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Convergence error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs

Science China Mathematics 2017年第5期60卷 923-948页

作者： LI Yang YANG Jie ZHAO WeiDong College of Science University of Shanghai for Science and Technology School of Mathematics Shandong University

In this work, we theoretically analyze the convergence error estimates of the Crank-Nicolson （C-N） scheme for solving decoupled FBSDEs. Based on the Taylor and ItS-Taylor expansions, the Malliavin calculus theory （e.g., the multiple Malliavin integration-by-parts formula）, and our new truncation error cancelation techniques, we rigorously prove that the strong convergence rate of the C-N scheme is of second order for solving decoupled FBSDEs, which fills the gap between the second-order numerical and theoretical analysis of the C-N scheme.

关键词： convergence analysis Crank-Nicolson scheme decoupled forward backward stochastic differentialequations Malliavin calculus trapezoidal rule

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CONVERGENCE AND MEAN-SQUARE STABILITY OF EXPONENTIAL EULER METHOD FOR SEMI-LINEAR STOCHASTIC DELAY INTEGRO-DIFFERENTIAL EQUATIONS

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Journal of Computational Mathematics 2022年第2期40卷 177-204页

作者： Haiyan Yuan Department of Mathematics Heilongjiang Institute of TechnologyHarbin 158100China

In this paper,the numerical methods for semi-linear stochastic delay integro-difFerential equations are *** uniqueness,existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also *** the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are *** is proved that the exponential Euler method is convergent with strong order 1/2 and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step *** addition,numerical experiments are presented to confirm the theoretical results.

关键词： Semi-linear stochastic delay integro-differential equation Exponential Euler method Mean-square exponential stability trapezoidal rule

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Extrapolation methods to compute hypersingular integral in boundary element methods

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Science China Mathematics 2013年第8期56卷 1647-1660页

作者： LI Jin ZHANG XiaoPing YU DeHao School of Science Shandong Jianzhu University School of Mathematics and Statistics Wuhan University School of Mathematical Sciences Xiamen University LSEC Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing

The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel I/sin2（x- s） is discussed, and the main part of the asymptotic expansion of error function is obtained. Based on the main part of the asymptotic expansion, a series is constructed to approach the singular point. An extrapolation algorithm is presented and the convergence rate is proved. Some numerical results are also presented to confirm the theoretical results and show the efficiency of the algorithms.

关键词： hypersingular integrals trapezoidal rule asymptotic error expansion extrapolation algorithm

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An FFT Based Fast Poisson Solver on Spherical Shells

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Communications in Computational Physics 2011年第3期9卷 649-667页

作者： Yin-Liang Huang Jian-Guo Liu Wei-Cheng Wang Department of Applied Mathematics National University of TainanTainan 70005Taiwan Department of Physics and Department of Mathematics Duke UniversityDurhamNC 27708USA Department of Mathematics National Tsing Hua UniversityHsinchu 30013Taiwan

We present a fast Poisson solver on spherical *** a special change of variable,the radial part of the Laplacian transforms to a constant coefficient differential *** a result,the Fast Fourier Transform can be applied to solve the Poisson equation with O(N^(3) logN)*** examples have confirmed the accuracy and robustness of the new scheme.

关键词： Poisson equation spherical coordinate FFT spectral-finite difference method fast diagonalization high order accuracy error estimate trapezoidal rule Euler-Maclaurin formula Bernoulli numbers

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广义Black-Scholes方程的Adomian级数解及其数值计算

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高等学校计算数学学报 2016年第1期38卷 1-17页

作者：傅强朱哲殷涛重庆大学经济与工商管理学院重庆大学数学与统计学院

In this paper we present a numerical method for a generalized BlackScholes partial differential equation for option *** first derive its series solution whose coefficients are determined by recursion formulas involving integrals by means of the Adomian *** we present a numerical scheme based on the trapezoidal rule instead of direct integral method for the Adomain series *** results show the effectiveness of the present method.

关键词： Option pricing Black-Scholes equation Adomian decomposition trapezoidal rule

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