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The maximum likelihood type estimator of SDEs with fractional Brownian motion under small noise asymptotics in the rough case

Shohei Nakajima

TL;DR

This work addresses parametric estimation for high-dimensional SDEs driven by fractional Brownian motion with Hurst indices in $(\tfrac{1}{3},\tfrac{1}{2})$ under small noise $\varepsilon\to0$. It develops a maximum likelihood-type estimator by embedding the model in rough path theory, transforming the process via a Girsanov change of measure, and defining a tractable log-likelihood $\mathbb{L}_{H,\varepsilon}(\theta)$. The authors establish an apriori estimate for the solution, prove the estimator is asymptotically normal with rate $\varepsilon^{-1}$ and covariance $\Gamma_H(\theta_0)^{-1}$ under identifiability conditions, and rely on Yoshida’s large deviation framework to validate the local asymptotic normality. These results enable rigorous statistical inference for rough stochastic systems where classical semimartingale methods fail, leveraging rough path analysis to handle fractional noise and small-dispersion asymptotics.

Abstract

We study the problem of parametric estimation for continuously observed stochastic differential equation driven by fractional Brownian motion. Under some assumptions on drift and diffusion coefficients, we construct maximum likelihood estimator and establish its the asymptotic normality and moment convergence of the drift parameter when a small dispersion coefficient vanishes.

The maximum likelihood type estimator of SDEs with fractional Brownian motion under small noise asymptotics in the rough case

TL;DR

This work addresses parametric estimation for high-dimensional SDEs driven by fractional Brownian motion with Hurst indices in under small noise . It develops a maximum likelihood-type estimator by embedding the model in rough path theory, transforming the process via a Girsanov change of measure, and defining a tractable log-likelihood . The authors establish an apriori estimate for the solution, prove the estimator is asymptotically normal with rate and covariance under identifiability conditions, and rely on Yoshida’s large deviation framework to validate the local asymptotic normality. These results enable rigorous statistical inference for rough stochastic systems where classical semimartingale methods fail, leveraging rough path analysis to handle fractional noise and small-dispersion asymptotics.

Abstract

We study the problem of parametric estimation for continuously observed stochastic differential equation driven by fractional Brownian motion. Under some assumptions on drift and diffusion coefficients, we construct maximum likelihood estimator and establish its the asymptotic normality and moment convergence of the drift parameter when a small dispersion coefficient vanishes.
Paper Structure (5 sections, 12 theorems, 106 equations)

This paper contains 5 sections, 12 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Apriori estimate
  4. Construction of the maximum likelihood type estimator
  5. Proof of Theorem \ref{['main']}

Key Result

Proposition 1

Let $(Y,\mathbb{Y})\in\mathscr{C}^\alpha([0,T],V)$ and $(Z,Z^\prime)\in\mathcal{D}_Y^{2\alpha}([0,T],\mathcal{L}(V,W))$. Then there exists a constant $C>0$ such that for any $s,t\in[0,T]$ holds.

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Remark 2
  • Definition 4
  • Lemma 1
  • Lemma 2
  • ...and 16 more