RAPPORT D ACTIVITE 2009

RAPPORT D ACTIVITE 2009

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LES PARTENAIRES Les institutions : L entreprise partenaire : Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 3

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Rapport d activité 2009 Chaire GROUPAMA-ENSAE-DAUPHINE Les Particuliers Face aux Risques Chaire de la Fondation du Risque Rapport d'activité - Année 2009 - INTRODUCTION... 7 1. AXES DE RECHERCHES DE LA CHAIRE... 9 1.1. Comportements individuels et collectifs face aux risques... 10 1.2. Décision en Finance et en Assurance... 10 1.3. Retraites, épargne, cycle de vie... 10 2. LISTE DES CHERCHEURS ASSOCIES A LA CHAIRE ET GOUVERNANCE... 11 2.1. Liste des chercheurs associés à la chaire... 12 2.2. Gouvernance de la chaire... 12 3. LISTE DES TRAVAUX DE RECHERCHE... 13 3.1. Articles publiés dans des revues à comité de lecture... 14 3.2. Working Papers rédigés ayant donné lieu à des communications dans des colloques ou séminaires... 20 a. Communications, colloques et séminaires internationaux... 20 b. Communications, colloques et séminaires nationaux... 25 3.3. Cahiers de la Chaire «Les particuliers face aux risques»... 27 4. WORKSHOP ORGANISES PAR LA CHAIRE... 31 4.1. Workshop «Dynamic Risk Sharing»... 32 Programme... 32 Book of abstracts... 34 4.2. Organisation d un Workshop intitulé «Econometrics of Hedge Funds»... 37 Programme... 37 Book of abstracts... 39 4.3. Organisation du colloque "7th International Workshop on Pension, Insurance and Saving... 42 Programme... 42 Book of abstracts... 44 5. AUTRES ACTIVITES DE VALORISATION... 49 5.1. Conférences... 50 5.2. Séminaires... 50 a. Séminaire "Economie du Risque", année 2009... 50 b. Séminaire Ageing and Risk, année 2009... 52 5.3. Invitation de chercheurs dans le cadre de la chaire sauf colloques et seminaires... 53 a. Invitation à Paris de chercheurs par des chercheurs de la Chaire... 53 b. Invitation de chercheurs de la chaire... 53 Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 5

5.4. Outils de valorisation et de diffusion auprès du grand public des cahiers de la chaire «Les particuliers face aux risques»... 54 a. Cahiers de la chaire «Les particuliers face aux risques» n 1, Janvier 2010... 54 b. Cahiers de la chaire «Les particuliers face aux risques» n 2, Avril 2010... 55 5.5. Revue Mathematics and Financial Economics... 55 a. Mathematics and Financial Economics, Volume 2, Number 2 / juillet 2009... 55 b. Mathematics and Financial Economics Volume 2, Number 3 / septembre 2009... 57 6. ENSEIGNEMENTS... 59 6.1. Partenariat avec les Masters 2... 60 6.2. Enseignements dans le cadre de la chaire en 2009... 61 7. NOUVEAUX PROJETS DE RECHERCHE LANCES EN 2010... 63 7.1. Demande d assurance et modèles non-additifs... 64 7.2. Analyse micro-économique des choix individuels en matière de prévention faits par des individus supportant un risque santé... 64 7.3. Nouvelles techniques pour le partage de risques multi-dimensionnels et applications à l Assurance... 65 7.4. L épargnant au bord de la crise financière... 66 7.5. Lancement de deux nouveaux cours à la rentrée 2010... 68 a) Behavioral Economics and Finance... 68 b) Méthodes de provisionnement en assurance... 68 8. FICHE BUDGETAIRE 2009... 69 ANNEXE 1: LISTE DES REVUES DONNANT LIEU A RACHAT DE DROITS D AUTEURS DANS LE CADRE DES CAHIERS DE LA CHAIRE GROUPAMA... 71 ANNEXE 2: CAHIERS DE LA CHAIRE «LES PARTICULIERS FACE AUX RISQUES» n 1 et n 2... 77 Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 6

INTRODUCTION Le but de la chaire «Les particuliers face aux risques» est de développer un pôle de réflexion et de recherche autour d une équipe de chercheurs confirmés. L objectif de la chaire est de consolider le niveau d'excellence en recherche en matière de théorie financière et plus précisément en finance des particuliers. Dans ce but il a été prévu de mobiliser et de solliciter les meilleurs chercheurs dans ce domaine. La thématique choisie est large de par la nécessité de diffuser les résultats auprès du plus grand nombre et d impliquer de nombreux chercheurs et opérateurs. L objectif cependant n est pas seulement de développer les aspects purement théoriques de la recherche, mais aussi de générer de nouvelles connaissances potentiellement utiles pour l industrie. La chaire se veut ainsi un lieu d'échange et de réflexion entre ses partenaires, les institutions académiques et les autres acteurs du secteur. L objectif est également de développer une culture spécifique de gestion des risques grâce à des enseignements spécialisés liés au thème de la chaire et qui seront produits dans le cadre de la chaire en partenariat avec des masters de dauphine ou de l Ensae. L effort financier sur le long terme effectué par Groupama garantit la pérennité et la permanence de la problématique choisie, ce qui permet d adapter les financements de la chaire au timing de la recherche qui est un temps long. Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 7

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1. AXES DE RECHERCHES DE LA CHAIRE Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 9

Les axes de recherche de la chaire ont été divisés en cinq grandes thématiques. Les délimitations de ces thématiques ne sont pas à prendre au sens strict, en effet certains thèmes peuvent se rapprocher de deux thématiques, voire plus. 1.1. Comportements individuels et collectifs face aux risques Dans cette thématique, il s agit d étudier les comportements d investissement des particuliers en matière d appréhension du risque : aversion au risque, pessimisme, optimisme, sur-confiance... Se pose alors le problème de l agrégation du comportement de ces agents ayant des croyances hétérogènes et de la formation et de l évolution des croyances des agents quand ceux-ci rentrent en interaction sur le marché. Les agents modifient-ils leur croyance par mimétisme, par apprentissage.? Répondre à ces questions permettra de modéliser les prévisions des agents à long terme en matière de risque, de déterminer le degré de divergence de ces prévisions et de rendre compte de manière théorique du principe de précaution. La recherche sur ce thème s effectue en partenariat avec des chercheurs de l IFD : Elyès Jouini, du laboratoire de gestion de dauphine DRM : Clotilde Napp et Selima Ben Mansour et de chercheurs du CEREMADE : Jean-Michel Marin, Filipe Martins et Jérôme Renault. 1.2. Décision en Finance et en Assurance Dans cette thématique de recherche il s agit de s intéresser à des nouveaux modèles d utilité (aversion à l ambiguïté, aux risques extrêmes, ) et de mesure des risques afin de déterminer des contrats optimaux de partage des risques entre les demandeurs et les offreurs. Il s agit aussi de travailler sur des nouveaux outils mathématiques de mesure des risques. La recherche sur ce thème s effectue surtout en partenariat avec le Ceremade: Luciano Campi, Rose-Anne Dana, Guillaume Carlier 1.3. Retraites, épargne, cycle de vie Il s agit ici de s intéresser à la gestion du risque de long terme de l épargne des particuliers: règles de gestion, impact des dispositifs d épargne privée de long terme sur l épargne des ménages, comportement d accumulation des ménages et risques sociaux (structure par âge, risques sur le financement des retraites, la santé, chômage, ), inégalités face à l espérance de vie, comportements d épargne et rendements différenciés des régimes de retraites, dynamique des risques et offre de fond prêtable au cours du cycle de vie. La recherche sur ce thème s effectue surtout en partenariat avec EURISCO autour de Najat el Mekkaoui de Freitas. Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 10

2. LISTE DES CHERCHEURS ASSOCIES A LA CHAIRE ET GOUVERNANCE Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 11

2.1. Liste des chercheurs associés à la chaire Head : Elyès Jouini, Université Paris-Dauphine Associates : Rose-Anne Dana, Université Paris-Dauphine (Economics of Risks Seminar) Najat El Mekkaoui de Freitas, Université Paris-Dauphine (Ageing and Risk Seminar) Affiliates : Kevin Beaubrun, Université Paris-Dauphine Selima Ben Mansour, Université Paris-Dauphine Bruno Bouchard, Université Paris-Dauphine Luciano Campi, Université Paris-Dauphine Guillaume Carlier, Université Paris-Dauphine Emmanuel Denis, Université Paris-Dauphine Françoise Forges, Université Paris-Dauphine Jean-Michel Marin, INRIA Clotilde Napp, Université Paris-Dauphine Filipe Martins da Rocha, Getulio Fargas Foundation Joaquim Oliveira Martins, OCDE Jérôme Renault, Université Paris-Dauphine Tristan Tomala, HEC Yannick Viossat, Université Paris-Dauphine 2.2. Gouvernance de la chaire Comité d Orientation : -Le président du Directoire : Jean-Michel BEACCO) -Le représentant de l Université Paris Dauphine : Edith GINGLINGER -Le représentant de l ENSAE : Sylviane GASTALDO -Le représentant de GROUPAMA : Thierry MARTEL -Personnalités extérieures : Christian GOLLIER et Pierre-André CHIAPPORI Réunion du Comité d Orientation le 9 février 2009 dans les locaux de la Fondation du Risque à 17h Comité de pilotage : -Le responsable du Projet : Elyes JOUINI -Le représentant de GROUPAMA : Thierry MARTEL -Le représentant de l Université Paris-Dauphine : Yves SIMON -Le représentant de l ENSAE : Laurence LESCOURRET -Personnalité extérieure : Christian GOLLIER Réunion du Comité de Pilotage de la chaire le 29 Avril 2009 dans les locaux de Groupama de 11h à 13h Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 12

3. LISTE DES TRAVAUX DE RECHERCHE Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 13

3.1. Articles publiés dans des revues à comité de lecture - Unbiased Disagreement in Financial Markets, Waves of Pessimism and the Risk- Return Tradeoff Elyes Jouini and Clotilde Napp to appear Review of Finance Abstract: Can investors with irrational beliefs be neglected as long as they are rational on average? Do their trades cancel out with no consequences on prices, as implicitly assumed by traditional models? We consider a model with irrational investors, who are rational on average. We obtain waves of pessimism and optimism that lead to countercyclical market prices of risk and procyclical risk-free rates. The variance of the state price density is greatly increased. The long run risk-return relation is modified in particular the long run market price of risk might be higher than both the instantaneous and the rational ones. - Discounting and Divergence of Opinion Elyes Jouini, Jean-Michel Marin, Clotilde Napp to appear Journal of Economic Theory Abstract: Agents impatience rate and their anticipations about the future of the economy, are two essential determinants of the equilibrium discount rate, as illustrated by the Ramsey formula. Heterogeneity in time preference rates and in anticipations is widely acknowledged. Our objective is to determine the equilibrium discount rate when this heterogeneity is taken into account. Among others we tackle the following questions: As an additional risk or uncertainty, can dispersion in agents' characteristics lead to lower discount rates? What is the asymptotic behavior of the discount rate in such a setting? More generally, what is the shape of the yield curve? - Multivariate utility maximization under proportional transaction costs Luciano Campi et Mark Owen à paraitre dans Finance and Stochastics Abstract: We present an optimal investment theorem for a currency exchange model with random and possibly discontinuous proportional transaction costs. The investor's preferences are represented by a multivariate utility function, allowing for simultaneous consumption of any prescribed selection of the currencies at a given terminal date. We prove the existence of an optimal portfolio process under the assumption of asymptotic satiability of the value function. Sufficient conditions for asymptotic satiability of the value function include reasonable asymptotic elasticity of the utility function, or a growth condition on its dual function. We show that the portfolio optimization problem can be reformulated in terms of maximization of a terminal liquidation utility function, and that both problems have a common optimizer. - Demand for concave Rank-Linear Utilities Rose-Anne Dana et Guillaume Carlier à paraître dans Mathematical Finance Abstract: We consider a class of law invariant utilities which contains the Rank Dependent Expected Utility (RDU) and the cumulative prospect theory (CPT). We show that the computation of demand for a contingent claim when utilities are within that class, although not as simple as in the Expected Utility (EU) case, is still tractable. Specific attention is given to the RDU and to the CPT cases. Numerous examples are fully solved. - Time Average Replicator and Best Reply Dynamics" Joseph Hofbauer, Sylvain Sorin et Yannick Viossat Mathematics of Operation Research, 34, 263-269 (2009) Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 14

Abstract: Using an explicit representation in terms of the logit map, we show, in a unilateral framework, that the time average of the replicator dynamics is a perturbed solution of the best-reply dynamics. - Properties and Applications of Dual Reduction" Yannick Viossat à paraître dans Economic Theory Abstract: The dual reduction process, introduced by Myerson, allows to reduce a finite game into a smaller dimensional game such that any equilibrium of the reduced game is an equilibrium of the original game. This holds both for Nash equilibrium and correlated equilibrium. We present examples of applications of dual reduction and argue that this is a useful tool to study Nash equilibria and correlated equilibria. We then investigate its properties. - Double Kernel estimation of sensitivities Romuald Elie Journal of Applied Probability, 46, 3, 781-811, (2009) Abstract: In this paper we address the general issue of estimating the sensitivity of the expectation of a random variable with respect to a parameter characterizing its evolution. In finance, for example, the sensitivities of the price of a contingent claim are called the Greeks. A new way of estimating the Greeks has recently been introduced in Elie, Fermanian and Touzi (2007) through a randomization of the parameter of interest combined with nonparametric estimation techniques. In this paper we study another type of estimator that turns out to be closely related to the score function, which is well known to be the optimal Greek weight. This estimator relies on the use of two distinct kernel functions and the main interest of this paper is to provide its asymptotic properties. Under a slightly more stringent condition, its rate of convergence is the same as the one of the estimator introduced in Elie, Fermanian and Touzi (2007) and outperforms the finite differences estimator. In addition to the technical interest of the proofs, this result is very encouraging in the dynamic of creating new types of estimator for the sensitivities. - Optimal control under stochastic target constraints Romuald Elie, Bruno Bouchard et Cyril Imbert to appear in SIAM Journal on Control and Optimization. Abstract: We study a class of Markovian optimal stochastic control problems in which the controlled v v d 1 process Z is constrained to satisfy an almost sure constraint Z ( T ) G R P a.s. at some final d time T>0. When the set is of the form G : x, y) R R : g x, y) 0, with g nondecreasing in y, we provide a Hamilton Jacobi Bellman characterization of the associated value function. It gives rise to a state constraint problem, where the constraint can be expressed in terms of an auxiliary value v d 1 v function w which characterizes the set D:= t, Z ( t)) [ 0, T ] R : Z ( T ) G. Contrary to standard state constraint problems, the domain D is not given a priori and we do not need to impose conditions on its boundary. It is naturally incorporated in the auxiliary value function w, which itself is a viscosity solution of a nonlinear parabolic PDE. Applying ideas recently developed in Bouchard, Elie, and Touzi [SIAM J. Control Optim., 48 (2009), pp. 3123 3150], our general result also allows us to consider optimal control problems with moment constraints of the form E[g(Z v ( T ))] 0 or v P[ g( Z ( T )) 0] p. - Stochastic target problems with control loss Bruno Bouchard, Romuald Elie et Nizar Touzi SIAM Journal on Control and Optimization, 48, 5, 3123-3150, (2009) Abstract: We consider the problem of finding the minimal initial data of a controlled process which guarantees to reach a controlled target with a given probability of success or, more generally, with a Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 15

given level of expected loss. By suitably increasing the state space and the controls, we show that this problem can be converted into a stochastic target problem, i.e. find the minimal initial data of a controlled process which guarantees to reach a controlled target with probability one. Unlike the existing literature on stochastic target problems, our increased controls are valued in an unbounded set. In this paper, we provide a new derivation of the dynamic programming equation for general stochastic target problems with unbounded controls, together with the appropriate boundary conditions. These results are applied to the problem of quantile hedging in Financial mathematics, and are shown to recover the explicit solution of Follmer and Leukert [1999]. - Discrete-Time Approximation of BSDEs and Probabilistic Schemes for Fully Nonlinear PDEs Bruno Bouchard, Romuald Elie and Nizar Touzi Radon Series Comp. Appl. Math., Advanced Financial Modelling, Ed. H. Albrecher, W. Runggaldier and W. Schachermayer, 2009. Abstract: The aim of this paper is to provide a survey on recent advances on probabilistic numerical methods for nonlinear PDEs, which serve as an alternative to classical deterministic schemes and allow to handle a large class of multidimensional nonlinear problems. These probabilistic schemes are based on the stochastic representation of semilinear PDEs by means of backward SDEs, which can be viewed as an extension of the well-known Feynman Kac formula to the semilinear case. In this context, we first explain how smoothness properties can be obtained for non-reflected BSDEs set in the whole domain by using purely probabilistic techniques introduced in Ma and Zhang, and how they can be exploited to provide convergence rates of discrete time Euler-type approximation schemes. We then present some recent extensions to BSDEs with jumps, reflected BSDEs, BSDEs with horizon given by a finite stopping time, and second order BSDEs. The extension to fully nonlinear PDEs requires to enlarge the definition of backward SDEs. However, a natural probabilistic numerical scheme can be introduced formally by evaluating the nonlinear PDE along the trajectory of some chosen underlying diffusion. This point of view shows an intimate connection between the probabilistic numerical schemes reviewed in this paper and the standard finite differences methods. In particular, convergence and error estimates are provided by using the monotone schemes methods from the theory of viscosity solutions. - Uniform value in Dynamic Programming Jérôme Renault to appear in Journal of the European Mathematical Society Abstract: We consider dynamic programming problems with a large time horizon, and give sufficient conditions for the existence of the uniform value. As a consequence, we obtain an existence result when the state space is precompact, payoffs are uniformly continuous and the transition correspondence is non expansive. In the same spirit, we give an existence result for the limit value. We also apply our results to Markov decision processes and obtain a few generalizations of existing results. - The obstacle version of the Geometric Dynamic Programming Principle: Application to the pricing of American options under constraints Bruno Bouchard et Than Nam Vu Applied Mathematics and Optimization, 61, 2, 235-265, (2009). Abstract: We provide an American version of the Geometric Dynamic Programming Principle of Soner and Touzi (2002) for stochastic target problems. This opens the doors to a wide range of applications, particularly in risk control in finance and insurance, in which a controlled stochastic process has to be maintained in a given set on a time interval [0,T]. As an example of application, we show how it can be used to provide a viscosity characterization of the super-heging price of American options under portfolio constraints, without appealing to the standard dual formulation from mathematical finance. In particular, we allow for a degenerate volatility, a case which does not seem to have been studied so far in this context. Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 16

- Representation of continuous linear forms on the set of ladlag processes and the pricing of American claims under proportional costs Bruno Bouchard et Jean-Francois Chassagneux Electronic Journal of Probability, 14, 612-632 (2009) Abstract: We discuss a d-dimensional version of a duality result by Meyer (1976) between bounded càdlàg adapted processes and random measures. We show that it allows to establish, in a very natural way, a dual representation for the set of initial endowments which allow to super-hedge a given American claim in a continuous time model with proportional transaction costs. It generalizes a previous result of Bouchard and Temam (2005) who considered a discrete time setting. It also completes the very recent work of Denis, De Vallière and Kabanov (2008) who restricted to càdlàg American claims and used a completely different approach. - Strong Approximations of BSDEs in a domain Bruno Bouchard et Stéphane Menozzi Bernoulli, 15, 4, 1117-1147 (2009) Abstract: We study the strong approximation of a Backward SDE with finite stopping time horizon, namely the first exit time of a forward SDE from a cylindrical domain. We use the Euler scheme approach of Bouchard and Touzi, Zhang. When the domain is piecewise smooth and under a non- 4 characteristic boundary condition, we show that the associated strong error is at most of order h where h denotes the time step and e is any positive parameter. This rate corresponds to the strong exit 1 2 time approximation. It is improved to h when the exit time can be exactly simulated or for a weaker form of the approximation error. Importantly, these results are obtained without uniform ellipticity condition. 1 - Equilibrium theory with asymmetric information and infinitely many states Victor-Filipe Martins da Rocha, Carlos Hervé-Beloso et Paulo Monteiro Economic Theory, 38, 2, 295-320, (2009). Abstract: Radner (1968) proved existence of a competitive equilibrium for differential information economies with finitely many states. We extend this result to economies with infinitely many states of nature. Each agent observes a public and a private signal. The publicly observed signal may take infinitely many values but, in order to get existence, we assume that private signals only take finitely many values. Actually, there is no hope to get a general existence result since Podczeck, Tourky and Yannelis (2008) already proposed non-existence results. - Large economies with differential information and without free disposal Victor-Filipe Martins da Rocha et Laura Angeloni Economic Theory, 38, 2, 263-286, (2009). Abstract: We consider exchange economies with a continuum of agents and differential information about finitely many states of nature. It was proved in Einy et al. (Econ Theory 18, 321 332, 2001) that if we allow for free disposal in the market clearing (feasibility) constraints then an irreducible economy has a competitive (or Walrasian expectations) equilibrium, and moreover, the set of competitive equilibrium allocations coincides with the private core. However when feasibility is defined with free disposal, competitive equilibrium allocations may not be incentive compatible and contracts may not be enforceable (see e.g. Glycopantis et al. in Econ Theory 21, 495 526, 2002). This is the main motivation for considering equilibrium solutions with exact feasibility. We first prove that the results in Einy et al. (Econ Theory 18, 321 332, 2001) are still valid without free-disposal. Then, motivated by the issue of contracts execution, we adapt the incentive compatibility property introduced in Krasa and Yannelis (Econometrica 62, 881 900, 1994) and we prove that every Pareto Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 17

optimal exact feasible allocation is incentive compatible, implying that contracts of competitive or core allocations are enforceable. - Secure Communication: A Mechanism Design Approach Louis Renou et Tristan Tomala A paraitre dans Proceedings of the 2009 International Conference on Game Theory for Networks, IEEE Catalog, number: CFP0982G-CDR. Abstract: This paper considers a mechanism design model where a designer, or receiver, takes an action based on the information received by multiple players, or senders. The agents, senders and receiver, communicate in a fixed directed network. We characterize the communication networks for which, in any environment (utilities and beliefs) with a worst action, every incentive compatible social choice function is implementable on the network. We show that this holds true if and only if the network is weakly 2-connected. A network is weakly 2-connected if for each player i, who is not directly connected to the designer, there exists a player k and two vertex-disjoint paths from k to the designer such that i lies on one of the two paths. We couple encryption techniques together with appropriate incentives to secure the transmission of each player's private information to the designer. - Unbounded exchange economies with satiation: How far can we go? Victor-Filipe Martins da Rocha et Paulo Monteiro Journal of Mathematical Economics, 45, 7-8, 465-478, (2009) Abstract: We unify and generalize the existence results in Werner (1987), Dana, Le Van and Magnien (1999), Allouch, Le Van and Page (2006) and Allouch and Le Van (2008). We also show that, in erms of weakening the set of assumptions, we cannot go too far. - Interim efficiency with MEU-preferences Victor Filipe Martins-da-Rocha A paraître dans Journal of Economic Theory Abstract: Recently Kajii and (2008) proposed to characterize interim efficient allocations in an exchange economy under asymmetric information when uncertainty is represented by multiple posteriors.when agents have Bewley's incomplete preferences, Kajii and Ui (2008) proposed a necessary and sufficient condition on the set of posteriors.however, when agents have Gilboa-- Schmeidler's MaxMin expected utility preferences, they only propose a sufficient condition.the objective of this paper is to complete Kajii and Ui's work by proposing a necessary and sufficient condition for interim efficiency for various models of ambiguity aversion and in particular MaxMin expected utility.our proof is based on a direct application of some results proposed by Rigotti, Shannon and Stralecki (2008). - Derivatives with Respect to Metrics and Applications: Subgradient Marching Algorithm Fetallah Benmansour, Guillaume Carlier, Gabriel Peyré et Fillipo Santambrogio à paraître dans Numerische Mathematike. Abstract: This paper describes the Subgradient Marching algorithm to compute the derivative of the geodesic distance with respect to the metric. The geodesic distance being a concave function of the metric, this algorithm computes an element of the subgradient in O(N^2 log(n)) operations on a discrete grid of N points. It performs a front propagation that computes the subgradient of a discrete geodesic distance. Equipped with this Subgradient Marching, a Riemannian metric can be designed through an optimization process. We show applications to landscape modeling and to traffic congestion. Both applications require the maximization of geodesic distances under convex constraints, and are solved by subgradient descent computed with our Subgradient Marching. We also show application to the inversion of travel time tomography, where the recovered metric is the local minimum of a non-convex variational problem involving geodesic distances. Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 18

- A strategy for non-strictly convex transport costs and the example of x-y p in MathbbR2 Guillaume Carlier, Luigui De Pascale et Fillipo Santambrogio à paraître dans Communications in Mathematical Sciences. Abstract: This paper deals with the existence of optimal transport maps for some optimal transport problems with a convex but non strictly convex cost. We give a decomposition strategy to address this issue. As part of our strategy, we have to treat some transport problems, of independent interest, with a convex constraint on the displacement. As an illustration of our strategy, we prove existence of optimal transport maps in the case where the source measure is absolutely continuous with respect to the Lebesgue measure and the transportation cost is of the form h( x y ) with h strictly convex increasing and x y an arbitrary norm in R 2. - From Knothe s transport to Brenier s map and a continuation method for optimal transport Guillaume Carlier, Alfred Galichon et Fillipo Santambrogio à paraître dans SIAM Journal on Mathematical Analysis. Abstract: A simple procedure to map two probability measures in R^d is the so-called Knothe- Rosenblatt rearrangement, which consists in rearranging monotonically the marginal distributions of the first coordinate, and then the conditional distributions, iteratively. We show that this mapping is the limit of solutions to a class of Monge-Kantorovich mass transportation problems with quadratic costs, with the weights of the coordinates asymptotically dominating one another. This enables us to design a continuation method for numerically solving the optimal transport problem. - Optimal demand for contingent claims when agents have law invariant utilities, Guillaume Carlier, Rose-Anna Dana, à paraître dans Mathematical Finance. Abstract: We consider a class of law invariant utilities which contains the Rank Dependent Expected Utility (RDU) and the cumulative prospect theory (CPT). We show that the computation of demand for a contingent claim when utilities are within that class, although not as simple as in the Expected Utility (EU) case, is still tractable. Specific attention is given to the RDU and to the CPT cases. Numerous examples are fully solved. - Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory Guillaume Carlier, Rabah Tahraoui à paraître dans ESAIM COCV Abstract: This article is devoted to the optimal control of state equations with memory of the form:. 0 y x s ( x, z) : infu V e L( yx, z, u ( s), u( s)) ds 0 x( t) F( x( t), u( t), A( s) x( t s) ds), t>0, with initial conditions x(0)=x, x(-s)=z(s),s>0. Denoting by, z, u the solution of the previous Cauchy problem and: u. where V is a class of admissible controls, we prove that V is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form: u( x, z) H ( x, z, xu( x, z)) ( Dzu( x, z), z) 0 in the sense of the theory of viscosity solutions in infinite-dimensions of M. Crandall and P.-L. Lions. - Numerical Approximation of Continuous Traffic Congestion Equilibria Fetallah Benmansour, Gabriel Peyré et Fillipo Santambrogio à paraître dans Networks and Heterogeneous Media. Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 19

Abstract: Starting from a continuous congested traffic framework recently introduced in [Carlier, Jimenez, Santambrogio, 2008], we present a consistent numerical scheme to compute equilibrium metrics. We show that equilibrium metric is the solution of a variational problem involving geodesic distances. Our discretization scheme is based on the Fast Marching Method. Convergence is proved via a $\Gamma$-convergence result and numerical results are given. - Congested traffic dynamics, weak flows and very degenerate elliptic equations, Lorenzo Brasco, Guillaume Carlier et Filipo Santambrogio à paraître dans Journal de Mathematiques Pures et Appliquées. Abstract: Starting from a model of traffic congestion, we introduce a minimal-flow-like variational problem whose solution is characterized by a very degenerate elliptic PDE. We precisely investigate the relations between these two problems, which can be done by considering some weak notion of flow for a related ODE. We also prove regularity results for the degenerate elliptic PDE, which enables us in some cases to apply the DiPerna-Lions theory. 3.2. Working Papers rédigés ayant donné lieu à des communications dans des colloques ou séminaires a. Communications, colloques et séminaires internationaux - Discounting and Divergence of opinion Elyes Jouini, Jean-Michel Marin, Clotilde Napp American Economic Association Annual Meeting, San Francisco, 2009. North American Summer Meetings of the Econometric Society, Boston, 2009. Abstract: Voir articles publiés - Behavioral Biaises and Representative Agent Elyes Jouini et Clotilde Napp, Neuro-économie, évaluation et décision Workshop Collège de France, 2009 Abstract: In this paper, we show that behavioral features can be obtained at a group level when the individuals of the group are heterogeneous enough. More precisely, starting from a standard model of Pareto optimal allocations, with expected utility maximizers and exponential discounting, but allowing for heterogeneity among agents.beliefs and time preference rates, we show that the representative agent exhibits interesting behavioral properties. In particular, we obtain an inverse S-shaped probability distribution weighting function and hyperbolic discounting. We provide possible interpretation as well as applications for this result. - Overlapping sets of priors and the existence of efficient allocations and equilibria for risk measures Rose-Anne Dana et Cuong Le Van Séminaire d Economie, Université de Warwick (Royaume-Uni), février 2009. Abstract: The overlapping expectations and the collective absence of arbitrage conditions introduced in the economic literature to insure existence of Pareto optima and equilibria when short-selling is allowed and investors hold a single belief about future returns, is reconsidered. Investors use measures of risk. The overlapping sets of priors and the Pareto equilibrium conditions introduced by Heath and Ku for coherent risk measures are reinterpreted as a weak no-arbitrage and a weak collective absence of arbitrage conditions and shown to imply existence of Pareto optima and Arrow Debreu equilibria. - No-arbitrage, Overlapping sets of priors and the existence of efficient allocations and equilibria for variational preferences Rose-Anne Dana et Cuong Le Van Rapport d activité 2009 Chaire «Les particuliers faces aux risques» 20