Wiener-Hopf Research Articles

Efficient Evaluation of Expectations of Functions of a Lévy Process and Its Extremum

We prove simple general formulas for expectations of functions of a L\'evy process and its running extremum. Under additional conditions, we derive analytical formulas using the Fourier/Laplace inversion and Wiener-Hopf factorization, and discuss efficient numerical methods for realization of these formulas. As applications, the cumulative probability distribution function of the process and its running maximum and the price of the option to exchange the power of a stock for its maximum are calculated. The most efficient numerical methods use the sinh-acceleration technique and simplified trapezoid rule. The program in Matlab running on a Mac with moderate characteristics achieves the precision E-7 and better in several milliseconds, and E-14 - in a fraction of a second.

The generalized Wiener–Hopf equations for the elastic wave motion in angular regions

In this work, we introduce a general method to deduce spectral functional equations in elasticity and thus, the generalized Wiener–Hopf equations (GWHEs), for the wave motion in angular regions filled by arbitrary linear homogeneous media and illuminated by sources localized at infinity. The work extends the methodology used in electromagnetic applications and proposes for the first time a complete theory to get the GWHEs in elasticity. In particular, we introduce a vector differential equation of first-order characterized by a matrix that depends on the medium filling the angular region. The functional equations are easily obtained by a projection of the reciprocal vectors of this matrix on the elastic field present on the faces of the angular region. The application of the boundary conditions to the functional equations yields GWHEs for practical problems. This paper extends and applies the general theory to the challenging canonical problem of elastic scattering in angular regions.

УСТОЙЧИВОСТЬ ФАКТОРИЗАЦИОННЫХ МНОЖИТЕЛЕЙ ФАКТОРИЗАЦИИ ВИНЕРА–ХОПФА МАТРИЦ-ФУНКЦИЙ

We consider the Wiener–Hopf factorization of two matrix functions A(t) and B(t) that are quite close in the norm of the Wiener algebra. The aim of this work is to study the question when the factorization factors of A(t), B(t) will be close enough to each other. This problem is of considerable interest in connection with the development of methods for approximate factorization of matrix functions. There are two main obstacles in the study of this problem: the instability of the partial indices of matrix functions and the non-uniqueness of their factorization factors. The problem was previously studied by M.A. Shubin, who showed that the stability of factorization factors takes place only in the case when A(t) and B(t) have the same partial indices. Then there is a factorization B(t) for which the factorization factors are sufficiently close to the factors of A(t). Theorem M.A. Shubin is non-constructive since it is not known when the partial indices of two close matrix functions will be the same, and the method for choosing the required Wiener–Hopf factorization of the matrix function B(t) is not indicated. To overcome these shortcomings, in the present paper we study the problem of normalization of the factorization in the stable case, describe all possible types of normalizations, and prove their stability under a small perturbation A(t). Now it is possible to find a constructive way of choosing the factorization of the perturbed matrix function, which guarantees the stability of the factorization factors.

Solvability of a Class of Setvalued Implicit Quasi-Variational Inequalities: A Wiener-Hopf Equation Method

Solvability of a Class of Setvalued Implicit Quasi-Variational Inequalities: A Wiener-Hopf Equation Method

Levy Models Amenable to Efficient Calculations

In our previous publications (IJTAF 2019, Math. Finance 2020), we introduced a general class of SINH-regular processes and demonstrated that efficient numerical methods for the evaluation of the Wiener-Hopf factors and various probability distributions (prices of options of several types) in L\'evy models can be developed using only a few general properties of the characteristic exponent $\psi$. Essentially all popular L\'evy processes enjoy these properties. In the present paper, we define classes of Stieltjes-L\'evy processes (SL-processes) as processes with completely monotone L\'evy densities of positive and negative jumps, and signed Stieltjes-L\'evy processes (sSL-processes) as processes with densities representable as differences of completely monotone densities. We demonstrate that 1) all crucial properties of $\psi$ are consequences of the representation $\psi(\xi)=(a^+_2\xi^2-ia^+_1\xi)ST(\cG_+)(-i\xi)+(a^-_2\xi^2+ia^-_1\xi)ST(\cG_-)(i\xi)+(\sg^2/2)\xi^2-i\mu\xi$, where $ST(\cG)$ is the Stieltjes transform of the (signed) Stieltjes measure $\cG$ and $a^\pm_j\ge 0$; 2) essentially all popular processes other than Merton's model and Meixner processes areSL-processes; 3) Meixner processes are sSL-processes; 4) under a natural symmetry condition, essentially all popular classes of L\'evy processes are SL- or sSL-subordinated Brownian motion.

Barrier billiard and random matrices

The barrier billiard is the simplest example of pseudo-integrable models with interesting and intricate classical and quantum properties. Using the Wiener–Hopf method it is demonstrated that quantum mechanics of a rectangular billiard with a barrier in the centre can be reduced to the investigation of a certain unitary matrix. Under heuristic assumptions this matrix is substituted by a special low-complexity random unitary matrix of independent interest. The main results of the paper are (i) spectral statistics of such billiards is insensitive to the barrier height and (ii) it is well described by the semi-Poisson distributions.

Moments and polynomial expansions in discrete matrix-analytic models

Calculation of factorial moments and point probabilities is considered in integer-valued matrix-analytic models at a finite horizon T. Two main settings are considered, maxima of integer-valued downward skipfree Lévy processes and Markovian point process with batch arrivals (BMAPs). For the moments of the finite-time maxima, the procedure is to approximate the time horizon T by an Erlang distributed one and solve the corresponding matrix Wiener–Hopf factorization problem. For the BMAP, a structural matrix-exponential representation of the factorial moments of N(T) is derived. Moments are then used as a computational vehicle to provide a converging Gram–Charlier series for the point probabilities. Topics such as change-of-measure techniques and time inhomogeneity are also discussed.

Some questions on the relationship of the factorization problem of matrix functions and the truncated Wiener–Hopf equation in the Wiener algebra

Some questions on the relationship of the factorization problem of matrix functions and the truncated Wiener–Hopf equation in the Wiener algebra

Analysis of high frequency EM-waves diffracted by a finite strip with impedance in anisotropic medium

This article presents the rigorous analysis of the EM-plane wave diffraction by a conductible strip of finite length in an anisotropic medium. Leontovich boundary conditions are imposed along the strip to consider the effects of impedance. The presence of geomagnetic field in ionosphere containing cold plasma makes anisotropic medium. Mathematical formulation of this model using Maxwell's equations along with dielectric permittivity of cold plasma gives Helmholtz equation which gets transformed by the application of Fourier transform. Two coupled Wiener–Hopf equations are formulated which are then solved by the standard Wiener–Hopf procedure. The modified stationary phase method is used to find the asymptotic result of separated field in an anisotropic medium. The problem turns out to be for isotropic medium on assigning the particular values to the components of permittivity tensor as under the consideration of large operating frequency as compared to cyclotron frequency while it is of same order with plasma frequency. Graphical analysis is performed to investigate the effects of physical parameter on separated (diffracted by finite plate) field for both isotropic and anisotropic medium.

On the uniqueness of the solution to the Wiener–Hopf equation with probability kernel

On the uniqueness of the solution to the Wiener–Hopf equation with probability kernel

The Wiener-Hopf technique, its generalizations and applications: constructive and approximate methods.

This paper reviews the modern state of the Wiener–Hopf factorization method and its generalizations. The main constructive results for matrix Wiener–Hopf problems are presented, approximate methods are outlined and the main areas of applications are mentioned. The aim of the paper is to offer an overview of the development of this method, and demonstrate the importance of bringing together pure and applied analysis to effectively employ the Wiener–Hopf technique.

Theory of plasmonic edge states in chiral bilayer systems

We analytically describe the plasmonic edge modes for an interface that involves the twisted bilayer graphene (TBG) or other similar Moire van der Waals heterostructure. For this purpose, we employ a spatially homogeneous, isotropic and frequency-dependent tensor conductivity which in principle accounts for electronic and electrostatic interlayer couplings. We predict that the edge mode dispersion relation explicitly depends on the chiral response even in the nonretarded limit, in contrast to the collective bulk plasmonic excitations in the TBG. We obtain a universal function for the dispersion of the optical edge plasmon in the paramagnetic regime. This implies a correspondence of the chiral-TBG optical plasmon to a magnetoplasmon of a single sheet, and chirality is interpreted as an effective magnetic field. The chirality also opens up the possibility of nearly undamped acoustic modes in the paramagnetic regime. Our results may guide future near-field nanoscopy for van der Waals heterostructures. In our analysis, we retain the long-range electrostatic interaction, and apply the Wiener-Hopf method to a system of integral equations for the scalar potentials of the two layers.

Semi-infinite moving crack between two orthotropic strips

The problem of a semi-infinite moving crack at the interface of two bonded dissimilar strips of finite width has been solved. The crack is subjected to two constant normal displacements applied on the boundaries. The problem has been converted to one that is suitable for applying the Wiener–Hopf technique. Fourier transform technique has been used to reduce the mixed boundary value problem to the standard form of the Wiener–Hopf equation which has been solved in asymptotic form to obtain the analytical expressions of stress intensity factor and crack opening displacement. Finally the numerical values of stress intensity factor and crack opening displacement have been plotted graphically against various parameters to show the effects of these parameters on stress intensity factor and crack opening displacement.

Sound radiation and suppression of an unbaffled long enclosure using Helmholtz resonators

Theoretical, numerical, and experimental investigations are presented to predict and suppress the noise radiated from monopole point sources inside an unbaffled long enclosure including the ground. First, a mathematical model is established to calculate the acoustical fields. The modal superposition method is adopted to express the sound pressure inside the long enclosure, while the radiated noise is described by applying the Wiener-Hopf (W-H) technique. Subsequently, the interior and exterior acoustical fields are coupled using the continuity equations of sound pressure and particle velocity at the opening. After that, the theoretical model is validated through the finite element method. The formation mechanisms of sound peaks, lobes, the shadow, and illuminated zones are explained from the perspective of mode theory. Meanwhile, Helmholtz resonators (HRs) are proposed to control the dominant modal responses at the opening so that the radiated noise near the resonant frequencies is attenuated. Afterwards, the relationship between acoustical modes and radiation patterns is analyzed. The HR locations, optimized to reduce the radiated noise, are obtained. Besides, the influences of different noise sources on the radiated sound field are explored. Finally, a quasi-two-dimensional experiment is carried out to verify the proposed model and examine the feasibility of HRs in suppressing the noise radiated from an unbaffled long enclosure including the ground. This study facilitates the understanding of physics behind the sound radiation phenomenon and provides new insights into noise control strategies.

Discrete scattering and meta-arrest of locally resonant elastic wave metamaterials with a semi-infinite crack

Previous investigations on wave scattering and crack propagation in the discrete periodic structure are concentrated on the conditional mass–spring model, in which the internal mass is not included. In this work, elastic wave metamaterials with local resonators are studied to show the scattering of elastic waves by a semi-infinite crack and the arrest behaviour. The influences of internal mass–spring structure are analysed and the discrete Wiener–Hopf method is used to obtain the displacement solution. Numerical calculations are performed to show that the dynamic negative effective mass and band gaps can be observed owing to the local resonance of the internal mass. Therefore, the scattering of an elastic wave with a specific frequency by a semi-infinite crack can be avoided by tuning the structural parameters. Moreover, the energy release ratio which characterizes the splitting resistance is presented and the meta-arrest performance is found. It is expected that this study will increase understanding of how to control the scattering characteristics of elastic waves by a semi-infinite crack in locally resonant metamaterials and also help to improve their fracture resistance.

The generalized Wiener-Hopf equations for wave motion in angular regions: electromagnetic application.

In this work, we introduce a general method to deduce spectral functional equations and, thus, the generalized Wiener–Hopf equations (GWHEs) for wave motion in angular regions filled by arbitrary linear homogeneous media and illuminated by sources localized at infinity with application to electromagnetics. The functional equations are obtained by solving vector differential equations of first order that model the problem. The application of the boundary conditions to the functional equations yields GWHEs for practical problems. This paper shows the general theory and the validity of GWHEs in the context of electromagnetic applications with respect to the current literature. Extension to scattering problems by wedges in arbitrarily linear media in different physics will be presented in future works.

Heuristic solution to the problem of diffraction of a TE-polarized electromagnetic wave on a semitransparent half-plane

Using the method of fundamental components for the problem of diffraction of a TE-polarized wave on a half-plane with two-sided impedance boundary conditions a heuristic solution is obtained that approximately describes the scattered field. The verification and adjustment of heuristic formulas using a rigorous solution by the Wiener–Hopf method are carried out. A physical interpretation of the rigorous solution based on the obtained heuristic relations is proposed.

Radiation of sound waves from a coaxial duct with perforated screen

Abstract Radiation of sound waves by a coaxial rigid duct with perforated screen is investigated by using the Mode Matching technique in conjunction with the Jones’ Method. The geometry of the problem consists semi-infinite outer duct and infinite inner duct. It is assumed that the duct walls are fully rigid. The solution of current study contains an infinite sets of coefficients satisfying an infinite systems of linear algebraic equations. These systems are truncated at a certain number and then solved numerically. The effects of open and perforated case, frequency and porosity on the radiation phenomenon are shown graphically. In the present study, perforated screen makes the problem more interesting when it is compared with the unperforated screen. In this context, solution of the problem is compered numerically with similar studies, which are used different method to obtain Wiener–Hopf equation, existing in the literature. As a result, it is observed that in the absence of a perforated screen, there is a perfect agreement between the two results.

Semi-infinite crack between two dissimilar orthotropic strips

The problem of a semi-infinite crack between two dissimilar orthotropic strips has been solved. Using the Fourier transform technique the problem has been converted to a standard Wiener-Hopf equation which has been solved for asymptotic cases only to deduce the expressions for the stress intensity factor and crack opening displacement. Finally the values of the stress intensity factor and crack opening displacement have been plotted graphically to show the effects of strip width and material orthotropy.

The Solution of the Problem of Scattering by a Semitransparent Half-Plane Using the Malyuzhinets Method

The rigorous solution of the problem of scattering of a plane electromagnetic wave by the semitransparent half-plane is presented. The solution is obtained using the method of the Sommerfeld—Malyuzhinets integral for two polarizations of the incident field. The analytical representation of the cylindrical wave scattered by the edge of the semitransparent half-plane is obtained with the use of the rigorous solution mentioned above. The accuracy of this analytical representation is tested by the comparison of numerical results obtained using the Wiener—Hopf method.