The Lq-bounds for Derivatives of Unitary Developments of Random Continuous Geometric Rough Paths
Abstract
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 Lq-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.
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BibTeX
@misc{liu2026lqbounds,
title = {{The $L^q$-bounds for Derivatives of Unitary Developments of Random Continuous Geometric Rough Paths}},
author = {Chong Liu and Zijiu Lyu and Hao Ni},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06980},
eprint = {2609.06980},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06980},
}
RIS
TY - UNPB
TI - The L^q-bounds for Derivatives of Unitary Developments of Random Continuous Geometric Rough Paths
AU - Chong Liu
AU - Zijiu Lyu
AU - Hao Ni
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.06980
UR - https://arxiv.org/abs/2609.06980
AB - 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.
ER -