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Parameter estimation for fractional Ornstein-Uhlenbeck processes. (English) Zbl 1187.62137
Summary: We study a least squares estimator $\widehat{\theta }_T$ for the Ornstein-Uhlenbeck process, $\text dX_t = \theta X_t \text dt+\sigma \text dB_t^H$, driven by fractional Brownian motion $B^H$ with Hurst parameter $H \geq 1/2$. We prove the strong consistence of $\widehat{\theta }_T$ (the almost surely convergence of $\widehat{\theta }_T$ to the true parameter $\theta $). We also obtain the rate of this convergence when $1/2\leq H<3/4$, applying a central limit theorem for multiple Wiener integrals. This least squares estimator can be used to study other more simulation friendly estimators such as the estimator $\tilde \theta _T$ obtained by a function of $\int _0^T X_t^2 \text dt$.

MSC:
62M05Markov processes: estimation
62F12Asymptotic properties of parametric estimators
60F05Central limit and other weak theorems
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References:
[1] Biagini, F.; Hu, Y.; øksendal, B.; Zhang, T.: Stochastic calculus for fractional Brownian motion and applications, (2008) · Zbl 1157.60002
[2] Cheridito, P.; Kawaguchi, H.; Maejima, M.: Fractional Ornstein--Uhlenbeck processes, Electron. J. Probab. 8, 1-14 (2003) · Zbl 1065.60033 · emis:journals/EJP-ECP/_ejpecp/EjpVol8/paper3.abs.html
[3] Duncan, T. E.; Hu, Y.; Pasik-Duncan, B.: Stochastic calculus for fractional Brownian motion. I. theory, SIAM J. Control optim. 38, 582-612 (2000) · Zbl 0947.60061 · doi:10.1137/S036301299834171X
[4] Hu, Y.: Integral transformations and anticipative calculus for fractional Brownian motions, Mem. amer. Math. soc. 175, 825 (2005) · Zbl 1072.60044
[5] Hu, Y.; Long, H.: Parameter estimation for Ornstein--Uhlenbeck processes driven by ${\alpha}$-stable Lévy motions, Commun. stoch. Anal. 1, 175-192 (2007) · Zbl 1328.62505
[6] Hu, Y.; Long, H.: Least squares estimator for Ornstein--Uhlenbeck processes driven by ${\alpha}$-stable motions, Stochastic process. Appl. 119, No. 8, 2465-2480 (2009) · Zbl 1171.62045 · doi:10.1016/j.spa.2008.12.006
[7] Hu, Y.; øksendal, B.: Fractional white noise calculus and applications to finance, Infin. dimens. Anal. quantum probab. Relat. top. 6, 1-32 (2003) · Zbl 1045.60072 · doi:10.1142/S0219025703001110
[8] Kleptsyna, M. L.; Le Breton, A.: Statistical analysis of the fractional Ornstein--Uhlenbeck type process, Stat. inference stoch. Process. 5, 229-248 (2002) · Zbl 1021.62061 · doi:10.1023/A:1021220818545
[9] Kutoyants, Yu.A.: Statistical inference for ergodic diffusion processes, (2004) · Zbl 1038.62073
[10] Liptser, R. S.; Shiryaev, A. N.: Statistics of random processes: II applications, Applications of mathematics (2001) · Zbl 1008.62073
[11] Nualart, D.: The Malliavin calculus and related topics, (2006) · Zbl 1099.60003
[12] Nualart, D.; Ortiz-Latorre, S.: Central limit theorems for multiple stochastic integrals and Malliavin calculus, Stochastic process. Appl. 118, 614-628 (2008) · Zbl 1142.60015 · doi:10.1016/j.spa.2007.05.004
[13] Nualart, D.; Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals, Ann. probab. 33, 177-193 (2005) · Zbl 1097.60007 · doi:10.1214/009117904000000621
[14] Pickands, J.: Asymptotic properties of the maximum in a stationary Gaussian process, Trans. amer. Math. soc. 145, 75-86 (1969) · Zbl 0206.18901 · doi:10.2307/1995059
[15] Skorohod, A. V.: On a generalization of a stochastic integral, Theory probab. Appl. 20, 219-233 (1975)