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Variability and the existence of rough integrals with irregular coefficients

Michael Hinz, Jonas M. Tölle, Lauri Viitasaari

TL;DR

The paper addresses the problem of defining pathwise integrals with irregular coefficients in rough-path settings by introducing a variability framework for functions with BV-type derivatives. It extends a fractional-calculus-based integration theory to Lipschitz coefficients whose first-order partial derivatives lie in $BV$, establishing a new existence result for $\int \varphi(X)\,dY$ under a weighted variability condition and a controlled second-level term $X\otimes Y$. The results yield explicit bounds and apply to Gaussian processes, notably fractional Brownian motion with $H\in(\tfrac{1}{3},\tfrac{1}{2}]$, where almost-sure full-probability events ensure integrability for broad classes of $\varphi$. This advances the theory of pathwise integration for non-smooth coefficients and broadens probabilistic applicability to rough Gaussian paths.

Abstract

Within the context of rough path analysis via fractional calculus, we show how variability can be used to prove the existence of integrals with respect to Hölder continuous multiplicative functionals in the case of Lipschitz coefficients with first order partial derivatives of bounded variation. We discuss applications to certain Gaussian processes, in particular, fractional Brownian motions with Hurst index $\frac13<H\leq \frac12$.

Variability and the existence of rough integrals with irregular coefficients

TL;DR

The paper addresses the problem of defining pathwise integrals with irregular coefficients in rough-path settings by introducing a variability framework for functions with BV-type derivatives. It extends a fractional-calculus-based integration theory to Lipschitz coefficients whose first-order partial derivatives lie in , establishing a new existence result for under a weighted variability condition and a controlled second-level term . The results yield explicit bounds and apply to Gaussian processes, notably fractional Brownian motion with , where almost-sure full-probability events ensure integrability for broad classes of . This advances the theory of pathwise integration for non-smooth coefficients and broadens probabilistic applicability to rough Gaussian paths.

Abstract

Within the context of rough path analysis via fractional calculus, we show how variability can be used to prove the existence of integrals with respect to Hölder continuous multiplicative functionals in the case of Lipschitz coefficients with first order partial derivatives of bounded variation. We discuss applications to certain Gaussian processes, in particular, fractional Brownian motions with Hurst index .
Paper Structure (9 sections, 9 theorems, 43 equations)

This paper contains 9 sections, 9 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Pathwise integrals via fractional calculus
  3. Function spaces and fractional derivatives
  4. Pathwise integrals via fractional calculus
  5. Rough path analysis via fractional calculus
  6. Variability and composition with $BV$-functions
  7. Existence of integrals in the elementary case
  8. Existence of integrals in the rough case
  9. Applications to Gaussian processes

Key Result

Proposition 2.1

Let $0<\gamma,\delta<1$ and $1\leq p,q\leq \infty$ be such that Suppose that $X,Y:[a,b]\to \mathbb{R}^m$ are Borel functions, $X$ is continuous at $a$ and $Y$ is continuous at $a$ and $b$. If $X \in W^{\gamma,p}([a,b];\mathbb{R}^m)$ and $Y\in W^{\delta,q}([a,b];\mathbb{R}^m)$, then $\int_a^b X\,dY$ in E:basicint exists and satisfies where $c>0$ is a constant depending only on $a$, $b$, $\gamma$,

Theorems & Definitions (24)

  • Proposition 2.1
  • Remark 2.2
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Example 2.5
  • Theorem 2.6
  • proof
  • Theorem 3.1
  • Corollary 3.2
  • ...and 14 more