发表机构
ShanghaiTech University; University of California, Berkeley; University College London（上海科技大学; 加州大学伯克利分校; 伦敦大学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推导随机连续几何粗糙路径酉发展导数的显式公式并建立$L^q$界，定量研究特征函数解析性，证明在温和条件下签名分布由期望签名决定，部分回答了Lyons和Ni的开放问题。
AI 中文摘要
本文推导了随机连续几何粗糙路径的所有阶酉发展的导数的显式公式，并建立了它们适当的$L^q$界，这使我们能够以定量的方式研究其特征函数的解析性，进而证明在温和条件下，随机连续几何粗糙路径的签名的分布由其期望签名决定，前提是后者具有正收敛半径。特别地，我们对T. Lyons和H. Ni在“Expected signature of Brownian motion up to the first exit time from a bounded domain”（《概率年刊》，43(5)，2729-2762，2015）中提出的一个开放问题给出了部分肯定回答，该问题涉及停止布朗运动的期望签名是否决定其签名的分布。
英文摘要
In this paper we derive an explicit formula for derivatives of unitary developments of random continuous geometric rough paths of all orders and establish proper $L^q$-bounds for them, which allow us to study the analyticity of their characteristic functions in a quantitative manner and then prove that under some mild conditions the distributions of the signatures of random continuous geometric rough paths are determined by their expected signatures, provided the latter have positive radius of convergence. In particular, we give a partial affirmative answer to an open question posed by T. Lyons and H. Ni in ``Expected signature of Brownian motion up to the first exit time from a bounded domain'' (The Annals of Probability, 43(5), 2729-2762, 2015), concerning whether the expected signature of stopped Brownian motion determines the law of its signature.
