On some non standard control problems

Pierre Cardaliaguet
U. Paris-Dauphine

We discuss some unusual control problems which appear in the analysis of partial differential equations or calculus of variation problems.

Computing and characterizing the cost spread of a financial structure with bid/ask

Bernard Cornet
U. of Paris I and University of Kansas, Lawrence, USA

The aim is to give an explicit formula (easy to compute) of the (super-hedging) cost spread of a financial structure in the case of bid/ask event securities, hence also of bid/ask (incomplete) Arrow securities. First we show that the set of discounting measures (risk-neutral probabilities) can be characterized as the core of its associated risk-neutral capacity (a set-function defined on events, also called characteristic function in TU cooperative games). Second, the risk-neutral capacity is shown to be concave, with an explicit formula, easy to compute for each event. Finally, the super-hedging cost of every payoff is proved to be equal to its Choquet integral with respect to the risk-neutral capacity, thus giving an explicit formula for the cost spread using the Chain formula of the Choquet integral.

Capture basin algorithms and their applications

Anya Désilles (ENSTA),
Alexandra Fronville (European Center for Virtual Reality, Brest)

T In this talk we focus on the capture basin algorithm on the viability theory. We present some numerical and theoretical aspects of the calculation of capture basins for controlled dynamical systems. In particular, we show how the notion of the capture basin and the viability theory can be applied to solve optimal control problems and some equations of Hamilton-Jacobi type. This approach is illustrated by an example of a traffic modeling problem. In the macroscopic LWR model the traffic evolution is described by a conservation law with some initial and boundary conditions. It is well known that such problems are difficult to solve analytically and numerically because of presence of shocks. To avoid these difficulties the problem can be stated in an integrated form, as a Hamilton-Jacoby type equation. We use the viability theory framework to define and compute the solution of such problems. The viability theory allows to give sense to a solution of the problem with a very general conditions including boundary, initial and lagrangian (or internal) conditions. The numerical computation of the solution can by made using the capture basin algorithm. Another example, is the morphogenesis of multicellular organisms. It is a dynamical system of cells (cell multiplication, cell migration, apoptosis) with local interactions between cells and with the extracellular matrix. Cells have the same genome and the orientation or direction of cellular division is not random. We formalize the cell dynamic in a proper metric space to find conditions (decisions, states) in which operational constraints are always satisfied and therefore in which the system is viable and maintain its shape while renewing. We use the viability theory and the mutational analysis framework to define the cell dynamic in order to think about form as a cellular growth designing a capture basin of the morphological dynamic which drives to a target (the final form) characterized as the fixed point.

Viabilist and Tychastic Approaches to Guaranteed ALM Problem

Olivier Dordan¹ and Luxi Chen²
1. U. Bordeaux 22. VIMADES

This study reconsiders the problem of hedging a liability by a portfolio made of a riskless asset and an underlying. Given a process forecasting the lower bounds of the returns of the underlying, the software computes both the Minimum Guaranteed Investment (or Solveny Requirement Capital), and a management rule VPPI, (Viabilist Portfolio Performance and Insurance) ensuring that, at each date, the value of the portfolio is "always": exceeding liabilities. Examples of management during the crisis of the summer 2011 are provided.

Sustainability and intergenerational equity

Marie-Hélène Durand
IRD, Montpellier

Economic studies of sustainable development deal with intergenerational equity considered as a conflict of interests arising between present and future generations. To define and to assess intergenerational equity in the setting of an infinite horizon, two separate approaches are undertaken. The first one is axiomatic and purely formal. It is devoted to the establishment of ethical properties that an inter-temporal choice criterion needs to fulfill, in order to adequately represent intergenerational equity. The second one is operational and devoted to the construction of choice criteria in long term economic studies or in an infinite horizon setting. All of the proposed criteria are criticized for not fulfilling the prescribed intergenerational equity conditions and because their results are highly dependent of an actualization parameter. Using viability theory, we propose an equity function that is not balancing present and future (and unknown) states as an improbable social planner but that provides each present generation the best way to fix their consumption such that consumption inequalities between generations are minimized. This equity function provides an indicator that respects both the finite anonymity and Pareto efficiency principles and that is computable.

Co-viability modelling for the sustainable management of biodiversity

Luc Doyen
CNRS, Museum national d'histoire naturelle

A fundamental issue for the sustainable management of renewable resources and biodiversity is the reconciliation of ecological and economic requirements with an equity perspective. The presence of numerous uncertainties in the systems at stake complicates achievement of such a goal. Fisheries and agro-ecological systems constitute challenging case studies in this respect. Viable control under uncertainty is here proposed as a relevant modelling framework to deal with such issues. The approach does not strive to determine optimal or steady-state paths for the joint dynamics of resources and exploitations, but rather aims at maintaining the trajectories of systems within satisfying normative bounds that mix ecological, economic and social requirements. Hence the approach offers a multi-criteria perspective and provides ways to analyse and control the risks and vulnerability of dynamic bio-economic systems. It has major conceptual links with Population Viability Analysis (PVA) and maximin or Rawlsian approaches. Given the flexibility it allows in defining management problems, the approach offers a fruitful modelling framework to address issues of adaptive management and management strategy evaluation. Examples inspired by the management of fisheries or farming worldwide will be proposed to illustrate the interest of such a general approach.

Division Impartiale d'un Euro

Hervé Moulin
Rice University, USA

Chaque participant donne son opinion sur les parts relatives des autres participants, et l'impartialité exige que son message n'ait aucune influence sur sa propre part. S'il est possible de partager l'euro d'une façon compatible avec les messages individuels, la méthode doit entériner ce consensus. S'il y a au moins quatre participants on peut construire une famille naturelle, et en un sens canonique, de méthodes de partage avec ces deux propriétés. Ce n'est pas possible avec deux ou trois agents.

Basé sur Impartial division of a Dollar, G. De Clippel, H. Moulin, N. Tideman, Journal of Economic Theory, 139 (2008) 176-191.

Discrete set-valued interpolation in the Alexandrov topology (with Thierry Géraud)

Laurent Najman
ESIEE, Paris

The main question of this paper is to retrieve continuity on a discrete grid. One possible application, which will guide us, is the construction of the "tree of shapes" (intuitively, the tree of level lines). This tree, which processes in the same way maxima and minima, faces quite a number of theoretical difficulties. These problems exist whether we start from discrete or continuous data, but we will show that an approach both discrete and set-valued makes solving them, with some advantages with respect to continuous ones. For this, two tools are needed:

• The (discrete) Alexandrov spaces. These are topological spaces in which the intersection of any family of open sets is open. In such spaces, we can express classical theorems such as the Jordan's one, which states that any simple closed curve separates the plane into an interior and an exterior. We will show the interest to build the contours on the Khalimsky grid, a grid equipped with the Alexandrov topology.
• Set-valued analysis. Specifically, we show that it is extremely natural in the discrete case for the value of a function at a given point to be an interval, and this will allow us to express properties of continuity of set-valued functions defined on the Khalimsky grid. Such properties will be essential to obtain a linear algorithm for computing the tree of shapes.

Average Long Time Behaviour for Optimal Control and Differential Games

Marc Quincampoix
Université de Bretagne Occidentale

We investigate the long run behaviour for an optimal control or a differential game with a cost which is a Cesaro mean of an integral cost.

The main issue of our approach is the ability to cope with cases where the limit value could depend of the initial condition of the control system as it is usually assumed or proved in the ergodic control literature.

Title to be Announced

Nicolas Seube
ENSTA Bretagne