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Convergence rate of numerical solutions to SFDEs with jumps

Bao, Jianhai and Bottcher, Bjorn and Mao, Xuerong and Yuan, Chenggui (2011) Convergence rate of numerical solutions to SFDEs with jumps. Journal of Computational and Applied Mathematics, 236 (2). pp. 119-131. ISSN 0377-0427

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Abstract

In this paper, we are interested in numerical solutions of stochastic functional differential equations with jumps. Under a global Lipschitz condition, we show that the pth-moment convergence of Euler–Maruyama numerical solutions to stochastic functional differential equations with jumps has order 1/p for any p ≥ 2. This is significantly different from the case of stochastic functional differential equations without jumps, where the order is 1/2 for any p ≥ 2. It is therefore best to use the mean-square convergence for stochastic functional differential equations with jumps. Moreover, under a local Lipschitz condition, we reveal that the order of mean-square convergence is close to 1/2, provided that local Lipschitz constants, valid on balls of radius j, do not grow faster than log j.