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Averaging principle for slow-fast systems of rough differential equations via controlled paths

Yuzuru Inahama

TL;DR

The strong averaging principle is proved for a slow-fast system of rough differential equations that is driven by a rather general random rough path and Brownian rough path, respectively.

Abstract

In this paper we prove the strong averaging principle for a slow-fast system of rough differential equations. The slow and the fast component of the system are driven by a rather general random rough path and Brownian rough path, respectively. These two driving noises are assumed to be independent. A prominent example of the driver of the slow component is fractional Brownian rough path with Hurst parameter between 1/3 and 1/2. We work in the framework of controlled path theory, which is one of the most widely-used frameworks in rough path theory. To prove our main theorem, we carry out Khas'minskii's time-discretizing method in this framework.

Averaging principle for slow-fast systems of rough differential equations via controlled paths

TL;DR

The strong averaging principle is proved for a slow-fast system of rough differential equations that is driven by a rather general random rough path and Brownian rough path, respectively.

Abstract

In this paper we prove the strong averaging principle for a slow-fast system of rough differential equations. The slow and the fast component of the system are driven by a rather general random rough path and Brownian rough path, respectively. These two driving noises are assumed to be independent. A prominent example of the driver of the slow component is fractional Brownian rough path with Hurst parameter between 1/3 and 1/2. We work in the framework of controlled path theory, which is one of the most widely-used frameworks in rough path theory. To prove our main theorem, we carry out Khas'minskii's time-discretizing method in this framework.
Paper Structure (14 sections, 24 theorems, 178 equations)

This paper contains 14 sections, 24 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main result
  3. Review of controlled path theory
  4. Rough paths and controlled paths
  5. Rough path integration of controlled paths
  6. Rough differential equations: The case of bounded and globally Lipschitz drift vector field
  7. Rough differential equations: The case of bounded and locally Lipschitz drift vector field
  8. Slow-fast system of rough differential equations
  9. Deterministic aspects
  10. Probabilistic aspects
  11. Proof of main result
  12. Gronwall's inequality
  13. A simple application of Itô's formula
  14. Frozen SDE

Key Result

Theorem 2.1

Assume ${\bf (A)}$, ${\bf (H1)}$--${\bf (H4)}$, ${\bf (H5)}_r$, ${\bf (H6)}_q$ and ${\bf (H7)}$ for some $q \ge 2$ and $r \ge 0$ such that $q> 2r$. Then, for every $p\in [1,\infty)$ and $\beta \in (\tfrac{1}{3}, \alpha_0)$, we have

Theorems & Definitions (58)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Example 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 48 more