ERC FUNDED PROJECTS

The main objective of this research project is to contribute at bridging the gap between the two main branches of financial theory, namely corporate finance and asset pricing. It is motivated by the conviction that these two aspects of financial activity should and can be analyzed within a unified framework. This research will borrow from these two approaches in order to construct theoretical models that allow one to analyze the design and issuance of financial securities, as well as the dynamics of their valuations. Unlike asset pricing, which takes as given the price of the fundamentals, the goal is to derive security price processes from a precise description of firm’s operations and internal frictions. Regarding the latter, and in line with traditional corporate finance theory, the analysis will emphasize the role of agency costs within the firm for the design of its securities. But the analysis will be pushed one step further by studying the impact of these agency costs on key financial variables such as stock and bond prices, leverage, book-to-market ratios, default risk, or the holding of liquidities by firms. One of the contributions of this research project is to show how these variables are interrelated when firms and investors agree upon optimal financial arrangements. The final objective is to derive a rich set of testable asset pricing implications that would eventually be brought to the data.

1 000 000 €

Start date: 2008-11-01, End date: 2014-10-31

The numerical simulation of complex compressible flow problem is still a challenge nowaday even for simple models. In our opinion, the most important hard points that need currently to be tackled and solved is how to obtain stable, scalable, very accurate, easy to code and to maintain schemes on complex geometries. The method should easily handle mesh refinement, even near the boundary where the most interesting engineering quantities have to be evaluated. Unsteady uncertainties in the model, for example in the geometry or the boundary conditions should represented efficiently.This proposal goal is to design, develop and evaluate solutions to each of the above problems. Our work program will lead to significant breakthroughs for flow simulations. More specifically, we propose to work on 3 connected problems: 1-A class of very high order numerical schemes able to easily deal with the geometry of boundaries and still can solve steep problems. The geometry is generally defined by CAD tools. The output is used to generate a mesh which is then used by the scheme. Hence, any mesh refinement process is disconnected from the CAD, a situation that prevents the spread of mesh adaptation techniques in industry! 2-A class of very high order numerical schemes which can utilize possibly solution dependant basis functions in order to lower the number of degrees of freedom, for example to compute accurately boundary layers with low resolutions. 3-A general non intrusive technique for handling uncertainties in order to deal with irregular probability density functions (pdf) and also to handle pdf that may evolve in time, for example thanks to an optimisation loop. The curse of dimensionality will be dealt thanks Harten's multiresolution method combined with sparse grid methods. Currently, and up to our knowledge, no scheme has each of these properties. This research program will have an impact on numerical schemes and industrial applications.

1 432 769 €

Start date: 2008-12-01, End date: 2013-11-30

The understanding of spatial, social and dynamical organization of large numbers of agents is presently a fundamental issue in modern science. ADORA focuses on problems motivated by biology because, more than anywhere else, access to precise and many data has opened the route to novel and complex biomathematical models. The problems we address are written in terms of nonlinear partial differential equations. The flux-limited Keller-Segel system, the integrate-and-fire Fokker-Planck equation, kinetic equations with internal state, nonlocal parabolic equations and constrained Hamilton-Jacobi equations are among examples of the equations under investigation. The role of mathematics is not only to understand the analytical structure of these new problems, but it is also to explain the qualitative behavior of solutions and to quantify their properties. The challenge arises here because these goals should be achieved through a hierarchy of scales. Indeed, the problems under consideration share the common feature that the large scale behavior cannot be understood precisely without access to a hierarchy of finer scales, down to the individual behavior and sometimes its molecular determinants. Major difficulties arise because the numerous scales present in these equations have to be discovered and singularities appear in the asymptotic process which yields deep compactness obstructions. Our vision is that the complexity inherent to models of biology can be enlightened by mathematical analysis and a classification of the possible asymptotic regimes. However an enormous effort is needed to uncover the equations intimate mathematical structures, and bring them at the level of conceptual understanding they deserve being given the applications motivating these questions which range from medical science or neuroscience to cell biology.

2 192 500 €

Start date: 2017-09-01, End date: 2022-08-31

In the road towards quantum technologies capable of exploiting the revolutionary potential of quantum theory for information technology, a major bottleneck is the large overhead needed to correct errors caused by unwanted noise. Despite important research activity and great progress in designing better error correcting codes and fault-tolerant schemes, the fundamental limits of communication/computation over a quantum noisy medium are far from being understood. In fact, no satisfactory quantum analogue of Shannon’s celebrated noisy coding theorem is known. The objective of this project is to leverage tools from mathematical optimization in order to build an algorithmic theory of optimal information processing that would go beyond the statistical approach pioneered by Shannon. Our goal will be to establish efficient algorithms that determine optimal methods for achieving a given task, rather than only characterizing the best achievable rates in the asymptotic limit in terms of entropic expressions. This approach will address three limitations — that are particularly severe in the quantum context — faced by the statistical approach: the non-additivity of entropic expressions, the asymptotic nature of the theory and the independence assumption. Our aim is to develop efficient algorithms that take as input a description of a noise model and output a near-optimal method for reliable communication under this model. For example, our algorithms will answer: how many logical qubits can be reliably stored using 100 physical qubits that undergo depolarizing noise with parameter 5%? We will also develop generic and efficient decoding algorithms for quantum error correcting codes. These algorithms will have direct applications to the development of quantum technologies. Moreover, we will establish methods to compute the relevant uncertainty of large structured systems and apply them to obtain tight and non-asymptotic security bounds for (quantum) cryptographic protocols.

1 492 733 €

Start date: 2021-01-01, End date: 2025-12-31