The Experts below are selected from a list of 6159 Experts worldwide ranked by ideXlab platform
Siqing Gan - One of the best experts on this subject based on the ideXlab platform.
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Convergence and stability of the Backward Euler method for jump–diffusion SDEs with super-linearly growing diffusion and jump coefficients
Journal of Computational and Applied Mathematics, 2020Co-Authors: Ziheng Chen, Siqing GanAbstract:Abstract This paper firstly investigates convergence of the Backward Euler method for stochastic differential equations (SDEs) driven by Brownian motion and compound Poisson process. Different from existing works, we do not impose a globally Lipschitz condition on the jump coefficient but formulate a more relaxed condition to allow for its super-linear growth. It is shown that the mean square convergence order of this method can be arbitrarily close to 1 2 under mild assumptions imposed on SDEs, allowing for possibly super-linearly growing drift, diffusion and jump coefficients. An exact order 1 2 is recovered when further differentiability assumption is put on the coefficients. Furthermore, the considered method is able to inherit the mean square stability of a wider class of Levy noise driven SDEs for all stepsizes. These results are finally supported by some numerical experiments.
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convergence and stability of the Backward Euler method for jump diffusion sdes with super linearly growing diffusion and jump coefficients
Journal of Computational and Applied Mathematics, 2020Co-Authors: Ziheng Chen, Siqing GanAbstract:Abstract This paper firstly investigates convergence of the Backward Euler method for stochastic differential equations (SDEs) driven by Brownian motion and compound Poisson process. Different from existing works, we do not impose a globally Lipschitz condition on the jump coefficient but formulate a more relaxed condition to allow for its super-linear growth. It is shown that the mean square convergence order of this method can be arbitrarily close to 1 2 under mild assumptions imposed on SDEs, allowing for possibly super-linearly growing drift, diffusion and jump coefficients. An exact order 1 2 is recovered when further differentiability assumption is put on the coefficients. Furthermore, the considered method is able to inherit the mean square stability of a wider class of Levy noise driven SDEs for all stepsizes. These results are finally supported by some numerical experiments.
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Dissipativity of the Backward Euler method for nonlinear Volterra functional differential equations in Banach space
Advances in Difference Equations, 2015Co-Authors: Siqing GanAbstract:This paper concerns the dissipativity of nonlinear Volterra functional differential equations (VFDEs) in Banach space and their numerical discretization. We derive sufficient conditions for the dissipativity of nonlinear VFDEs. The general results provide a unified theoretical treatment for dissipativity analysis to ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type appearing in practice. Then the dissipativity property of the Backward Euler method for VFDEs is investigated. It is shown that the method can inherit the dissipativity of the underlying system. The close relationship between the absorbing set of the numerically discrete system generated by the Backward Euler method and that of the underlying system is revealed.
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The improved split-step Backward Euler method for stochastic differential delay equations
arXiv: Numerical Analysis, 2011Co-Authors: Xiaojie Wang, Siqing GanAbstract:A new, improved split-step Backward Euler (SSBE) method is introduced and analyzed for stochastic differential delay equations(SDDEs) with generic variable delay. The method is proved to be convergent in mean-square sense under conditions (Assumption 3.1) that the diffusion coefficient $g(x,y)$ is globally Lipschitz in both $x$ and $y$, but the drift coefficient $f(x,y)$ satisfies one-sided Lipschitz condition in $x$ and globally Lipschitz in $y$. Further, exponential mean-square stability of the proposed method is investigated for SDDEs that have a negative one-sided Lipschitz constant. Our results show that the method has the unconditional stability property in the sense that it can well reproduce stability of underlying system, without any restrictions on stepsize $h$. Numerical experiments and comparisons with existing methods for SDDEs illustrate the computational efficiency of our method.
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The improved split-step Backward Euler method for stochastic differential delay equations
International Journal of Computer Mathematics, 2011Co-Authors: Xiaojie Wang, Siqing GanAbstract:A new, improved split-step Backward Euler method is introduced and analysed for stochastic differential delay equations (SDDEs) with generic variable delay. The method is proved to be convergent in the mean-square sense under conditions (Assumption 3.1) that the diffusion coefficient g(x, y) is globally Lipschitz in both x and y, but the drift coefficient f(x, y) satisfies the one-sided Lipschitz condition in x and globally Lipschitz in y. Further, the exponential mean-square stability of the proposed method is investigated for SDDEs that have a negative one-sided Lipschitz constant. Our results show that the method has the unconditional stability property, in the sense, that it can well reproduce stability of the underlying system, without any restrictions on stepsize h. Numerical experiments and comparisons with existing methods for SDDEs illustrate the computational efficiency of our method.
Yi Shen - One of the best experts on this subject based on the ideXlab platform.
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A NOTE ON STABILITY OF THE SPLIT-STEP Backward Euler METHOD FOR LINEAR STOCHASTIC DELAY INTEGRO-DIFFERENTIAL EQUATIONS
Journal of Systems Science and Complexity, 2012Co-Authors: Feng Jiang, Yi Shen, Xiaoxin LiaoAbstract:In the literature (Tan and Wang, 2010), Tan and Wang investigated the convergence of the split-step Backward Euler (SSBE) method for linear stochastic delay integro-differential equations (SDIDEs) and proved the mean-square stability of SSBE method under some condition. Unfortunately, the main result of stability derived by the condition is somewhat restrictive to be applied for practical application. This paper improves the corresponding results. The authors not only prove the mean-square stability of the numerical method but also prove the general mean-square stability of the numerical method. Furthermore, an example is given to illustrate the theory.
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Stability of the split-step Backward Euler scheme for stochastic delay integro-differential equations with Markovian switching
Communications in Nonlinear Science and Numerical Simulation, 2011Co-Authors: Feng Jiang, Yi ShenAbstract:Abstract In this paper, we concentrate on the numerical approximation of solutions of stochastic delay integro-differential equations with Markovian switching (SDIDEsMS). We establish the split-step Backward Euler (SSBE) scheme for solving linear SDIDEsMS and discuss its convergence and stability. Moreover, the SSBE method is convergent with strong order γ = 1/2 in the mean-square sense. The conditions under which the SSBE method is mean-square stable and general mean-square stable are obtained. Some illustrative numerical examples are presented to demonstrate the stability of the numerical method and show that SSBE method is superior to Euler method.
Feng Jiang - One of the best experts on this subject based on the ideXlab platform.
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A NOTE ON STABILITY OF THE SPLIT-STEP Backward Euler METHOD FOR LINEAR STOCHASTIC DELAY INTEGRO-DIFFERENTIAL EQUATIONS
Journal of Systems Science and Complexity, 2012Co-Authors: Feng Jiang, Yi Shen, Xiaoxin LiaoAbstract:In the literature (Tan and Wang, 2010), Tan and Wang investigated the convergence of the split-step Backward Euler (SSBE) method for linear stochastic delay integro-differential equations (SDIDEs) and proved the mean-square stability of SSBE method under some condition. Unfortunately, the main result of stability derived by the condition is somewhat restrictive to be applied for practical application. This paper improves the corresponding results. The authors not only prove the mean-square stability of the numerical method but also prove the general mean-square stability of the numerical method. Furthermore, an example is given to illustrate the theory.
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Stability of the split-step Backward Euler scheme for stochastic delay integro-differential equations with Markovian switching
Communications in Nonlinear Science and Numerical Simulation, 2011Co-Authors: Feng Jiang, Yi ShenAbstract:Abstract In this paper, we concentrate on the numerical approximation of solutions of stochastic delay integro-differential equations with Markovian switching (SDIDEsMS). We establish the split-step Backward Euler (SSBE) scheme for solving linear SDIDEsMS and discuss its convergence and stability. Moreover, the SSBE method is convergent with strong order γ = 1/2 in the mean-square sense. The conditions under which the SSBE method is mean-square stable and general mean-square stable are obtained. Some illustrative numerical examples are presented to demonstrate the stability of the numerical method and show that SSBE method is superior to Euler method.
Xueyin Tao - One of the best experts on this subject based on the ideXlab platform.
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Almost Sure and Convergence of Split-Step Backward Euler Method for Stochastic Delay Differential Equation
Abstract and Applied Analysis, 2014Co-Authors: Qian Guo, Xueyin TaoAbstract:The convergence of the split-step Backward Euler (SSBE) method applied to stochastic differential equation with variable delay is proven in -sense. Almost sure convergence is derived from the convergence by Chebyshev’s inequality and the Borel-Cantelli lemma; meanwhile, the convergence rate is obtained.
Xiaojie Wang - One of the best experts on this subject based on the ideXlab platform.
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On the Backward Euler method for a generalized Ait-Sahalia-type rate model with Poisson jumps
Numerical Algorithms, 2020Co-Authors: Yuying Zhao, Xiaojie Wang, Mengchao WangAbstract:This article aims to reveal the mean-square convergence rate of the Backward Euler method (BEM) for a generalized Ait-Sahalia interest rate model with Poisson jumps. The main difficulty in the analysis is caused by the non-globally Lipschitz drift and diffusion coefficients of the model. We show that the BEM preserves the positivity of the original problem. Furthermore, we successfully recover the mean-square convergence rate of order one-half for the BEM. The theoretical findings are accompanied by several numerical examples.
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On the Backward Euler method for a generalized Ait-Sahalia-type rate model with Poisson jumps.
arXiv: Numerical Analysis, 2020Co-Authors: Yuying Zhao, Xiaojie Wang, Mengchao WangAbstract:This article aims to reveal the mean-square convergence rate of the Backward Euler method (BEM) for a generalized Ait-Sahaliz interest rate model with Poisson jumps. The main difficulty in the analysis is caused by the non-globally Lipschitz drift and diffusion coefficients of the model. We show that the BEM preserves positivity of the original problem. Furthermore, we successfully recover the mean-square convergence rate of order one-half for the BEM. The theoretical findings are accompanied by several numerical examples.
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A transformed jump-adapted Backward Euler method for jump-extended CIR and CEV models
Numerical Algorithms, 2016Co-Authors: Xu Yang, Xiaojie WangAbstract:A novel time-stepping scheme, called transformed jump-adapted Backward Euler method, is developed in this paper to simulate a class of jump-extended CIR and CEV models. The proposed scheme is able to preserve the positivity of the underlying problems. Furthermore, its strong convergence rate of order one is recovered for the considered models with non-Lipschitz diffusion coefficients. Numerical examples are finally reported to confirm our theoretical findings.
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The improved split-step Backward Euler method for stochastic differential delay equations
arXiv: Numerical Analysis, 2011Co-Authors: Xiaojie Wang, Siqing GanAbstract:A new, improved split-step Backward Euler (SSBE) method is introduced and analyzed for stochastic differential delay equations(SDDEs) with generic variable delay. The method is proved to be convergent in mean-square sense under conditions (Assumption 3.1) that the diffusion coefficient $g(x,y)$ is globally Lipschitz in both $x$ and $y$, but the drift coefficient $f(x,y)$ satisfies one-sided Lipschitz condition in $x$ and globally Lipschitz in $y$. Further, exponential mean-square stability of the proposed method is investigated for SDDEs that have a negative one-sided Lipschitz constant. Our results show that the method has the unconditional stability property in the sense that it can well reproduce stability of underlying system, without any restrictions on stepsize $h$. Numerical experiments and comparisons with existing methods for SDDEs illustrate the computational efficiency of our method.
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The improved split-step Backward Euler method for stochastic differential delay equations
International Journal of Computer Mathematics, 2011Co-Authors: Xiaojie Wang, Siqing GanAbstract:A new, improved split-step Backward Euler method is introduced and analysed for stochastic differential delay equations (SDDEs) with generic variable delay. The method is proved to be convergent in the mean-square sense under conditions (Assumption 3.1) that the diffusion coefficient g(x, y) is globally Lipschitz in both x and y, but the drift coefficient f(x, y) satisfies the one-sided Lipschitz condition in x and globally Lipschitz in y. Further, the exponential mean-square stability of the proposed method is investigated for SDDEs that have a negative one-sided Lipschitz constant. Our results show that the method has the unconditional stability property, in the sense, that it can well reproduce stability of the underlying system, without any restrictions on stepsize h. Numerical experiments and comparisons with existing methods for SDDEs illustrate the computational efficiency of our method.
