Flores, Elvia; León, José R. Deviation of order \(p\) for estimators of the variance in first-order stochastic differential equation (SDE). (English) Zbl 1283.60036 Statistics 44, No. 5, 431-454 (2010). Summary: In this work, we consider a non-parametric estimator of the variance in one-dimensional diffusion models or, more generally, in Itô processes with a deterministic diffusion term and a general non-anticipative drift. The estimation is based on the quadratic variation of discrete time observations over a finite interval. In particular, a central limit theorem (CLT) is proved for the deviation in \(L^p\) norm \((p\geq 1)\) between the variance and this estimator. The method of the proof consists in writing the \(L^p\) norm of the deviation, when the drift term is equal to zero, as a sum of 4-dependent random variables. The moments are then computed by means of a Gaussian approximation and a CLT for \(m\)-dependent random variables is applied. The convergence is stable in law, this allows the result for processes with general drifts to be obtained, by using Girsanov’s formula. MSC: 60F05 Central limit and other weak theorems 60F25 \(L^p\)-limit theorems 60J60 Diffusion processes 60H05 Stochastic integrals 62M02 Markov processes: hypothesis testing 62M05 Markov processes: estimation; hidden Markov models Keywords:variance estimator; deviation in norm \(p\); central limit theorem PDFBibTeX XMLCite \textit{E. Flores} and \textit{J. R. León}, Statistics 44, No. 5, 431--454 (2010; Zbl 1283.60036) Full Text: DOI References: [1] DOI: 10.1198/016214505000000169 · Zbl 1117.62461 [2] DOI: 10.1016/j.mbs.2006.10.009 · Zbl 1255.60132 [3] DOI: 10.1016/j.jmarsys.2003.11.023 [4] DOI: 10.1016/S0927-5398(97)00003-0 [5] Genon-Catalot V., Scand. J. Statist. 19 pp 317– (1992) [6] DOI: 10.1111/1467-9469.00180 · Zbl 0938.62085 [7] DOI: 10.1080/07362999808809525 · Zbl 0894.62093 [8] DOI: 10.1016/S0304-4149(00)00068-5 · Zbl 1047.60082 [9] DOI: 10.1051/ps:2004005 · Zbl 1186.60080 [10] Revuz D., Continuous Martigales and Brownian motion (1999) · Zbl 0917.60006 [11] Gihman I. J., Stochastic Differential Equations (1972) · Zbl 0242.60003 [12] DOI: 10.1007/s11203-005-0059-6 · Zbl 1110.62110 [13] DOI: 10.1214/aop/1176995577 · Zbl 0376.60026 [14] Billingsley, P.Weak Convergence of Measures. Regional conference Series in Applied Mathematics. Philadelphia, PA. SIAM. · Zbl 0271.60009 [15] DOI: 10.2307/3318650 · Zbl 0882.60017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.