## Periodic Reporting for period 1 - MOCT (Spectral Theory of Non-Selfadjoint Markov Processes with Applications in Self-Similarity, Branching Processes and Financial Mathematics)

Reporting period: 2015-07-01 to 2017-06-30

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.

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.

"The properties of a general class of Markov processes have been thoroughly investigated. This has been achieved as a combination of two approaches. First, we have introduced and used a novel, general framework which allows to link the properties of every such Markov process to the ones of a well-known classical Markov process. Second, for this particular study we have established a link between the investigated Markov processes and a group of Markov processes, that we have a priori understood. Despite these useful links the work performed is based on detailed yet challenging studies. These results are included in three papers:

1.""Spectral expansions of non-selfadjoint generalized Laguerre semigroups"", Patie, P. and Savov, M.,162 pages, second round of refereeing;

2.""Cauchy problem of the non-selfadjoint Gauss-Laguerre semigroups and uniform bounds for generalized Laguerre polynomials"", Patie, P. and Savov, M., Journal of Spectral Theory, 33 pages, accepted;

3.""Intertwining, Excursion Theory and Krein Theory of Strings for Non-selfadjoint Markov Semigroups"", Patie, P., Savov, M. and Zhao Y., 33 pages, submitted.

As an application of these results and methods we have developed the spectral theory for an important class of self-similar Markov processes. This gives a wealth of new information for these objects and includes in its domain a number of results that appear in the literature.

The derivation of the results for the Markov processes also requires a thorough investigation of the class of random variables called the exponential functionals of Levy processes. To achieve this we introduce a class of special functions that we call Bernstein-Gamma functions, whose analytic properties reveal the most important features of the exponential functionals. For this purpose a separate and self-contained paper has been written on the topic:

4. ""Bernstein-Gamma functions and exponential functionals of Levy processes"", Patie, P. and Savov, M., 84 pages, submitted.

We note that the Bernstein-Gamma functions appear in various probabilistic contexts. For example, the pricing of Asian options and perpetuities are investigated with the help of these functions and the development of other novel tools.

As a result of networking two separate publications beyond the scope of the project have been refined and published. First, with the better understanding and employment of spectral theory an open problem has been resolved in :

5. ""Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case"", M. Kolb and M. Savov, Electronic Journal of Probability, (2017), 1--27.

Also, we have managed to improve on results on the behaviour of the classical Brownian motion with limited return to the origin, now published under the title:

6. ""Transience and recurrence of a Brownian path with limited local time"", Kolb, M. and Savov, M., Annals of Probability, (2016), Vol. 44, No. 6, 4083--4132.

"

1.""Spectral expansions of non-selfadjoint generalized Laguerre semigroups"", Patie, P. and Savov, M.,162 pages, second round of refereeing;

2.""Cauchy problem of the non-selfadjoint Gauss-Laguerre semigroups and uniform bounds for generalized Laguerre polynomials"", Patie, P. and Savov, M., Journal of Spectral Theory, 33 pages, accepted;

3.""Intertwining, Excursion Theory and Krein Theory of Strings for Non-selfadjoint Markov Semigroups"", Patie, P., Savov, M. and Zhao Y., 33 pages, submitted.

As an application of these results and methods we have developed the spectral theory for an important class of self-similar Markov processes. This gives a wealth of new information for these objects and includes in its domain a number of results that appear in the literature.

The derivation of the results for the Markov processes also requires a thorough investigation of the class of random variables called the exponential functionals of Levy processes. To achieve this we introduce a class of special functions that we call Bernstein-Gamma functions, whose analytic properties reveal the most important features of the exponential functionals. For this purpose a separate and self-contained paper has been written on the topic:

4. ""Bernstein-Gamma functions and exponential functionals of Levy processes"", Patie, P. and Savov, M., 84 pages, submitted.

We note that the Bernstein-Gamma functions appear in various probabilistic contexts. For example, the pricing of Asian options and perpetuities are investigated with the help of these functions and the development of other novel tools.

As a result of networking two separate publications beyond the scope of the project have been refined and published. First, with the better understanding and employment of spectral theory an open problem has been resolved in :

5. ""Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case"", M. Kolb and M. Savov, Electronic Journal of Probability, (2017), 1--27.

Also, we have managed to improve on results on the behaviour of the classical Brownian motion with limited return to the origin, now published under the title:

6. ""Transience and recurrence of a Brownian path with limited local time"", Kolb, M. and Savov, M., Annals of Probability, (2016), Vol. 44, No. 6, 4083--4132.

"

The work on spectral theory of generalized Laguerre semigroups goes beyond the state of the art in several directions. First, it is based upon a novel framework which deals with non-symmetric operators. Second, the results for the generalized Laguerre semigroups are comprehensive and new. Third, the particular tools developed to complete this work are novel and concern areas of analysis, complex analysis, probability theory, etc. Forth, as an immediate by-product, positive self-similar Markov processes have been analysed to a precision well-beyond the standard one in the area. The potential for a wider impact of this work is found in the fact that self-similarity is broadly used in cutting-edge models of the physical world and economics.

The work on the Bernstein-Gamma functions is beyond the state of the art as it investigates and delivers detailed information for a natural generalization of the Gamma function. Their immediate impact is in probability theory since they allow for the thorough understanding of the random variables called exponential functionals, which in turn offers new information for various areas of probability theory. We expect that the Bernstein-Gamma functions will contribute to other areas of mathematics and modelling. An example is their role in providing a sound way to evaluate the price of Asian options with a possible exploitation that is the development of new and rigorous numerical schemes. This is very important especially in view of the fact that financial instruments are often used without mathematical underpinning or do not reflect the main assumptions of the markets.

Research networking has resulted in progress beyond the state of the art in two areas of probability theory beyond the scope of the project. We have answered an open problem on the behaviour of the survival of a Brownian particle moving in Poissonian traps and we have shed light on the phenomenon called entropic repulsion, which means that upon imposing a restriction on the behaviour of a stochastic process, then to satisfy this restriction the process ought to fulfil even a stringer restriction. In the case of Brownian motion with restriction on the rate of its return to zero we have answered several open problems. Having treated the most basic scenario these results are a stepping stone for the future understanding of the entropic repulsion.

The work on the Bernstein-Gamma functions is beyond the state of the art as it investigates and delivers detailed information for a natural generalization of the Gamma function. Their immediate impact is in probability theory since they allow for the thorough understanding of the random variables called exponential functionals, which in turn offers new information for various areas of probability theory. We expect that the Bernstein-Gamma functions will contribute to other areas of mathematics and modelling. An example is their role in providing a sound way to evaluate the price of Asian options with a possible exploitation that is the development of new and rigorous numerical schemes. This is very important especially in view of the fact that financial instruments are often used without mathematical underpinning or do not reflect the main assumptions of the markets.

Research networking has resulted in progress beyond the state of the art in two areas of probability theory beyond the scope of the project. We have answered an open problem on the behaviour of the survival of a Brownian particle moving in Poissonian traps and we have shed light on the phenomenon called entropic repulsion, which means that upon imposing a restriction on the behaviour of a stochastic process, then to satisfy this restriction the process ought to fulfil even a stringer restriction. In the case of Brownian motion with restriction on the rate of its return to zero we have answered several open problems. Having treated the most basic scenario these results are a stepping stone for the future understanding of the entropic repulsion.