The present work deals with the parameter estimation problem for an nth-order mixed fractional Brownian motion (fBm) of the form \(X(t)=\theta \mathcal {P}(t)+\alpha W(t)+\sigma B_H^n(t)\), where W(t) is a Wiener process and \(B_H^n(t)\) is the nth-order fBm (\(n\ge 2\)) with Hurst index \(H\in (n-1,n)\). By using power-variations method we estimate \(\alpha\), then we build maximum likelihood estimators of the parameters \(\theta\) and \(\sigma\). Both weak and almost sure behaviour of the proposed estimators are established.

中文翻译：

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多项式漂移的n阶混合分数布朗运动的参数估计

目前的工作处理形式为\(X(t)=\theta \mathcal {P}(t)+\alpha W(t)+ \sigma B_H^n(t)\)，其中W ( t ) 是维纳过程，\(B_H^n(t)\)是n阶 fBm ( \(n\ge 2\) ) 和 Hurst索引\(H\in (n-1,n)\)。通过使用幂变化方法，我们估计\(\alpha\)，然后我们构建参数\(\theta\)和\(\sigma\)的最大似然估计。建立了所提出的估计量的弱行为和几乎确定的行为。