AI 中文总结
本文针对非自治随机微分方程的Milstein格式，给出李雅普诺夫指数的精确上下估计，证明连续场景的稳定效应可保留至离散场景，核心是利用Milstein格式的随机噪声二阶项完成泰勒展开估计。
AI 中文摘要
乘性噪声对随机微分方程的稳定效应虽违背直觉，但在过去数十年已被广泛观测和研究。实践中，希望这种稳定效应在离散化场景中也成立。本文通过对非自治随机微分方程Milstein格式的李雅普诺夫指数给出精确的上下估计来解决该问题，这些估计在几乎必然和p阶矩意义下提供了精确的长时间行为。特别地，结果表明连续场景的稳定效应可保留至离散场景。分析的核心思路是利用Milstein格式中与随机噪声相关的二阶项，对对数函数和幂函数的泰勒展开获得精确估计。
英文摘要
The stabilising effect of multiplicative noise for stochastic differential equations, though counterintuitive, has been observed and investigated extensively in last decades. In practice, it is desirable to know if such stabilisation holds also for the discretised setting. In this paper, we address this problem by means of sharp upper and lower estimates for Lyapunov exponents of Milstein schemes for non-autonomous stochastic differential equations. These estimates provide precise large time behaviour in both almost sure and $p$-moment sense. In particular, our results show the preservation of stabilisation from the continuum setting to the discretised setting. One main idea of our analysis is to exploit the second order term concerning the stochastic noise from the Milstein scheme to obtain precise estimates for Taylor expansions of logarithmic and power functions.
