Paper: arXiv 2504.06164

Authors: Christa Cuchiero, Xin Guo, Francesca Primavera

Abstract

We derive a functional Itô-formula for non-anticipative maps of rough paths, based on the approximation properties of the signature of càdlàg rough paths. This result is a functional extension of the Itô-formula for càdlàg rough paths (by Friz and Zhang (2018)), which coincides with the change of variable formula formulated by Dupire (2009) whenever the functionals’ representations, the notions of regularity, and the integration concepts can be matched. Unlike these previous works, we treat the vertical (jump) pertubation via the Marcus transformation, which allows for incorporating path functionals where the second order vertical derivatives do not commute, as is the case for typical signature functionals. As a byproduct, we show that sufficiently regular non-anticipative maps admit a functional Taylor expansion in terms of the path’s signature, leading to an important generalization of the recent results by Dupire and Tissot-Daguette (2022).

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This is a purely theoretical probability paper establishing a pathwise functional Itô-formula and functional Taylor expansion for non-anticipative maps of càdlàg rough paths using the Marcus transformation and path signatures. While mathematically rigorous and highly novel in its theoretical setup, it contains no empirical analysis, backtesting, or code implementations.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Derive Functional Itô Formula & Taylor Expansions for Càdlàg Rough Paths"] --> B["Inputs & Foundations:<br>Non-anticipative Path Functionals, Càdlàg Rough Paths & Signatures"]
    
    B --> C["Core Methodology:<br>Apply Marcus Transformation for Non-Commutative Vertical Jumps"]
    
    C --> D["Computational Process:<br>Approximate Path Functionals via Rough Path Signatures"]
    
    D --> E["Outcome 1:<br>Functional Itô Formula<br><i>(Extends Friz-Zhang & Dupire)</i>"]
    D --> F["Outcome 2:<br>Functional Taylor Expansion<br><i>(Generalizes Dupire & Tissot-Daguette)</i>"]