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Moderate deviations for two-time scale systems with mixed fractional Brownian motion

Xiaoyu Yang, Yuzuru Inahama, Yong Xu

Abstract

This work focuses on moderate deviations for two-time scale systems with mixed fractional Brownian motion. Our proof uses the weak convergence method which is based on the variational representation formula for mixed fractional Brownian motion. Throughout this paper, Hurst parameter of fractional Brownian motion is larger than $1/2$ and the integral along the fractional Brownian motion is understood as the generalized Riemann-Stieltjes integral. First, we consider single-time scale systems with fractional Brownian motion. The key of our proof is showing the weak convergence of the controlled system. Next, we extend our method to show moderate deviations for two-time scale systems. To this goal, we combine the Khasminskii-type averaging principle and the weak convergence approach.

Moderate deviations for two-time scale systems with mixed fractional Brownian motion

Abstract

This work focuses on moderate deviations for two-time scale systems with mixed fractional Brownian motion. Our proof uses the weak convergence method which is based on the variational representation formula for mixed fractional Brownian motion. Throughout this paper, Hurst parameter of fractional Brownian motion is larger than and the integral along the fractional Brownian motion is understood as the generalized Riemann-Stieltjes integral. First, we consider single-time scale systems with fractional Brownian motion. The key of our proof is showing the weak convergence of the controlled system. Next, we extend our method to show moderate deviations for two-time scale systems. To this goal, we combine the Khasminskii-type averaging principle and the weak convergence approach.
Paper Structure (12 sections, 9 theorems, 183 equations)

This paper contains 12 sections, 9 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fractional integrals and Spaces
  4. Fractional Brownian motion
  5. Mixed Fractional Brownian motion
  6. MDP for the single-time scale system
  7. Assumptions and Statement of Main Result
  8. Main Proof
  9. MDP for the two-time scale system
  10. Preliminaries, Assumptions and Main Result
  11. Preliminary Lemmas
  12. Proof of Main Theorem(Theorem \ref{['thm']})

Key Result

Proposition 2.1

Let $1/2<H<1$ and $1- H < \alpha < 1/2$. Then, $(B^H_t)_{t\in [0,T]}\in W_T^{1-\alpha, \infty} ([0,T], \mathbb{R}^{d_1})$ almost surely. Moreover, if a stochastic process ${(v_t)_{t \in [0, T ]}}\in W_0^{\alpha, 1}([0,T], \mathbb{R}^{m\times d_1})$, then the generalized Riemann-Stieltjes integral $\ Here, $\Lambda_\alpha\left(B^H\right):=\frac{1}{\Gamma(1-\alpha)} \sup _{0<s<t<T}\left|\left(D_{t-}

Theorems & Definitions (16)

  • Proposition 2.1
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Example 4.1
  • ...and 6 more