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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.

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 Mat\'{e}rn 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., Comment: 15 pages, 4 figures

This paper reports a cellular automata (CA) study of the transient dynamic responses of anti-plane shear Lamb’s problems on random fields (RFs) with fractal and Hurst effects. Both Cauchy and Dagum random field models are employed to capture the combined effects of spatial randomness in both mass density and stiffness tensor fields. First, with a dyadic representation, we formulate a second-rank anti-plane stiffness tensor random field (TRF) model with full anisotropy. Its statistical, fractal, and Hurst properties are investigated, leading to introduction of a so-called MOSP model of TRF. Then, we generalize the CA approach to incorporate the inhomogeneity in mass density as well as stiffness fields. Through parametric studies for both Cauchy and Dagum TRFs, the sensitivity of wave propagation on random fields is assessed for a wide range of fractal and Hurst parameters. In general, the mean response amplitude is lowered by the presence of randomness, and the Hurst parameter (especially, for $$\beta < 0.5$$ ) is found to have a stronger influence than the fractal dimension on the response. The results are compared with two simpler random fields: (1) randomness is present only in the mass density field; (2) randomness is present in the mass density field and in a locally isotropic stiffness tensor field. Overall, the results show that a second-rank anti-plane stiffness TRF with full anisotropy leads to the strongest fluctuation in displacement responses followed by a locally isotropic RF model.

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.

In risk management, foreign investors or multinational corporations are highly interested in knowing how volatile acurrency is in order to hedge risk. In this paper, using daily exchange rates and ...

The double-mean-reverting model by Gatheral [1] is motivated by empirical dynamics of the variance of the stock price. No closed-form solution for European option exists in the above model. We stud ...

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.

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 constructing 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., 15 pages, no figures, small typos fixed

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 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 convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix., Comment: 2 figures

The Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in Rd of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed forms expressions for the Dagum spectral density in terms of the Fox{Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities., Comment: 15 Pages

The general form of the one- and two-point correlation tensor of a homogeneous and \((K,\theta )\)-isotropic random field and the spectral expansion of such a field in terms of stochastic integrals with respect to certain random measures depend on the choice of a basis in the linear space where the field takes its values. We choose a basis for 11 different fields. It turns out that the basis depends only on the crystal system of the group K.

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.

Following the motivation of this work, this chapter introduces the basic concepts of continuum mechanics and electromagnetism. Attention is then focused on linear piezoelectricity, elaborating two ways of writing the governing equations: the displacement approach and the stress approach. This leads to variational principles. The final section provides a basis for generalising piezoelectricity—and, by mathematical analogy, piezomagnetism—to random media whose description necessitates tensor-valued random fields.

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 ...

We construct classes of homogeneous random fields on a three-dimensional Euclidean space that take values in linear spaces of tensors of a fixed rank and are isotropic with respect to a fixed orthogonal representation of the group of $3\times 3$ orthogonal matrices. The constructed classes depend on finitely many isotropic spectral densities. We say that such a field belong to either the Mat\'{e}rn or the dual Mat\'{e}rn class if all of the above densities are Mat\'{e}rn or dual Mat\'{e}rn. Several examples are considered., Comment: 37 pages, no figures

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).

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.

For each of the $8$ isotropy classes of elastic materials, we consider a homogeneous random field taking values in the fixed point set $\mathsf{V}$ 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., Modified version, typos corrected, 100 pages, 1 figure

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.

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 involve Jacobi polynomials and the distance defined on the compact two-point homogeneous space., 17 pages, no figures

Under the change of basis in the three-dimensional space by means of an orthogonal matrix g, a matrix A of a linear operator is transformed as \(A\mapsto gAg^{-1}\). Mathematically, the stationary subgroup of a symmetric matrix under the above action can be either \(D_2\times Z^c_2\), when all three eigenvalues of A are different, or \(\mathrm {O}(2)\times Z^c_2\), when two of them are equal, or \(\mathrm {O}(3)\), when all three eigenvalues are equal. Physically, one typical application relates to dependent quantities like a second-order symmetric stress (or strain) tensor. Another physical setting is that of dependent fields, such as conductivity with such three cases is the conductivity (or, similarly, permittivity, or anti-plane elasticity) second-rank tensor, which can be either orthotropic, transversely isotropic, or isotropic. For each of the above symmetry classes, we consider a homogeneous random field taking values in the fixed point set of the class that is invariant with respect to the natural representation of a certain closed subgroup of the orthogonal group. Such fields may model stochastic heat conduction, electric permittivity, etc. We find the spectral expansions of the introduced random fields.

After Black-Scholes proposed a model for pricing European Option in 1973, Cox, Ross and Rubinstein in 1979, and Heston in 1993, showed that the constant volatility assumption in the Black-Scholes m ...

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.

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...

Let V be a real finite-dimensional vector space. We introduce some physical problems that may be described by V-valued homogeneous and isotropic random fields on R 3 . We propose a general method f ...

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.