Mathematical Research

Finance-Applied Mathematics

This research is carried out within an international collaboration (Canada: University of British Columbia, Vancouver; USA: University of Texas, Austin; Austria: Universität Wien) as well as a French one (University of Paris Dauphine, University of Bourgogne)

Mandatory management

• - Mandatory management by methods of maximization of utility function, this introduces portfolio management, dealing with shares and bonds, using the same principles and in a unified way. This allows us to control the problem of portfolio choice within the financial markets with both shares and bonds.
• Roll-Overs and FDR (Production of Finite Dimension). FDR for Zero-Coupon Bonds is a big drawback which only allows the use of refined models. In order to produce models of a greater size and rate, we are researching a way to reduce these constraints, taking Roll-Overs as primitive objects in modeling, instead of Zero-Coupon Bonds. This means that we have to characterize the Zero-Coupon Bonds models as if they were an EDPS within a certain Hilbert space, for which the unpredictable development of Roll-Overs is confined to a sub-variety of finite dimensions.
• Extension to the general markets of zero-coupons. This deals with establishing the necessary and sufficient conditions for the existence of optimal portfolios, in the semi-martingales zero-coupons markets (semi-martingales Kramkov-Schachermayer theory for shares).
• Balance of bond markets. This deals with solving the problem of one market’s balance with a unique equivalent martingale equivalent (MME). The methods used when dealing with shares are not appropriate here, as a complete bond market is not equivalent to a unique MME.

Mutual Fund Theorems (MFT)

• Characterization of the general markets in continuous time, in which MFT is met. This is one of the best parts of mathematical finance, already developed in the mono-period (Markowitz 1953, Tobin 1958), in Discrete Time (Cass and Stiglitz 1970) and for certain simple models, Continuous Time (Merton 1973).
• Economic characterization of the markets where the MFT is real and true.

Balancing markets with shares and bonds

This deals with solving the problem of balancing in the case of a market with a non-unique MME.

Calibration and digital treatment

Physics-Applied Mathematics

Non-linear EDP (electronic data-processing) and non-linear group representations

Determining the properties for the solutions of non-linear EDP, accepting groups of covariance, and finding theories of diffusion for classic relativist equation diffusion theories (c.f. Maxwell-Dirac) and quantum.

Theory of Quantum Fields

Within the framework of the constructive theory of quantum fields, setting up the non-triviality of model 4, in 4 dimensions of time-space.

Research in statistics and optimisation methods for digitals simulation

This research is being performed with TOTAL, in Pau.