This document discusses the generalization of comonotonicity to multivariate risks.
[1] Comonotonicity in one dimension means two risks are maximally correlated through a common underlying risk factor. The document explores generalizing this concept to multiple dimensions when risks have several components.
[2] -Comonotonicity is introduced as a generalization where two multivariate risks are -comonotonic if they can be expressed as functions of a common underlying risk vector through convex functions.
[3] -Comonotonicity reduces to classical comonotonicity in one dimension but depends on the baseline distribution - in higher dimensions. Applications to risk measures and efficient risk sharing are discussed.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
Recently, there has been a surge in activity at the interface of optimal transport and statistics (with special emphasis on machine learning applications). The talk will summarize new results and challenges in this active area. For example, we will show how many of the most popular estimators in machine learning (such as Lasso and svm's) can be interpreted as games. This interpretation opens the door for new and potentially better estimators and algorithms, as well as questions about the underlying complexity of these new class of estimators.
(This talk is based on joint work with F. He, Y. Kang, K. Murthy, and F. Zhang)
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
Recently, there has been a surge in activity at the interface of optimal transport and statistics (with special emphasis on machine learning applications). The talk will summarize new results and challenges in this active area. For example, we will show how many of the most popular estimators in machine learning (such as Lasso and svm's) can be interpreted as games. This interpretation opens the door for new and potentially better estimators and algorithms, as well as questions about the underlying complexity of these new class of estimators.
(This talk is based on joint work with F. He, Y. Kang, K. Murthy, and F. Zhang)
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
This paper focuses on showing that much of the theoretical part of linear algebra works fairly well without determinants and provides proofs for most of the major structure, theorems of linear algebra without resorting to determinants.
A brief discussion of Multivariate Gaussin, Rayleigh & Rician distributions
Prof. H.Amindavar complementary notes for the first session of "Advanced communications theory" course, Spring 2021
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
This paper focuses on showing that much of the theoretical part of linear algebra works fairly well without determinants and provides proofs for most of the major structure, theorems of linear algebra without resorting to determinants.
A brief discussion of Multivariate Gaussin, Rayleigh & Rician distributions
Prof. H.Amindavar complementary notes for the first session of "Advanced communications theory" course, Spring 2021
A Note on “ Geraghty contraction type mappings”IOSRJM
In this paper, a fixed point result for Geraghty contraction type mappings has been proved. Karapiner [2] assumes to be continuous. In this paper, the continuity condition of has been replaced by a weaker condition and fixed point result has been proved. Thus the result proved generalizes many known results in the literature [2-7].
Noise is unwanted sound considered unpleasant, loud, or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arises when the brain receives and perceives a sound.
Talk at the modcov19 CNRS workshop, en France, to present our article COVID-19 pandemic control: balancing detection policy and lockdown intervention under ICU sustainability
SAP Sapphire 2024 - ASUG301 building better apps with SAP Fiori.pdfPeter Spielvogel
Building better applications for business users with SAP Fiori.
• What is SAP Fiori and why it matters to you
• How a better user experience drives measurable business benefits
• How to get started with SAP Fiori today
• How SAP Fiori elements accelerates application development
• How SAP Build Code includes SAP Fiori tools and other generative artificial intelligence capabilities
• How SAP Fiori paves the way for using AI in SAP apps
Dr. Sean Tan, Head of Data Science, Changi Airport Group
Discover how Changi Airport Group (CAG) leverages graph technologies and generative AI to revolutionize their search capabilities. This session delves into the unique search needs of CAG’s diverse passengers and customers, showcasing how graph data structures enhance the accuracy and relevance of AI-generated search results, mitigating the risk of “hallucinations” and improving the overall customer journey.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
Climate Impact of Software Testing at Nordic Testing DaysKari Kakkonen
My slides at Nordic Testing Days 6.6.2024
Climate impact / sustainability of software testing discussed on the talk. ICT and testing must carry their part of global responsibility to help with the climat warming. We can minimize the carbon footprint but we can also have a carbon handprint, a positive impact on the climate. Quality characteristics can be added with sustainability, and then measured continuously. Test environments can be used less, and in smaller scale and on demand. Test techniques can be used in optimizing or minimizing number of tests. Test automation can be used to speed up testing.
Climate Impact of Software Testing at Nordic Testing Days
Galichon jds
1. Multivariate comonotonicity,
stochastic orders and risk
measures
Alfred Galichon
(Ecole polytechnique)
Brussels, May 25, 2012
Based on collaborations with:
– A. Charpentier (Rennes) – G. Carlier (Dauphine)
– R.-A. Dana (Dauphine) – I. Ekeland (Dauphine)
– M. Henry (Montréal)
2. This talk will draw on four papers:
[CDG]. “Pareto e¢ ciency for the concave order and mul-
tivariate comonotonicity”. Guillaume Carlier, Alfred Gali-
chon and Rose-Anne Dana. Journal of Economic Theory,
2012.
[CGH] “"Local Utility and Multivariate Risk Aversion”.
Arthur Charpentier, Alfred Galichon and Marc Henry.
Mimeo.
[GH] “Dual Theory of Choice under Multivariate Risks”.
Alfred Galichon and Marc Henry. Journal of Economic
Theory, forthcoming.
[EGH] “Comonotonic measures of multivariate risks”. Ivar
Ekeland, Alfred Galichon and Marc Henry. Mathematical
Finance, 2011.
3. Introduction
Comonotonicity is a central tool in decision theory, insur-
ance and …nance.
Two random variables are « comonotone » when they are
maximally correlated, i.e. when there is a nondecreasing
map from one to another. Applications include risk mea-
sures, e¢ cient risk-sharing, optimal insurance contracts,
etc.
Unfortunately, no straightforward extension to the multi-
variate case (i.e. when there are several numeraires).
The goal of this presentation is to investigate what hap-
pens in the multivariate case, when there are several di-
mension of risk. Applications will be given to:
– Risk measures, and their aggregation
– E¢ cient risk-sharing
– Stochastic ordering.
4. 1 Comonotonicity and its general-
ization
1.1 One-dimensional case
Two random variables X and Y are comonotone if there
exists a r.v. Z and nondecreasing maps TX and TY such
that
X = TX (Z ) and Y = TY (Z ) :
For example, if X and Y are sampled from empirical
distributions, X (! i) = xi and Y (! i) = yi, i = 1; :::; n
where
x1 ::: xn and y1 ::: yn
then X and Y are comonotonic.
5. By the rearrangement inequality (Hardy-Littlewood),
n
X n
X
max xiy (i) = xi y i :
permutation
i=1 i=1
More generally, X and Y are comonotonic if and only if
h i
~
max E X Y = E [XY ] :
~
Y =d Y
6. Example. Consider
! !1 !2
P (! ) 1=2 1=2
X (! ) +1 1
Y (! ) +2 2
~
Y (! ) 2 +2
X and Y are comonotone.
~
Y has the same distribution as Y but is not comonotone
with X .
One has
h i
E [XY ] = 2 > ~
2 = E XY :
7. Hardy-Littlewood inequality. The probability space is
now [0; 1]. Assume U (t) = (t), where is nonde-
creasing.
Let P a probability distribution, and let
X (t) = FP 1(t):
~ ~
For X : [0; 1] ! R a r.v. such that X P , one has
Z 1 h i
1 ~
E [XU ] = (t)FP (t)dt E XU :
0
Thus, letting
Z 1 n o
1 ~ ~
%( X ) = (t)FX (t)dt = max E[XU ]; X =d X
0 n o
~ ~
= max E[X U ]; U =d U :
8. A geometric characterization. Let be an absolutely
continuous distribution; two random variables X and Y
are comonotone if for some random variable U , we
have
n o
~
~ ~
U 2 argmaxU E[X U ]; U , and
n o
~
~ ~
U 2 argmaxU E[Y U ]; U :
Geometrically, this means that X and Y have the same
projection of the equidistribution class of =set of r.v.
with distribution .
9. 1.2 Multivariate generalization
Problem: what can be done for risks which are multidi-
mensional, and which are not perfect substitutes?
Why? risk usually has several dimension (price/liquidity;
multicurrency portfolio; environmental/…nancial risk, etc).
Concepts used in the univariate case do not directly ex-
tend to the multivariate case.
10. The variational characterization given above will be the
basis for the generalized notion of comonotonicity given
in [EGH].
De…nition ( -comonotonicity). Let be an atomless
probability measure on Rd. Two random vectors X and
Y in L2 are called -comonotonic if for some random
d
vector U , we have
n o
~
~
U 2 argmaxU E[X U ]; ~
U , and
n o
~
U 2 argmax ~ E[Y U ]; ~
U
U
equivalentely:
X and Y are -comonotonic if there exists two convex
functions V1 and V2 and a random variable U such
that
X = rV1 (U )
Y = rV2 (U ) :
Note that in dimension 1, this de…nition is consistent with
the previous one.
11. Monge-Kantorovich problem and Brenier theorem
Let and P be two probability measures on Rd with
second moments, such that is absolutely continuous.
Then
sup E [hU; Xi]
U ;X P
where the supremum is over all the couplings of and P if
attained for a coupling such that one has X = rV (U )
almost surely, where V is a convex function Rd ! R
which happens to be the solution of the dual Kantorovich
problem
Z Z
inf V (u) d (u) + W (x) dP (x) :
V (u)+W (x) hx;ui
Call QP (u) = rV (u) the -quantile of distribution P .
12. Comonotonicity and transitivity.
Puccetti and Scarsini (2010) propose the following de…n-
ition of comonotonicity, called c-comonotonicity: X and
Y are c-comonotone if and only if
n o
~
~ ~
Y 2 argmaxY E[X Y ]; Y Y
or, equivalently, i¤ there exists a convex function u such
that
Y 2 @u (X )
that is, whenever u is di¤erentiabe at X ,
Y = ru (X ) :
However, this de…nition is not transitive: if X and Y are
c-comonotone and Y and Z are c-comonotone, and if the
distributions of X , Y and Z are absolutely continuous,
then X and Z are not necessarily c-comonotome.
This transivity (true in dimension one) may however be
seen as desirable.
13. In the case of -comonotonicity, transitivity holds: if X
and Y are -comonotone and Y and Z are -comonotone,
and if the distributions of X , Y and Z are absolutely con-
tinuous, then X and Z are -comonotome.
Indeed, express -comonotonicity of X and Y : for some
U ,
X = rV1 (U )
Y = rV2 (U )
~
and by -comonotonicity of Y and Z , for some U ,
Y ~
= rV2 U
~
Z = rV3 U
~
this implies U = U , and therefore X and Z are -
comonotone.
14. Importance of . In dimension one, one recovers the
classical notion of comotonicity regardless of the choice of
. However, in dimension greater than one, the comonotonic-
ity relation crucially depends on the baseline distribution
, unlike in dimension one. The following lemma from
[EGH] makes this precise:
Lemma. Let and be atomless probability measures
on Rd. Then:
- In dimension d = 1, -comonotonicity always implies
-comonotonicity.
- In dimension d 2, -comonotonicity implies -comonotonicity
if and only if = T # for some location-scale transform
T (u) = u + u0 where > 0 and u0 2 Rd. In other
words, comonotonicity is an invariant of the location-
scale family classes.
15. 2 Applications to risk measures
2.1 Coherent, regular risk measures (uni-
variate case)
Following Artzner, Delbaen, Eber, and Heath, recall the
classical risk measures axioms:
Recall axioms:
De…nition. A functional % : L2 ! R is called a coherent
d
risk measure if it satis…es the following properties:
- Monotonicity (MON): X Y ) %(X ) %(Y )
- Translation invariance (TI): %(X +m) = %(X )+m%(1)
- Convexity (CO): %( X + (1 )Y ) %(X ) + (1
)%(Y ) for all 2 (0; 1).
- Positive homogeneity (PH): %( X ) = %(X ) for all
0.
16. De…nition. % : L2 ! R is called a regular risk measure
if it satis…es:
~
- Law invariance (LI): %(X ) = %(X ) when X X . ~
- Comonotonic additivity (CA): %(X + Y ) = %(X ) +
%(Y ) when X; Y are comonotonic, i.e. weakly increasing
transformation of a third randon variable: X = 1 (U )
and Y = 2 (U ) a.s. for 1 and 2 nondecreasing.
Result (Kusuoka, 2001). A coherent risk measure % is
regular if and only if for some increasing and nonnegative
function on [0; 1], we have
Z 1
%(X ) := (t)FX 1(t)dt;
0
where FX denotes the cumulative distribution functions
of the random variable X (thus QX (t) = FX 1(t) is the
associated quantile).
% is called a Spectral risk measure. For reasons explained
later, also called Maximal correlation risk measure.
17. Leading example: Expected shortfall (also called Con-
ditional VaR or TailVaR): (t) = 1 1 1ft g: Then
Z 1
1
%(X ) := FX 1(t)dt:
1
18. Kusuoka’ result, intuition.
s
Law invariance ) %(X ) = FX 1
Comonotone additivity+positive homogeneity )
is linear w.r.t. FX 1:
R1
FX 1 = 0 (t)FX 1(t)dt.
Monotonicity ) is nonnegative
Subadditivity ) is increasing
Unfortunately, this setting does not extend readily to mul-
tivariate risks. We shall need to reformulate our axioms in
a way that will lend itself to easier multivariate extension.
19. 2.2 Alternative set of axioms
Manager supervising several N business units with risk
X1; :::; XN .
Eg. investments portfolio of a fund of funds. True
economic risk of the fund X1 + ::: + XN .
Business units: portfolio of (contingent) losses Xi report
a summary of the risk %(Xi) to management.
Manager has limited information:
1) does not know what is the correlation of risks - and
more broadly, the dependence structure, or copula be-
tween X1; :::; XN . Maybe all the hedge funds in the
portfolio have the same risky exposure; maybe they have
independent risks; or maybe something inbetween.
2) aggregates risk by summation: reports %(X1) + ::: +
%(XN ) to shareholders.
20. Reported risk: %(X1)+ ::: + %(XN ); true risk: %(X1 +
::: + XN ).
Requirement: management does not understate risk to
shareholders. Summarized by
%(X1) + ::: + %(XN ) ~ ~
%(X1 + ::: + XN ) (*)
whatever the joint dependence (X1; :::; XN ) 2 (L1)2.
d
But no need to be overconservative:
%(X1)+:::+%(XN ) = sup %(X1+:::+XN )
~ ~
X1 X1 ;:::;XN XN
where denotes equality in distribution.
De…nition. A functional % : L2 ! R is called a strongly
d
coherent risk measure if it is convex continuous and for
N
all (Xi)i N 2 L2 ,
d
n o
~ ~ ~
%(X1)+:::+%(XN ) = sup %(X1 + ::: + XN ) : Xi Xi :
21. A representation result.
The following result is given in [EGH].
Theorem. The following propositions about the func-
tional % on L2 are equivalent:
d
(i) % is a strongly coherent risk measure;
(ii) % is a max correlation risk measure, namely there
exists U 2 L2 , such that for all X 2 L2 ,
d d
n o
~ ~
%(X ) = sup E[U X ] : X X ;
(iii) There exists a convex function V : Rd ! R such
that
%(X ) = E [U rV (U )]
22. n
~
Idea of the proof . One has %(X )+ %(Y ) = sup %(X + Y ) :
~ ~
But %(X + Y ) = %(X ) + D%X (Y ) + o ( )
h i
~ ~
By the Riesz theorem (vector case) D%X (Y ) = E mX :Y ,
thus
n h i o
~ ~
%(X )+ %(Y ) = sup %(X ) + E mX :Y + o ( ) : Y Y
thus
n h i o
~ ~
%(Y ) = sup E mX :Y : Y Y
therefore % is a maximum correlation measure.
23. 3 Application to e¢ cient risk-sharing
Consider a risky payo¤ X (for now, univariate) to be
shared between 2 agents 1 and 2, so that in each contin-
gent state:
X = X1 + X2
X1 and X2 are said to form an allocation of X.
Agents are risk averse in the sense of stochastic domi-
nance: Y is preferred to X if every risk-averse expected
utility decision maker prefers Y to X:
X cv Y i¤ E[u(X )] E[u(Y )] for all concave u
Agents are said to have concave order preferences. These
are incomplete preferences: it can be impossible to rank
X and Y.
24. One wonders what is the set of e¢ cient allocations, i.e.
allocations that are not dominated w.r.t. the concave
order for every agent.
Dominated allocations. Consider a random variable X
(aggregate risk). An allocation of X among p agents is
a set of random variables (Y1; :::; Yp) such that
X
Yi = X:
i
Given two allocations of X , Allocation (Yi) dominates
allocation (Xi) whenever
2 3 2 3
X X
E 4 ui (Yi)5 E 4 ui (Xi)5
i i
for every continuous concave functions u1; :::; up. The
domination is strict if the previous inequality is strict
whenever the ui’ are strictly concave.
s
Comonotone allocations. In the single-good case, it is
intuitive that e¢ cient sharing rules should be such that in
25. “better”states of the world, every agent should be better
of than in “worse” state of the world – otherwise there
would be some mutually agreeable transfer.
This leads to the concept of comonotone allocations. The
precise connection with stochastic dominance is due to
Landsberger and Meilijson (1994). Comonotonicity has
received a lot of attention in recent years in decision the-
ory, insurance, risk management, contract theory, etc.
(Landsberger and Meilijson, Ruschendorf, Dana, Jouini
and Napp...).
Theorem (Landsberger and Meilijson). Any allocation
of X is dominated by a comonotone allocation. More-
over, this dominance can be made strict unless X is al-
ready comonotone. Hence the set of e¢ cient allocations
of X coincides with the set of comonotone allocations.
This result generalizes well to the multivariate case. Up
to technicalities (see [CDG] for precise statement), ef-
…cient allocations of a random vector X is the set of
26. -comonotone allocations of X , hence (Xi) solves
Xi = rui (U )
X
Xi = X
i
for convex functions ui : Rd ! R, with U . Hence
X = ru (U )
P
with u = i ui. That is
U = ru (X ) ;
hence e¢ cient allocations are such that
Xi = rui ru (X ) :
This result opens the way to the investigation of testable
implication of e¢ ciency in risk-sharing in an risky endow-
ment economy.
27. 4 Application to stochastic orders
Quiggin (1992) shows that the notion of monotone mean
preserving increases in risk (hereafter MMPIR) is the
weakest stochastic ordering that achieves a coherent rank-
ing of risk aversion in the rank dependent utility frame-
work. MMPIR is the mean preserving version of Bickel-
Lehmann dispersion, which we now de…ne.
De…nition. Let QX and QY be the quantile functions
of the random variables X and Y . X is said to be
Bickel-Lehmann less dispersed, denoted X %BL Y , if
QY (u) QX (u) is a nondecreasing function of u on
(0; 1). The mean preserving version is called monotone
mean preserving increase in risk (MMPIR) and denoted
-M M P IR.
MMPIR is a stronger ordering than concave ordering in
the sense that X %M M P IR Y implies X %cv Y .
28. The following result is from Landsberger and Meilijson
(1994):
Proposition (Landsberger and Meilijson). A random
variable X has Bickel-Lehmann less dispersed distribution
than a random variable Y if and only i¤ there exists Z
comonotonic with X such that Y =d X + Z .
The concept of -comonotonicity allows to generalize this
notion to the multivariate case as done in [CGH].
De…nition. A random vector X is called -Bickel-Lehmann
less dispersed than a random vector Y , denoted X % BL
Y , if there exists a convex function V : Rd ! R such
that the -quantiles QX and QY of X and Y satisfy
QY (u) QX (u) = rV (u) for -almost all u 2 [0; 1]d.
As de…ned above, -Bickel-Lehmann dispersion de…nes a
transitive binary relation, and therefore an order. Indeed,
if X % BL Y and Y % BL Z , then QY (u) QX (u) =
29. rV (u) and QZ (u) QY (u) = rW (u). Therefore,
QZ (u) QX (u) = r(V (u) + W (u)) so that X % BL
Z . When d = 1, this de…nition simpli…es to the classical
de…nition.
[CGH] propose the following generalization of the Landsberger-
Meilijson characterization .
Theorem. A random vector X is -Bickel-Lehmann less
dispersed than a random vector Y if and only if there
exists a random vector Z such that:
(i) X and Z are -comonotonic, and
(ii) Y =d X + Z .
30. Conclusion
We have introduced a new concept to generalize comonotonic-
ity to higher dimension: “ -comonotonicity”. This con-
cept is based on Optimal Transport theory and boils down
to classical comonotonicity in the univariate case.
We have used this concept to generalize the classical ax-
ioms of risk measures to the multivariate case.
We have extended existing results on equivalence between
e¢ ciency of risk-sharing and -comonotonicity.
We have extended existing reults on functions increasing
with respect to the Bickel-Lehman order.
Interesting questions for future research: behavioural in-
terpretation of mu? computational issues? empirical
testability? case of heterogenous beliefs?