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Branched Itô formula and natural Itô-Stratonovich isomorphism

Carlo Bellingeri, Emilio Ferrucci, Nikolas Tapia

TL;DR

The paper develops a canonical, algebraic framework for branched rough paths by exploiting the Connes–Kreimer Hopf algebra and its Grossman–Larson duality, yielding a natural Itô–type change-of-variable formula for rough differential equations driven by branched rough paths. It proves a cofree-structure via a cofreeness projection to primitives and establishes a Log–Exp isomorphism, linking CK to the shuffle algebra over primitives and generalizing the Itô–Stratonovich correction through an Itô–Stratonovich transformation that is natural in the decorating space. In the one-dimensional setting, the authors derive branched Kailath–Segall polynomials and construct stochastic examples up to order four, including truly branched cases that fail quasi-geometricity, thus illustrating the broad reach of the framework beyond geometric rough paths. The results unify and extend prior approaches (e.g., Hoffman's exponential, bracket extensions) under a functorial, canonical viewpoint, with potential implications for SPDEs and regularity structures. Overall, the work provides structural theorems and explicit constructions that reveal the intrinsic algebraic content of Itô–type corrections in branched rough-path theory and clarifies when and how branched lifts can be translated into geometric ones via natural isomorphisms.

Abstract

Branched rough paths, defined as paths with values in the character group of the Connes-Kreimer Hopf algebra $\mathcal{H}_\mathrm{CK}$, constitute integration theories that may fail to satisfy the usual integration by parts identity. Using known results on the primitive elements of $\mathcal{H}_\mathrm{CK}$ we can view it as a commutative cofree Hopf algebra (i.e. a commutative $\mathbf{B}_\infty$-algebra) and thus write an explicit change-of-variable formula for solutions to rough differential equations. This formula restricts to the well-known Itô formula in the very special case of semimartingales. In addition, we establish an isomorphism between $\mathcal{H}_\mathrm{CK}$ and the shuffle algebra over its primitives, which extends Hoffman's exponential for the quasi-shuffle algebra, and can therefore be viewed as a far-reaching generalisation of the usual Itô-Stratonovich correction formula for semimartingales. Indeed, this can be stated as a characterisation of the algebra structure of any commutative $\mathbf{B}_\infty$-algebra. Compared to previous approaches, this transformation has the key property of being natural in the decorating vector space. We study the one-dimensional case more closely, by introducing the branched analogue of the Kailath-Segall polynomials and Doléans-Dade exponential, and conclude with some examples of branched rough path lifts of a stochastic process which are not quasi-geometric.

Branched Itô formula and natural Itô-Stratonovich isomorphism

TL;DR

The paper develops a canonical, algebraic framework for branched rough paths by exploiting the Connes–Kreimer Hopf algebra and its Grossman–Larson duality, yielding a natural Itô–type change-of-variable formula for rough differential equations driven by branched rough paths. It proves a cofree-structure via a cofreeness projection to primitives and establishes a Log–Exp isomorphism, linking CK to the shuffle algebra over primitives and generalizing the Itô–Stratonovich correction through an Itô–Stratonovich transformation that is natural in the decorating space. In the one-dimensional setting, the authors derive branched Kailath–Segall polynomials and construct stochastic examples up to order four, including truly branched cases that fail quasi-geometricity, thus illustrating the broad reach of the framework beyond geometric rough paths. The results unify and extend prior approaches (e.g., Hoffman's exponential, bracket extensions) under a functorial, canonical viewpoint, with potential implications for SPDEs and regularity structures. Overall, the work provides structural theorems and explicit constructions that reveal the intrinsic algebraic content of Itô–type corrections in branched rough-path theory and clarifies when and how branched lifts can be translated into geometric ones via natural isomorphisms.

Abstract

Branched rough paths, defined as paths with values in the character group of the Connes-Kreimer Hopf algebra , constitute integration theories that may fail to satisfy the usual integration by parts identity. Using known results on the primitive elements of we can view it as a commutative cofree Hopf algebra (i.e. a commutative -algebra) and thus write an explicit change-of-variable formula for solutions to rough differential equations. This formula restricts to the well-known Itô formula in the very special case of semimartingales. In addition, we establish an isomorphism between and the shuffle algebra over its primitives, which extends Hoffman's exponential for the quasi-shuffle algebra, and can therefore be viewed as a far-reaching generalisation of the usual Itô-Stratonovich correction formula for semimartingales. Indeed, this can be stated as a characterisation of the algebra structure of any commutative -algebra. Compared to previous approaches, this transformation has the key property of being natural in the decorating vector space. We study the one-dimensional case more closely, by introducing the branched analogue of the Kailath-Segall polynomials and Doléans-Dade exponential, and conclude with some examples of branched rough path lifts of a stochastic process which are not quasi-geometric.
Paper Structure (16 sections, 48 theorems, 287 equations, 1 figure)

This paper contains 16 sections, 48 theorems, 287 equations, 1 figure.

Table of Contents

  1. Algebraic preliminaries
  2. The Connes-Kreimer Hopf algebra over a vector space
  3. Duality with Grossman-Larson
  4. Pre-Lie algebras and higher-order differential operators
  5. The Itô formula
  6. Branched rough paths and the change-of-variables formula
  7. A criterion for quasi-geometricity and the simple change-of-variables formula
  8. The Itô-Stratonovich correction formula
  9. Commutative B∞-algebras are shuffle algebras
  10. The quasi-geometric case: Hoffman's exponential
  11. Relationship with previous approaches
  12. The one-dimensional case
  13. Kailath-Segall polynomials
  14. Construction of branched rough paths: stochastic examples at order 4
  15. Calculation in \ref{['prop:failed']}
  16. ...and 1 more sections

Key Result

proposition 1.2

Let $\Phi$ be a collection of real numbers indexed by bijections of vertex sets with the property that $\phi_b = \phi_{b \circ \sigma}$ for any $\sigma \in \mathbb S_{\mathscr f_1} \times \cdots \times \mathbb S_{\mathscr f_m}$ (viewed as a subgroup of $\mathbb S_{\{\mathscr f_1\} \sqcup \ldots \sqcup \{\mathscr f_m\}}$). For any finite-dimensional vector space $U$ we set where each $\mathscr f_

Figures (1)

  • Figure 1: Visual representation for left endpoint and midpoint Riemann-Stieltjes sum approximations.

Theorems & Definitions (119)

  • definition 1.1
  • proposition 1.2
  • proof : Proof of \ref{['prop:functor']}
  • definition 1.3
  • proposition 1.4
  • proof
  • remark 1.5
  • definition 1.6: Natural growth, BK2000
  • theorem 1.7: Foi02c
  • proposition 1.8: Foi2002
  • ...and 109 more