4.6 Article

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

Journal

MATHEMATICAL METHODS IN THE APPLIED SCIENCES
Volume 45, Issue 10, Pages 6005-6029

Publisher

WILEY
DOI: 10.1002/mma.8153

Keywords

convergence rate; Euler-Maruyama method; fractional substantial derivative; non-Lipschitz condition; stochastic Volterra integral equations

Funding

  1. National Natural Science Foundation of China [11601206]
  2. Natural Science Foundation of Hunan Province of China [2018JJ3491]
  3. Project of Scientific Research Fund of Hunan Provincial Science and Technology Department [2018WK4006]
  4. Research Foundation of Education Commission of Hunan Province of China [19B565]
  5. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University

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In this paper, the Euler-Maruyama method is used to solve a class of nonlinear stochastic Volterra integral equations (SVIEs). The existence and uniqueness of the solution are proved for the SVIEs under the non-Lipschitz condition. Convergence estimates are established for the SVIEs, and numerical experiments are provided to illustrate the effectiveness of the method.
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 under the non-Lipschitz condition. Moreover, the sharp convergence estimate with O ( h..+ 12) if... ( 0, 12), O(h center dot v | ln h|) if.. = 12 and O(h) if... ( 12, 1], 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 thatwas left on the convergence analysis in (J ComputAppl Math 317 (2017) 447-457). Finally, numerical experiments are given to illustrate the effectiveness of the presented method.

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