Workshop “Exploring the World of Mathematics II” Mälardalen University, Västerås, 28 November 2024

Organized by MAM (Mathematics and Applied Mathematics research environment) https://mammath.wordpress.com/ External link.

Division of Mathematics and Physics, The School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden, with financial support from Nordplus project “Network FinEng2024”, project ID NPHE-2024/10408

Organizing committee: Associate Professor Ying Ni, Professor Sergei Silvestrov, Professor Anatoliy Malyarenko, Professor Yuliya Mishura

Program

Each talk consists of 25 minutes lecture, plus 5 minutes for questions/discussion.

12:00-13:00 Lunch in MDU Restaurant Rosenhill

13:15-13:30 Registration and welcome from organizers, room U2-129

Session 1. Probabilistic

Chair: Ying Ni

13.30-14.00 Prof. Yuliya Mishura, Taras Shevchenko National University of Kyiv, Mälardalen University

Entropies of Poisson distribution: unexpected phenomena

Abstract. This is the common talk with Kostiantyn Ralchenko, Anatoliy Malyarenko and Dmitrii Finkelstein. We consider various examples of entropies (Shannon, Renyi, generalized Renyi, Tsallis, Sharma-Mittal) of Poisson distribution. It is natural to assume in advance that all these entropies increase with intensity of distribution. And indeed, this fact is established by us for Shannon, Renyi, Tsallis and Sharma-Mittal entropies. However, we observed “anomalous“ behavior of generalized Renyi entropies and described the domains of anomality partially analytically and partially with the help of simulations.

14.00-14.30 Prof. Anatoliy Malyarenko, Mälardalen University

Isotropic random cross-sections of homogeneous vector bundles

Abstract. Tianshi Lu defined isotropic and reflexive random vector fields over spheres and described their correlation operators and spectral expansions. We put his results into a broader context of isotropic and reflexive random cross-sections of homogeneous vector bundles attached to irreducible representations of compact Lie groups. The spectral expansions of several cross-sections are explicitly calculated.

14.30-15.00 Asoc. Prof. Oksana Pavlenko, Riga Technical University

STAR time series models, their estimation using R software and applications

Abstract. Smooth transition autoregressive models are said to be useful if data have structural breaks. STAR models with different transition functions suit time series with different patterns of changes in their structure. In this talk I present adjustment of the estimation procedure for LSTAR2 and ESTAR models in R software depending on the transition function, and analysis of suitability of these models for data with different patterns of structural breaks.

15.00-15.30 PhD student Ayoub Haida, Mälardalen University

An Almost Exact Mixed Scheme for Gatheral Double-Mean-Reverting Model

Abstract. This talk is on a joint paper with Mara K. Dimitrov, Tilemachos Marmaras, and Ying Ni. We consider the American option pricing problem using the Least-squares Monte Carlo method and propose two novel simulation schemes for the three-factor Gatheral Double-Mean-Reverting model. We name these two new schemes as Almost-Exact Mixed Scheme (AEMS) and the AEMS-Second Order Refinement (AEMS-SOR) respectively. Traditional techniques like simulating using the standard Euler-Maruyama discretization scheme often produce biases for this type of asset price models, which motivates the present study. Our proposed schemes are based on a combination of the exact simulation of square root variance processes and a higher-order time discretization for such processes and is inspired by the almost-exact simulation approach under Heston-type models. We conduct extensive numerical studies on these two schemes on American style options including both plain vanilla and several exotic options e.g. barrier options, Asian options and Basket options. The effectiveness of our schemes is evaluated by comparing their performance to the traditional Euler-Maruyama simulation scheme.

15.30-16.00 Coffee Break and Scientific Discussions

Session 2. Deterministic

Chair: Sergei Silvestrov

16.00-16.30, Prof. Viktor Abramov, University of Tartu, Estonia

Ternary Associativity and Ternary Lie Algebra at Cube Roots of Unity

Abstract. In this talk we propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear combination of six triple products. The coefficients of this linear combination are the cube roots of unity. We find an identity for the ternary commutator that holds due to the ternary associativity of either the first or second kind. The form of this identity is determined by the permutations of the general affine group GA(1,5). We consider this identity as a ternary analog of the Jacobi identity. Based on the results obtained, we introduce the concept of a ternary Lie algebra at cube roots of unity and provide examples of such algebras constructed using ternary multiplications of rectangular and three-dimensional matrices.

16.30-17.00 Dr. Lars Hellström, Mälardalen University

The Lanczos Moments of a Random Variable

Abstract. The Lanczos algorithm in linear algebra involves computing a sequence of scalar quantities that become elements of a tridiagonal matrix. These scalars may also be interpreted as moments of a probability distribution (or more general measure), which provides an alternative characterisation of that distribution. In particular, it is straightforward to construct finite discrete approximations from these moments. Moreover, there exists an associated sequence of orthogonal polynomials which may be constructed as characteristic polynomials of truncations of the tridiagonal matrices; the standard families of orthogonal polynomials turn out to be such sequences for certain standard probability distributions.

From the descriptive statistics point of view, it may be mentioned that the first and second Lanczos moments natively become the mean and standard deviation, whereas the third and fourth provide interpretations for skewness and kurtosis. Another interesting feature of the Lanczos moments is that they are all in the same unit as the underlying random variable.

17.00-17.30 Prof. Predrag M. Rajković, Faculty of Mechanical Engineering, University of Niš, Serbia

Title. Various modifications of the Gamma function - the connections and numerical computation

Abstract. The gamma function is a special function with a lot of applications in different sciences (combinatorics, statistics, integral transforms, physics, quantum mechanics, fluid dynamics, etc.) The trials to expand their definition on the negative integers were done by a new kind of the limit. A lot of different formulas exist for its numerical computation. We can notice that the exact and numerical computations are included in modern software. A few generalizations and deformations of the gamma function are known. The purpose of the present paper is to give an overview lecture in recent research of the gamma function and our own contributions from these different points of view.

REFERENCES

1. Rajković, P.M., Marinković, S.D., Stanković, M.S. 2008, Differential and integral Calculus of Basic Hypergeometric Functions, Faculty of Mechanical Engineering at University of Niš.
2. Koepf, W., Rajković, P.M., Marinković, S.D. (2016) On a connection between formulas about q-gamma functions, Journal of Nonlinear Mathematical Physics, Vol. 23, No. 3, pp. 343–350.
3. Rajković, P.M., Stanković, M.S., Marinković, S.D. (2018) The Laplace transform induced by the deformed exponential function of two variables, Fractional Calculus and Applied Analysis, Vol. 21, Issue 3, pp. 775-785.