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Deviation of order \(p\) for estimators of the variance in first-order stochastic differential equation (SDE). (English) Zbl 1283.60036

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
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