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Quasi-geometric rough paths and rough change of variable formula

Carlo Bellingeri

Abstract

Using some basic notions from the theory of Hopf algebras and quasi-shuffle algebras, we introduce rigorously a new family of rough paths: the quasi-geometric rough paths. We discuss their main properties. In particular, we will relate them with iterated Brownian integrals and the concept of "simple bracket extension", developed in the PhD thesis of David Kelly. As a consequence of these results, we have a sufficient criterion to show for any $γ\in (0,1)$ and any sufficiently smooth function $\varphi \colon \mathbb{R}^d\to \mathbb{R}$ a rough change of variable formula on any $γ$-Hölder continuous path $x\colon [0, T]\to \mathbb{R}^d$, i.e. an explicit expression of $\varphi(x_t)$ in terms of rough integrals.

Quasi-geometric rough paths and rough change of variable formula

Abstract

Using some basic notions from the theory of Hopf algebras and quasi-shuffle algebras, we introduce rigorously a new family of rough paths: the quasi-geometric rough paths. We discuss their main properties. In particular, we will relate them with iterated Brownian integrals and the concept of "simple bracket extension", developed in the PhD thesis of David Kelly. As a consequence of these results, we have a sufficient criterion to show for any and any sufficiently smooth function a rough change of variable formula on any -Hölder continuous path , i.e. an explicit expression of in terms of rough integrals.

Paper Structure

This paper contains 13 sections, 26 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Rough paths and Hopf algebras
  3. Algebraic preliminaries
  4. Weighted rough paths and controlled rough paths
  5. Examples of rough paths
  6. Quasi-geometric rough paths
  7. Quasi-shuffle algebras
  8. Main properties
  9. Examples of quasi-geometric rough paths
  10. Rough change of variable formula
  11. Kelly's change of variable formula
  12. Quasi-geometric bracket extensions
  13. Quasi-geometric change of variable formula

Key Result

Proposition 2.8

Let $X$ be an $\gamma$-weighted rough path over $\mathcal{H}$ with respect to some compatible weight. For any $Y\in (\mathcal{D}^{N\gamma}_X)^e$ and $\varphi\in C^{N}_b(\mathbb{R}^{e}, \mathbb{R})$, $\Phi(Y_t)$ belongs to $\mathcal{D}^{N\gamma}_{X}$.

Theorems & Definitions (80)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Proposition 2.8
  • proof
  • Definition 2.9
  • ...and 70 more