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Existence, uniqueness, almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the Euler-Maruyama approximation

机译：一类高度非线性受电弓随机微分方程和Euler-Maruyama近似解的存在性，唯一性，几乎确定的多项式稳定性

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摘要

In the first part of this paper the existence, uniqueness and almost sure polynomial stability of solutions for pantograph stochastic differential equations are considered, under nonlinear growth conditions. The results are obtained using the idea from (Mao and Rassias, 2005) [10], where stochastic differential equation with constant delay are considered. However, the presence of the unbounded delay in stochastic pantograph differential equations required certain modification of that idea. Moreover, the convergence in probability of the appropriate Euler-Maruyama solution is proved under the same nonlinear growth conditions. Adding the linear growth condition, we show that the almost sure polynomial stability of the Euler-Maruyama solution implies the almost sure polynomial stability of the exact solution. This part of the paper represents the extension of the idea from (Wu et al., 2010) [17]. The main novelty in this part of the paper is also related to the treatment of the unbounded delay in pantograph stochastic differential equations.
机译：在本文的第一部分中，考虑了在非线性增长条件下受电弓随机微分方程解的存在性，唯一性和几乎确定的多项式稳定性。使用（Mao和Rassias，2005）[10]的思想获得了结果，其中考虑了具有恒定延迟的随机微分方程。但是，随机受电弓微分方程中存在无界延迟的情况需要对该想法进行某些修改。此外，在相同的非线性增长条件下，证明了适当的Euler-Maruyama解的概率收敛。加上线性增长条件，我们证明了Euler-Maruyama解的几乎确定的多项式稳定性暗示了精确解的几乎确定的多项式稳定性。本文的这一部分代表了这一思想的延伸（Wu等，2010）[17]。本文这一部分的主要新颖之处还涉及到受电弓随机微分方程中无界延迟的处理。

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