Convergence rate of numerical solutions to SFDEs with jumps

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 ppth-moment convergence of Euler–Maruyama numerical solutions to stochastic functional differential equations with jumps has order 1/p1/p for any p≥2p≥2. This is significantly different from the case of stochastic functional differential equations without jumps, where the order is 1/21/2 for any p≥2p≥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 1/21/2, provided that local Lipschitz constants, valid on balls of radius jj, do not grow faster than logjlogj.

► We are interested in numerical solutions of SFDEs with jumps. ► Under a global Lipschitz condition, we show the strong convergence of EM scheme has order 1/p1/p for p≥2p≥2. ► It is best to use the mean-square convergence for SFDEs with jumps. ► Under a local Lipschitz condition, we reveal that the order of mean-square convergence is close to 1/21/2.