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

Chunhong Li; Sanxing Liu; Chunhong Li; Sanxing Liu

doi:10.3934/math.2020234

AIMS Mathematics

2020, Volume 5, Issue 4: 3612-3633. doi: 10.3934/math.2020234

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Research article

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

Chunhong Li ^1,2,
Sanxing Liu ^{1,2
,
,}

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

Received: 23 December 2019 Accepted: 16 March 2020 Published: 14 April 2020
MSC : 60J27, 60H27, 60H28

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

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Abstract

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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