zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 $\int _0^T X_t^2 \text dt$.

62M05Markov processes: estimation
62F12Asymptotic properties of parametric estimators
60F05Central limit and other weak theorems
Full Text: DOI arXiv
[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)