Projects / Programmes
Methods of algebra and functional analysis in theory and practice of financial mathematics
| Code |
Science |
Field |
Subfield |
| 1.01.00 |
Natural sciences and mathematics |
Mathematics |
|
| Code |
Science |
Field |
| P160 |
Natural sciences and mathematics |
Statistics, operations research, programming, actuarial mathematics |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
affine processes, symmetric cones, one-parameter semigroups of linear maps, positive Maps, positive polynomials, semialgebraic sets, pricing of derivative financial instruments, multi-asset options.
Researchers (13)
Organisations (3)
Abstract
In this project we intend to study one-parameter semigroups of positive maps, which are an important tool in financial mathematics, for example, in the study of affine processes. We would like to determine whether on each symmetric cone there exists a one-parameter semigroup of positive maps whose generators can not be written as a sum of a positive map and a generator of one-parameter group of positive maps. If such semigroups do exist, we intend to study their structure. Since the generators of one-parameter semigroup of positive maps define biquadratic forms which are nonnegative on an appropriate algebraic set, we intend to utilize methods of real algebraic geometry and functional analysis in our study. We will use the findings about affine processes to create improved pricing models for financial derivatives such as multi-asset options.
Significance for science
The structure of one-parameter semigroups of linear mappings which preserve a cone, was well researched before the start of the project in the case of a polyhedron cone, but in all other cases there was little known about such semigroups. In the case of a symmetric cone, only very partial results were known, and even those were obtained only for semigroups satisfying some special restrictive conditions. Therefore, our results present major progress on the structure of such semigroups and indicate the directions of further research. As we have expected in the application phase of the project, the methods we have developed have proven useful in the wider field of real algebraic geometry, where studying polynomials that are non-negative on a semialgebraic set is a core topic. It turned out that our methods are very fruitful in the study of biquadratic forms that are non-negative everywhere. Such biquadratic forms correspond to positive linear mappings on symmetric matrices. We have shown that few positive linear mappings are completely positive in the sense that the ratio of their volumes of cones tends to zero, when the size of the matrices grows. We have also developed a general algorithm for constructing examples of positive but not completely positive mappings on matrices of size at least 3. We expect our ideas to be powerful in solving some interesting related questions in the future. This will add to our understanding of positive mappings and construct further examples of positive mappings with additional properties. These are useful in mathematical physics and quantum information theory, where our results have peaked interest among the experts.
Significance for the country
The significance for the Slovenian company Ektimo, which is the co-financer of the project, is presented in greater detail under point 12. We will now describe the importance for the Slovenian financial sector. The project is an important step towards the transfer of financial and mathematical knowledge from the scientific and research sphere to the economy. The topic of valuation of derivative financial instruments is very relevant for the Slovenian financial sector, since they form the basis for increasingly widespread structured products, in which the Slovenian companies are almost entirely dependent on foreign financial service providers. Greater integration of domestic knowledge means increasing added value in the sale of such products in both domestic and foreign markets, and the control of the risks of these complex products is also improved. Financial mathematics is a relatively new field of science in Slovenia. The implementation of the project is of great importance to the further development of this area as it was the first major research project with such content in Slovenia. Successful implementation of project goals thus contributes to the positioning of Slovenia on the map of European and global financial mathematics. At the same time, the project consolidated the international contacts of Slovenian researchers. This is especially true for contacts with researchers from ETH in Zurich, who are leading experts in financial mathematics and working on problems related to the content of the proposed project. The members of the project team were also the administrators of the study program Financial Mathematics at the Faculty of Mathematics and Physics at the University of Ljubljana, which began at the 1st Bologna level in the academic year 2007/2008, and at the second Bologna level in 2010/2011. At the same time, the holders were also subject to the subject of verified statistical and financial-mathematical contents at the doctoral study Mathematics and Interdisciplinary Doctoral Studies Statistics. This allowed the students to enrol in the project through seminar, graduate and doctoral theses of the mentioned study programs.
Most important scientific results
Annual report
2014,
2015,
final report
Most important socioeconomically and culturally relevant results
Annual report
2014,
2015,
final report
