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The geometry of controlled rough paths

Author

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

Abstract

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.

Suggested Citation

  • Ghani Varzaneh, Mazyar & Riedel, Sebastian & Schmeding, Alexander & Tapia, Nikolas, 2025. "The geometry of controlled rough paths," Stochastic Processes and their Applications, Elsevier, vol. 184(C).
    • Handle: RePEc:eee:spapps:v:184:y:2025:i:c:s0304414925000353
    DOI: 10.1016/j.spa.2025.104594
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