Trapezoidal Quadrature Research Articles

Option price computation under binary control regime switching triple-factor stochastic volatility model

This study presents an efficient pricing framework for European call options under a binary control regime that switches to a triple-factor stochastic volatility model, tailored for recessionary and stable market phases. The model captures regime transitions via binary controls and incorporates triple volatility sources. We derive the characteristic function and implement a semi-analytical pricing formula using trapezoidal and Gauss-Laguerre quadrature in MATLAB. The economic recovery process is influenced by the control parameter α, while the impacts θ₃ are considered secondary to other factors driving recovery. The results show that the option prices under recessionary conditions were lower compared to the recession-free regime, thereby validating the model’s sensitivity to macroeconomic uncertainty. It further confirms that the binary control regime switching triple-factor stochastic volatility model offers greater accuracy and adaptability across economic states, making it a promising tool for option pricing in dynamic financial environments.

Численная схема для одной интегро-дифференциальной системы, связанной с задачей космического динамо

Статья посвящена описанию разработанной численной схемы для моделирования эредитарной динамической системой, являющейся моделью двумодового гидромагнитного динамо. Модели включают в себя два генератора магнитного поля — крупномасштабный и турбулентный (α-эффект). Влияние магнитного поля на движения среды представлено через подавление α-эффекта функционалом от компонент поля, что вводит в модель память (эредитарность). Модель описывается интегро-дифференциальной системой уравнений.В работе представлена сама численная схема и исследован порядок точности на вложенных сетках. Численная схема состоит из двух частей, для дифференциальной части используется метод трапеций, а для интегральной квадратурная формула трапеций. В результате сопряжения схем получаем нелинейную алгебраическую систему уравнений. Для решения такой системы необходимо привлечение методов для нелинейных алгебраических систем. В работе был выбран метод Ньютона. Показано, что в случае экспоненциального ядра функционала подавления модель может быть сведена к классической системе Лоренца. Известный характер динамики системы Лоренца при различных параметрах позволил верифицировать численную схему. Показано, что численная схема позволяет решать на качественном уровне интегро-дифференциальную систему уравнений, которая является моделью космического динамо. Данная численная схема была разработана для конкретной модели, но может быть легко обобщена для других квадратично-нелинейных интегро-дифференциальных систем. The article is devoted to the description of the developed numerical scheme for modeling a hereditary dynamic system, which is a model of a two-mode hydromagnetic dynamo. The models include two magnetic field generators – large-scale and turbulent (α-effect). The influence of the magnetic field on the motion of the medium is presented through the suppression of the α-effect by the functional of the field components, which introduces memory into the model (hereditary). The model is described by an integro-differential system of equations. The paper presents the numerical scheme itself and investigates the order of accuracy on nested grids. The numerical scheme consists of two parts, the trapezoid method is used for the differential part, and the trapezoid quadrature formula is used for the integral part. As a result of conjugation of the schemes, we obtain a nonlinear algebraic system of equations. To solve such a system, it is necessary to involve methods for nonlinear algebraic systems. In this paper, the Newton method was chosen. It is shown that in the case of an exponential kernel of the suppression functional, the model can be reduced to the classical Lorenz system. The known nature of the dynamics of the Lorenz system for various parameters allowed us to verify the numerical scheme. It is shown that the numerical scheme allows us to qualitatively solve the integro-differential system of equations, which is a model of a cosmic dynamo. This numerical scheme was developed for a specific model, but can be easily generalized for other quadratic-nonlinear integro-differential systems.

A highly parallelized multiscale preconditioner for Darcy flow in high-contrast media

A highly parallelized multiscale preconditioner for Darcy flow in high-contrast media

Computing free convolutions via contour integrals

This work proposes algorithms for computing additive and multiplicative free convolutions of two given probability measures. We consider probability measures with compact support whose free convolution results in a probability measure with a density function that exhibits a square-root decay at the boundary (for example, the semicircle distribution or the Marchenko–Pastur distribution). A key ingredient of our method is rewriting the intermediate quantities of the free convolution using the Cauchy integral formula and then discretizing these integrals using the trapezoidal quadrature rule, which converges exponentially fast under suitable analyticity properties of the functions to be integrated.

Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods

The pursuit of more efficient and reliable numerical methods to solve nonlinear systems of equations has long intrigued many researchers. Among these, the Broyden method has stood out since its introduction, serving as a foundational technique from which various derivative methods have evolved. These derivative methods, commonly referred to as Broyden-like iterative methods, often surpass the traditional Broyden method in terms of both the number of iterations required and the computational time needed. This study aimed to develop new Broyden-like methods by incorporating weighted combinations of different quadrature rules. Specifically, the research focused on leveraging the Composite Trapezoidal rule with n=3n=3, and comparing it against the Midpoint, Trapezoidal, and Simpson quadrature rules. By integrating these approaches, three novel methods were formulated. The findings revealed that several of these new methods demonstrated enhanced efficiency and robustness compared to their established counterparts. In a detailed comparative analysis with the classical Broyden method and other improved versions, the Midpoint–Composite Trapezoidal (M<i>T_3</i>) method emerged as the top performer. This method consistently provided superior numerical outcomes across all benchmark problems examined in the study. The results highlight the potential of these new methods to significantly advance the field of numerical analysis, offering more powerful tools for researchers and practitioners dealing with complex nonlinear systems of equations. Through this innovative approach, the study not only broadens the understanding of Broyden-like methods but also sets the stage for further advancements in the development of efficient numerical solutions.

Composite trapezoidal quadrature for computing hypersingular integrals on interval

<p>In this paper, composite trapezoidal quadrature for numerical evaluation of hypersingular integrals was first introduced. By Taylor expansion at the singular point $ y $, error functional was obtained. We know that the divergence rate of $ O(h^{-p}), p = 1, 2 $, and there were no roots of the special function for the first part in the error functional. Meanwhile, for the second part of the error functional, the divergence rate was $ O(h^{-p+1}), p = 1, 2 $, but there were roots of the special function. We proved that the convergence rate could reach $ O(h^{2}) $ at superconvergence points far from the end of the interval. Two modified trapezoidal quadratures are presented and their convergence rate can reach $ O(h^{2}) $ at certain superconvergence points or any local coordinate point. At last, several examples were presented to test our theorem.</p>

A Unified Trapezoidal Quadrature Method for Singular and Hypersingular Boundary Integral Operators on Curved Surfaces

This paper describes a trapezoidal quadrature method for the discretization of singular and hypersingular boundary integral operators (BIOs) that arise in solving boundary value problems for elliptic partial differential equations. The quadrature is based on a uniform grid in parameter space coupled with the standard punctured Trapezoidal rule. A key observation is that the error incurred by the singularity in the kernel can be expressed exactly using generalized Euler-Maclaurin formulae that involve the Riemann zeta function in 2D and the Epstein zeta functions in 3D. These expansions are exploited to correct the errors via local stencils at the singular point using a novel systematic moment-fitting approach. This new method provides a unified treatment of all common BIOs (Laplace, Helmholtz, Stokes, etc.). We present numerical examples that show convergence of up to 32 nd -order in 2D and 9 th -order in 3D with respect to the mesh size.

Some bounds for the elliptical integral of the first kind

Abstract In this article we present a method to get upper bound for some integrals. The method is in connection with on a generalized trapezoidal quadrature formula. Applications for upper bounds of the elliptical integral of the first kind are presented. We compare these bounds with some previous results from the literature.

Reconstruction of Timewise Dependent Coefficient and Free Boundary in Nonlocal Diffusion Equation with Stefan and Heat Flux as Overdetermination Conditions

The problem of reconstruction of a timewise dependent coefficient and free boundary at once in a nonlocal diffusion equation under Stefan and heat Flux as nonlocal overdetermination conditions have been considered. A Crank–Nicolson finite difference method (FDM) combined with the trapezoidal rule quadrature is used for the direct problem. While the inverse problem is reformulated as a nonlinear regularized least-square optimization problem with simple bound and solved efficiently by MATLAB subroutine lsqnonlin from the optimization toolbox. Since the problem under investigation is generally ill-posed, a small error in the input data leads to a huge error in the output, then Tikhonov’s regularization technique is applied to obtain regularized stable results.

Multi-Point Flux MFE Decoupled Method for Compressible Miscible Displacement Problem

In this paper, a multi-point flux mixed-finite-element decoupled method was considered for the compressible miscible displacement problem. For this compressible problem, a fully discrete backward Euler scheme was proposed, in which the velocity and pressure equations were decoupled by a multi-point flux MFE method using BDM1 elements combined with a trapezoidal quadrature rule. The concentration equation was handled by a standard FE method. The error analysis for velocity, pressure, and concentration were rigorously derived. Numerical experiments to verify the convergence rates and simulate the miscible displacement problem of a water–oil system were presented.

A mixed finite element method for a class of evolution differential equations with p-Laplacian and memory

A mixed finite element method for a class of evolution differential equations with p-Laplacian and memory

Tosoh reports its consolidated results for fiscal 2022: Speciality Group

Tosoh reports its consolidated results for fiscal 2022: Speciality Group

Adaptive density tracking by quadrature for stochastic differential equations

Adaptive density tracking by quadrature for stochastic differential equations

Splitting extrapolation algorithms for solving linear delay Volterra integral equations with a spatial variable

Splitting extrapolation algorithms for solving linear delay Volterra integral equations with a spatial variable

New Fractional Integral Inequalities for Convex Functions Pertaining to Caputo–Fabrizio Operator

In this article, a generalized midpoint-type Hermite–Hadamard inequality and Pachpatte-type inequality via a new fractional integral operator associated with the Caputo–Fabrizio derivative are presented. Furthermore, a new fractional identity for differentiable convex functions of first order is proved. Then, taking this identity into account as an auxiliary result and with the assistance of Hölder, power-mean, Young, and Jensen inequality, some new estimations of the Hermite-Hadamard (H-H) type inequality as refinements are presented. Applications to special means and trapezoidal quadrature formula are presented to verify the accuracy of the results. Finally, a brief conclusion and future scopes are discussed.

Solving a System of Fractional-Order Volterra Integro-Differential Equations Based on the Explicit Finite Difference Approximation via the Trapezoid Method with Error Analysis

The well-known central finite difference approximation was combined with the trapezoid quadrature method in this study to provide a numerical solution of the linear system of Volterra integro-fractional differential equations (LSVI-FDEs) of arbitrary orders, where the fractional derivative is described in the Caputo sense and the orders are between zero and one. The method works by first using the central finite difference approximation to approximate the Caputo derivative at any fixed point and then using the trapezoidal rule to obtain a finite difference expression for our fractional equation, while recalling the linear spline approximation for the first steps. This new, more efficient method involves converting sets of equations and conditions into matrix relationships, from which symmetry matrices can be created in some cases. We also present a new approach for error analysis of the discrete numerical scheme and the explicit numerical technique for LSVI-FDEs. The multi-level explicit finite difference approximation’s stability and convergence were explored, and a MatLab application was created to explain the results. Finally, several numerical examples are offered to demonstrate the technique’s application.

New inequalities for strongly exponentially generalized functions with applications

The aim of this paper is to introduce a new class of functions called strongly exponentially generalized $(m,\nu_{1},\nu_{2},g_{1},g_{2})$.Some new integral inequalities of trapezium-type for strongly exponentially generalized $(m,\nu_{1},\nu_{2},g_{1},g_{2})$ functions with modulus $c$ via Riemann-Liouville fractional integral are established. Also, some new estimates with respect to trapezium-type integral inequalities for strongly exponentially generalized $(m,\nu_{1},\nu_{2},g_{1},g_{2})$ functions with modulus $c$ via general fractional integrals are obtained. We show that the strongly exponentially generalized $(m,\nu_{1},\nu_{2},g_{1},g_{2})$ functions with modulus $c$ includes several other classes of functions. At the end, some new error estimates for trapezoidal quadrature formula are provided as well. This results may stimulate further research in different areas of pure and applied sciences.

A Note on Hermite–Hadamard–Fejer Type Inequalities for Functions Whose n-th Derivatives Are m-Convex or (α,m)-Convex Functions

In this paper, we develop some Hermite–Hadamard–Fejér type inequalities for n-times differentiable functions whose absolute values of n-th derivatives are (α,m)-convex function. The results obtained in this paper are extensions and generalizations of the existing ones. As a special case, the generalization of the remainder term of the midpoint and trapezoidal quadrature formulas are obtained.

The determination of the aggregate loss distribution through the numerical inverse of the characteristic function using the trapezoidal quadrature rule

Aggregate loss is the total loss suffered by an insured in a certain period. The aggregate loss depends on the claim frequency and the amount of the claim each time the insured makes a claim. The distribution of aggregate losses must be known to calculate motor vehicle insurance premiums. In general, there are two methods that can be used in determining the distribution of aggregate losses, namely exact and numerical. When an exact solution is difficult to find, numerical methods such as Monte Carlo, Panjer Recursion, and Fast Fourier Transform can be used. This research will discuss the determination of the distribution of aggregate losses through the numerical inverse of the characteristic function using the trapezoidal quadrature rule, on the data of motor vehicle insurance category 7 in Indonesia. The estimated cumulative distribution function for the largest aggregate loss is 0.999993. When x=0, it means that if someone does not file a claim, the estimated value of the cumulative distribution function is 0.9293. This value is close to the percentage of the number of insured, which is 0.9241.

Corrected trapezoidal rules for boundary integral equations in three dimensions

The manuscript describes a quadrature rule that is designed for the high order discretization of boundary integral equations (BIEs) using the Nyström method. The technique is designed for surfaces that can naturally be parameterized using a uniform grid on a rectangle, such as deformed tori, or channels with periodic boundary conditions. When a BIE on such a geometry is discretized using the Nyström method based on the Trapezoidal quadrature rule, the resulting scheme tends to converge only slowly, due to the singularity in the kernel function. The key finding of the manuscript is that the convergence order can be greatly improved by modifying only a very small number of elements in the coefficient matrix. Specifically, it is demonstrated that by correcting only the diagonal entries in the coefficient matrix, \(O(h^{3})\) convergence can be attained for the single and double layer potentials associated with both the Laplace and the Helmholtz kernels. A nine-point correction stencil leads to an \(O(h^5)\) scheme. The method proposed can be viewed as a generalization of the quadrature rule of Duan and Rokhlin, which was designed for the 2D Lippmann–Schwinger equation in the plane. The techniques proposed are supported by a rigorous error analysis that relies on Wigner-type limits involving the Epstein zeta function and its parametric derivatives.