AIMS Mathematics, 2020, 5(4): 3612-3633. doi: 10.3934/math.2020234.

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Stochastic invariance for hybrid stochastic differential equation with non-Lipschitz coefficients

1 School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, Guangdong 510006, PR China
2 School of civil Engineering, Jiaying University, Meizhou, Guangdong 514015, PR China

In this paper, by using of the martingale property and positive maximum principle, we investigate the stochastic invariance for a class of hybrid stochastic differential equations (HSDEs) and provide necessary and sufficient conditions for the invariance of closed sets of $\mathbb{R}^d$ with non-Lipschitz coefficients. Moreover, an example of the most probable phase portrait is given to illustrate the effectiveness of the main results.
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Keywords linear growth condition; martingale problem; hybrid stochastic differential equations; stochastic invariance

Citation: Chunhong Li, Sanxing Liu. Stochastic invariance for hybrid stochastic differential equation with non-Lipschitz coefficients. AIMS Mathematics, 2020, 5(4): 3612-3633. doi: 10.3934/math.2020234

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