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The truncated Euler-Maruyama method for highly nonlinear stochastic differential equations with multiple time delays

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Abstract

The main aim of this paper is to investigate the strong convergence order for the truncated Euler-Maruyama (TEM) method to solve stochastic differential delay equations (SDDEs) with multiple time delays and super-linearly growing coefficients. The strong Lp (1 ≤ p < 2) convergence rate of the TEM method under the one-sided polynomial growth condition is first established. Imposing additional conditions on the diffusion coefficient, the p-th moment uniform boundedness of both the exact and approximate solutions is then proved. Next, we show that the strong order of Lq-convergence can be arbitrarily close to 1/2 for 2 ≤ q ≤ p. Several examples and a numerical simulation are provided to illustrate the main results at the end.

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Funding

This research was supported in part by the National Natural Science Foundation of China (Grant No. 12171173; 11701161) and by grants from the National Key Research and Development Project of China (Grant No. 2020YFC2006205).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, People’s Republic of China

    Shaobo Zhou

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, People’s Republic of China

    Shaobo Zhou

  3. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Hai Jin

  4. Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Hai Jin

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  1. Shaobo Zhou
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  2. Hai Jin
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Correspondence to Hai Jin.

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Zhou, S., Jin, H. The truncated Euler-Maruyama method for highly nonlinear stochastic differential equations with multiple time delays. Numer Algor 94, 581–617 (2023). https://doi.org/10.1007/s11075-023-01512-1

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