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Rough path theory

Ilya Chevyrev

TL;DR

Rough path theory provides a pathwise framework to give meaning to differential equations driven by rough signals beyond semimartingales by enriching signals with higher order iterated integrals and a robust solution map $\hat I$. The Universal Limit Theorem establishes existence, uniqueness, and continuity of solutions to rough differential equations driven by geometric rough paths via two complementary approaches, Gubinelli's controlled paths and Davie's Euler estimates. The survey highlights core concepts such as sewing and Young integration, the path signature, and extensions to branched rough paths and regularity structures, illustrating how the framework separates probabilistic construction from deterministic analysis and extends to SPDEs. These tools yield new insights for SDEs driven by irregular signals, including fractional Brownian motion and Gaussian processes, and they underpin pathwise methods with applications in machine learning through the signature transform.

Abstract

The theory of rough paths arose from a desire to establish continuity properties of ordinary differential equations involving terms of low regularity. While essentially an analytic theory, its main motivation and applications are in stochastic analysis, where it has given a new perspective on Itô calculus and a meaning to stochastic differential equations driven by irregular paths outside the setting of semi-martingales. In this survey, we present some of the main ideas that enter rough path theory. We discuss complementary notions of solutions for rough differential equations and the related notion of path signature, and give several applications and generalisations of the theory.

Rough path theory

TL;DR

Rough path theory provides a pathwise framework to give meaning to differential equations driven by rough signals beyond semimartingales by enriching signals with higher order iterated integrals and a robust solution map . The Universal Limit Theorem establishes existence, uniqueness, and continuity of solutions to rough differential equations driven by geometric rough paths via two complementary approaches, Gubinelli's controlled paths and Davie's Euler estimates. The survey highlights core concepts such as sewing and Young integration, the path signature, and extensions to branched rough paths and regularity structures, illustrating how the framework separates probabilistic construction from deterministic analysis and extends to SPDEs. These tools yield new insights for SDEs driven by irregular signals, including fractional Brownian motion and Gaussian processes, and they underpin pathwise methods with applications in machine learning through the signature transform.

Abstract

The theory of rough paths arose from a desire to establish continuity properties of ordinary differential equations involving terms of low regularity. While essentially an analytic theory, its main motivation and applications are in stochastic analysis, where it has given a new perspective on Itô calculus and a meaning to stochastic differential equations driven by irregular paths outside the setting of semi-martingales. In this survey, we present some of the main ideas that enter rough path theory. We discuss complementary notions of solutions for rough differential equations and the related notion of path signature, and give several applications and generalisations of the theory.
Paper Structure (7 sections, 2 theorems, 58 equations, 4 figures)

This paper contains 7 sections, 2 theorems, 58 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Sewing and Young integration
  3. Iterated integrals
  4. Integration and differential equations
  5. Extensions
  6. Signature
  7. Generalisations

Key Result

Lemma 2.1

Consider $\beta>1$, a real Banach space $V$, and a control function $\omega$. Let $\Xi\colon \Delta_T\to V$ be a continuous map such that, for all $0\leq s\leq u\leq t\leq T$, where $\delta\Xi_{s,u,t} = \Xi_{s,t}-\Xi_{s,u}-\Xi_{u,t}$. Then there exists a unique function $\xi\colon [0,T]\to V$ such that $\xi_0=0$ and where we denote $\xi_{s,t}=\xi_t-\xi_s$ and $C$ is a constant depending only on

Figures (4)

  • Figure 1:
  • Figure 2: Maps from Theorem \ref{['thm:ULT']}. Both $\Psi$ and $I$ are discontinuous in $p$-variation norm for $p\geq 2$.
  • Figure 3: Linear (Marcus) interpolation of càdlàg driver $X$ (left) and the corresponding geometric solution to $\mathop{}\!\mathrm{d} Y = f(Y)\mathop{}\!\mathrm{d} X$ (right).
  • Figure 4: (a): the path $X_t = \frac{\pi}{2}\mathbf{1}_{t\geq 1}$. (b) and (c): solutions to $\mathop{}\!\mathrm{d} Y^1 _t\mathop{}\!\mathrm{d} Y^2_t = - Y_t^2Y^1_t \mathop{}\!\mathrm{d} X_t$ in forward and geometric sense respectively.

Theorems & Definitions (9)

  • Lemma 2.1: Sewing
  • Example 2.2
  • Definition 3.1: Rough path
  • Definition 3.2: Geometric rough path
  • Definition 3.3: Weakly geometric rough path
  • Theorem 4.1: Universal limit theorem
  • Example 4.2
  • Example 4.3
  • Definition 4.4: Davie's RDE solution