SCIENCE CHINA Information Sciences, Volume 61, Issue 7: 070214(2018) https://doi.org/10.1007/s11432-017-9301-y

## Exponential stability of the Euler-Maruyama method for neutral stochasticfunctional differential equations with jumps

• AcceptedNov 19, 2017
• PublishedJun 4, 2018
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### Abstract

The exponential stability of trivial solution and numerical solution for neutral stochastic functional differential equations (NSFDEs) with jumps is considered. The stability includes thealmost sure exponential stability and the mean-square exponential stability.New conditions for jumps are proposed by means of the Borel measurable function toensure stability. It is shown that if the drift coefficientsatisfies the linear growth condition,the Euler-Maruyama method can reproduce the corresponding exponentialstability of the trivial solution.A numerical example is constructed to illustrate our theory.

### Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61573156, 61503142) and Key Youth Research Fund of Guangdong University of Technology (Grant No. 17ZK0010).

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• Figure 1

(Color online) Mean square stability of EM numerical solution $X_{k}$ with $\Delta=0.004$.

• Figure 2

(Color online) A sample path of EM numerical solution $X_{k}$ with $\Delta=0.004$.

• Figure 3

(Color online) Mean square stability of EM numerical solution $X_{k}$ with $\Delta=0.01$.

• Figure 4

(Color online) Mean square stability of EM numerical solution $X_{k}$ with $\Delta=0.001$.

• Figure 5

(Color online) Mean square stability of EM numerical solution $X_{k}$ with $\Delta=0.3$.

• Figure 6

(Color online) A sample path of EM numerical solution $X_{k}$ with $\Delta=0.3$.

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