Department of Mathematics

The subject of scientific research of the Mathematics Department is found within major mathematical disciplines– mathematical analysis, algebra, geometry, probability and statistics and methodology.

Research based on statistical analyses of various types of time series with a significant practical application, mostly in econometrics. Special importance is given to the research of stochastic properties of such series, as well as to the construction of new, mathematically based, stochastic financial models which can describe different properties of the dynamics of real financial series and sequences.

The problem of estimating the parameters of the obtained models is studied extensivelly and for that purpose new, modern statistical techniques are introduced, primarily based on numerical integration and the application of various software tools. The focus here is on the method of empirical characteristic functions (ECF-method) and the method of probabilistic functions of generatrise (PGF-method), which have proven to be very effective in modeling the so-called nonlinear time series with pronounced fluctuations.

Application of methods of homotopy perturbations (HPM), as well as methods of homotopy analysis (HAM) in finding (exact or approximate) solutions of equations and problems of different types. By applying these two mathematical methods, for example, approximate solutions of differential equations and systems of partial equations are obtained, which occur in a wide group of physically based problems. On the other hand, the HPM and HAM methods have also been applied in solving various stochastic problems, such as the approximation of stochastic distributions that do not have a closed form or the determination of invariant probability measures of different types of chaotic maps.

The concept of a fixed point, its existence in Menger spaces, as well as in ordered and partially arranged b-metric spaces. Namely, special attention is given to the fixed point theorem for increasing mappings, as well as its direct consequences in metric spaces. We work on the concept of statistical causality and stopped statistical causality, which includes stopping times as well as their application to martingale theory, weak solutions of stochastic differential equations generated by semimartingales and the application of obtained results in the field of financial mathematics. Namely, the given concept of causality can be related to the orthogonality of the martingale, the stability of space of right-continuous martingales, the predictable representation property, extreme measures, optional and predictable projections, as well as purely discontinuous martingales. Using the concept of causality, we can describe weak solutions and local weak solutions of stochastic differential equations generated by semimartingales, solutions of the martingale problem (MP), stopped MP and local MP. The weak uniqueness and local uniqueness of these solutions can also be described by the concept of causality. All these concepts have a wide application in practice, especially in econometrics, financial mathematics, risk minimization of optimal stock trading strategies. ...