Power variations and limit theorems for stochastic processes controlled by fractional Brownian motions

TL;DR

This paper establishes limit theorems for power variations of stochastic processes driven by fractional Brownian motions with Hurst parameter $H \\leq 1/2$, revealing different convergence behaviors depending on $H$.

Contribution

It provides a detailed decomposition of power variations and determines their convergence rates and limits for processes controlled by fractional Brownian motions.

Findings

Centered power variation converges stably at rate $n^{-1/2}$ for $H \\geq 1/4$.

Power variation converges in probability at rate $n^{-2H}$ for $H<1/4$.

Limit of mixed weighted sum is characterized using a rough path approach.

Abstract

In this paper we establish limit theorems for power variations of stochastic processes controlled by fractional Brownian motions with Hurst parameter $H\leq 1/2$. We show that the power variations of such processes can be decomposed into the mix of several weighted random sums plus some remainder terms, and the convergences of power variations are dominated by different combinations of those weighted sums depending on whether $H<1/4$, $H=1/4$, or $H>1/4$. We show that when $H\geq 1/4$ the centered power variation converges stably at the rate $n^{-1/2}$, and when $H<1/4$ it converges in probability at the rate $n^{-2H}$. We determine the limit of the mixed weighted sum based on a rough path approach developed in \cite{LT20}.