标题： 具有粗糙乘法分数噪声驱动的随机微分方程的漂移参数的定点估计 显示英文标题

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

摘要： 我们研究从$N$个独立的随机微分方程解中估计漂移参数的问题，该随机微分方程由具有 Hurst 参数$H\in (1/3,1)$的乘法分数布朗噪声驱动。 基于涉及 Skorokhod 积分的最小二乘类型目标函数，一个关键挑战在于用可计算的固定点估计量来近似这个不可观测的数量，这需要处理用路径积分替代 Skorokhod 积分所引起的修正项。 为此，本文的一个关键技术贡献是将过程的 Malliavin 导数重新表述为不显式依赖于驱动噪声的形式，从而在乘法情况下控制近似误差。 对于情况$H\in (1/3,1/2]$，我们进一步利用二维 Young 积分的结果来处理出现的更复杂的修正项。 结果，我们建立了对于任何$H\in (1/3,1)$的固定点估计量的适定性，同时给出了渐近置信区间和非渐近风险界。 最后，数值研究展示了所提出估计量的良好实际性能。 显示英文摘要

摘要： 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.