EKELAND Ivar

No summary available.

In this paper we introduce a new algorithm for solving perturbed nonlinear functional equations which admit a right-invertible linearization, but with an inverse that loses derivatives and may blow up when the perturbation parameter $\epsilon$ goes to zero. These equations are of the form $F_\epsilon(u)=v$ with $F_\epsilon(0)=0$, $v$ small and given, $u$ small and unknown. The main difference with the by now classical Nash-Moser algorithm is that, instead of using a regularized Newton scheme, we solve a sequence of Galerkin problems thanks to a topological argument.

We give new insights into the dynamic behavior of commodity prices with an infinite horizon rational expectations equilibrium model for spot and futures commodity prices. Numerical simulations of the model emphasize the heterogeneity that exists in the behavior of commodity prices by showing the link between the physical characteristics of a market and some stylized facts of commodity futures prices. They show the impact of storage costs on both the variability of the basis and on the Samuelson effect. Finally, the simulations of the model show that an increase in the speculative activity on commodity futures markets has an overall positive effect on risk premia. However, not all of the agents benefit from it.

No summary available.

No summary available.

"This easy-to-read comic book fascinates with the stories of the discoveries, the genial finds of their authors. Algorithms, the discovery of the number Pi, the discovery of the 13-knot rope, the calculation of the earth's circumference, the invention of the chess game, the resolution of equations. Ten stories, each more extraordinary than the last, are presented for the pleasure of young and old alike.

No summary available.

No summary available.

No summary available.

The back cover states: "An unfortunate frog put in a cooking pot tolerates a steady rise in water temperature, whereas a sudden scalding would make it react immediately. In the same way, global warming is insidious: it is only perceptible on the scale of a decade, or even a century, does not imply any urgent decision and, in fact, is regularly pushed back on the agenda of policies whose horizon rarely exceeds a few years. However, in the environmental field, the delay between action and its impact is at least fifty years. Only an ethical and anthropological point of view taking into account the survival of the human species could solve the dilemma, but as Homo oeconomicus we are calculating individuals acting out of self-interest, and for whom the environment is an infinite and free resource. In the ordinary economic game, there is no "ecological rate of interest", as shown by the inevitable disappearance, under the effect of economic laws, of fish resources. It is thus to a broader conception of humanity and to a renewal of ethics that the author invites us, failing to see the human species, victim of the economic thought, share the sad fate of the cod, the bluefin tuna and the frog.

No summary available.

This is a talk delivered at the conference “Mathematics in a Complex World”, on the occasion of the 150th anniversary of the Politecnico di Milano. Asymmetry of information, i.e. the possibility for human beings to hide their information, or not to keep their promises, is a fundamental fact of social life, and must be taken into account. I will show how this creates complexity, even in the very simple situation of a contract between two parties, one of whom commits to work for the other, but cannot be monitored.

No summary available.

We revisit the neo-classical model of optimal growth by introducing an additional term, due to Chichilnisky, which embodies society's concern for the long-term welfare of future generations. We show that the additional term causes time inconsistency, so there is no such thing as an optimal solution. However, we introduce a notion of equilibrium solution and we prove that there is a continuum of such solutions.

This paper exhibits a duality between the theory of revealed preference of Afriat and the housing allocation problem of Shapley and Scarf. In particular, it is shown that Afriat's theorem can be interpreted as a second welfare theorem in the housing problem. Using this duality, the revealed preference problem is connected to an optimal assignment problem, and a geometrical characterization of the rationalizability of experiment data is given. This allows in turn to give new indices of rationalizability of the data and to define weaker notions of rationalizability, in the spirit of Afriat's efficiency index.

We imbed a classic fishery model, where the optimal policy follows a Most Rapid Approach Path to a steady state, into an overlapping generations setting. The current generation discounts future generations׳ utility flows at a rate possibly different from the pure rate of time preference used to discount their own utility flows. The resulting model has non-constant discount rates, leading to time inconsistency. The unique Markov Perfect equilibrium to this model has a striking feature: provided that the current generation has some concern for the not-yet born, the equilibrium policy does not depend on the degree of that concern.

This thesis is devoted to some optimization problems arising in contract theory. We first consider some one-dimensional problems. In this simple case, the principal's program consists in minimizing a certain cost on the cone of increasing functions. We consider non-convex cases and use a variant of the Hardy-Littlewood inequality. We then consider variational problems under convexity constraints. After giving existence results for non-convex Lagrangians, we introduce a penalty to recover the Euler equation due to Pierre-Louis Lions. In a collaborative work with Thomas Lachand-Robert, we then establish C1 regularity results for minimizers in various situations (Dirichlet, Neumann, Choné and Rochet model. . . ). Finally, in a paper co-authored with Thomas Lachand-Robert and Bertrand Maury, we present a method for numerical approximation of quadratic problems under convexity constraints. The presented algorithm is, to our knowledge, the only one whose convergence is established. In the second part, we characterize incentive contracts in a general way by means of abstract convexity and subdifferentiability notions used in the theory of mass transportation and we prove the existence of optimal incentive contracts without any particular functional specification on the agents' preferences. We then use the link between the characterization of incentive contracts and a class of optimal transportation problems. First, existence, uniqueness, and duality results are established for mass transportation problems where the cost verifies a generalizing Spence-Mirrlees hypothesis, which is classical in the economic literature in dimension 1. This allows us, returning to incentive problems, to de��monstrate a reallocation principle: any measurable allocation profile can be uniquely rearranged into an implementable profile, via a suitable optimal transfer problem. The final section is devoted to two specific economic problems. The first one is motivated by insurance fraud issues that give rise to a non-convex problem and consists of a paper co-authored with Rose-Anne Dana and Maxime Renaudin. The second one is related to the design of labor contracts in the presence of two-dimensional adverse selection. It is a paper co-authored with Damien Gaumont.

This work is divided into three parts. The first one presents a strictly financial application . the second one deals with the specific domain of reinsurance . finally the third one studies an example of interaction between these two domains. The first part, specific to the stochastic domain of finance in the context of a complete market, provides explicit expressions for the generalization of the Black-Scholes model to the particular case of the valuation and management of an option on a homographic function of two correlated underlyings that pay dividends. The second part presents the construction and econometric analysis of market price indices for excess catastrophe reinsurance treaties in the United States. Based on a significant data collection, the empirical part covers the period 1975 to 1993. The two types of price indicators provided (one qualitative, the other quantitative) make it possible to set up, in turn, two types of efficiency measurement protocols: that intrinsic to the market, and that of the company's portfolio relative to the market. The third part, specific to stochastic calculus and econometrics, focuses on the organized financial market for natural catastrophe reinsurance futures (traded at the Chicago Board of Trade since 1993). Various proposals for functional optimizations are presented. The idea is to retain a market price index representative of the financial market that allows the introduction of: a family of information tribes converging to a continuous information tribe, and a reference index, in the context of this incomplete market. Finally, we propose, as an example, a valuation and management method in line with the management of a reinsurance portfolio.

This work consists of two parts. The first part deals with the existence of periodic Hamiltonian trajectories on certain r#2#n subvarieties, intersections of hyper level surfaces of several Hamiltonian functions in involution. We define the notion of periodic trajectory for such subvarieties and we define a contact condition which ensures the existence of at least one periodic trajectory. We also prove that a subvariety satisfying the contact condition has a strictly positive symplectic capacity. The second part studies the periodic solutions for the billiard problem in a boundary open of r#n limits of regular solutions of Lagrangian systems with potential wells. The main result establishes a precise link between the Morse index of approximate solutions (considered as critical points of Lagrangian functionals) and some properties of the periodic trajectory of boundary billiards.

In this thesis, we propose a variational approach to the problem of the existence and multiplicity of homoclinic orbits of Hamiltonian systems. The Hamiltonians considered are defined on an even dimensional real vector space with the usual symplectic form. They have a hyperbolic equilibrium, and we call homoclinic orbits, non-constant solutions which tend to this equilibrium in the two infinite directions of time. We obtain the existence of one or more of these orbits under global assumptions on the Hamiltonian. In the first part of this thesis, the Hamiltonian depends periodically on time. Under convexity and superquadraticity assumptions, we show the existence of an infinite number of homoclinic orbits, without the classical transversality condition. Our result takes the form of an alternative: either the homoclinic orbits are uncountable, or there is an infinite set of homoclinic orbits, possessing a particular structure that we call approximate Bernoulli shift. In this second case, the topological entropy of the system is strictly positive. In the second part, the Hamiltonian system is autonomous. We present the first result on the existence of homoclinic orbits under assumptions invariant by symplectic transformations.

The present work is oriented in two research directions. It has been realized under the direction of Ivar Ekeland. The first part is placed in the framework of the theory of Hamiltonian systems. We use a variational approach and the principle of Clarke duality, the aim of the study being to develop numerical methods to find the solutions of first and second order equations in order to study their index. Two methods are presented, one giving a large number of solutions but this at the price of the absence of control on the obtained solutions (in particular on their index), the other one linked to the neck theorem, of Ambrosetti and Rabinowitz, ensuring to find solutions of index 1. In this last case we have looked for asymptotic estimates of the solutions found on an example. The second part deals with problems of financial mathematics, concerning the valuation of contingent assets, a field that has developed strongly in recent years. This part was co-directed by Guy Barles. On the one hand, we consider the study of a two-factor Markov model for contingent assets. We present the implementation of a numerical method. On the other hand, we consider an option pricing model depending on the history of the underlying price. The numerical implementation of the model is also presented.

We present here a variational approach to almost-periodic trajectories of Euler-Lagrange differential equations. The computation of mean variations consists in studying nonlinear time-averaged functionals defined on spaces of almost-periodic functions. The critical points of these functionals are then exactly the almost-periodic solutions of Euler-Lagrange equations. We thus obtain results on the structure of the set of almost-periodic and periodic solutions of autonomous equations with convex Lagrangians. We construct spaces of almost-periodic functions of the Sobolev space type, which induces a notion of weak almost-periodic solution. With these spaces, which are Hilbert spaces, we obtain existence theorems of almost-periodic solutions for almost-periodically forced equations.

In this work, we are interested in the study of periodic solutions of the Hill equation: q+k q/q#32j q3#2aq=0. We show that for a fixed period t>0, there are at least two periodic anti-t/2 solutions of the Hill equation in the neighborhood of a variety of circular solutions of the Kepler equation, and this for small values of . Then, we relate these anti-t/2 periodic solutions to the circular solutions of the Kepler equation by applying the implicit function theorem. This leads us to a Taylor development of the anti-t/2 periodic solutions of the Hill equation as a function of the parameter and the period t. We use this development to make a numerical experiment whose goal is to verify the theoretical results. We end with a study of the Floquet multipliers of the anti-t/2 periodic solutions of the Hill equation, in particular we show that the Floquet multipliers of the circular solutions of the Kepler equation are all equal to 1.