Wiener-Hopf Research Articles

A Wiener-Hopf Integral Equation with a Nonsymmetric Kernel in the Supercritical Case

The paper is devoted to the solvability questions of the Wiener-Hopf integral equation in the case where the kernel K satisfies the conditions 0 ≤ K ∈ L1(ℝ), \(\int_{-\infty}^{\infty} K(t)dt>1\), K(±x) ∈ C(3)(ℝ+), (−1)nK(±x)(n)(x) ≥ 0, x ∈ ℝ+, n =1, 2, 3. Based on Volterra factorization of the Wiener-Hopf operator, and invoking the technique of nonlinear functional equations, we construct real-valued solutions both for homogeneous and non-homogeneous Wiener-Hopf equations, assuming that the function g is real-valued and summable, and the corresponding conditions are satisfied. The behavior at infinity of the corresponding solutions is also studied.

Scattering by two staggered semi-infinite cracks on square lattice: an application of asymptotic Wiener–Hopf factorization

Scattering of time-harmonic plane wave by two parallel semi-infinite rows, but with staggered edges, is considered on square lattice. The condition imposed on the semi-infinite rows is a discrete analogue of Neumann boundary condition. A physical interpretation assuming an out-of-plane displacement for the particles arranged in the form of a square lattice and interacting with nearest-neighbours, associates the scattering problem to lattice wave scattering due to the presence of two staggered but parallel crack tips. The discrete scattering problem is reduced to the study of a pair of Wiener-Hopf equation on an annulus in complex plane, using Fourier transforms. Due to the offset between the crack edges, the Wiener-Hopf kernel, a 2x2 matrix, is not amenable to factorization in a desirable form and an asymptotic method is adapted. Further, an approximation in the far field is carried out using the stationary phase method. A graphical comparison between the far-field approximation based on asymptotic Wiener-Hopf method and that obtained by a numerical solution is provided. Also included is a graphical illustration of the low frequency approximation, where it has been found that the numerical solution of the scattering problem coincides with the well known formidable solution in the continuum framework.

Transmission and reflection of waves at an interface between ordinary material and metamaterial

The propagation of waves though bulk composite material (or metamaterial) is to a large extent understood, but the problem of how waves are transmitted and reflected at any interface between two different composites has hardly been considered, particularly if the composites have random microgeometry. The present work makes a contribution towards the solution of this problem by studying a configuration of particular simplicity. A randomly inhomogeneous composite which occupies the half-space x2 > 0 is bonded at x2=0 to a half-space x2 < 0 of a homogeneous medium, and an acoustic wave is incident on the interface from below. The composite has two constituents which have the same elastic modulus but different densities. A variational approximation of Hashin–Shtrikman type yields an explicit formula for the Fourier/Laplace transform of its effective density and a corresponding dispersion relation for the plane mean waves that the composite can support. When its two-point correlation is chosen as h(r)=e−r/a, the dispersion relation admits two distinct mean acoustic waves, and the only directly available conditions at the interface x2=0 are continuity of displacement and traction, which are insufficient in number to fix the two transmission coefficients and one reflection coefficient. It is possible, however, to apply the variational approach directly to an infinite medium comprising the composite in x2 > 0 and the homogeneous medium in x2 < 0, leading to a problem of Wiener–Hopf type. When the homogeneous half-space has the same properties as one of the constituents of the composite, the Wiener–Hopf problem involves only a single unknown “polarization” supported in x2 > 0 and is solved explicitly.

Sound Radiation from the Perforated End of a Lined Duct

Radiation of sound wave through a lined duct with perforated end is analyzed rigorously. The problem considered is axisymmetric. By using the Fourier transform technique in conjunction with the Mode Matching method, the related boundary value problem is formulated as a Wiener-Hopf (W-H) equation. The Mode-Matching technique allows us to express the field component defined in the waveguide region in terms of normal modes. The solution involves a set of infinitely many expansion coefficients satisfying an infinite system of linear algebraic equations. The numerical solution of this system is obtained for different parameters of the problem such as the surface impedances, specific impedance of the perforated screen and their effects on the radiation phenomenon are shown graphically.

Calculation of Probability of the Exit of a Stochastic Process from a Band by Monte-Carlo Method: A Wiener-Hopf Factorization

This paper considers a method of the calculation of probability of the exit from a band of the solution of a stochastic differential equation. The method is based on the approximation of the solution of the considered equation by a process which is received as a concatenation of Gauss processes, random partition of the interval, Girsanov transform and Wiener-Hopf factorization, and the Monte-Carlo method. The errors of approximation are estimated. The proposed method is illustrated by numerical examples.

Alternating Strain Regimes for Failure Propagation in Flexural Systems

Summary We consider both analytical and numerical studies of a steady-state fracture process inside a discrete mass-beam structure, composed of periodically placed masses connected by Euler–Bernoulli beams. A fault inside the structure is assumed to propagate with a constant speed and this occurs as a result of the action of a remote sinusoidal, mechanical load. The established regime of fracture corresponds to the case of an alternating generalised strain regime. The model is reduced to a Wiener–Hopf equation and its solution is presented. We determine the minimum feeding wave energy required for the steady-state fracture process to occur. In addition, we identify the dynamic features of the structure during the steady-state fracture regime. A transient analysis of this problem is also presented, where the existence of steady-state fracture regimes, revealed by the analytical model, are verified and the associated transient features of this process are discussed.

The Discrete Wiener-Hopf Equation with Probability Kernel of Oscillating Type

We prove the existence of a solution to the discrete inhomogeneous Wiener-Hopf equation whose kernel is an arithmetic probability distribution generating an oscillating random walk. Asymptotic properties of the solution are established depending on the properties of the inhomogeneous term of the equation and its kernel.

The Wiener–Hopf equation in measures with probability kernel

The Wiener–Hopf equation in measures with probability kernel

Pricing Methods for Alpha-Quantile and Perpetual Early Exercise Options Based on Spitzer Identities

We present new numerical schemes for pricing perpetual Bermudan and American options as well as α-quantile options. This includes a new direct calculation of the optimal exercise barrier for early-exercise options. Our approach is based on the Spitzer identities for general Levy processes and on the Wiener-Hopf method. Our direct calculation of the price of alpha-quantile options combines for the first time the Dassios-Port-Wendel identity and the Spitzer identities for the extrema of processes. Our results show that the new pricing methods provide excellent error convergence with respect to computational time when implemented with a range of Levy processes.

Semi-infinite moving crack between two bonded dissimilar isotropic strips

The problem of a moving semi-infinite crack between two bonded dissimilar isotropic strips has been considered. The mixed boundary value problem has been reduced to a standard Wiener–Hopf equation by the use of Fourier transform technique. The Wiener–Hopf equation has been solved in asymptotic cases to obtain the required expressions of the stress intensity factor and crack opening displacement. The effects of various parameters on the stress intensity factor and crack opening displacement have been shown by virtue of the graphs.

On using convolutions with exponential distributions for solving a Kolmogorov backward equation

We propose a numerical method to solve 3-dimensional partial differential equations with variable coefficients, which appear in applications focused on studying random walks in the presence of an absorbing barrier. We restrict ourselves to the case of the Kolmogorov backward equation, which most commonly arises in mathematical finance. A probabilistic interpretation of the problem we use allows to apply Carr’s randomization for dimensionality reduction and employ the Wiener-Hopf factorization to solve the arising 1-dimensional problems. The key idea of the approach proposed is to realize the expected present value operators, which appear in the factorization identity, as convolutions with certain exponential distributions. We generalize our method for the case of exponentially distributed jumps presence using the Russian splitting method. The Wiener-Hopf factors related to the jump component of the operator again involve a convolution with exponential kernels. The numerical iterative scheme suggested to evaluate Wiener-Hopf operators is much less computationally expensive than a standard Fast Fourier transform-based approach.

On Distributions Determined by Their Upward, Space–Time Wiener–Hopf Factor

According to the Wiener–Hopf factorization, the characteristic function $$\varphi $$ of any probability distribution $$\mu $$ on $$\mathbb {R}$$ can be decomposed in a unique way as $$\begin{aligned} 1-s\varphi (t)=[1-\chi _-(s,it)][1-\chi _+(s,it)],\quad |s|\le 1,\,t\in \mathbb {R}\,, \end{aligned}$$where $$\chi _-(e^{iu},it)$$ and $$\chi _+(e^{iu},it)$$ are the characteristic functions of possibly defective distributions in $$\mathbb {Z}_+\times (-\infty ,0)$$ and $$\mathbb {Z}_+\times [0,\infty )$$, respectively. We prove that $$\mu $$ can be characterized by the sole data of the upward factor $$\chi _+(s,it)$$, $$s\in [0,1)$$, $$t\in \mathbb {R}$$ in many cases including the cases where: We conjecture that any probability distribution is actually characterized by its upward factor. This conjecture is equivalent to the following: Any probability measure$$\mu $$on$$\mathbb {R}$$whose support is not included in$$(-\,\infty ,0)$$is determined by its convolution powers$$\mu ^{*n}$$, $$n\ge 1$$restricted to$$[0,\infty )$$. We show that in many instances, the sole knowledge of $$\mu $$ and $$\mu ^{*2}$$ restricted to $$[0,\infty )$$ is actually sufficient to determine $$\mu $$. Then we investigate the analogous problem in the framework of infinitely divisible distributions.

A predictive deconvolution method for non-white-noise reflectivity

Conventional predictive deconvolution assumes that the reflection coefficients of the earth conform to an uncorrelated white noise sequence. The Wiener-Hopf (WH) equation is constructed to solve the filter and eliminate the correlated components of the seismic records, attenuate multiples, and improve seismic resolution. However, in practice, the primary reflectivity series of field data rarely satisfy the white noise sequence assumption, with the result that the correlated components of the primary reflectivity series are also eliminated by traditional deconvolution. This results in signal distortion. To solve this problem, we have proposed an improved method for deconvolution. First, we estimated the wavelet correlation from seismic records using the spectrum-modeling method. Second, this wavelet autocorrelation was used to construct a new autocorrelation function which contains the correlated components caused by the existence of multiples and avoids the correlated components of the primary reflectivity series. Finally, the new autocorrelation function was brought into the WH equation, and the predictive filter operator was calculated for deconvolution. In this paper, we have applied this new method to simulated and field data processing, and we have compared its performance with that of traditional predictive deconvolution. Our results show that the new method can adapt to non-white reflectivity series without changing the statistical characteristics of the primary reflection coefficient series. Compared with traditional predictive deconvolution, the new method reduces processing noise and improves fidelity, all while maintaining the ability to attenuate multiples and enhance seismic resolution.

Standardizability and H2-Optimization of Sampled-Data Systems with Multiple Delays

This paper introduces the concept of generalized standard sampled-data systems with delay Sg and considers a class of multidimensional sampled-data systems that are equivalent to a certain system Sg. Such systems are called standardizable. Necessary and sufficient standardizability conditions are established. For the system Sg and standardizable systems, the H2-optimization problem is solved using the Wiener–Hopf method. As demonstrated below, the H2-optimization problem for a standardizable system with multiple delays is reduced to a similar problem for a standard system with single delay, which significantly simplifies calculations and qualitative analysis of the H2-optimal system.

Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits

Phase-type (PH) distributions are defined as distributions of lifetimes of finite continuous-time Markov processes. Their traditional applications are in queueing, insurance risk, and reliability, but more recently, also in finance and, though to a lesser extent, to life and health insurance. The advantage is that PH distributions form a dense class and that problems having explicit solutions for exponential distributions typically become computationally tractable under PH assumptions. In the first part of this paper, fitting of PH distributions to human lifetimes is considered. The class of generalized Coxian distributions is given special attention. In part, some new software is developed. In the second part, pricing of life insurance products such as guaranteed minimum death benefit and high-water benefit is treated for the case where the lifetime distribution is approximated by a PH distribution and the underlying asset price process is described by a jump diffusion with PH jumps. The expressions are typically explicit in terms of matrix-exponentials involving two matrices closely related to the Wiener-Hopf factorization, for which recently, a Lévy process version has been developed for a PH horizon. The computational power of the method of the approach is illustrated via a number of numerical examples.

Skip-free Markov chains

The aim of this paper is to develop a general theory for the class of skip-free Markov chains on denumerable state space. This encompasses their potential theory via an explicit characterization of their potential kernel expressed in terms of family of fundamental excessive functions, which are defined by means of the theory of Martin boundary. We also describe their fluctuation theory generalizing the celebrated fluctuations identities that were obtained by using the Wiener-Hopf factorization for the specific skip-free random walks. We proceed by resorting to the concept of similarity to identify the class of skip-free Markov chains whose transition operator has only real and simple eigenvalues. We manage to find a set of sufficient and easy-to-check conditions on the one-step transition probability for a Markov chain to belong to this class. We also study several properties of this class including their spectral expansions given in terms of Riesz basis, derive a necessary and sufficient condition for this class to exhibit a separation cutoff, and give a tighter bound on its convergence rate to stationarity than existing results.

Riemann-Hilbert factorization of matrices invariant under inversion in a circle

We consider matrix functions with certain invariance under inversion in the unit circle. If such a function satisfies a positivity assumption on the unit circle, then only zero partial indices appear in its Riemann-Hilbert (Wiener-Hopf) factorization. It implies the unique solvability of a certain class of Riemann-Hilbert boundary value problems. It includes the ones associated with the inverse scattering transform of the focusing/defocusing integrable discrete nonlinear Schrodinger equations.

Fluctuation theory for Lévy processes with completely monotone jumps

We study the Wiener–Hopf factorization for Levy processes $X_{t}$ with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener–Hopf factors are complete Bernstein functions of both the spatial and the temporal variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the supremum of $X_{t}$ up to an independent exponential time; (b) the Laplace transform of the supremum of $X_{t}$ up to a fixed time $T$, as a function of $T$. The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent $f(\xi )$ of $X_{t}$, including a peculiar structure of the curve along which $f(\xi )$ takes real values.

On electronic conductance of partially unzipped armchair nanotubes: further analysis

An investigation of the electronic conductance, based on the nearest-neighbor tight-binding approximation for non-interacting π-electrons, is presented for a junction generated by partial unzipping along the zigzag direction of armchair nanotube. An exact solution of the transmission problem has been found through the scalar discrete Wiener–Hopf method, by exploiting a symmetry of the structure for the reduction of 2 × 2 matrix kernel. The electronic probability distribution, far away from the junction on either side of it, is obtained in closed form via mode-matching. As the main result of the paper, an analytical expression for the reflection and transmission coefficients, as well as the scattering matrix, previously obtained via computational methods in earlier studies, is provided using the Chebyshev polynomials. It is found that the highly simplified nearest neighbor tight binding model of the ribbon/tube junction does not allow a mixing of the parity based on even and odd modes. The paper includes several graphical illustrations and analytical derivations.

Static and Semi-Static Hedging as Contrarian or Conformist Bets

In this paper, we argue that, once the costs of maintaining the hedging portfolio are properly taken into account, semi-static portfolios should more properly be thought of as separate classes of derivatives, with non-trivial, model-dependent payoff structures. We derive new integral representations for payoffs of exotic European options in terms of payoffs of vanillas, different from Carr-Madan representation, and suggest approximations of the idealized static hedging/replicating portfolio using vanillas available in the market. We study the dependence of the hedging error on a model used for pricing and show that the variance of the hedging errors of static hedging portfolios can be sizeably larger than the errors of variance-minimizing portfolios. We explain why the exact semi-static hedging of barrier options is impossible for processes with jumps, and derive general formulas for variance-minimizing semi-static portfolio. We show that hedging using vanillas only leads to larger errors than hedging using vanillas and first touch digitals. In all cases, efficient calculations of the weights of the hedging portfolios are in the dual space using new efficient numerical methods for calculation of the Wiener-Hopf factors and Laplace-Fourier inversion.