AI 中文总结
研究粗糙路径信号分布变化的最快检测，通过将信号建模为变化前后粗糙路径的连接，转化为最优停止问题，给出最优停止规则及统计保证，扩展到分布鲁棒公式，经数值评估，性能在布朗设置中可比最优方法，在分数布朗设置中更优且抗扰动。
AI 中文摘要
我们开发了一个用于最快检测建模为粗糙路径的信号中分布变化的框架。通过将变化前和变化后的动态表示为两个独立的粗糙路径，并将观测信号建模为它们在未知随机变化点处的连接，我们将最快检测表述为使用粗糙路径的最优停止问题。我们表明最优停止规则采取粗糙路径签名的线性泛函首次击中半空间的形式，并确定相同的结构形式独立于观测路径的内在几何结构出现。推导了检测延迟和误报概率的统计保证，并将该框架扩展到分布鲁棒公式，其中变化前和变化后的模型都是不确定的。所提出的规则通过对截断签名系数的零阶随机逼近实现，并在布朗动力学和分数布朗动力学下进行数值评估，在布朗设置中实现了与最优方法相当的性能，同时在分数布朗设置中优于它们并对对抗性路径扰动保持鲁棒性。
英文摘要
We develop a framework for quickest detection of distributional change in signals modeled as rough paths. By representing the pre-change and post-change dynamics as two independent rough paths and modeling the observed signal as their concatenation at an unknown random change-point, we formulate quickest detection as an optimal stopping problem using rough paths. We show that the optimal stopping rule takes the form of the first hitting time to a half-space by a linear functional of the rough path signature, and establish that the same structural form arises independently from the intrinsic geometry of the observed path. Statistical guarantees on detection delay and false alarm probability are derived, and the framework is extended to a distributionally robust formulation in which both the pre-change and post-change models are uncertain. The proposed rules are implemented via zeroth-order stochastic approximation over truncated-signature coefficients and evaluated numerically under Brownian and fractional Brownian dynamics, achieving performance comparable to optimal methods in the Brownian setting while outperforming them and remaining robust to adversarial path perturbations in the fractional Brownian setting.
