Fundamental Properties of Nonlinear Stochastic Differential Equations
Fundamental Properties of Nonlinear Stochastic Differential Equations
复制标题
非线性随机微分方程的基本性质
DOI：
10.3390/math10152690
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发表时间：
2022-07
期刊：
影响因子：
2.4
通讯作者：
Yanhong Meng
中科院分区：
文献类型：
--
作者：
Linna Liu;Feiqi Deng;Boyang Qu;Yanhong Meng
解的存在性是讨论动力系统其它性质的前提.利用Khasminskii判别法研究了一类非线性随机微分方程解的局部存在性和整体存在性.首先，作为引理给出了一个基本结果，证明了所考虑方程解的局部存在性。然后，形式化地建立了整体存在性的等价命题和Khasminskii检验的基本原理。此外，经典的Khasminskii测试推广到高阶估计和重非线性的李雅普诺夫函数的随机导数的情况。特别研究了噪声在这方面的作用，得到了一些具体的判据，并相应地给出了噪声在金融系统持续性中的应用。作为基本原理的另一个应用，本文对时滞随机系统建立了一种新的Khasminskii检验方法。最后通过仿真验证了本文的结论。结果表明，在较弱的条件下，可以得到随机系统的整体存在比现有文献更好的解。
The existence of solutions is used the premise of discussing other properties of dynamic systems. The goal of this paper is to investigate the fundamental properties of nonlinear stochastic differential equations via the Khasminskii test, including the local existence and global existence of the solutions. Firstly, a fundamental result is given as a lemma to verify the local existence of solutions to the considered equation. Then, the equivalent proposition for the global existence and the fundamental principle for the Khasminskii test are formally established. Moreover, the classical Khasminskii test is generalized to the cases with high-order estimates and heavy nonlinearity for the stochastic derivatives of the Lyapunov functions. The role of the noise in this aspect is especially investigated, some concrete criteria are obtained, and an application for the role of the noise in the persistence of financial systems is accordingly provided. As another application of the fundamental principle, a new version of the Khasminskii test is established for the delayed stochastic systems. Finally the conclusions obtained in the paper are verified by simulation. The results show that, under weaker conditions, the global existence of better solutions to stochastic systems to those in the existing literature can be obtained.