Drift parameter estimation for tempered fractional Ornstein–Uhlenbeck processes based on discrete observations
Pub. online: 18 November 2025
Type: Research Article
Open Access
Open Access
Received
25 June 2025
25 June 2025
Revised
6 November 2025
6 November 2025
Accepted
8 November 2025
8 November 2025
Published
18 November 2025
18 November 2025
Abstract
The problem of estimating the drift parameter is considered for an Ornstein–Uhlenbeck-type process driven by a tempered fractional Brownian motion (tfBm) or tempered fractional Brownian motion of the second kind (tfBmII). Unlike most existing studies, which assume continuous-time observations, a more realistic setting of discrete-time data is in focus. The strong consistency of a discretized least squares estimator is established under an asymptotic regime where the observation interval tends to zero while the total time horizon increases. A key step in the analysis involves deriving almost sure upper bounds for the increments of both tfBm and tfBmII.
References
Azmoodeh, E., Mishura, Y., Sabzikar, F.: How does tempering affect the local and global properties of fractional Brownian motion? J. Theor. Probab. 35(1), 484–527 (2022). MR4379473. https://doi.org/10.1007/s10959-020-01068-z
Azmoodeh, E., Sottinen, T., Viitasaari, L., Yazigi, A.: Necessary and sufficient conditions for Hölder continuity of Gaussian processes. Stat. Probab. Lett. 94, 230–235 (2014). MR3257384. https://doi.org/10.1016/j.spl.2014.07.030
Berzin, C., Latour, A., León, J.R.: Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion. Lecture Notes in Statistics, vol. 216. Springer, Cham (2014). MR3289986. https://doi.org/10.1007/978-3-319-07875-5
Bojdecki, T., Gorostiza, L.G., Talarczyk, A.: Sub-fractional Brownian motion and its relation to occupation times. Stat. Probab. Lett. 69(4), 405–419 (2004). MR2091760. https://doi.org/10.1016/j.spl.2004.06.035
Boniece, B.C., Didier, G., Sabzikar, F.: Tempered fractional Brownian motion: wavelet estimation, modeling and testing. Appl. Comput. Harmon. Anal. 51, 461–509 (2021). MR4196450. https://doi.org/10.1016/j.acha.2019.11.004
Brouste, A., Iacus, S.M.: Parameter estimation for the discretely observed fractional Ornstein-Uhlenbeck process and the Yuima R package. Comput. Stat. 28(4), 1529–1547 (2013). MR3120827. https://doi.org/10.1007/s00180-012-0365-6
Cheridito, P., Kawaguchi, H., Maejima, M.: Fractional Ornstein-Uhlenbeck processes. Electron. J. Probab. 8, 3–14 (2003). MR1961165. https://doi.org/10.1214/EJP.v8-125
Davies, R.B., Harte, D.S.: Tests for Hurst effect. Biometrika 74(1), 95–101 (1987). MR0885922. https://doi.org/10.1093/biomet/74.1.95
Dietrich, C.R., Newsam, G.N.: Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput. 18(4), 1088–1107 (1997). MR1453559. https://doi.org/10.1137/S1064827592240555
Dozzi, M., Kozachenko, Y., Mishura, Y., Ralchenko, K.: Asymptotic growth of trajectories of multifractional Brownian motion, with statistical applications to drift parameter estimation. Stat. Inference Stoch. Process. 21(1), 21–52 (2018). MR3769831. https://doi.org/10.1007/s11203-016-9147-z
El Machkouri, M., Es-Sebaiy, K., Ouknine, Y.: Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes. J. Korean Stat. Soc. 45(3), 329–341 (2016). MR3527650. https://doi.org/10.1016/j.jkss.2015.12.001
El-Nouty, C., Zili, M.: On the sub-mixed fractional Brownian motion. Appl. Math. J. Chin. Univ. Ser. B 30(1), 27–43 (2015). MR3319622. https://doi.org/10.1007/s11766-015-3198-6
Houdré, C., Villa, J.: An example of infinite dimensional quasi-helix. In: Stochastic Models, Mexico City, 2002. Contemp. Math., vol. 336, pp. 195–201. Amer. Math. Soc., Providence, RI (2003). MR2037165. https://doi.org/10.1090/conm/336/06034
Kozachenko, Y., Melnikov, A., Mishura, Y.: On drift parameter estimation in models with fractional Brownian motion. Statistics 49(1), 35–62 (2015). MR3304366. https://doi.org/10.1080/02331888.2014.907294
Kubilius, K., Melichov, D.: On comparison of the estimators of the Hurst index of the solutions of stochastic differential equations driven by the fractional Brownian motion. Informatica (Vilnius) 22(1), 97–114 (2011). MR2885661. https://doi.org/10.15388/Informatica.2011.316
Kubilius, K., Mishura, Y., Ralchenko, K., Seleznjev, O.: Consistency of the drift parameter estimator for the discretized fractional Ornstein–Uhlenbeck process with Hurst index $H\in (0,\frac{1}{2})$. Electron. J. Stat. 9(2), 1799–1825 (2015). MR3391120. https://doi.org/10.1214/15-EJS1062
Lim, S., Eab, C.H.: Tempered fractional Brownian motion revisited via fractional Ornstein–Uhlenbeck processes. arXiv preprint arXiv:1907.08974 (2019).
Macioszek, K., Sabzikar, F., Burnecki, K.: Testing of tempered fractional brownian motions. arXiv preprint arXiv:2504.11906 (2025).
Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968). MR0242239. https://doi.org/10.1137/1010093
Meerschaert, M.M., Sabzikar, F.: Tempered fractional Brownian motion. Stat. Probab. Lett. 83(10), 2269–2275 (2013). MR3093813. https://doi.org/10.1016/j.spl.2013.06.016
Meerschaert, M.M., Sabzikar, F.: Stochastic integration for tempered fractional Brownian motion. Stoch. Process. Appl. 124(7), 2363–2387 (2014). MR3192500. https://doi.org/10.1016/j.spa.2014.03.002
Mishura, Y.S.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics, vol. 1929. Springer (2008). MR2378138. https://doi.org/10.1007/978-3-540-75873-0
Mishura, Y., Ralchenko, K.: Drift parameter estimation in the models involving fractional Brownian motion. In: Modern Problems of Stochastic Analysis and Statistics. Springer Proc. Math. Stat., vol. 208, pp. 237–268. Springer (2017). MR3747669. https://doi.org/10.1007/978-3-319-65313-6_10
Mishura, Y., Ralchenko, K.: Asymptotic growth of sample paths of tempered fractional Brownian motions, with statistical applications to Vasicek-type models. Fractal Fract. 8(2), 79 (2024). https://doi.org/10.3390/fractalfract8020079
Mishura, Y., Ralchenko, K., Dehtiar, O.: Asymptotic properties of parameter estimators in Vasicek model driven by tempered fractional Brownian motion. Austrian J. Stat. 54, 61–81 (2025). https://doi.org/10.17713/ajs.v54i1.1966
Mishura, Y., Ralchenko, K., Shklyar, S.: Gaussian Volterra processes: asymptotic growth and statistical estimation. Theory Probab. Math. Stat. 108, 149–167 (2023). MR4588243. https://doi.org/10.1090/tpms/1190
Sabzikar, F., Surgailis, D.: Tempered fractional Brownian and stable motions of second kind. Stat. Probab. Lett. 132, 17–27 (2018). MR3718084. https://doi.org/10.1016/j.spl.2017.08.015
Wood, A.T.A., Chan, G.: Simulation of stationary Gaussian processes in ${[0,1]^{d}}$. J. Comput. Graph. Stat. 3(4), 409–432 (1994). MR1323050. https://doi.org/10.2307/1390903
Zili, M.: On the mixed fractional Brownian motion. J. Appl. Math. Stoch. Anal., 32435–9 (2006). MR2253522. https://doi.org/10.1155/JAMSA/2006/32435
Zili, M.: Generalized fractional Brownian motion. Mod. Stoch. Theory Appl. 4(1), 15–24 (2017). MR3633929. https://doi.org/10.15559/16-VMSTA71
