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From January 1, 2026, I return back to the Mälardalen University in Sweden, this time as a Professor Emeritus.
I am working on distributions of scalar-, vector-, tensor-, and spinor-valued functions which are invariant with respect to different groups of transformations. In this area, I proposed to use the theory of induced representations, Clebsch-Gordan coefficients, and tensor products of representations. New links to continuum physics, geometry of convex compacta, and cosmology have been established by myself and in collaboration with colleagues from USA, UK and other countries. Four research monographs published by Springer, Cambridge University Press, CRC Press, and several tens of research papers and chapters in edited books are devoted to this area.
Books
Books
Combining research methods from various areas of mathematics and physics, Probabilistic Models of Cosmic Backgrounds describes the isotropic random sections of certain fiber bundles and their applications to creating rigorous mathematical models of both discovered and hypothetical cosmic backgrounds.
Previously scattered and hard-to-find mathematical and physical theories have been assembled from numerous textbooks, monographs, and research papers, and explained from different or even unexpected points of view. This consists of both classical and newly discovered results necessary for understanding a sophisticated problem of modelling cosmic backgrounds.
The book contains a comprehensive description of mathematical and physical aspects of cosmic backgrounds with a clear focus on examples and explicit calculations. Its reader will bridge the gap of misunderstanding between the specialists in various theoretical and applied areas who speak different scientific languages.
The audience of the book consists of scholars, students, and professional researchers. A scholar will find basic material for starting their own research. A student will use the book as supplementary material for various courses and modules. A professional mathematician will find a description of several physical phenomena at the rigorous mathematical level. A professional physicist will discover mathematical foundations for well-known physical theories.
Malyarenko, Anatoliy; Ostoja-Starzewski, Martin; Amiri-Hezaveh, Amirhossein Random fields of piezoelectricity and piezomagnetism—correlation structures. SpringerBriefs in Applied Sciences and Technology. SpringerBriefs in Mathematical Methods. Springer, Cham, [2020], ©2020. xi+97 pp. ISBN: 978-3-030-60064-8; 978-3-030-60063-1.
Random fields are a necessity when formulating stochastic continuum theories. In this book, a theory of random piezoelectric and piezomagnetic materials is developed. First, elements of the continuum mechanics of electromagnetic solids are presented. Then the relevant linear governing equations are introduced, written in terms of either a displacement approach or a stress approach, along with linear variational principles. On this basis, a statistical description of second-order (statistically) homogeneous and isotropic rank-3 tensor-valued random fields is given. With a group-theoretic foundation, correlation functions and their spectral counterparts are obtained in terms of stochastic integrals with respect to certain random measures for the fields that belong to orthotropic, tetragonal, and cubic crystal systems. The target audience will primarily comprise researchers and graduate students in theoretical mechanics, statistical physics, and probability.
Malyarenko, Anatoliy; Ostoja-Starzewski, Martin Tensor-valued random fields for continuum physics. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 2019. x+302 pp. ISBN: 978-1-108-42985-6.
Many areas of continuum physics pose a challenge to physicists. What are the most general, admissible statistically homogeneous and isotropic tensor-valued random fields (TRFs)? Previously, only the TRFs of rank 0 were completely described. This book assembles a complete description of such fields in terms of one- and two-point correlation functions for tensors of ranks 1 through 4. Working from the standpoint of invariance of physical laws with respect to the choice of a coordinate system, spatial domain representations, as well as their wavenumber domain counterparts are rigorously given in full detail. The book also discusses, an introduction to a range of continuum theories requiring TRFs, an introduction to mathematical theories necessary for the description of homogeneous and isotropic TRFs, and a range of applications including a strategy for simulation of TRFs, ergodic TRFs, scaling laws of stochastic constitutive responses, and applications to stochastic partial differential equations. It is invaluable for mathematicians looking to solve problems of continuum physics, and for physicists aiming to enrich their knowledge of the relevant mathematical tools.
Malyarenko, Anatoliy Invariant random fields on spaces with a group action. With a foreword by Nikolai Leonenko. Probability and its Applications (New York). Springer, Heidelberg, 2013. xviii+261 pp. ISBN: 978-3-642-33405-4; 978-3-642-33406-1 (Reviewer: Yimin Xiao)
The author describes the current state of the art in the theory of invariant random fields. This theory is based on several different areas of mathematics, including probability theory, differential geometry, harmonic analysis, and special functions. The present volume unifies many results scattered throughout the mathematical, physical, and engineering literature, as well as it introduces new results from this area first proved by the author. The book also presents many practical applications, in particular in such highly interesting areas as approximation theory, cosmology and earthquake engineering. It is intended for researchers and specialists working in the fields of stochastic processes, statistics, functional analysis, astronomy, and engineering.
Chapters in books
Chapters in books
Muhumuza, Asaph Keikara; Lundengård, Karl; Malyarenko, Anatoliy; Silvestrov, Sergei; Mango, John Magero; Kakuba, Godwin The Wishart distribution on symmetric cones. Non-commutative and non-associative algebra and analysis structures. SPAS 2019, Västerås, Sweden, September 30–October 2. Papers related to the international conference "Stochastic Processes and Algebraic Structures—From Theory Towards Applications''. Edited by Sergei Silvestrov and Anatoliy Malyarenko. 661–684, Springer Proc. Math. Stat., 426, Springer, Cham, [2023], ©2023. (Reviewer: Houida Mohammed Ahmed)
Anatoliy Malyarenko, Yuliya Mishura, Kostiantyn Ralchenko and Yevheniia Anastasiia Rudyk, "Entropy, Gaussian distribution and fractional processes", 2023 2nd International Conference on Innovative Solutions in Software Engineering (ICISSE), Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine, Nov. 29-30, 2023, pp. 31-34.
Muhumuza, Asaph Keikara; Malyarenko, Anatoliy; Lundengård, Karl; Silvestrov, Sergei; Mango, John Magero; Kakuba, Godwin Extreme points of the Vandermonde determinant and Wishart ensemble on symmetric cones. Stochastic processes, statistical methods, and engineering mathematics. SPAS 2019, Västerås, Sweden, September 30–October 2. Papers related to the conference "Stochastic Processes and Algebraic Structures—From Theory Towards Applications''. Edited by Anatoliy Malyarenko, Ying Ni, Milica Rančić and Sergei Silvestrov. 625–649, Springer Proc. Math. Stat., 408, Springer, Cham, [2022], ©2022.
Muhumuza, Asaph Keikara; Lundengård, Karl; Malyarenko, Anatoliy; Silvestrov, Sergei; Mango, John Magero; Kakuba, Godwin Connections between the extreme points for Vandermonde determinants and minimizing risk measure in financial mathematics. Stochastic processes, statistical methods, and engineering mathematics. SPAS 2019, Västerås, Sweden, September 30–October 2. Papers related to the conference "Stochastic Processes and Algebraic Structures—From Theory Towards Applications''. Edited by Anatoliy Malyarenko, Ying Ni, Milica Rančić and Sergei Silvestrov. 587–623, Springer Proc. Math. Stat., 408, Springer, Cham, [2022], ©2022.
The extreme points of Vandermonde determinants when optimized on surfaces like spheres and cubes have various applications in random matrix theory, electrostatics and financial mathematics. In this study, we apply the extreme points Vandermonde determinant when optimized on various surfaces to risk minimization in financial mathematics. We illustrate this by constructing the efficient frontiers represented by spheres, cubes and other general surfaces as applies to portfolio theory. The extreme points of Vandermonde determinant lying on such surfaces as efficient frontier would be used to determine the set of assets with minimum risk and maximum returns. This technique can also be applied in optimal portfolio selection and asset pricing.
Malyarenko, Anatoliy; Nohrouzian, Hossein Testing cubature formulae on Wiener space versus explicit pricing formulae. Stochastic processes, statistical methods, and engineering mathematics. SPAS 2019, Västerås, Sweden, September 30–October 2. Papers related to the conference "Stochastic Processes and Algebraic Structures—From Theory Towards Applications''. Edited by Anatoliy Malyarenko, Ying Ni, Milica Rančić and Sergei Silvestrov. 223–248, Springer Proc. Math. Stat., 408, Springer, Cham, [2022], ©2022.
Cubature is an effective way to calculate integrals in a finite dimensional space. Extending the idea of cubature to the infinite-dimensional Wiener space would have practical usages in pricing financial instruments. In this paper, we calculate and use cubature formulae of degree 5 and 7 on Wiener space to price European options in the classical Black–Scholes model. This problem has a closed form solution and thus we will compare the obtained numerical results with the above solution. In this procedure, we study some characteristics of the obtained cubature formulae and discuss some of their applications to pricing American options.
Albuhayri, Mohammed; Engström, Christopher; Malyarenko, Anatoliy; Ni, Ying; Silvestrov, Sergei An improved asymptotics of implied volatility in the Gatheral model. Stochastic processes, statistical methods, and engineering mathematics. SPAS 2019, Västerås, Sweden, September 30–October 2. Papers related to the conference "Stochastic Processes and Algebraic Structures—From Theory Towards Applications''. Edited by Anatoliy Malyarenko, Ying Ni, Milica Rančić and Sergei Silvestrov. 3–13, Springer Proc. Math. Stat., 408, Springer, Cham, [2022], ©2022.
We study the double-mean-reverting model by Gatheral. Our previous results concerning the asymptotic expansion of the implied volatility of a European call option, are improved up to order 3, that is, the error of the approximation is ultimately smaller that the 1.5th power of time to maturity plus the cube of the absolute value of the difference between the logarithmic security price and the logarithmic strike price.
Mohammed Albuhayri, Marko Dimitrov, Ying Ni, Anatoliy Malyarenko. Numerical Studies of Implied Volatility Expansions under the Gatheral Model. Chapter 10 in Data Analysis and Related Applications 1: Computational, Algorithmic and Applied Economic Data Analysis. Konstantinos N. Zafeiris, Christos H. Skiadas, Yiannis Dimotikalis, Alex Karagrigoriou, Christiana Karagrigoriou-Vonta (eds), Wiley, 2022, 133-148. Print ISBN:9781786307712 Online ISBN:9781394165513 DOI:10.1002/9781394165513.
The Gatheral model is a three factor model with mean-reverting stochastic volatility that reverts to a stochastic long run mean. This chapter reviews previous analytical results on the first and second order implied volatility expansions under this model. Using the Monte Carlo simulation as the benchmark method, numerical studies are conducted to investigate the accuracy and properties of these analytical expansions. The classical Black–Scholes option pricing model assumes that the underlying asset follows a geometric Brownian motion with constant volatility. The chapter discusses partial calibration procedure is proposed and synthetic and real data calibration. If a full calibration is desired, we can use the results from the partial calibration as inputs for the final local optimization over all model parameters. In implementing the calibration procedure, the effect of the Covid-19 pandemic on the model calibration is high.
Hossein Nohrouzian, Anatoliy Malyarenko, Ying Ni. Pricing Financial Derivatives in the Hull–White Model Using Cubature Methods on Wiener Space. Chapter 25 in Data Analysis and Related Applications 1: Computational, Algorithmic and Applied Economic Data Analysis. Konstantinos N. Zafeiris, Christos H. Skiadas, Yiannis Dimotikalis, Alex Karagrigoriou, Christiana Karagrigoriou-Vonta (eds), Wiley, 2022, 133-148. Print ISBN:9781786307712 Online ISBN:9781394165513 DOI:10.1002/9781394165513.
This chapter discusses the idea of cubature formulae on Wiener space. It briefly explains the theory behind the idea of the cubature method and compares and contrasts it with the classical Monte Carlo simulation. The chapter reviews how the cubature formulae of degree 5 can be used in security market models, namely in the Samuelson price process to estimate the price of European call and put options. It also reviews the idea of constructing a recombining trinomial tree in the Black–Scholes model to price path-dependent derivatives. The chapter focuses on the interest-rate models and explores the Hull–White one-factor model to study the application of the cubature formula in the fixed-income market models. In order to price financial derivatives via cubature method, more possible random values are needed to be accessed each time. Iteration of the cubature formula which results in obtaining a non-recombining trinomial tree is performed.
Mohammed Albuhayri, Anatoliy Malyarenko, Sergei Silvestrov, Ying Ni, Christopher Engström, Finnan Tewolde, Jiahui Zhang Asymptotics of Implied Volatility in the Gatheral Double Stochastic Volatility Model. Chapter 2 in Applied Modeling Techniques and Data Analysis 2: Financial, Demographic, Stochastic and Statistical Models and Methods, Volume 8. Yannis Dimotikalis, Alex Karagrigoriou, Christina Parpoula, Christos H. Skiadas (eds), Wiley, 2021, 27-38. Print ISBN:9781786306746 Online ISBN:9781119821724 DOI:10.1002/9781119821724
Gatheral's double-mean-reverting model by is motivated by empirical dynamics of the variance of stock price. In this chapter, the authors study the behavior of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. Using the method by Pagliarani and Pascucci, they explicitly calculate the first few terms of the asymptotic expansion of the implied volatility within a parabolic region. The DMR model can be consistently calibrated to both the SPX options and the VIX options. However, due to the lack of an explicit formula for both the European option price and the implied volatility, the calibration is usually done using time-consuming methods like the Monte Carlo simulation or the finite difference method. The authors provide an explicit solution to the implied volatility under this model.
Hossein Nohrouzian, Ying Ni, Anatoliy Malyarenko An arbitrage‐free large market model for forward spread curves. Chapter 6 in Applied Modeling Techniques and Data Analysis 2: Financial, Demographic, Stochastic and Statistical Models and Methods, Volume 8. Yannis Dimotikalis, Alex Karagrigoriou, Christina Parpoula, Christos H. Skiadas (eds), Wiley, 2021, 75-89. Print ISBN:9781786306746 Online ISBN:9781119821724 DOI:10.1002/9781119821724
Before the financial crisis started in 2007, the forward rate agreement contracts could be perfectly replicated by overnight indexed swap zero coupon bonds. Using an approach proposed by Cuchiero, Klein and Teichmann, the authors construct an arbitrage-free market model, where the forward spread curves for a given finite tenor structure are described as a mild solution to a boundary value problem for a system of infinite-dimensional stochastic differential equations. They also construct a unique risk-neutral probability measure (equivalent martingale probability measure). The authors investigate the necessary conditions for existence, uniqueness and non-negativity of solutions to some desirable stochastic equations within the Heath–Jarrow–Morton (HJM) framework. The HJM and LMM models also allow us to specify more realistic volatility structures to construct an interest rate model. The authors review the theories of constructing a large financial market model.
Jean‐Paul Murara, Anatoliy Malyarenko, Milica Rancic, Sergei Silvestrov Forecasting Stochastic Volatility for Exchange Rates using EWMA . Chapter 6 in Applied Modeling Techniques and Data Analysis 2: Financial, Demographic, Stochastic and Statistical Models and Methods, Volume 8. Yannis Dimotikalis, Alex Karagrigoriou, Christina Parpoula, Christos H. Skiadas (eds), Wiley, 2021, 65-74. Print ISBN:9781786306746 Online ISBN:9781119821724 DOI:10.1002/9781119821724
In risk management, foreign investors or multinational corporations are highly interested in knowing how volatile a currency is in order to hedge risk. In this chapter, using daily exchange rates and the exponential weighted moving average (EWMA) model, the authors perform volatility forecasting. They investigate how the use of the available time series affects the forecasting, i.e. how reliable our forecasting is depending on the period of available data used. Then, two important questions are investigated, the first being the best value of the decay factor in the EWMA model when forecasting volatility of exchange rates and the second being the optimal out-of-sample forecasting period.
Canhanga, Betuel; Malyarenko, Anatoliy; Murara, Jean-Paul; Ni, Ying; Silvestrov, Sergei Advanced Monte Carlo pricing of European options in a market model with two stochastic volatilities. Algebraic structures and applications. SPAS 2017, Västerås and Stockholm, Sweden, October 4–6. Volume 2 of contributions based on the International Conference "Stochastic Processes and Algebraic Structures—From Theory Towards Applications". Edited by Sergei Silvestrov, Anatoliy Malyarenko and Milica Rančić. 857–874, Springer Proc. Math. Stat., 317, Springer, Cham, [2020], ©2020. 91G60 (65C05 91G20 91G30)
We consider a market model with four correlated factors and two stochastic volatilities, one of which is rapid-changing, while another one is slow-changing in time. An advanced Monte Carlo method based on the theory of cubature in Wiener space is used to find the no-arbitrage price of the European call option in the above model.
Malyarenko, Anatoliy; Nohrouzian, Hossein; Silvestrov, Sergei An algebraic method for pricing financial instruments on post-crisis market. Algebraic structures and applications. SPAS 2017, Västerås and Stockholm, Sweden, October 4–6. Volume 2 of contributions based on the International Conference "Stochastic Processes and Algebraic Structures—From Theory Towards Applications". Edited by Sergei Silvestrov, Anatoliy Malyarenko and Milica Rančić. 839–856, Springer Proc. Math. Stat., 317, Springer, Cham, [2020], ©2020.
After the financial crisis of 2007, significant spreads between interbank rates associated to different maturities have emerged. To model them, we apply the Heath–Jarrow–Morton framework. The price of a financial instrument can then be approximated using cubature formulae on Wiener space in the infinite-dimensional setting. We present a short introduction to the area and illustrate the methods by examples.
Malyarenko, Anatoliy; Ostoja-Starzewski, Martin Random fields related to the symmetry classes of second-order symmetric tensors. Stochastic processes and applications. SPAS2017, Västerås and Stockholm, Sweden, October 4–6, 2017. Based on the International Conference "Stochastic processes and algebraic structures—from theory towards applications''. Edited by Sergei Silvestrov, Anatoliy Malyarenko and Milica Rančić. 173–185, Springer Proc. Math. Stat., 271, Springer, Cham, 2018. (Reviewer: David Ginsbourger)
Malyarenko, A., Ostoja-Starzewski, M. (2018). Tensor Random Fields in Continuum Mechanics. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_71-1
Canhanga, Betuel; Malyarenko, Anatoliy; Murara, Jean-Paul; Silvestrov, Sergei Pricing European options under stochastic volatilities models. Engineering mathematics. I. Electromagnetics, fluid mechanics, material physics and financial engineering. Edited by Sergei Silvestrov and Milica Rančić. 315–338, Springer Proc. Math. Stat., 178, Springer, Cham, 2016.
Interested by the volatility behavior, different models have been developed for option pricing. Starting from constant volatility model which did not succeed on capturing the effects of volatility smiles and skews; stochastic volatility models appear as a response to the weakness of the constant volatility models. Constant elasticity of volatility, Heston, Hull and White, Schöbel–Zhu, Schöbel–Zhu–Hull–White and many others are examples of models where the volatility is itself a random process. Along the chapter we deal with this class of models and we present the techniques of pricing European options. Comparing single factor stochastic volatility models to constant factor volatility models it seems evident that the stochastic volatility models represent nicely the movement of the asset price and its relations with changes in the risk. However, these models fail to explain the large independent fluctuations in the volatility levels and slope. Christoffersen et al. (Manag Sci 22(12):1914–1932, 2009, [4]) proposed a model with two-factor stochastic volatilities where the correlation between the underlying asset price and the volatilities varies randomly. In the last section of this chapter we introduce a variation of Chiarella and Ziveyi model, which is a subclass of the model presented in [4] and we use the first order asymptotic expansion methods to determine the price of European options.
Malyarenko, Anatoliy; Röman, Jan; Schyberg, Oskar Sensitivity analysis of catastrophe bond price under the Hull-White interest rate model. Engineering mathematics. I, Electromagnetics, fluid mechanics, material physics and financial engineering. Edited by Sergei Silvestrov and Milica Rančić. 301–314, Springer Proc. Math. Stat., 178, Springer, Cham, 2016.
We consider a model, where the natural risk index is described by the Merton jump-diffusion while the risk-free interest rate is governed by the Hull–White stochastic differential equation. We price a catastrophe bond with payoff depending on finitely many values of the underlying index. The sensitivities of the bond price with respect to the initial condition, volatility of the diffusion component, and jump amplitude, are calculated using the Malliavin calculus approach.
Malyarenko, Anatoliy; Ostoja-Starzewski, Martin Spectral expansion of three-dimensional elasticity tensor random fields. Engineering mathematics. I. Electromagnetics, fluid mechanics, material physics and financial engineering. Edited by Sergei Silvestrov and Milica Rančić. 281–300, Springer Proc. Math. Stat., 178, Springer, Cham, 2016. (Reviewer: Paolo Vannucci)
We consider a random field model of the 21-dimensional elasticity tensor. Representation theory is used to obtain the spectral expansion of the model in terms of stochastic integrals with respect to random measures.
Malyarenko, A., Ostoja-Starzewski, M. (2016). Tensor-Valued Random Fields in Continuum Physics. In: Trovalusci, P. (eds) Materials with Internal Structure. Springer Tracts in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-21494-8_6
This article reports progress on homogeneous isotropic tensor random fields (TRFs) for continuum mechanics. The basic thrust is on determining most general representations of the correlation functions as well as their spectral expansions. Once this is accomplished, the second step is finding the restrictions dictated by a particular physical application. Thus, in the case of fields of material properties (like conductivity and stiffness), the restriction resides in the positive-definiteness, whereby a connection to experiments and/or computational micromechanics can be established. On the other hand, in the case of fields of dependent properties (e.g., stress, strain and displacement), restrictions are due to the respective field equations.
M. Ostoja-Starzewski, S. Kale, P. Karimi, A. Malyarenko, B. Raghavan, S.I. Ranganathan, J. Zhang, Chapter Two - Scaling to RVE in Random Media, Editor(s): Stéphane P.A. Bordas, Daniel S. Balint, Advances in Applied Mechanics, Elsevier, Volume 49, 2016, 111-211, ISSN 0065-2156, ISBN 9780128047798, https://www.sciencedirect.com/science/article/pii/S0065215616300011.
The problem of effective properties of material microstructures has received considerable attention over the past half a century. By effective (or overall, macroscopic, global) is meant the response assuming the existence of a representative volume element (RVE) on which a homogeneous continuum is being set up. Since the efforts over the past quarter century have been shifting to the problem of the size of RVE, this chapter reviews the results and challenges in this broad field for a wide range of materials. For the most part, the approach employed to assess the scaling to the RVE is based on the Hill–Mandel macrohomogeneity condition. This leads to bounds that explicitly involve the size of a mesoscale domain—this domain also being called a statistical volume element (SVE)—relative to the microscale and the type of boundary conditions applied to this domain. In general, the trend to pass from the SVE to RVE depends on random geometry and mechanical properties of the microstructure, and displays certain, possibly universal tendencies. This chapter discusses that issue first for linear elastic materials, where a scaling function plays a key role to concisely grasp the SVE-to-RVE scaling. This sets the stage for treatment of nonlinear and or/inelastic random materials, including elasto-plastic, viscoelastic, permeable, and thermoelastic classes. This methodology can be extended to homogenization of random media by micropolar (Cosserat) rather than by classical (Cauchy) continua as well as to homogenization under stationary (standing wave) or transient (wavefront) loading conditions. The final topic treated in this chapter is the formulation of continuum mechanics accounting for the violations of second law of thermodynamics, which have been studied on a molecular level in statistical physics over the past two decades. We end with an overview of open directions and challenges of this research field.
Malyarenko, Anatoliy A family of series representations of the multiparameter fractional Brownian motion. Seminar on Stochastic Analysis, Random Fields and Applications VI. Proceedings of the 6th Seminar held in Ascona, May 19–23, 2008. Edited by Robert C. Dalang, Marco Dozzi and Francesco Russo. 209–226, Progr. Probab., 63, Birkhäuser/Springer Basel AG, Basel, 2011. (Reviewer: Anthony Réveillac)
We derive a family of series representations of the multiparameter fractional Brownian motion in the centred ball of radius R in the N-dimensional space RN. Some known examples of series representations are shown to be the members of the family under consideration.
Edited conference proceedings
Edited conference proceedings
Non-commutative and non-associative algebra and analysis structures. SPAS 2019, Västerås, Sweden, September 30–October 2. Papers related to the international conference "Stochastic Processes and Algebraic Structures—From Theory Towards Applications''. Edited by Sergei Silvestrov and Anatoliy Malyarenko. Springer Proceedings in Mathematics & Statistics, 426. Springer, Cham, [2023], ©2023. xvii+830 pp. ISBN: 978-3-031-32008-8; 978-3-031-32009-5.
This book highlights the latest advances in algebraic structures and applications focused on mathematical notions, methods, structures, concepts, problems, algorithms, and computational methods for the natural sciences, engineering, and modern technology. In particular, the book features mathematical methods and models from non-commutative and non-associative algebras and rings associated to generalizations of differential calculus, quantumdeformations of algebras, Lie algebras, Lie superalgebras, color Lie algebras, Hom-algebras and their n-ary generalizations, semi-groups and group algebras, non-commutative and non-associative algebras and computational algebra interplay with q-special functions and q-analysis, topology, dynamical systems, representation theory, operator theory and functional analysis, applications of algebraic structures in coding theory, information analysis, geometry and probability theory.
Stochastic processes, statistical methods, and engineering mathematics. SPAS 2019, Västerås, Sweden, September 30–October 2. Papers related to the conference "Stochastic Processes and Algebraic Structures—From Theory Towards Applications''. Edited by Anatoliy Malyarenko, Ying Ni, Milica Rančić and Sergei Silvestrov. Springer Proceedings in Mathematics & Statistics, 408. Springer, Cham, [2022], ©2022. xix+914 pp. ISBN: 978-3-031-17819-1; 978-3-031-17.
The conference brought together a select group of scientists, researchers, and practitioners from the industry who are actively contributing to the theory and applications of stochastic, and algebraic structures, methods, and models. The conference provided early stage researchers with the opportunity to learn from leaders in the field, to present their research, as well as to establish valuable research contacts in order to initiate collaborations in Sweden and abroad. New methods for pricing sophisticated financial derivatives, limit theorems for stochastic processes, advanced methods for statistical analysis of financial data, and modern computational methods in various areas of applied science can be found in this book. The principal reason for the growing interest in these questions comes from the fact that we are living in an extremely rapidly changing and challenging environment. This requires the quick introduction of new methods, coming from different areas of applied science.
Algebraic structures and applications. SPAS 2017, Västerås and Stockholm, Sweden, October 4–6. Volume 2 of contributions based on the International Conference "Stochastic Processes and Algebraic Structures—From Theory Towards Applications". Edited by Sergei Silvestrov, Anatoliy Malyarenko and Milica Rančić. Springer Proceedings in Mathematics & Statistics, 317. Springer, Cham, [2020], ©2020. xxx+968 pp. ISBN: 978-3-030-41850-2; 978-3-030-41849-6
This book explores the latest advances in algebraic structures and applications, and focuses on mathematical concepts, methods, structures, problems, algorithms and computational methods important in the natural sciences, engineering and modern technologies. In particular, it features mathematical methods and models of non-commutative and non-associative algebras, hom-algebra structures, generalizations of differential calculus, quantum deformations of algebras, Lie algebras and their generalizations, semi-groups and groups, constructive algebra, matrix analysis and its interplay with topology, knot theory, dynamical systems, functional analysis, stochastic processes, perturbation analysis of Markov chains, and applications in network analysis, financial mathematics and engineering mathematics.
Stochastic processes and applications. SPAS2017, Västerås and Stockholm, Sweden, October 4–6, 2017. Based on the International Conference "Stochastic processes and algebraic structures—from theory towards applications''. Edited by Sergei Silvestrov, Anatoliy Malyarenko and Milica Rančić. Springer Proceedings in Mathematics & Statistics, 271. Springer, Cham, 2018. xix+475 pp. ISBN: 978-3-030-02825-1; 978-3-030-02824-4.
This book highlights the latest advances in stochastic processes, probability theory, mathematical statistics, engineering mathematics and algebraic structures, focusing on mathematical models, structures, concepts, problems and computational methods and algorithms important in modern technology, engineering and natural sciences applications.
It comprises selected, high-quality, refereed contributions from various large research communities in modern stochastic processes, algebraic structures and their interplay and applications. The chapters cover both theory and applications, illustrated by numerous figures, schemes, algorithms, tables and research results to help readers understand the material and develop new mathematical methods, concepts and computing applications in the future.
Selected peer-reviewed research papers
Selected peer-reviewed research papers
Finkelshtein, Dmitri; Malyarenko, Anatoliy; Mishura, Yuliya; Ralchenko, Kostiantyn Entropies of the Poisson distribution as functions of intensity: "normal'' and "anomalous'' behavior. Methodol. Comput. Appl. Probab. 27 (2025), no. 2, Paper No. 45, 32 pp. (Reviewer: Suchandan Kayal).
Jetti, Yaswanth Sai; Malyarenko, Anatoliy; Ostoja-Starzewski, Martin Fractal and Hurst effects in solenoidal and irrotational vector random fields. SIAM J. Appl. Math. 85 (2025), no. 2, 1006–1021.
Many phenomena in physics are described by a vector-valued homogeneous and isotropic random function in two or three space variables, or a random field. On the one hand, such a field can be uniquely represented as a sum of two components: a solenoidal one without divergence, and an irrotational one without curl. On the other hand, it may be represented as a sum of a longitudinal part, parallel to a fixed space direction, and a lateral part, orthogonal to that direction. If the longitudinal part of the solenoidal component has certain fractal and memory properties, what are the corresponding properties of its lateral part? Similarly, if the lateral part of the irrotational component has certain fractal and memory properties, what are the corresponding properties of its longitudinal part? We give an answer to those questions using well-known tools of classical real analysis.
Ostoja-Starzewski, M., Jetti, Y., Malyarenko, A., and Porcu, E. (April 7, 2025). "Fractal and Hurst effects in stochastic mechanics." ASME. Appl. Mech. Rev. doi: https://doi.org/10.1115/1.4068387
Myriad random phenomena in nature possess fractal and Hurst characteristics. Random processes/fields, such as those with Cauchy or Dagum correlations, enable modeling such stochastic structures in time and space. In the first place, this paper provides a compact review of these models, including their spectral properties, for wide ranges of the fractal dimension and Hurst parameter. The Cauchy and Dagum models can be used to determine stochastic responses of dynamical systems and/or spatial problems in 1d, 2d, or 3d in the presence of fractal and Hurst characteristics. The paper surveys various examples ranging from vibration problems, rods and beams with random properties under random loadings, waves and wavefronts, fracture, homogenization of random media, and statistical turbulence, to stochastically evolving spontaneous violations of the entropy inequality in granular flows. The latter case shows the route to examine whether a mechanical system gives rise to stochastics with such intriguing features. Common features brought out in this survey show what and how can be achieved with Cauchy and Dagum models in mechanics.
Anatoliy Malyarenko. What Is the Spectral Theory of Random Fields? Austrian Journal of Statistics 54 (2025), no. 1, 17-32.
We review the current state of the spectral theory of random functions of several variables created by Professor M. I. Yadrenko at the end of 1950s. It turns out that the spectral expansions of multi-dimensional homogeneous and isotropic random fields are governed by a pair of convex compacts and are especially simple when these compacts are simplexes. Our new result gives necessary and sufficient conditions for such a situation in terms of the group representation that defines the field.
Braiman, Volodymyr; Malyarenko, Anatoliy; Mishura, Yuliya; Rudyk, Yevheniia Anastasiia Properties of Shannon and Rényi entropies of the Poisson distribution as the functions of intensity parameter. Nonlinear Anal. Model. Control 29 (2024), no. 4, 802–815.
We consider two types of entropy, namely, Shannon and Rényi entropies of the Poisson distribution, and establish their properties as the functions of intensity parameter. More precisely, we prove that both entropies increase with intensity. While for Shannon entropy the proof is comparatively simple, for Rényi entropy, which depends on additional parameter α > 0, we can characterize it as nontrivial. The proof is based on application of Karamata’s inequality to the terms of Poisson distribution.
Malyarenko, A.; Mishura, Y.; Ralchenko, K.; Rudyk, Y.A. Properties of Various Entropies of Gaussian Distribution and Comparison of Entropies of Fractional Processes. Axioms 2023, 12, 1026. https://doi.org/10.3390/axioms12111026
We consider five types of entropies for Gaussian distribution: Shannon, Rényi, generalized Rényi, Tsallis and Sharma–Mittal entropy, establishing their interrelations and their properties as the functions of parameters. Then, we consider fractional Gaussian processes, namely fractional, subfractional, bifractional, multifractional and tempered fractional Brownian motions, and compare the entropies of one-dimensional distributions of these processes.
Malyarenko, Anatoliy; Porcu, Emilio Multivariate random fields evolving temporally over hyperbolic spaces. J. Theoret. Probab. 37 (2024), no. 2, 975–1000.
Gaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the n-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below).
Malyarenko, Anatoliy; Mishura, Yuliya; Ralchenko, Kostiantyn; Shklyar, Sergiy Entropy and alternative entropy functionals of fractional Gaussian noise as the functions of Hurst index. Fract. Calc. Appl. Anal. 26 (2023), no. 3, 1052–1081.
This paper is devoted to the study of the properties of entropy as a function of the Hurst index, which corresponds to the fractional Gaussian noise. Since the entropy of the Gaussian vector depends on the determinant of the covariance matrix, and the behavior of this determinant as a function of the Hurst index is rather difficult to study analytically at high dimensions, we also consider simple alternative entropy functionals, whose behavior, on the one hand, mimics the behavior of entropy and, on the other hand, is not difficult to study. Asymptotic behavior of the normalized entropy (so called entropy rate) is also studied for the entropy and for the alternative functionals.
Malyarenko, Anatoliy; Ostoja-Starzewski, Martin Tensor- and spinor-valued random fields with applications to continuum physics and cosmology. Probab. Surv. 20 (2023), 1–86.
In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes the physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points. The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp., rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp., its tensor square), making the field isotropic.
Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random cross-section models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic.
For the convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix.
Leonenko, Nikolai; Malyarenko, Anatoliy; Olenko, Andriy On spectral theory of random fields in the ball. Theory Probab. Math. Statist. No. 107 (2022), 61–76. (Reviewer: José Rafael León)
The paper investigates random fields in the ball. It studies three types of such fields: restrictions of scalar random fields in the ball to the sphere, spin, and vector random fields. The review of the existing results and new spectral theory for each of these classes of random fields are given. Examples of applications to classical and new models of these three types are presented. In particular, the Matérn model is used for illustrative examples. The derived spectral representations can be utilised to further study theoretical properties of such fields and to simulate their realisations. The obtained results can also find various applications for modelling and investigating ball data in cosmology, geosciences and embryology.
Faouzi, Tarik; Porcu, Emilio; Kondrashuk, Igor; Malyarenko, Anatoliy A deep look into the Dagum family of isotropic covariance functions. J. Appl. Probab. 59 (2022), no. 4, 1026–1041. (Reviewer: Enzo Orsingher)
Malyarenko, Anatoliy; Ostoja-Starzewski, Martin Polyadic random fields. Z. Angew. Math. Phys. 73 (2022), no. 5, Paper No. 204, 21 pp.
The paper considers mean-square continuous, wide-sense homogeneous, and isotropic random fields taking values in a linear space of polyadics. We find a set of such fields whose values are symmetric and positive-definite dyadics, and outline a strategy for their simulation.
Zhang, Xian; Malyarenko, Anatoliy; Porcu, Emilio; Ostoja-Starzewski, Martin Elastodynamic problem on tensor random fields with fractal and Hurst effects. Meccanica 57 (2022), no. 4, 957–970.
Malyarenko, A., Mishura, Y. S., & Rudyk, Y. A. O. (2023). Approximation of fractional integrals of H¨older functions. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics, 4, 18-25. https://doi.org/10.17721/1812-5409.2022/4.2
The paper is devoted to the rate of convergence of integral sums of two different types to fractional integrals. The first theorem proves the Hölder property of fractional integrals of functions from various integral spaces. Then we estimate the rate of convergence of the integral sums of two types corresponding to the Hölder functions, to the respective fractional integrals. We illustrate the obtained results by several figures.
Malyarenko, A., & Nohrouzian, H. (2021). Evolution of forward curves in the Heath–Jarrow–Morton framework by cubature method on Wiener space. Communications in Statistics: Case Studies, Data Analysis and Applications, 7(4), 717–735. https://doi.org/10.1080/23737484.2021.2010622
The multi-curve extension of the Heath–Jarrow–Morton framework is a popular method for pricing interest rate derivatives and overnight indexed swaps in the post-crisis financial market. That is, the set of forward curves is represented as a solution to an initial boundary value problem for an infinite-dimensional stochastic differential equation. In this paper, we review the post-crisis market proxies for interest rate models. Then, we consider a simple model that belongs to the above framework. This model is driven by a single Wiener process, and we discretize the space of trajectories of its driver by cubature method on Wiener space. After that, we discuss possible methods for numerical solution of the resulting deterministic boundary value problem in the finite-dimensional case. Finally, we compare the obtained numerical solutions of cubature method with the classical Monte Carlo simulation.
Ma, Chunsheng; Malyarenko, Anatoliy Time-varying isotropic vector random fields on compact two-point homogeneous spaces. J. Theoret. Probab. 33 (2020), no. 1, 319–339. (Reviewer: Jiancang Zhuang)
A general form of the covariance matrix function is derived in this paper for a vector random field that is isotropic and mean square continuous on a compact connected two-point homogeneous space and stationary on a temporal domain. A series representation is presented for such a vector random field which involves Jacobi polynomials and the distance defined on the compact two-point homogeneous space.
Anatoliy Malyarenko, Martin Ostoja-Starzewski,Towards stochastic continuum damage mechanics, International Journal of Solids and Structures,184 (2020), 202-210, ISSN 0020-7683,https://www.sciencedirect.com/science/article/pii/S0020768319301040.
In classical continuum damage mechanics, the distribution of cracks over differently oriented planes is an even deterministic function defined on the unit sphere. The coefficients of its Fourier expansion are completely symmetric and completely traceless tensors of even rank, the so-called fabric or damage tensors. We propose a stochastic generalisation of the above described mathematical model, where damage tensors are mean-square continuous wide-sense homogeneous and isotropic random fields.
Karimi, Pouyan; Malyarenko, Anatoliy; Ostoja-Starzewski, Martin; Zhang, Xian RVE problem: mathematical aspects and related stochastic mechanics. Internat. J. Engrg. Sci. 146 (2020), 103169, 16 pp.
The paper examines (i) formulation of field problems of mechanics accounting for a random material microstructure and (ii) solution of associated boundary value problems. The adopted approach involves upscaling of constitutive properties according to the Hill--Mandel condition, as the only method yielding hierarchies of scale-dependent bounds and their statistics for a wide range of (non)linear elastic and inelastic, coupled-field, and even electromagnetic problems requiring (a) weakly homogeneous random fields and (b) corresponding variational principles. The upscaling leads to statistically homogeneous and isotropic mesoscale tensor random fields (TRFs) of constitutive properties, whose realizations are, in general, everywhere anisotropic. A summary of most general admissible correlation tensors for TRFs of ranks 1, …, 4 is given. A method of solving boundary value problems based on the TRF input is discussed in terms of the torsion of a randomly structured rod. Given that many random materials encountered in nature (e.g., in biological and geological structures) are fractal and possess long-range correlations, we also outline a method for simulating such materials, accompanied by an application to wave propagation.
Canhanga, Betuel; Malyarenko, Anatoliy; Ni, Ying; Rančić, Milica; Silvestrov, Sergei Analytical and numerical studies on the second-order asymptotic expansion method for European option pricing under two-factor stochastic volatilities. Comm. Statist. Theory Methods 47 (2018), no. 6, 1328–1349.
The celebrated Black–Scholes model made the assumption of constant volatility but empirical studies on implied volatility and asset dynamics motivated the use of stochastic volatilities. Christoffersen in 2009 showed that multi-factor stochastic volatilities models capture the asset dynamics more realistically. Fouque in 2012 used it to price European options. In 2013, Chiarella and Ziveyi considered Christoffersen’s ideas and introduced an asset dynamics where the two volatilities of the Heston type act separately and independently on the asset price, and using Fourier transform for the asset price process and double Laplace transform for the two volatilities processes, solved a pricing problem for American options. This paper considers the Chiarella and Ziveyi model and parameterizes it so that the volatilities revert to the long-run-mean with reversion rates that mimic fast (for example daily) and slow (for example seasonal) random effects. Applying asymptotic expansion method presented by Fouque in 2012, we make an extensive and detailed derivation of the approximation prices for European options. We also present numerical studies on the behavior and accuracy of our first- and second-order asymptotic expansion formulas.
Malyarenko, A. Spectral expansions of random sections of homogeneous vector bundles. Teor. Ĭmovīr. Mat. Stat. No. 97 (2017), 142–156; reprinted in Theory Probab. Math. Statist. No. 97 (2018), 151–165 (Reviewer: Maurizia Rossi)
Tiny fluctuations of the Cosmic Microwave Background as well as various observable quantities obtained by spin raising and spin lowering of the effective gravitational lensing potential of distant galaxies and galaxy clusters are described mathematically as isotropic random sections of homogeneous spin and tensor bundles. We consider the three existing approaches to rigourous construction of the above objects, emphasising an approach based on the theory of induced group representations. Both orthogonal and unitary representations are treated in a unified manner. Several examples from astrophysics are included.
Malyarenko, Anatoliy; Ostoja-Starzewski, Martin Fractal planetary rings: energy inequalities and random field model. Internat. J. Modern Phys. B 31 (2017), no. 30, 1750236, 14 pp.
Canhanga, Betuel; Malyarenko, Anatoliy; Murara, Jean-Paul; Ni, Ying; Silvestrov, Sergei Numerical studies on asymptotics of European option under multiscale stochastic volatility. Methodol. Comput. Appl. Probab. 19 (2017), no. 4, 1075–1087.
Multiscale stochastic volatilities models relax the constant volatility assumption from Black-Scholes option pricing model. Such models can capture the smile and skew of volatilities and therefore describe more accurately the movements of the trading prices. Christoffersen et al. Manag Sci 55(2):1914–1932 (2009) presented a model where the underlying price is governed by two volatility components, one changing fast and another changing slowly. Chiarella and Ziveyi Appl Math Comput 224:283–310 (2013) transformed Christoffersen’s model and computed an approximate formula for pricing American options. They used Duhamel’s principle to derive an integral form solution of the boundary value problem associated to the option price. Using method of characteristics, Fourier and Laplace transforms, they obtained with good accuracy the American option prices. In a previous research of the authors (Canhanga et al. 2014), a particular case of Chiarella and Ziveyi Appl Math Comput 224:283–310 (2013) model is used for pricing of European options. The novelty of this earlier work is to present an asymptotic expansion for the option price. The present paper provides experimental and numerical studies on investigating the accuracy of the approximation formulae given by this asymptotic expansion. We present also a procedure for calibrating the parameters produced by our first-order asymptotic approximation formulae. Our approximated option prices will be compared to the approximation obtained by Chiarella and Ziveyi Appl Math Comput 224:283–310 (2013).
Leonenko, Nikolai; Malyarenko, Anatoliy Matérn class tensor-valued random fields and beyond. J. Stat. Phys. 168 (2017), no. 6, 1276–1301. (Reviewer: Mark Kelbert)
Malyarenko, Anatoliy; Ostoja-Starzewski, Martin A random field formulation of Hooke's law in all elasticity classes. J. Elasticity 127 (2017), no. 2, 269–302. (Reviewer: Vladimir Mityushev)
For each of the 8 symmetry classes of elastic materials, we consider a homogeneous random field taking values in the fixed point set V of the corresponding class, that is isotropic with respect to the natural orthogonal representation of a group lying between the isotropy group of the class and its normaliser. We find the general form of the correlation tensors of orders 1 and 2 of such a field, and the field’s spectral expansion.
Ying Ni, Betuel Canhanga, Anatoliy Malyarenko, Sergei Silvestrov; Approximation methods of European option pricing in multiscale stochastic volatility model. AIP Conf. Proc. 27 January 2017; 1798 (1): 020112. https://doi.org/10.1063/1.4972704
In the classical Black-Scholes model for financial option pricing, the asset price follows a geometric Brownian motion with constant volatility. Empirical findings such as volatility smile/skew, fat-tailed asset return distributions have suggested that the constant volatility assumption might not be realistic. A general stochastic volatility model, e.g. Heston model, GARCH model and SABR volatility model, in which the variance/volatility itself follows typically a mean-reverting stochastic process, has shown to be superior in terms of capturing the empirical facts. However in order to capture more features of the volatility smile a two-factor, of double Heston type, stochastic volatility model is more useful as shown in Christoffersen, Heston and Jacobs [12]. We consider one modified form of such two-factor volatility models in which the volatility has multiscale mean-reversion rates. Our model contains two mean-reverting volatility processes with a fast and a slow reverting rate respectively. We consider the European option pricing problem under one type of the multiscale stochastic volatility model where the two volatility processes act as independent factors in the asset price process. The novelty in this paper is an approximating analytical solution using asymptotic expansion method which extends the authors earlier research in Canhanga et al. [5, 6]. In addition we propose a numerical approximating solution using Monte-Carlo simulation. For completeness and for comparison we also implement the semi-analytical solution by Chiarella and Ziveyi [11] using method of characteristics, Fourier and bivariate Laplace transforms.
Anatoliy Malyarenko; Spectral expansions of tensor-valued random fields. AIP Conf. Proc. 27 January 2017; 1798 (1): 020095. https://doi.org/10.1063/1.4972687
In this paper, we review the theory of random fields that are defined on the space domain ℝ3, take values in a real finite-dimensional linear space V that consists of tensors of a fixed rank, and are homogeneous and isotropic with respect to an orthogonal representation of a closed subgroup G of the group O(3). A historical introduction, the statement of the problem, some current results, and a sketch of proofs are included.
Betuel Canhanga, Ying Ni, Milica Rančić, Anatoliy Malyarenko, Sergei Silvestrov; Numerical methods on European option second order asymptotic expansions for multiscale stochastic volatility. AIP Conf. Proc. 27 January 2017; 1798 (1): 020035. https://doi.org/10.1063/1.4972627
After Black–Scholes proposed a model for pricing European Options in 1973, Cox, Ross and Rubinstein in 1979, and Heston in 1993, showed that the constant volatility assumption made by Black-Scholes was one of the main reasons for the model to be unable to capture some market details. Instead of constant volatilities, they introduced stochastic volatilities to the asset dynamic modeling. In 2009, Christoffersen empirically showed “why multifactor stochastic volatility models work so well”. Four years later, Chiarella and Ziveyi solved the model proposed by Christoffersen. They considered an underlying asset whose price is governed by two factor stochastic volatilities of mean reversion type. Applying Fourier transforms, Laplace transforms and the method of characteristics they presented a semi-analytical formula to compute an approximate price for American options. The huge calculation involved in the Chiarella and Ziveyi approach motivated the authors of this paper in 2014 to investigate another methodology to compute European Option prices on a Christoffersen type model. Using the first and second order asymptotic expansion method we presented a closed form solution for European option, and provided experimental and numerical studies on investigating the accuracy of the approximation formulae given by the first order asymptotic expansion. In the present paper we will perform experimental and numerical studies for the second order asymptotic expansion and compare the obtained results with results presented by Chiarella and Ziveyi.
Andrejs Matvejevs, Jegors Fjodorovs, Anatoliy Malyarenko, Algorithms of the Copula Fit to the Nonlinear Processes in the Utility Industry, Procedia Computer Science 104 (2017), 572-577, ISSN 1877-0509, https://www.sciencedirect.com/science/article/pii/S1877050917301758.
Our research studies the construction and estimation of copula-based semi parametric Markov model for the processes, which involved in water flows in the hydro plants. As a rule analyzing the dependence structure of stationary time series regressive models defined by invariant marginal distributions and copula functions that capture the temporal dependence of the processes is considered. This permits to separate out the temporal dependence (such as tail dependence) from the marginal behavior (such as fat tails) of a time series. Dealing with utility company data we have found the best copula describing data - Gumbel copula. As a result constructed algorithm was used for an imitation of low probability events (in a hydro power industry) and predictions.
Malyarenko, Anatoliy; Ostoja-Starzewski, Martin Spectral expansions of homogeneous and isotropic tensor-valued random fields. Z. Angew. Math. Phys. 67 (2016), no. 3, Art. 59, 20 pp. (Reviewer: Ingemar Kaj)
Ostoja-Starzewski, Martin; Shen, Lihua; Malyarenko, Anatoliy Tensor random fields in conductivity and classical or microcontinuum theories. Math. Mech. Solids 20 (2015), no. 4, 418–432. (Reviewer: Angelica Malaspina)
We study the basic properties of tensor random fields (TRFs) of the wide-sense homogeneous and isotropic kind with generally anisotropic realizations. Working within the constraints of small strains, attention is given to antiplane elasticity, thermal conductivity, classical elasticity and micropolar elasticity, all in quasi-static settings albeit without making any specific statements about the Fourier and Hooke laws. The field equations (such as linear and angular momentum balances and strain–displacement relations) lead to consequences for the respective dependent fields involved. In effect, these consequences are restrictions on the admissible forms of the correlation functions describing the TRFs.
Malyarenko, Anatoliy; Ostoja-Starzewski, Martin Statistically isotropic tensor random fields: correlation structures. Math. Mech. Complex Syst. 2 (2014), no. 2, 209–231. (Reviewer: Serge Cohen)
Anatoliy Malyarenko, Martin Ostoja-Starzewski; The spectral expansion of the elasticity random field. AIP Conf. Proc. 10 December 2014; 1637 (1): 647–655. https://doi.org/10.1063/1.4904635
We consider a deformable body that occupies a region D in the plane. In our model, the body’s elasticity tensor H(x) is the restriction to D of a second-order mean-square continuous random field. Under translation, the expected value and the correlation tensor of the field H(x) do not change. Under action of an arbitrary element k of the orthogonal group O(2), they transform according to the reducible orthogonal representation k ⟼ S2(S2(k)) of the above group. We find the spectral expansion of the correlation tensor R(x) of the elasticity field as well as the expansion of the field itself in terms of stochastic integrals with respect to a family of orthogonal scattered random measures.
Ying Ni, Christopher Engström, Anatoliy Malyarenko, Fredrik Wallin; Investigating the added values of high frequency energy consumption data using data mining techniques. AIP Conf. Proc. 10 December 2014; 1637 (1): 734–743. https://doi.org/10.1063/1.4904645
In this paper we apply data-mining techniques to customer classification and clustering tasks on actual electricity consumption data from 350 Swedish households. For the classification task we classify households into different categories based on some statistical attributes of their energy consumption measurements. For the clustering task, we use average daily load diagrams to partition electricity-consuming households into distinct groups. The data contains electricity consumption measurements on each 10-minute time interval for each light source and electrical appliance. We perform the classification and clustering tasks using four variants of processed data sets corresponding to the 10-minute total electricity consumption aggregated from all electrical sources, the hourly total consumption aggregated over all 10-minute intervals during that clock hour, the total consumption over each four-hour intervals and finally the daily total consumption. The goal is to see if there are any differences in using data sets of various frequency levels. We present the comparison results and investigate the added value of the high-frequency measurements, for example 10-minute measurements, in terms of its influence on customer clustering and classification.
M. Ostoja-Starzewski, A. Malyarenko; Continuum mechanics beyond the second law of thermodynamics. Proc. A 1 November 2014; 470 (2171): 20140531. https://doi.org/10.1098/rspa.2014.0531
The results established in contemporary statistical physics indicating that, on very small space and time scales, the entropy production rate may be negative, motivate a generalization of continuum mechanics. On account of the fluctuation theorem, it is recognized that the evolution of entropy at a material point is stochastically (not deterministically) conditioned by the past history, with an increasing trend of average entropy production. Hence, the axiom of Clausius–Duhem inequality is replaced by a submartingale model, which, by the Doob decomposition theorem, allows classification of thermomechanical processes into four types depending on whether they are conservative or not and/or conventional continuum mechanical or not. Stochastic generalizations of thermomechanics are given in the vein of either thermodynamic orthogonality or primitive thermodynamics, with explicit models formulated for Newtonian fluids with, respectively, parabolic or hyperbolic heat conduction. Several random field models of the martingale component, possibly including spatial fractal and Hurst effects, are proposed. The violations of the second law are relevant in those situations in continuum mechanics where very small spatial and temporal scales are involved. As an example, we study an acceleration wavefront of nanoscale thickness which randomly encounters regions in the medium characterized by a negative viscosity coefficient.
Ostoja-Starzewski M, Shen L, Malyarenko A. Tensor random fields in conductivity and classical or microcontinuum theories. Mathematics and Mechanics of Solids. 2013;20(4):418-432. doi:10.1177/1081286513498524
We study the basic properties of tensor random fields (TRFs) of the wide-sense homogeneous and isotropic kind with generally anisotropic realizations. Working within the constraints of small strains, attention is given to antiplane elasticity, thermal conductivity, classical elasticity and micropolar elasticity, all in quasi-static settings albeit without making any specific statements about the Fourier and Hooke laws. The field equations (such as linear and angular momentum balances and strain–displacement relations) lead to consequences for the respective dependent fields involved. In effect, these consequences are restrictions on the admissible forms of the correlation functions describing the TRFs.
Malyarenko, Anatoliy Invariant random fields in vector bundles and application to cosmology. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 4, 1068–1095. (Reviewer: H. Heyer)
We develop the theory of invariant random fields in vector bundles. The spectral decomposition of an invariant random field in a homogeneous vector bundle generated by an induced representation of a compact connected Lie group G is obtained. We discuss an application to the theory of relic radiation, where G = SO(3). A theorem about equivalence of two different groups of assumptions in cosmological theories is proved.
Malyarenko, Anatoliy An optimal series expansion of the multiparameter fractional Brownian motion. J. Theoret. Probab. 21 (2008), no. 2, 459–475. (Reviewer: Erick Herbin)
We derive a series expansion for the multiparameter fractional Brownian motion. The derived expansion is proven to be rate optimal.
Malyarenko, Anatoliy Functional limit theorems for multiparameter fractional Brownian motion. J. Theoret. Probab. 19 (2006), no. 2, 263–288. (Reviewer: Ciprian A. Tudor)
We prove a general functional limit theorem for multiparameter fractional Brownian motion. The functional law of the iterated logarithm, functional Lévy’s modulus of continuity and many other results are its particular cases. Applications to approximation theory are discussed.
Malyarenko, A. A. Abelian and Tauberian theorems for random fields on two-point homogeneous spaces. (Ukrainian) Teor. Ĭmovīr. Mat. Stat. No. 69 (2003), 106–118; translation in Theory Probab. Math. Statist. No. 69 (2004), 115–127 (2005) (Reviewer: Victoria Majkowska)
We consider centered mean-square continuous random fields for which the variance of increments between two points depends only on the distance between these points. Relations between the asymptotic behavior of the variance of increments near zero and the asymptotic behavior of the spectral measure of the field near infinity are investigated. We prove several Abelian and Tauberian theorems in terms of slowly varying functions.
Malyarenko, A. A. Local properties of Gaussian random fields on compact symmetric spaces, and Jackson-type and Bernstein-type theorems. (Ukrainian) Ukraïn. Mat. Zh. 51 (1999), no. 1, 60–68; translation in Ukrainian Math. J. 51 (1999), no. 1, 66–75
We consider local properties of smaple functions of Gaussian isotropic random fields on compact Riemannian symmetric spaces M of rank 1. We give conditions under which the sample functions of a field almost surely possess logarithmic and power modulus of continuity. As a corollary, we prove a theorem of the Bernstein type for optimal approximations of functions of this sort by harmonic polynomials in the metric of the space L2(M). We use theorems of the Jackson-Bernstein-type to obtain sufficient conditions for the sample functions of a field to almost surely belong to the classes of functions associated with the Riesz and Cesàro means.
Malyarenko, A. A.; Olenko, A. Ya. Multidimensional covariant random fields on commutative locally compact groups. (Russian) Ukraïn. Mat. Zh. 44 (1992), no. 11, 1505–1510; translation in Ukrainian Math. J. 44 (1992), no. 11, 1384–1389 (1993)
Homogeneous in the wide sense, covariant random fields on commutative local compact groups with values in finite-dimensional complex Hilbert spaces are considered. The general formula for the correlation operator of such a field is proved, as well as the spectral representation of the field itself in the form of a series of stochastic integrals with respect to orthogonal random measures.
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