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A Comparative Numerical Study of Parabolic partial integro-differential equation Arising from Convection-Diffusion

Computer Modeling in Engineering & Sciences 2021年第2期126卷 673-692页

作者： Kamil Khan Arshed Ali Fazal-i-Haq Iltaf Hussain Nudrat Amir Department of Mathematics Islamia CollegePeshawar25000Pakistan Department of Mathematics Statistics and Computer ScienceThe University of AgriculturePeshawar25000Pakistan Department of Basic Sciences&Islamiat University of Engineering&TechnologyPeshawar25000Pakistan Department of Basic Sciences&Humanities CECOS University of Information Technology and Emerging SciencesPeshawar25000Pakistan

This article studies the development of two numerical techniques for solving convection-diffusion type partial integro-differential equation(PIDE)with a weakly singular *** trigonometric B-spline(CTBS)functions are used for interpolation in both *** first method is CTBS based collocation method which reduces the PIDE to an algebraic tridiagonal system of linear *** other method is CTBS based differential quadrature method which converts the PIDE to a system of ODEs by computing spatial derivatives as weighted sum of function *** efficient tridiagonal solver is used for the solution of the linear system obtained in the first method as well as for determination of weighting coefficients in the second *** explicit scheme is employed as time integrator to solve the system of ODEs obtained in the second *** methods are tested with three nonhomogeneous problems for their ***,computational efficiency and numerical convergence of the methods are *** of errors in approximations produced by the present methods versus different values of discretization parameters and convection-diffusion coefficients are *** and diffusion dominant cases are discussed in terms of Peclet *** results are also compared with cubic B-spline collocation method.

关键词： partial integro-differential equation convection-diffusion collocation method differential quadrature cubic trigonometric B-spline functions weakly singular kernel

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A differential Quadrature Based Approach for Volterra partial integro-differential equation with a Weakly Singular Kernel

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Computer Modeling in Engineering & Sciences 2020年第9期124卷 915-935页

作者： Siraj-ul-Islam Arshed Ali Aqib Zafar Iltaf Hussain Department of Basic Sciences and Islamiat University of Engineering and Technology PeshawarKhyber PakhtunkhwaPakistan Department of Mathematics Islamia College PeshawarKhyber PakhtunkhwaPakistan

differential quadrature method is employed by numerous researchers due to its numerical accuracy and computational efficiency,and is mentioned as potential alternative of conventional numerical *** this paper,a differential quadrature based numerical scheme is developed for solving volterra partial integro-differential equation of second order having a weakly singular *** scheme uses cubic trigonometric B-spline functions to determine the weighting coefficients in the differential quadrature approximation of the second order spatial *** advantage of this approximation is that it reduces the problem to a first order time dependent integro-differential equation(IDE).The proposed scheme is obtained in the form of an algebraic system by reducing the time dependent IDE through unconditionally stable Euler backward method as time *** scheme is validated using a homogeneous and two nonhomogeneous test *** of the system matrix and numerical convergence of the method are analyzed for spatial and temporal domain discretization *** of results of the present approach with Sinc collocation method and quasi-wavelet method are also made.

关键词： partial integro-differential equation differential quadrature cubic trigonometric B-spline functions weakly singular kernel

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The Global Behavior of Finite Difference-Spatial Spectral Collocation Methods for a partial integro-differential equation with a Weakly Singular Kernel

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Numerical Mathematics(Theory,Methods and Applications) 2013年第3期6卷 556-570页

作者： Jie Tang Da Xu College of Mathematics and Computer Science Hunan Normal UniversityChangsha 410081HunanP.R.China College of Science Hunan University of TechnologyZhuzhou 412008HunanP.R.China

The z-transform is introduced to analyze a full discretization method fora partial integro-differential equation (PIDE) with a weakly singular kernel. In thismethod, spectral collocation is used for the spatial discretization, and, for the time stepping, the finite difference method combined with the convolution quadrature rule isconsidered. The global stability and convergence properties of complete discretizationare derived and numerical experiments are reported.

关键词： partial integro-differential equation weakly singular kernel spectral collocation methods z-transform convolution quadrature

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Modelling and Numerical Valuation of Power Derivatives in Energy Markets

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Advances in Applied Mathematics and Mechanics 2012年第3期4卷 259-293页

作者： Mai Huong Nguyen Matthias Ehrhardt Institut fur Mathematik Technische Universitat BerlinStrasse des 17.Juni 13610623 BerlinGermany Lehrstuhl fur Angewandte Mathematik und Numerische Analysis Fachbereich C-Mathematik und NaturwissenschaftenBergische Universitat WuppertalGaußstr.2042119 WuppertalGermany

In this work we investigate the pricing of swing options in a model where the underlying asset follows a jump diffusion *** focus on the derivation of the partial integro-differential equation(PIDE)which will be applied to swing contracts and construct a novel pay-off function from a tree-based pay-off matrix that can be used as initial condition in the PIDE *** valuing swing type derivatives we develop a theta implicit-explicit finite difference scheme to discretize the PIDE using a Gaussian quadrature method for the integral *** on known results for the classical theta-method the existence and uniqueness of solution to the new implicit-explicit finite difference method is *** numerical examples illustrate the usability of the proposed method and allow us to analyse the sensitivity of swing options with respect to model *** particular the effects of number of exercise rights,jump intensities and dividend yields will be investigated in depth.

关键词： Swing options jump-diffusion process mean-reverting Black-Scholes equation energy market partial integro-differential equation theta-method Implicit-Explicit-Scheme

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Fourth Order Compact Boundary Value Method for Option Pricing with Jumps

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Advances in Applied Mathematics and Mechanics 2009年第6期1卷 845-861页

作者： Spike T.Lee Hai-Wei Sun Department of Mathematics University of MacaoMacaoChina

We consider pricing options in a jump-diffusion model which requires solving a partial integro-differential *** the spatial direction with a fourth order compact scheme leads to a linear system of ordinary differential *** the temporal direction,we utilize the favorable boundary value methods owing to their advantageous stability *** addition,the resulting large sparse system can be solved rapidly by the GMRES method with a circulant Strang-type *** results demonstrate the high order accuracy of our scheme and the efficiency of the preconditioned GMRES method.

关键词： partial integro-differential equation fourth order compact scheme boundary value method preconditioning Toeplitz matrix

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On the Convergence of a Crank-Nicolson Fitted Finite Volume Method for Pricing European Options under Regime-Switching Kou’s Jump-Diffusion Models

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Advances in Applied Mathematics and Mechanics 2023年第5期15卷 1290-1314页

作者： Xiaoting Gan Junfeng Yin Rui Li School of Mathematics and Computer Science Chuxiong Normal UniversityChuxiongYunnan 675000China School of Mathematical Sciences Tongji UniversityShanghai 200092China College of Mathematics Physics and Information Engineering Jiaxing UniversityJiaxingZhejiang 314001China

In this paper,we construct and analyze a Crank-Nicolson fitted finite volume scheme for pricing European options under regime-switching Kou’s jumpdiffusion model which is governed by a system of partial integro-differential equations(PIDEs).We show that this scheme is consistent,stable and monotone as the mesh sizes in space and time approach zero,hence it ensures the convergence to the solution of continuous ***,numerical experiments are performed to demonstrate the efficiency,accuracy and robustness of the proposed method.

关键词： European option pricing regime-switching Kou’s jump-diffusion model partial integro-differential equation fitted finite volume method Crank-Nicolson scheme

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A universal robust limit theorem for nonlinear Lévy processes under sublinear expectation

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Probability, Uncertainty and Quantitative Risk 2023年第1期8卷 1-32页

作者： Mingshang Hu Lianzi Jiang Gechun Liang Shige Peng Zhongtai Securities Institute for Financial Studies Shandong UniversityJinan 250100China College of Mathematics and Systems Science Shandong University of Science and TechnologyQingdao 266590China Department of Statistics University of WarwickCoventry CV47ALUK School of Mathematics Shandong UniversityJinan 250100China

This article establishes a universal robust limit theorem under a sublinear expectation *** moment and consistency conditions,we show that,forα∈(1,2),the ***{(1/√∑_(i=1)^(n)X_(i),1/n∑_(i=1)^(n)X_(i)Y_(i),1/α√n∑_(i=1)^(n)X_(i))}_(n=1)^(∞)converges in distribution to L_(1),where L_(t=(ε_(t),η_(t),ζ_(t))),t∈[0,1],is a multidimensional nonlinear Lévy process with an uncertainty■set as a set of Lévy *** nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation(PIDE){δ_(t)u(t,x,y,z)-sup_(F_(μ),q,Q)∈■{∫_(R^(d)δλu(t,x,y,z)(dλ)*** construct the limit process,we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation *** further prove a new type of Lévy-Khintchine representation formula to *** a byproduct,we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.

关键词： Universal robust limit theorem partial integro-differential equation Nonlinear Lévy process α-stable distribution Sublinear expectation

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