MOCT · Spectral Theory of Non-Selfadjoint Markov Processes with Applications in Self-Similarity, Branching Processes and Financial Mathematics
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2015-07-01 → 2017-06-30
- EU contribution
- €128,994
- Participants
- 1
- Scheme
- MSCA-IF-EF-ST
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Results in brief
Spectral Theory of Non-Selfadjoint Markov Processes with Applications in Self-Similarity, Branching Processes and Financial Mathematics
The main focus of the project is the theoretical development of the spectral theory of a class of Markov processes that are used in the stochastic modelling of the real world. Hence, the project contributes to the development of science and in particular allows for better understanding of the phenomenon of self-similarity. The main objective of the project is to develop a general framework to study the spectral properties of Markov processes and to test it on self-similar Markov processes. The tools developed to achieve this objective are general enough so as to be employed in various areas of mathematics, e.g. the newly developed class of special functions that we call Bernstein-Gamma functions appears in complex analysis, special functions, probability theory, etc. As an application of these tools, another objective of the project is to understand the Asian options and perpetuities, which are important quantities in financial and insurance mathematics. The novel framework has been developed and tested on important class of Markov processes and it has been used to derive new information for positive self-similar Markov processes. The Bernstein-Gamma functions have been thoroughly investigated and information about the pricing of Asian options and perpetuities has been extracted. As an illustration of the generality and wide applicability of the tools we have carried out a complete study of the important class of random variables called exponential functionals. The acquisition of new expertise in the area of spectral theory has also resulted in additional work beyond the scope of the project. The project has achieved a considerable amount of transfer of knowledge to the principle researcher and to the host institution in the areas of probability theory, functional analysis and complex analysis. The expanded research network of the investigator has contributed to new EU-US collaborations and a genuine inter-European transfer of frontier research in view of the secondment to the group of Mathematical Finance and Probability Theory at the University of Vienna.
Data: CORDIS, © European Union
Project objective
The project contextually sets up a novel framework to study the spectral-theoretical properties of classes of non-selfadjoint (NSA) operators related to Markov processes (MP) via their intertwining to a continuous path selfadjoint (SA) MP. Conceptually, this means that the jumps of each class of NSA MP can be considered a perturbation of one SA MP realized by an intertwining kernel. This approach can have far-reaching consequences for understanding classes of MP as the reduction to SA MP leads to well-studied objects whereas the spectral theory of NSA operators is far from understood. The price of that is the non-invertability of the intertwining kernels. This framework is explored and crystallized by a challenging,detailed spectral-theoretical study of an enormous class of NSA operators directly arising from the key phenomenon of self-similarity and in duality from branching. This is achieved by a synergy of research fields complementing each other to obtain the spectral properties of those operators culminating in the derivation of spectral expansions of the generated semigroups. As a result of this synergy, a number of tools and techniques with impact, including applications to fields beyond the scope of the project, are derived. A particular development in the area of recurrent equations and special functions will be unexpectedly exploited to the effect of a comprehensive theoretical and applied study, including numerical schemes, ofkey quantities in financial and insurance mathematics such as Asian options and perpetuities. A training-through-research in line with the fellow’s affiliation to the host institution and the proposed secondment will critically contribute to the optimal completion of the proposal in terms of time, scope and quality.
Original text from CORDIS.
Participants
- INSTITUTE OF MATHEMATICS AND INFORMATICS AT THE BULGARIAN ACADEMY OF SCIENCE · SofiaCoordinatorBulgaria
Links
Data: CORDIS, © European Union