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A note on order of convergence of numerical method for neutral stochastic functional differential equations |
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| Authors: | Feng Jiang Fuke Wu |
| Affiliation: | a School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China b Department of Control Science and Engineering, The Key Laboratory of Ministry of Education for Image Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan 430074, China c School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China |
| Abstract: | In this paper, we study the order of convergence of the Euler-Maruyama (EM) method for neutral stochastic functional differential equations (NSFDEs). Under the global Lipschitz condition, we show that the pth moment convergence of the EM numerical solutions for NSFDEs has order p/2 − 1/l for any p ? 2 and any integer l > 1. Moreover, we show the rate of the mean-square convergence of EM method under the local Lipschitz condition is 1 − ε/2 for any ε ∈ (0, 1), provided the local Lipschitz constants of the coefficients, valid on balls of radius j, are supposed not to grow faster than log j. This is significantly different from the case of stochastic differential equations where the order is 1/2. |
| Keywords: | Neutral stochastic functional differential equations Euler-Maruyama (EM) Local Lipschitz condition Order of convergence |
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