Estimating drift parameters in a non-ergodic Gaussian Vasicek-type model
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We study the problem of parameter estimation for a non-ergodic Gaussian Vasicek-type model defined as dX_t=(μ+θ X_t)dt+dG_t, t≥0 with unknown parameters θ>0 and μ∈, where G is a Gaussian process. We provide least square-type estimators θ_T and μ_T respectively for the drift parameters θ and μ based on continuous-time observations {X_t, t∈[0,T]} as T→∞. Our aim is to derive some sufficient conditions on the driving Gaussian process G in order to ensure that θ_T and μ_T are strongly consistent, the limit distribution of θ_T is a Cauchy-type distribution and μ_T is asymptotically normal. We apply our result to fractional Vasicek, subfractional Vasicek and bifractional Vasicek processes. In addition, this work extends the result of <cit.> studied in the case where μ=0.
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