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Moderate deviations for rough differential equations

Yuzuru Inahama, Yong Xu, Xiaoyu Yang

Abstract

Small noise problems are quite important for all types of stochastic differential equations. In this paper we focus on rough differential equations driven by scaled fractional Brownian rough path with Hurst parameter H between 1/4 and 1/2. We prove a moderate deviation principle for this equation as the scale parameter tends to zero.

Moderate deviations for rough differential equations

Abstract

Small noise problems are quite important for all types of stochastic differential equations. In this paper we focus on rough differential equations driven by scaled fractional Brownian rough path with Hurst parameter H between 1/4 and 1/2. We prove a moderate deviation principle for this equation as the scale parameter tends to zero.
Paper Structure (3 sections, 4 theorems, 37 equations)

This paper contains 3 sections, 4 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Deterministic Part
  3. Probabilistic Part

Key Result

Proposition 2.1

Let $\alpha \in (1/4, 1/2]$, $\varepsilon\in [0,1]$ and consider the system def.RDE_sys1--def.RDE_sys2 of RDEs driven by $\mathbf{x}\in G\Omega_{\alpha} ({\mathbb R}^d)$. (i) Suppose that $\sigma$ is of ${\rm Lip}^{\gamma +1}$ for some $\gamma > \alpha^{-1}$ and $b$ is of ${\rm Lip}^2$. Then, def.R for every $\varepsilon \in [0,1]$ and $\mathbf{x}$ with $\sum_{i=1}^{\lfloor 1/\alpha \rfloor} \|\m

Theorems & Definitions (10)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3