
(c) There is a law invariant, lower semi-continuous, convex function V :
L
1
(Ω, F, P) [0, ] such that dom(V ) P(Ω, F, P) and
U(X) = inf
Y L
1
{E[XY ] + V (Y )} for every X L
,
where P(Ω, F, P) denotes the set of Pabsolutely continuous probability mea-
sures on (Ω, F, P), which we identify with a subset of L
1
(Ω, F, P). For com-
pleteness, we report a proof of the equivalence between conditions (b) and (c)
in Section 3.
The equivalence of (a) and (c) is due to F. Delbaen in the framework of (not
necessarily law invariant) coherent risk measures [D 02], and was extended to
convex risk measures in [FS 04].
The first main contribution of this paper is to drop the Fatou property in
condition (a) of the above Theorem 1.1 by proving that it is automatically
satisfied by law-invariant monetary utility functions. In fact, we prove more
generally that the Fatou property is implied by the concavity, the L
-u.s.c.
and the law-invariance properties. This result is stated in Section 2 and proved
in Section 4. The reader only interested in this result may directly proceed to
these sections.
We next introduce the following natural notion.
Definition 1.2 A utility function U : L
R {−∞} satisfies the Lebesgue
property if for every uniformly bounded sequence (X
n
)
n=1
tending a.s. to X
we have
U(X) = lim
n→∞
U(X
n
) .
Clearly the Lebesgue property is a stronger condition than the Fatou prop-
erty defined in (1), as the inequality has b e en replaced by an equality. In
fact, this property was under different names already investigated in the
previous literature, as was kindly pointed out to us by A. Schied.
3

The second contribution of this paper is a characterization of the Lebesgue
property for a monetary utility function U in terms of the corresponding
Fenchel transform V introduced in condition (c) of Theorem 1.1. If in addition
U is law-invariant, this implies a characterization in terms of the function v
introduced in the above Theorem 1.1 (b). These results are stated in Section
2 and proved in Section 5.
2 AV@R representation of law-invariant mon-
etary utilities
2.1 Definitions
Let (Ω, F, P) be an atomless probability space, and assume that L
2
(Ω, F, P)
is separable. The assumption of F being free of atoms is crucial (otherwise
one is led to combinatorial problems which are irrelevant from the economic
point of view). On the other hand, the separability assumption is convenient
for the arguments below, but does not reduce the generality: indeed in all the
arguments below we shall only encounter (at most) countably many random
variables (X
n
)
n=1
; hence we may assume w.l.g. that the σ-algebra F is gener-
ated by countably many random variables, i.e., that L
2
(Ω, F, P) is separable.
We denote by P(Ω, F, P) the set of P-absolutely continuous probability
measures on (Ω, F, P), which we identify with a subset of L
1
(Ω, F, P). We also
denote by P([0, 1]) (resp. P(]0, 1])) the set of all Borel probability measures
on the compact space [0, 1] (resp. on the locally compact space ]0, 1]).
A measure preserving transformation of (Ω, F, P) is a bi-measurable bi-
jection τ : leaving P invariant, i.e., τ (P) = P. For 1 p ,
the transformation τ induces an isometric isomorphism, still denoted by τ, on
L
p
(Ω, F, P), mapping X to X τ.
A map f : L
R is called law invariant, if f(X) depends only on the
law of X for every X L
. The function f is called transformation invariant
if f τ = f for every measure preserving transformation τ, where we abuse
notations by writing f τ (X) := f(X τ).
We shall verify in Lemma A.4 that these notions of law invariance and
transformation invariance may be used in a synonymous way in the present
context of monetary utility function, as a consequence of the concavity and
the k . k
-continuity property of monetary utility functions.
An important example of law invariant and transformation invariant func-
tion is the so-called quantile function defined by
q
X
(α) := inf {x R | P[X x] α} , X L
, α [0, 1] . (3)
The functions U
α
, 0 α 1, introduced in (2) provide the simplest example
of law invariant monetary utility functions, which correspond to the so-called
4

average value at risk.
2.2 Strong and weak upper semi-continuity of law in-
variant maps
We now have assembled all the concepts that are needed to formulate our first
main result.
Theorem 2.1 Suppose that (Ω, F, P) is a standard probability space. For a
function U : L
(Ω, F, P) R the following are equivalent:
(i) U is a law invariant monetary utility function.
(ii) There is a law invariant, lower semi-continuous, convex function V :
L
1
(Ω, F, P) [0, ] such that dom(V ) P(Ω, F, P) and
U(X) = inf
Y L
1
{E[XY ] + V (Y )} for X L
.
(iii) There is a convex function v : P([0, 1]) [0, ] such that
U(X) = inf
m∈P([0,1])
Z
1
0
U
α
(X)dm(α) + v(m)
for X L
.
If any of these conditions is satisfied, then U satisfies the Fatou property.
This result shows that law invariant monetary utility functions admit a
representation in terms of the corresponding Fenchel transform without any
further assumption. In particular the AV@R representation of such utility
functions holds without any further condition. Our novel contribution is that
the Fatou property is automatically implied by the law invariance and the
strong upper semi-continuity; recall that monetary utility functions are L
-
Lipschitz continuous. We state this fact in a slightly more general framework:
Theorem 2.2 Suppose that (Ω, F, P) is a standard probability space. Let U :
L
R {−∞} be a concave function, which is law invariant and u.s.c. with
respect to the topology induced by k . k
. Then U is u.s.c. with respect to the
σ(L
, L
1
)-topology.
We prove this result, which we consider as the main contribution of this
paper, in section 4. The reader only interested in Theorem 2.2 may directly
proceed to this section.
Finally we observe that Theorem 2.1 implies in particular that in Theorem 7
of [K 01] the assumption of the Fatou property may also be dropped. For the
sake of completeness we formulate this result.
A monetary utility function U : L
R is called comonotone if U(X
1
+
X
2
) = U(X
1
) + U(X
2
), for comonotone X
1
, X
2
L
(compare [JST 05]).
Note that this implies in particular that U is positively homogeneous, i.e.,
5

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Frequently Asked Questions (11)
Q1. What are the contributions in this paper?

The latter property was introduced by F. Delbaen [ D 02 ]. Firstly the authors generalize — similarly as M. Fritelli and E. Rossaza Gianin [ FG05 ] — from the notion of coherent risk measures to the more general notion of convex risk measures as introduced by H. Föllmer and A. Schied [ FS 04 ]. Secondly — and more importantly — the authors show that the hypothesis of Fatou property may actually be dropped as it is automatically implied by the hypothesis of law invariance. The authors also introduce the notion of the Lebesgue property of a convex risk measure, where the inequality in the definition of the Fatou property is replaced by an equality, and give some dual characterizations of this property. 

Their novel contribution is that the Fatou property is automatically implied by the law invariance and the strong upper semi-continuity; recall that monetary utility functions are L∞Lipschitz continuous. 

Denoting by ΠN the set of permutations of {1, . . . , N}, the elementX := 1N ! ∑ π∈ΠN X ◦ τπis in D, as D is convex and transformation invariant (Lemma A.4). 

D↘: indeed, by considering f ∧ c, for c > 0 renormalizing and letting c → ∞, one reduces to the case of bounded f ; for general bounded f ∈ 

(3)The functions Uα, 0 ≤ α ≤ 1, introduced in (2) provide the simplest example of law invariant monetary utility functions, which correspond to the so-calledaverage value at risk. 

The authors define the map T : D↘ → M(]0, 1]) by T (f) = m where the measure m on the locally compact space ]0, 1] is defined bydm(x) = −xdf(x), x ∈]0, 1]. (6)To verify that (6) well-defines a probability measure on ]0, 1] suppose first that f is differentiable and bounded on ]0, 1]. 

In the above theorem, the only requirement on the probability space is that L1(Ω,F ,P) is separable (we need this assumption for the implication (v)⇒(iv)). 

As in [JST05] the authors denote by D↘ the set of non-increasing, right continuous, R+-valued functions f on ]0, 1] such that f(1) = 0 and‖f‖1 = ∫ 10f(x)dx = 1. 

Step 2: Now suppose that G is finite, hence generated by a partition {B1, . . . , Bn} of Ω with P[Bj] > 0.In this case it suffices to apply step 1 on each atom Bj to obtain the conclusion of the lemma. 

By adding a constant to U if necessary, the authors may and shall always assume that U(0) = 0.Defining ρ(X) = −U(X) the above definition of a monetary utility function yields the definition of a convex risk measure [FS 04]. 

The authors still have to verify that the function V in Theorem 2.1 (ii) may be assumed to be lower semi-continuous with respect to ‖ . ‖1. In fact, this is a triviality: the authors may always pass from a law invariant, convex function Ṽ : L1 → [0,∞] to its lower semi-continuous envelope V , i.e., the largest lower semi-continuous function dominated by V .