Statistical inference for Vasicek-type model driven by Hermite processes
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Let (Z^q, H_t)_t ≥ 0 denote a Hermite process of order q ≥ 1 and self-similarity parameter H ∈ (1/2, 1). This process is H-self-similar, has stationary increments and exhibits long-range dependence. When q=1, it corresponds to the fractional Brownian motion, whereas it is not Gaussian as soon as q≥ 2. In this paper, we deal with the following Vasicek-type model driven by Z^q, H: X_0=0, dX_t = a(b - X_t)dt +dZ_t^q, H, t ≥ 0, where a > 0 and b ∈R are considered as unknown drift parameters. We provide estimators for a and b based on continuous-time observations. For all possible values of H and q, we prove strong consistency and we analyze the asymptotic fluctuations.
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