Fixed-Point Estimation of the Drift Parameter in Stochastic Differential Equations Driven by Rough Multiplicative Fractional Noise

TL;DR

This paper develops a fixed-point estimator for the drift parameter in stochastic differential equations driven by rough fractional noise, addressing approximation challenges and providing theoretical guarantees and numerical validation.

Contribution

It introduces a novel fixed-point estimation method for the drift in rough SDEs driven by fractional Brownian motion, with new reformulations and error control techniques.

Findings

Estimator is well-posed for H in (1/3,1)

Provides asymptotic confidence intervals and risk bounds

Numerical results show good practical performance

Abstract

We investigate the problem of estimating the drift parameter from $N$ independent copies of the solution of a stochastic differential equation driven by a multiplicative fractional Brownian noise with Hurst parameter $H\in (1/3,1)$. Building on a least-squares-type object involving the Skorokhod integral, a key challenge consists in approximating this unobservable quantity with a computable fixed-point estimator, which requires addressing the correction induced by replacing the Skorokhod integral with its pathwise counterpart. To this end, a crucial technical contribution of this work is the reformulation of the Malliavin derivative of the process in a way that does not depend explicitly on the driving noise, enabling control of the approximation error in the multiplicative setting. For the case $H\in (1/3,1/2]$, we further exploit results on two-dimensional Young integrals to manage the more intricate correction term that appears. As a result, we establish the well-posedness of a fixed-point estimator for any $H\in (1/3,1)$, together with both an asymptotic confidence interval and a non-asymptotic risk bound. Finally, a numerical study illustrates the good practical performance of the proposed estimator.