Convergence rate of numerical solutions to SFDEs with jumps

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 p th-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 .