随机对数范数用于随机微分方程的稳定性分析
The Stochastic Logarithmic Norm for Stability Analysis of Stochastic Differential Equations
AI总结：
针对乘性噪声的Itô随机微分方程，引入随机对数范数，推导线性Itô SDE在l阶均值下的稳定性估计，并通过实例说明其应用。
摘要：
为了分析带乘性噪声的Itô随机微分方程的稳定性，我们引入了随机对数范数。对数范数最初由G. Dahlquist于1958年提出，用于研究常微分方程解的增长以及估计其近似解离散化方法的误差增长。我们将这一概念扩展到带乘性噪声的Itô随机微分方程的稳定性分析。推导了线性Itô SDE在$l$阶均值（其中$1 \leq l < \infty$）下使用一范数、二范数和无穷范数的稳定性估计，并通过实例说明了随机对数范数的应用。
英文摘要：
To analyze the stability of Itô stochastic differential equations with multiplicative noise, we introduce the stochastic logarithmic norm. The logarithmic norm was originally introduced by G. Dahlquist in 1958 as a tool to study the growth of solutions to ordinary differential equations and for estimating the error growth in discretization methods for their approximate solutions. We extend the concept to the stability analysis of Itô stochastic differential equations with multiplicative noise. Stability estimates for linear Itô SDEs using the one, two and $\infty$-norms in the $l$-th mean, where $1 \leq l < \infty $, are derived and the application of the stochastic logarithmic norm is illustrated with examples.
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