Adv. Appl. Math. Mech., 17 (2025), pp. 1654-1681.
Published online: 2025-09
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We study the strong convergence of the general one-step method for neutral stochastic delay differential equations with a variable delay. First, we give the notions of C-stability and B-consistency, and then establish a fundamental theorem of strong convergence for the general one-step method solving the nonlinear neutral stochastic delay differential equations, where the corresponding diffusion coefficient with respect to the non-delay variables is highly nonlinear. Then, we construct the split-step backward Euler method which is a special implicit one-step method, and prove that it is C-stable, B-consistent, and strongly convergent of order 1/2. Finally, we give some numerical experiments to support the obtained results.
}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2024-0017}, url = {http://global-sci.org/intro/article_detail/aamm/24490.html} }We study the strong convergence of the general one-step method for neutral stochastic delay differential equations with a variable delay. First, we give the notions of C-stability and B-consistency, and then establish a fundamental theorem of strong convergence for the general one-step method solving the nonlinear neutral stochastic delay differential equations, where the corresponding diffusion coefficient with respect to the non-delay variables is highly nonlinear. Then, we construct the split-step backward Euler method which is a special implicit one-step method, and prove that it is C-stable, B-consistent, and strongly convergent of order 1/2. Finally, we give some numerical experiments to support the obtained results.