Strong superconvergence of the Euler–Maruyama method for linear stochastic Volterra integral equations

Liang, Hui; Yang, Zhanwen; Gao, Jun

doi:10.1016/j.cam.2016.11.005

Cited by 36 publications

(15 citation statements)

References 21 publications

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“…Moreover, the related conclusions for the other classes of SVIEs by using Euler-type method can be obtained [20,31,32]. Similarly, if we add a jump term to the right side of the SVIEs, Khalaf et al [33] showed that the strong convergence order can reach up to order 1 if the kernel function is Lipschitz continuous and the diffusion coeffcient and the jump coeffcient satisfy a same additional assumption as in [30]. For the SVIEs with doubly singular kernels, Dai and Xiao [34] analysed the strong convergence order of EM method, and constructed the fast EM method to improve the computational effciency.…”

Section: Introductionmentioning

confidence: 90%

“…Indeed, for the strong convergence order of numerical methods for SVIEs, some interesting conclusions also have been obtained. For the linear SVIEs with convolution kernels, Liang et al [30] obtained the superconvergence order of EM method when the kernel function is Lipschitz continuous and satisfies an additional assumption. Moreover, the related conclusions for the other classes of SVIEs by using Euler-type method can be obtained [20,31,32].…”

Section: Introductionmentioning

confidence: 99%

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Optimal Convergence Rate of $\theta$--Maruyama Method for Stochastic Volterra Integro-Differential Equations with Riemann--Liouville Fractional Brownian Motion

sci¹

2022

AAMM

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Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Optimal Convergence Rate of $\theta$--Maruyama Method for Stochastic Volterra Integro-Differential Equations with Riemann--Liouville Fractional Brownian Motion

sci¹

2022

AAMM

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“…Under the same settings with the EM approximation (5), we can modify it by the left rectangle rule [21,26] as…”

Section: Continuous Dependence Of Solutions On the Initial Valuementioning

confidence: 99%

Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation

Dai

Xiao

2022

DCDS-B

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<p style='text-indent:20px;'>This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler–Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler–Maruyama method can reach strong first-order superconvergence when <inline-formula><tex-math id="M1">\begin{document}$ \alpha = 1 $\end{document}</tex-math></inline-formula>. Finally, several numerical tests are reported for verification of the theoretical findings.</p>

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“…In recent years, some important progresses for the numerical algorithm of the SVIE (1.1) with 𝜎 = 0 has been proposed, for example, Galerkin methods, 25 block pulse approximation, [26][27][28] spectral collocation method, [29][30][31] operational matrix method, [32][33][34] improved rectangular method, [35][36][37] B-spline collocation method, [38][39][40] Milstein method, 41,42 meshless discrete collocation method, 43 and Euler-Maruyama (EM) method. [44][45][46][47] In this paper, we are interested in numerically solving the nonlinear SVIE (1.1) for which 𝜎 > 0 by using EM method and devoting to two main goals.…”

Section: Introductionmentioning

confidence: 99%

A sharp error estimate of Euler‐Maruyama method for stochastic Volterra integral equations

Wang

Chen

Deng

et al. 2022

Math Methods in App Sciences

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In this paper, the Euler-Maruyama method is used to solve a class of nonlinear stochastic Volterra integral equations (SVIEs), which can be derived from stochastic fractional substantial integro-differential equations (SFSIDEs). The existence and uniqueness of the solution are proved for the nonlinear SVIEs ] , respectively, is established for the nonlinear SVIEs, which can be revised as supplemented theory results of our recent work (J Comput Appl Math 356 (2019) 377-390). In particular, it also closes a gap that was left on the convergence analysis in (J Comput Appl Math 317 (2017) 447-457). Finally, numerical experiments are given to illustrate the effectiveness of the presented method.

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Strong superconvergence of the Euler–Maruyama method for linear stochastic Volterra integral equations

Cited by 36 publications

References 21 publications

Optimal Convergence Rate of $\theta$--Maruyama Method for Stochastic Volterra Integro-Differential Equations with Riemann--Liouville Fractional Brownian Motion

Optimal Convergence Rate of $\theta$--Maruyama Method for Stochastic Volterra Integro-Differential Equations with Riemann--Liouville Fractional Brownian Motion

Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation

A sharp error estimate of Euler‐Maruyama method for stochastic Volterra integral equations

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