# American Institute of Mathematical Sciences

February  2019, 24(2): 587-613. doi: 10.3934/dcdsb.2018198

## Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation

 1 College of Information Sciences and Technology, Donghua University, Shanghai, 201620, China 2 School of mathematics and information technology, Jiangsu Second Normal University, Nanjing 210013, China 3 Department of Applied Mathematics, Donghua University, Shanghai 201620, China 4 Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK

* Corresponding author: Liangjian Hu

Received  May 2017 Revised  January 2018 Published  June 2018

Fund Project: The author Wei Mao is supported by the National Natural Science Foundation of China (11401261) and "333 High-level Personnel Training Project" of Jiangsu Province. The author Liangjian Hu is supported by the National Natural Science Foundation of China (11471071). The author Xuerong Mao is supported by the Leverhulme Trust (RF-2015-385), the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship), the EPSRC (EP/K503174/1).

In this paper, we are concerned with the asymptotic properties and numerical analysis of the solution to hybrid stochastic differential equations with jumps. Applying the theory of M-matrices, we will study the $p$th moment asymptotic boundedness and stability of the solution. Under the non-linear growth condition, we also show the convergence in probability of the Euler-Maruyama approximate solution to the true solution. Finally, some examples are provided to illustrate our new results.

Citation: Wei Mao, Liangjian Hu, Xuerong Mao. Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 587-613. doi: 10.3934/dcdsb.2018198
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