Law invariant risk measures have the Fatou property
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.
Did you find this useful? Give us your feedback
Citations
2,443 citations
299 citations
260 citations
231 citations
227 citations
References
8,651 citations
1,286 citations
748 citations
608 citations
Related Papers (5)
Frequently Asked Questions (11)
Q2. What is the novel contribution of the Fatou 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.
Q3. What is the proof of Lemma 4.2?
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).
Q4. what is the case for arbitrary f D?
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 ∈
Q5. What is the simplest example of law invariant monetary utility functions?
(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.
Q6. how do the authors verify that f is differentiable?
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].
Q7. What is the only requirement on the probability space?
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)).
Q8. How do the authors denote the set of non-increasing, right continuous,?
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.
Q9. What is the simplest way to conclude the lemma?
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.
Q10. What is the definition of a convex risk measure?
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].
Q11. What is the function V in the Theorem 2.1?
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 .