2024-04-29

# Workshop “Exploring the World of Mathematics”

MDU is hosting the workshop “Exploring the World of Mathematics” for researchers and doctoral students, 14-16 May 2024.

The workshop is organized by MAM (Mathematics and Applied Mathematics research environment), External link. Division of Mathematics and Physics at the School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden with financial support from the Nordplus project “Network FinEng2023”, project ID NPHE-2023/10197.

**Organising committee:** Professor Anatoliy Malyarenko, Professor Yuliya Mishura, Associate Professor Ying Ni, Professor Sergei Silvestrov.

## Program

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

### May 14

09.45-10.00 Registration and welcome from organizers, room **U2-016**

*Chair Yuliia Mishura*

10.00-10.30 **Prof. Jonas Šiaulys, Vilnius University **

Closure properties of several regularity classes of distributions.

**Abstract. **All distributions can be classified into classes based on the behaviour of their right tail. Fourteen different regularity classes are currently being considered by various authors. Interrelationships and closure properties of those classes are studied. During the presentation, heavy-tailed, regularly varying, consistently varying, dominatedly varying, long-tailed, generalized long-tailed, subexponential and O-subexponential distributions will be discussed.

10.30-11.00 **Prof. Kestutis Kubilius, Vilnius University**

Fractional SDEs with stochastic forcing

**Abstract**. We are interested in fractional stochastic differential equations (FSDEs) with stochastic forcing, i.e. to FSDE, we add a stochastic forcing term to the FSDE. The conditions for the existence and uniqueness of the FSDE solution with stochastic forcing and the convergence rate of the implicit Euler approximation scheme are established. Such equations can be applied to considering FSDEs with a permeable wall.

11.00-11.30 **Assoc. Prof. Martynas Manstavičius, Vilnius University**

On a few generalizations of popular concordance measures

**Abstract. **In this talk we will focus on the intrinsic meaning of various generalizations of Kendall’s τ in the bivariate case. This is motivated by a question what such generalizations really measure and how they are related to the concordance order on the set of copulas. Along the way, we will also discuss several methods to construct such generalizations.

11.30-12.00 **Assoc.** **Prof. Olga Liivapuu, Estonian University of Life Science**

Transposed Poisson superalgebra

**Abstract**. The notion of a transposed Poisson superalgebra is introduced, showcasing how it can be constructed by means of a commutative associative superalgebra and an even degree derivation of this algebra. Through this approach, two examples of transposed Poisson superalgebras are presented. One of them is the graded differential algebra of differential forms on a smooth finite dimensional manifold, where we use the Lie derivative as an even degree derivation. The second example is the commutative superalgebra of basic fields of the quantum Yang–Mills theory, where we use the BRST-supersymmetry as an even degree derivation to define a graded Lie bracket. We show that a transposed Poisson superalgebra has six identities that play an important role in the study of the structure of this algebra.

12.00-13.00 Lunch, restaurant Rosenhill

*Chair Anatoliy Malyarenko*

13.00-13.30** Associate Prof. Ying Ni, Mälardalen University**

Option Hedging using Explainable Artificial Intelligence (X-hedging)

**Abstract. **We develop a financial option hedging framework called X Hedging that utilizes new artificial intelligence methods, is inherently explainable, and is adaptable to different market models, market frictions, and hedging instruments. The topic of pricing and hedging options is broadly studied in financial literature. Recent methods use neural networks’ ability to map complex non-linear relationships to create general and versatile methods, however, at the expense of explainability. We propose a hedging framework that uses gradient-boosted decision trees to increase the explainability of the state-of-the-art frameworks without sacrificing performance. Along these lines, X Hedging complies with the current guidelines and regulations related to explainable artificial intelligence through local explainability, highlighting the practical usability of the hedging framework in the industry.

13.30-14.00 **Prof. Dmitrii Silvestrov, Stockholm University and Mälardalen University**

Semi-Markov modifications of MCMC algorithms

**Abstract**. Semi-Markov modifications of the Hastings-Metropolis MCMC algorithm are presented and compared with the classical version of this algorithm.

14.00-14.30 **Prof. Yuliia Mishura, Taras Shevchenko University of Kyiv and Mälardalen University**

Entropy and entropy risk measures of probability distributions

**Abstract**. We calculate the entropies and entropy risk measures for a wide class of probability distributions and investigate their properties with respect to the parameters of distributions. It is interesting to apply these results to fractional processes and compare their entropies.

14.30-15.00 **Assoc.** **Prof. Kostiantyn Ralchenko, Taras Shevchenko University of Kyiv**

Asymptotic growth of tempered fractional Brownian motions with statistical applications

**Abstract**. Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) modify the power-law kernel in the moving average representation of fractional Brownian motion by incorporating exponential tempering. We develop least-square estimators for the unknown drift parameters within Vasicek models driven by these processes. To demonstrate their strong consistency, we establish almost sure asymptotic bounds for the rate of growth of sample paths of tempered fractional processes.

15.00-15.30 **Coffee break**

15.30-16.00 **Prof. Andrejs Matvejevs, Riga Technical University**

Asymptotic Methods for Stability Analysis of Interacting Populations Dynamics

**Abstract. **A stochastic model of the Holling type II population model is presented and asymptotic methods for analyzing the stability of impulsive dynamic systems for this case are developed.

16.00-16.30 **Assoc.** **Prof. Oksana Pavlenko, Riga Technical University**

Regime-Switching Time Series Models based on Latvian Economic Data

**Abstract. **We discuss self-exciting threshold autoregressive models (SETAR) and smooth transition autoregressive time series models (STAR) with different smoothing functions, the choice between them for the popular economic time series and their advantages. The idea is to compare their usefulness depending on data properties.

16.30-17.00 **Dr.** **Lisa Niklasson, Mälardalen University**

Toric and non-toric ideals of Bayesian networks** **

**Abstract**. In this talk we will consider Bayesian networks from an algebraic perspective. A Bayesian network is a discrete statistical model which can be presented graphically by a directed acyclic graph (DAG). Algebraically such a model can be described as a variety of a homogeneous prime ideal.

A statistical model is called toric if it is given by a toric variety, or in other words consists of solutions to a system of binomial equations. We will discuss which DAG's give rise to toric models, and which do not.

17.00-17.30 **PhD student Marko Dimitrov, Mälardalen University**

Almost-Exact Scheme for Heston-type Models: American and Bermudan Option Pricing

**Abstract**. In this study, Monte Carlo simulation methods are employed for pricing American and Bermudan options, focusing on using an almost exact simulation scheme. This efficient scheme allows simulations to be conducted only on potential early exercise dates. For the classical Black-Scholes model, which follows a stochastic differential equation with a known closed-form solution, an exact scheme is applicable. However, for the more complex Heston model, characterized by a single stochastic volatility process, an Almost Exact Scheme (AES) is used. The AES leverages the non-central chi-square probability distribution to model the variance process. The research addresses the valuation of American and Bermudan options under Heston-type stochastic volatility models, implementing the AES for simulation. Two models are examined: the standard Heston model with a single stochastic volatility and the Double Heston model, which features dual stochastic volatilities. An analytical derivation of the AES for the Double Heston model is presented, along with a numerical analysis to assess the benefits of using AES in both the Heston and Double Heston models. The findings suggest that the AES is particularly effective when the number of simulation steps corresponds with the number of exercise dates, enhancing efficiency.

### May 15

Social program and scientific discussions.

### May 16

09.45-10.00 Registration and welcome from organizers, room **U2-158**

*Chair Sergei Silvestrov*

10.00-10.30** Prof. Sergei Silvestrov, Mälardalen University**

Hom-algebras and -ary Hom-algebras

**Abstract**. I will present a short introductory overview and some recent developments on Hom-algebras and -ary Hom-algebras.

10.30-11.00** PhD stud. German Garcia, Mälardalen University**

Lie-type construction based on twisted derivations.

**Abstract.** The Jacobian determinant, given derivations over a commutative associative algebra, gives an -Lie algebra structure to this algebra. In this talk, I will construct a generalized Jacobian determinant using twisted derivations and study certain properties of it with respect to twisted derivations over the algebra. Under certain conditions on the derivations, it is possible to obtain a series of generalized -ary hom-Lie structures and other more general structures, relying on commutation relations between derivations and twisting maps. I will discuss the construction and explore a particular case in which a new generalization of -ary hom-Lie algebras is found.

11.00-11.30** PhD stud. Stephen Mboya, University of Nairobi, Kenya, Mälardalen University**

Hom-Lie structures and ternary Hom-Lie structures of generalized -type

**Abstract.** This work is devoted to certain properties and structures of Hom-Lie algebras of generalized sl(2)-type. We construct classes of linear twisting maps that turn a skew-symmetric algebra of generalized sl(2)-type into a Hom-Lie algebra and describe the subclasses of such linear maps which yield multiplicative Hom-Lie algebras. We explore some properties of these Hom-algebras related to their ideals, Hom-ideals, subalgebras and Hom-subalgebras, with emphasis on their derived series, and central descending series, as well as their nilpotence and solvability properties. We investigate the invariance of these sub-algebras under the linear twisting maps, and determine whether these sub-algebras are weak subalgebras, Hom-subalgebras, weak ideals, or Hom-ideals. In particular, we investigate these sub-algebras and properties for the subfamilies of non-multiplicative Hom-Lie algebras of generalized sl(2)-type indicating the differences between non-multiplicative and multiplicative cases. We also construct and investigate the ternary Hom-Lie algebras induced by Hom-Lie algebras of generalized sl(2)-type.

This is a joint work with Abdennour Kitouni, Elvice Ongong’a, Jared Ongaro and Sergei Silvestrov.

11.30-12.00 **Dr.** **Per Bäck, Mälardalen University**

A new perspective on the Cayley—Dickson construction: flipped polynomial rings

**Abstract.** The Cayley—Dickson construction is a famous construction that generates new *-algebras out of old ones. It is perhaps best known for generating all the real, normed division algebras: the real numbers, the complex numbers, the quaternions, and the octonions. However, the construction is undoubtedly quite mysterious and seems to be a patchwork created ad hoc to make algebras like the above fit in a construction. In this talk, I will try to shed new light on the Cayley—Dickson construction with the purpose of illuminating the underlying patchwork. We introduce a new class of polynomial rings with a “flipped” multiplication which all Cayley—Dickson algebras naturally appear as quotients of. In particular, this extends the classical construction of the complex numbers as a quotient of a polynomial ring to the quaternions, the octonions, and beyond.

This is based on joint work with Masood Aryapoor (MDU).

12.00-12.30 **Dr.** **Thomas Westerbäck, Mälardalen University**

Matroidal and polymatroidal structures of algebraic objects and some connection to Shannon entropy

**Abstract.** Matroids and their generalizations, polymatroids, can be defined in several equivalent ways, such as certain submodular functions or as special classes of polytopes. These theories find applications in diverse fields of mathematics and computer science, where many properties of the associated objects are not solely facts about the objects themselves, but rather stem from their underlying matroidal or polymatroidal structure. Therefore, matroid and polymatroid theory can serve as common frameworks for studying such structures more naturally. In this talk, I will present some matroidal and polymatroidal structures and results of these structures for algebraic objects. Additionally, I will demonstrate how such structures can be associated with Shannon entropy for some finite algebraic objects.

12.30-13.30 Lunch, restaurant Rosenhill

*Chair Sergei Silvestrov, rum Zeta*

13.30-14.00 **Prof. Natan Kruglyak, Linköping University**

Mathematics and Neural Networks

**Abstract**. In recent years we have seen a tremendous success of artificial intelligence based on neural networks. It was achieved by significant progress in computer science and engineering. Surprisingly, we still don’t have a mathematical theory that would explain these results. During the talk, I plan to discuss one approach to understanding the success of neural networks based on geometry (affine transforms, convex polytopes) combined with approximation and statistics.

14.00-14.30 **Prof. Victor Abramov, University of Tartu**

Ternary algebras of hypermatrices and irreducible geometry in dimension 5

**Abstract. **A semiheap is a set equipped with ternary multiplication which satisfies a generalized (ternary) associativity. This structure was introduced by V. V. Wagner in order to develop an algebraic approach to the set of transition functions of a manifold. A semiheap is said to be a ternary algebra if is a vector space and a ternary multiplication is linear with respect to each argument. We study a ternary algebra of third-order hypermatrices (3-dimensional matrices), where a ternary multiplication is the fish product. We show that the subspace of traceless hypermatrices transforming under cyclic permutations by means of a third-order root of unity provides the right biunits for fish product.

14.30-15.00 **Dr.** **Lars Hellström, Mälardalen University**

Rewriting for a hom-Poisson operad

**Abstract:** In the programme of homifying various algebraic structures, one next step could be the Poisson algebras, which are simultaneously associative algebras and Lie algebras, both of which have established hom counterparts. This talk discusses the matter from the operadic point of view, and studies the matter of how rewriting techniques could be applied to systematically explore the consequences of different homification approaches.

15.00-15.30** Coffee break**

15.30-16.00 **Prof. Anatoliy Malyarenko, **Mälardalen University

Isotropic random vector fields on homogeneous spaces of certain compact Gelfand pairs.

**Abstract.** A centred random vector field on a sphere is called isotropic if its correlation operator transforms according to the tautological representation of the rotation group. We extend this definition to a wider class of manifolds including various projective spaces and calculate the Karhunen-Loéve expansions of this class of random fields.

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