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Universal limit theorem for rough differential equations driven by controlled rough paths

Nannan Li, Xing Gao

Abstract

In this article, we re-establish the existence of the level-$2$ rough integral of a controlled rough path against another controlled rough path via the point-removal method, and we derive a new estimate of a priori type for this rough integral. We further establish a universal limit theorem -- a central result in rough path theory -- for rough differential equations driven by controlled rough paths in the same level-$2$ regime, thereby extending the classical universal limit theorem for rough differential equations driven by rough paths.

Universal limit theorem for rough differential equations driven by controlled rough paths

Abstract

In this article, we re-establish the existence of the level- rough integral of a controlled rough path against another controlled rough path via the point-removal method, and we derive a new estimate of a priori type for this rough integral. We further establish a universal limit theorem -- a central result in rough path theory -- for rough differential equations driven by controlled rough paths in the same level- regime, thereby extending the classical universal limit theorem for rough differential equations driven by rough paths.
Paper Structure (7 sections, 14 theorems, 140 equations)

This paper contains 7 sections, 14 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Rough integrals beyond the classical case
  3. Motivation from rough differential equations driven by controlled rough paths
  4. Main results and outline
  5. Rough integrals
  6. Rough differential equations driven by controlled rough paths
  7. Universal limit theorem

Key Result

Theorem 2.3

Gu04 Let $\alpha \in(\frac{1}{3},\frac{1}{2}]$ and ${\bf X} =(X, \mathbb{X})\in \mathcal{D}^{\alpha}(\Delta_T, V)$. Let ${\bf Y} = (Y, Y')\in \mathcal{C}_{{\bf X}}^{\alpha}([0, T], \mathcal{L}(W, U))$ and ${\bf Z} = (Z, Z')\in \mathcal{C}_{{\bf X}}^{\alpha}([0, T], W)$. For each pair of $s<t$, the r is well-defined, where $\pi$ is an arbitrary partition of $[s, t]$.

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Corollary 3.3
  • ...and 21 more