AI 中文总结
该研究针对由布朗运动和对称α-稳定过程驱动的一维SDE，通过同步耦合建立了定量逐路稳定性估计，明确了不同Hölder指数下的误差率，还得到了含可预测强迫误差和时间一致尾界的估计。
AI 中文摘要
我们证明了由普通布朗运动和普通对称α-稳定过程（其中α∈(1,2)）驱动的一维随机微分方程的定量逐路稳定性估计。该比较在同步耦合下进行，通过sup_{0≤t≤T}E|X_t−X̃_t|^{α−1}来衡量。受扰漂移项和布朗扩散系数是空间Lipschitz连续的，而受扰稳定跳跃系数可以是达到临界指数1/α的Hölder连续的。该估计用初始误差和三个律加权的系数误差表示：漂移误差B、布朗扩散误差A和稳定跳跃系数误差S。布朗分量在伊藤公式中产生一个二阶修正项，该项通过对|x|^{α−1}的Komatsu型磨光的二阶导数估计来控制。对于Hölder指数为η̃>1/α的稳定跳跃系数，该估计给出B、A、S中的显式幂次率；当η̃=1/α时，给出对数率。相同的方法通过停时拟鞅产生具有可预测强迫误差和时间一致尾界的估计。
英文摘要
We prove quantitative pathwise stability estimates for one-dimensional stochastic differential equations driven by a common Brownian motion and a common symmetric $α$-stable process, where $α\in(1,2)$. The comparison is made under a synchronous coupling and is measured by $\sup_{0\le t\le T}\mathbb E|X_t-\wt X_t|^{α-1}$. The perturbed drift and Brownian diffusion coefficients are spatially Lipschitz, while the perturbed stable jump coefficient may be Hölder continuous down to the critical exponent $1/α$. The estimate is expressed in terms of the initial error and three law-weighted coefficient errors: the drift error $B$, the Brownian diffusion error $A$, and the stable jump-coefficient error $S$. The Brownian component produces a second-order correction term in Itô's formula. This term is controlled by a second-derivative estimate for a Komatsu-type mollification of $|x|^{α-1}$. For stable jump coefficients with Hölder exponent $\wtη>1/α$, the estimate gives explicit power rates in $B,A,S$; at $\wtη=1/α$, it gives a logarithmic rate. The same method yields estimates with predictable forcing errors and time-uniform tail bounds via stopped quasi-martingales.
