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Statistical inference for Vasicek-type model driven by Hermite processes

Abstract

Let (Ztq,H)t0(Z^{q, H}_t)_{t \geq 0} denote a Hermite process of order q1q \geq 1 and self-similarity parameter H(12,1)H \in (\frac{1}{2}, 1). This process is HH-self-similar, has stationary increments and exhibits long-range dependence. When q=1q=1, it corresponds to the fractional Brownian motion, whereas it is not Gaussian as soon as q2q\geq 2. In this paper, we deal with the following Vasicek-type model driven by Zq,HZ^{q, H}: \[ X_0=0,\quad dX_t = a(b - X_t)dt +dZ_t^{q, H}, \qquad t \geq 0, \] where a>0a > 0 and bRb \in \mathbb{R} are considered as unknown drift parameters. We provide estimators for aa and bb based on continuous-time observations. For all possible values of HH and qq, we prove strong consistency and we analyze the asymptotic fluctuations.

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