Estimating drift parameters in a non-ergodic Gaussian Vasicek-type model

We study a problem of parameter estimation for a non-ergodic Gaussian Vasicek-type model defined as $$dX_t=\theta (\mu + X_t)dt+dG_t,\ t\ge 0$$ d X t = θ ( μ + X t ) d t + d G t , t ≥ 0 with unknown parameters $$\theta >0$$ θ > 0 , $$\mu \in {\mathbb {R}}$$ μ ∈ R and $$\alpha :=\theta \mu $$ α : = θ μ , where G is a Gaussian process. We provide least square-type estimators $$(\widetilde{\theta }_T,\widetilde{\mu }_T)$$ ( θ ~ T , μ ~ T ) and $$(\widetilde{\theta }_T,\widetilde{\alpha }_T)$$ ( θ ~ T , α ~ T ) , respectively, for $$(\theta ,\mu )$$ ( θ , μ ) and $$(\theta ,\alpha )$$ ( θ , α ) based a continuous-time observation of $$\{X_t,\ t\in [0,T]\}$$ { X t , t ∈ [ 0 , T ] } as $$T\rightarrow \infty $$ T → ∞ . Our aim is to derive some sufficient conditions on the driving Gaussian process G in order to ensure the strongly consistency and the joint asymptotic distribution of $$(\widetilde{\theta }_T,\widetilde{\mu }_T)$$ ( θ ~ T , μ ~ T ) and $$(\widetilde{\theta }_T,\widetilde{\alpha }_T)$$ ( θ ~ T , α ~ T ) . Moreover, we obtain that the limit distribution of $$\widetilde{\theta }_T$$ θ ~ T is a Cauchy-type distribution, and $$\widetilde{\mu }_T$$ μ ~ T and $$\widetilde{\alpha }_T$$ α ~ T are asymptotically normal. We apply our result to fractional Vasicek, subfractional Vasicek and bifractional Vasicek processes. This work extends the results of El Machkouri et al. (J Korean Stat Soc 45:329–341, 2016) studied in the case where $$\mu =0$$ μ = 0 .