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

• Published in: Statistical methods & applications Vol. 30; no. 2; pp. 409 - 436
• Main Authors: Es-Sebaiy, Khalifa, Es.Sebaiy, Mohammed
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2021; Springer Nature B.V
• Subjects: Asymptotic properties; Chemistry and Earth Sciences; Computer Science; Drift estimation; Driving conditions; Economics; Ergodic processes; Finance; Gaussian process; Health Sciences; Humanities; Insurance; Law; Management; Mathematical models; Mathematics and Statistics; Medicine; Original Paper; Parameter estimation; Physics; Statistical Theory and Methods; Statistics; Statistics for Business; Statistics for Engineering; Statistics for Life Sciences; Statistics for Social Sciences; Gaussian Vasicek-type model; Joint asymptotic distribution; Young integral; Parameter estimation; 60G22; Strong consistency; 62M09; 62F12; Fractional Gaussian processes; 60G15; 62M86
• ISSN: 1618-2510; 1613-981X
• DOI: 10.1007/s10260-020-00528-4

We study a problem of parameter estimation for a non-ergodic Gaussian Vasicek-type model defined as d X t = θ ( μ + X t ) d t + d G t , t ≥ 0 with unknown parameters θ > 0 , μ ∈ R and α : = θ μ , where G is a Gaussian process. We provide least square-type estimators ( θ ~ T , μ ~ T ) and ( θ ~ T , α ~ T ) , respectively, for ( θ , μ ) and ( θ , α ) based a continuous-time observation of { 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 the strongly consistency and the joint asymptotic distribution of ( θ ~ T , μ ~ T ) and ( θ ~ T , α ~ T ) . Moreover, we obtain that the limit distribution of θ ~ T is a Cauchy-type distribution, and μ ~ T and 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 μ = 0 .