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A New Convergent Explicit Tree-Grid Method for HJB Equations in One Space Dimension

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Numerical Mathematics(Theory,Methods and Applications) 2018年第1期11卷 1-29页

作者： Igor Kossaczky matthias ehrhardt Michael Gunther Lehrstuhl fur Angewandte Mathematik und Numerische Analysis Fakultat fur Mathematik und NaturwissenschaftenBergische Universitat WuppertalGaußstrasse 2042119 WuppertalGermany

In this work we introduce a new unconditionally convergent explicit Tree-Grid Method for solving stochastic control problems with one space and one time dimension or equivalently,the corresponding Hamilton-Jacobi-Bellman *** prove the convergence of the method and outline the relationships to other numerical *** case of vanishing diffusion is treated by introducing an artificial diffusion *** illustrate the superiority of our method to the standardly used implicit finite difference method on two numerical examples from finance.

关键词： Tree-Grid Method Hamilton-Jacobi-Bellman equation Stochastic control problem

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

Numerical Simulation of Waves in Periodic Structures

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Communications in Computational Physics 2009年第5期5卷 849-870页

作者： matthias ehrhardt Houde Han Chunxiong Zheng Weierstraß-Institut fur Angewandte Analysis und Stochastik Mohrenstr.3910117 BerlinGermany Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China

In this work we improve and extend a technique named recursive doubling procedure developed by Yuan and Lu[*** Technology 25(2007),3649-3656]for solving periodic array *** turns out that when the periodic array contains an infinite number of periodic cells,our method gives a fast evaluation of the exact boundary Robin-to-Robin mapping if the wave number is complex,or real but in the stop *** technique is also used to solve the time-dependent Schr¨odinger equation in both one and two dimensions,when the periodic potential functions have some local defects.

关键词： Periodic media Helmholtz equation Schrodinger equation Dirichlet-to-Neumann maps Robin-to-Robin maps band structure Floquet-Bloch theory high-order finite elements

Efficient Numerical Valuation of Continuous Installment Options

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Advances in Applied Mathematics and Mechanics 2011年第2期3卷 141-164页

作者： Anton Mezentsev Anton Pomelnikov matthias ehrhardt IDE Halmstad UniversityBox 82330118HalmstadSweden Lehrstuhl fur Angewandte Mathematik und Numerische Analysis Fachbereich C-Mathematik und NaturwissenschaftenBergische Universitat WuppertalGaußstr.2042119 WuppertalGermany

In this work we investigate the novel Kryzhnyi method for the numerical inverse Laplace transformation and apply it to the pricing problem of continuous installment *** compare the results with the one obtained using other classical methods for the inverse Laplace transformation,like the Euler summation method or the Gaver-Stehfest method.

关键词： Installment options Black-Scholes equation numerical inverse Laplace transform Gaver-Stehfest method Euler summation method optimal stopping problem non-monotonic stopping boundary

Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schr¨odinger Equations with Unbounded Potentials and Nonlinearities

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Communications in Computational Physics 2011年第10期10卷 1280-1304页

作者： Pauline Klein Xavier Antoine Christophe Besse matthias ehrhardt Institut Elie CartanNancy Nancy-UniversiteCNRSUMR 7502INRIA CORIDA TeamBoulevard des Aiguillettes B.P.23954506 Vandoeuvre-les-NancyFrance Laboratoire Paul Painleve CNRS UMR 8524Simpaf Project Team-Inria CR Lille Nord EuropeUniversite des Sciences et Technologies de LilleCite Scientifique59655 Villeneuve d’Ascq CedexFrance Lehrstuhl fur Angewandte Mathematik und Numerische Analysis Fachbereich C-Mathematik und NaturwissenschaftenBergische Universitat WuppertalGaußstr.2042119 WuppertalGermany

We propose a hierarchy of novel absorbing boundary conditions for the onedimensional stationary Schr¨odinger equation with general(linear and nonlinear)*** accuracy of the new absorbing boundary conditions is investigated numerically for the computation of energies and ground-states for linear and nonlinear Schr¨odinger *** turns out that these absorbing boundary conditions and their variants lead to a higher accuracy than the usual Dirichlet boundary ***,we give the extension of these ABCs to N-dimensional stationary Schr¨odinger equations.

关键词： Absorbing boundary conditions stationary Schrodinger equations unbounded domain spatially dependent potential ground states computation

Conservative and Finite Volume Methods for the Convection-Dominated Pricing Problem

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Advances in Applied Mathematics and Mechanics 2013年第6期5卷 759-790页

作者： German I.Ramırez-Espinoza matthias ehrhardt EON Global Commodities Holzstraße 640221 DusseldorfGermany Lehrstuhl fur Angewandte Mathematik und Numerische Analysis Fachbereich C-Mathematik und NaturwissenschaftenBergische Universitat WuppertalGaußstr.2042119WuppertalGermany

This work presents a comparison study of different numerical methods to solve Black-Scholes-type partial differential equations(PDE)in the convectiondominated case,i.e.,for European options,if the ratio of the risk-free interest rate and the squared volatility-known in fluid dynamics as P´eclet number-is *** Asian options,additional similar problems arise when the"spatial"variable,the stock price,is close to *** we focus on three methods:the exponentially fitted scheme,a modification of Wang’s finite volume method specially designed for the Black-Scholes equation,and the Kurganov-Tadmor scheme for a general convection-diffusion equation,that is applied for the first time to option pricing *** emphasis is put in the Kurganov-Tadmor because its flexibility allows the simulation of a great variety of types of options and it exhibits quadratic *** the reduction technique proposed by Wilmott,a put-call parity is presented based on the similarity reduction and the put-call parity expression for Asian ***,we present experiments and comparisons with different(non)linear Black-Scholes PDEs.

关键词： Black-Scholes equation convection-dominated case exponential fitting methods fitted finite volume method Kurganov-Tadmor scheme minmod limiter

Stochastic Runge-Kutta–Munthe-Kaas Methods in the Modelling of Perturbed Rigid Bodies

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Advances in Applied Mathematics and Mechanics 2022年第2期14卷 528-538页

作者： Michelle Muniz matthias ehrhardt Michael Günther Renate Winkler Institute of Mathematical Modelling Analysis and Computational Mathematics(IMACM)Applied Mathematics and Numerical AnalysisBergische Universität WuppertalGaußstraße 2042119 WuppertalGermany

In this paper we present how nonlinear stochastic Itˆo differential equations arising in the modelling of perturbed rigid bodies can be solved numerically in such a way that the solution evolves on the correct *** this end,we formulate an approach based on Runge-Kutta–Munthe-Kaas(RKMK)schemes for ordinary differ-ential equations on ***,we provide a proof of the mean-square convergence of this stochastic version of the RKMK schemes applied to the rigid body problem and illustrate the effectiveness of our proposed schemes by demonstrating the structure preservation of the stochastic RKMK schemes in contrast to the stochastic Runge-Kutta methods.

关键词： Stochastic Runge-Kutta method Runge-Kutta–Munthe-Kaas scheme nonlinear Itˆo SDEs rigid body problem

A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrodinger Equations

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Communications in Computational Physics 2008年第9期4卷 729-796页

作者： Xavier Antoine Anton Arnold Christophe Besse matthias ehrhardt Achim Schadle Institut Elie Cartan Nancy(IECN) Universite Henri Poincare Nancy 1UMR 7502INRIA CoridaNancy-UniversiteFrance. Institut fur Analysis und Scientific Computing Technische Universitat WienWiedner Hauptstr.81040 WienAustria. Project-Team SIMPAF INRIA Lille Nord Europe Research Centre Laboratoire Paul Painleve U.M.R CNRS 8524 Universite des Sciences et Technologies de LilleFrance. Weierstrass Institute for Applied Analysis and Stochastics Mohrenstr.3910117 BerlinGermany. Zuse-Institut Berlin Takustrasse 714195 BerlinGermany.

In this review article we discuss different techniques to solve numerically the time-dependent Schrodinger equation on unbounded *** present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these *** conclude with several numerical examples from different application areas to compare the presented *** mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.

关键词： Schrodinger equation transparent boundary conditions discrete convolution unbounded domain finite difference schemes finite elements

Note on:“Ballistic model for the prediction of penetration depth and residual velocity in adobe:A new interpretation of the ballistic resistance of earthen masonry”

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Defence Technology（防务技术） 2024年第1期31卷 607-609页

作者： Andreas Heine matthias Wickert Fraunhofer EMI Ernst-Zermelo-Str.479104FreiburgGermany

A recently published modeling approach for the penetration into adobe and previous approaches implicitly criticized are reviewed and *** article contains a note on the paper titled“Ballistic model for the prediction of penetration depth and residual velocity in adobe:A new interpretation of the ballistic resistance of earthen masonry”(DOI:https://***/10.1016/***.2018.07.017).Reply to the Note from Li Piani et al is linked to this article.

关键词： adobe prediction earth