Work

The $L^q$-bounds for Derivatives of Unitary Developments of Random Continuous Geometric Rough Paths

Year: 2026

Type: preprint

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 $L^q$-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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.06980

Authors: Liu, Chong, Lyu, Zijiu, Ni, Hao

Institutions: ShanghaiTech University, University of California, Berkeley, University College London

Language: English

Topic: Stochastic processes and financial applications

Subfield: Finance

Field: Economics, Econometrics and Finance

Domain: Social Sciences

Open Access status: green

Funders: Norges Forskningsråd, Engineering and Physical Sciences Research Council

Award: High order mathematical and computational infrastructure for streamed data that enhance contemporary generative and large language models

OpenAlex ID: W7212053747

Cites: 1

Cited by:

Related to: 10