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Functional Itô-formula and Taylor expansions for non-anticipative maps of càdlàg rough paths

Christa Cuchiero, Xin Guo, Francesca Primavera

TL;DR

The paper develops a robust framework to extend the functional Itô-calculus to non-anticipative maps of càdlàg rough paths by exploiting the Marcus transformation and the path signature. It introduces a universal approximation theorem for vertically differentiable Marcus canonical path functionals, enabling a rigorous Itô-formula and a deterministic Taylor expansion expressed in terms of the signature. By distinguishing non-commutative higher-order vertical derivatives and leveraging a density-based approach, the authors unify and generalize classical Dupire-type results with rough-path theory, including connections to Föllmer’s pathwise integration. The results provide a deterministic, pathwise toolbox for functional calculus on rough paths, with clear implications for stochastic analysis, stochastic differential equations driven by jump processes, and pathwise Taylor expansions. The formalism yields both theoretical insights and potential applications in areas requiring non-anticipative path functionals and rough integration under càdlàg dynamics.

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).

Functional Itô-formula and Taylor expansions for non-anticipative maps of càdlàg rough paths

TL;DR

The paper develops a robust framework to extend the functional Itô-calculus to non-anticipative maps of càdlàg rough paths by exploiting the Marcus transformation and the path signature. It introduces a universal approximation theorem for vertically differentiable Marcus canonical path functionals, enabling a rigorous Itô-formula and a deterministic Taylor expansion expressed in terms of the signature. By distinguishing non-commutative higher-order vertical derivatives and leveraging a density-based approach, the authors unify and generalize classical Dupire-type results with rough-path theory, including connections to Föllmer’s pathwise integration. The results provide a deterministic, pathwise toolbox for functional calculus on rough paths, with clear implications for stochastic analysis, stochastic differential equations driven by jump processes, and pathwise Taylor expansions. The formalism yields both theoretical insights and potential applications in areas requiring non-anticipative path functionals and rough integration under càdlàg dynamics.

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).

Paper Structure

This paper contains 71 sections, 18 theorems, 195 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Preliminaries
  4. Algebraic setting
  5. Weakly geometric càdlàg rough paths
  6. Time-stretching of continuous weakly geometric rough paths
  7. Marcus transformation of weakly geometric càdlàg rough paths
  8. Signature of weakly geometric càdlàg rough paths
  9. Functionals of weakly geometric càdlàg rough paths
  10. Non-anticipative Marcus canonical path functionals
  11. Vertically differentiable path functionals
  12. Var-continuous path functionals
  13. Linear functionals of the signature
  14. Universal approximation theorem for vertically differentiable path functionals
  15. The subspace of the tracking-jumps-extended paths
  16. ...and 56 more sections

Key Result

Theorem 2.10

(Minimal jump extension theorem, Theorem 20 in FS:17) Let $\mathbb{N} \ni N>[p]$. Every $\mathbf{X}\in D^p([0,1],G^{[p]}(\mathbb{R}^d))$ with $\mathbf{X}_0=\mathbf{1}\in G^{[p]}(\mathbb{R}^d))$ admits a unique extension to a càdlàg path $\mathbb{X}^{N}:[0,1]\rightarrow G^{N}(\mathbb{R}^d)$, such th

Figures (21)

  • Figure 1: Step 0
  • Figure 2: Step I
  • Figure 3: Step II
  • Figure 4: Vertically perturbed $2$-dimensional path
  • Figure : Continuous 1-dimensional path $\mathbf{X}$
  • ...and 16 more figures

Theorems & Definitions (73)

  • Definition 2.1
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.9
  • Theorem 2.10
  • Definition 2.11
  • Corollary 2.13: Corollary 39 in FS:17
  • ...and 63 more