控制粗糙路径的几何形状

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications Pub Date : 2025-02-12 DOI:10.1016/j.spa.2025.104594

Mazyar Ghani Varzaneh , Sebastian Riedel , Alexander Schmeding , Nikolas Tapia

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引用次数: 0

摘要

证明了任意阶的可控（分支）粗糙路径空间构成了Banach空间的连续域。这种结构与（无限维）向量束有许多相似之处，并允许在总空间上定义拓扑，即所有受控路径空间的集合，这在几何情况下是很完美的。构造是内在的，并且基于一个新的控制粗糙路径的近似结果。该框架将粗糙集成映射和Itô-Lyons映射等众所周知的映射转换为连续（保持结构）映射。此外，它与粗糙积分稳定性理论中先前的构造是相容的。

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The geometry of controlled rough paths

We prove that the spaces of controlled (branched) rough paths of arbitrary order form a continuous field of Banach spaces. This structure has many similarities to an (infinite-dimensional) vector bundle and allows to define a topology on the total space, the collection of all controlled path spaces, which turns out to be Polish in the geometric case. The construction is intrinsic and based on a new approximation result for controlled rough paths. This framework turns well-known maps such as the rough integration map and the Itô–Lyons map into continuous (structure preserving) mappings. Moreover, it is compatible with previous constructions of interest in the stability theory for rough integration.

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来源期刊

Stochastic Processes and their Applications 数学-统计学与概率论

CiteScore

2.90

自引率

7.10%

发文量

180

审稿时长

23.6 weeks

期刊介绍： Stochastic Processes and their Applications publishes papers on the theory and applications of stochastic processes. It is concerned with concepts and techniques, and is oriented towards a broad spectrum of mathematical, scientific and engineering interests. Characterization, structural properties, inference and control of stochastic processes are covered. The journal is exacting and scholarly in its standards. Every effort is made to promote innovation, vitality, and communication between disciplines. All papers are refereed.