Statistical inference for n th-order mixed fractional Brownian motion with polynomial drift
Pub. online: 19 November 2024
Type: Research Article
Open Access
Open Access
Received
27 May 2024
27 May 2024
Revised
21 October 2024
21 October 2024
Accepted
29 October 2024
29 October 2024
Published
19 November 2024
19 November 2024
Abstract
The mixed model with polynomial drift of the form $X(t)=\theta \mathcal{P}(t)+\alpha W(t)+\sigma {B_{H}^{n}}(t)$ is studied, where ${B_{H}^{n}}$ is the nth-order fractional Brownian motion with Hurst index $H\in (n-1,n)$ and $n\ge 2$, independent of the Wiener process W. The polynomial function $\mathcal{P}$ is known, with degree $d(\mathcal{P})\in [1,n)$. Based on discrete observations and using the ergodic theorem estimates of H, ${\alpha ^{2}}$ and ${\sigma ^{2}}$ are given. Finally, a continuous time maximum likelihood estimator of θ is provided. Both strong consistency and asymptotic normality of the proposed estimators are established.
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