Rate-optimal estimation of mixed semimartingales

TL;DR

This paper develops optimal estimation methods for mixed semimartingales, a class of processes combining Brownian motion and fractional Brownian motion, especially when they are correlated, enabling consistent parameter estimation.

Contribution

It introduces a comprehensive statistical framework for mixed semimartingales, deriving consistent estimators and central limit theorems, and demonstrates optimal rates in a minimax sense.

Findings

Correlation between components allows Hurst parameter identification.

Estimates are consistent and asymptotically normal.

Achieves minimax optimal convergence rates.

Abstract

Consider the sum $Y=B+B(H)$ of a Brownian motion $B$ and an independent fractional Brownian motion $B(H)$ with Hurst parameter $H\in(0,1)$. Even though $B(H)$ is not a semimartingale, it was shown in [\textit{Bernoulli} \textbf{7} (2001) 913--934] that $Y$ is a semimartingale if $H>3/4$. Moreover, $Y$ is locally equivalent to $B$ in this case, so $H$ cannot be consistently estimated from local observations of $Y$. This paper pivots on another unexpected feature in this model: if $B$ and $B(H)$ become correlated, then $Y$ will never be a semimartingale, and $H$ can be identified, regardless of its value. This and other results will follow from a detailed statistical analysis of a more general class of processes called \emph{mixed semimartingales}, which are semiparametric extensions of $Y$ with stochastic volatility in both the martingale and the fractional component. In particular, we derive consistent estimators and feasible central limit theorems for all parameters and processes that can be identified from high-frequency observations. We further show that our estimators achieve optimal rates in a minimax sense.