In this paper, we study the least-squares estimation problem for the Vasicek model $\mathrm{d}{Z}_t=a(b-Z_t)\mathrm{d}t+\mathrm{d}{\xi }_t^H,t\ge 0,$ driven by the complex fractional Brownian motion ${\xi }_t^H=\frac{B_t^{H,1}+i{B}_t^{H,2}}{\sqrt{2}},$ where $(B_t^{H,1},B_t^{H,2})$ is a two-dimensional fractional Brownian motion with Hurst parameter $H\in (\frac{1}{2},\frac{3}{4}).$ We obtain the strong consistency and asymptotic normality of ${\hat{a}}_T$ and ${\hat{b}}_T$ using the Garsia–Rodemich–Rumsey inequality and complex fourth moment theorems.

中文翻译：

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由复分数布朗运动驱动的 Vasicek 模型的最小二乘估计

在本文中，我们研究了 Vasicek 模型的最小二乘估计问题$\mathrm{d}{Z}_{吨}=\mathrm{一个}(b-Z_{吨})\mathrm{d}吨+\mathrm{d}{\xi }_{吨}^H,吨\ge 0,$由复分数布朗运动驱动${\xi }_{吨}^H=\frac{{乙}_{吨}^{H,1}+\mathrm{一世}{乙}_{吨}^{H,2}}{\sqrt{2}},$在哪里$({乙}_{吨}^{H,1},{乙}_{吨}^{H,2})$是具有赫斯特参数的二维分数布朗运动$H\in (\frac{1}{2},\frac{3}{4}).$我们得到强一致性和渐近正态性${\widehat{\mathrm{一个}}}_{吨}$和${\hat{b}}_{吨}$使用 Garsia-Rodemich-Rumsey 不等式和复四阶矩定理。