Deviation of order p for estimators of the variance in first-order stochastic differential equation (SDE)

Abstract

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≥; 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.

Keywords:

• variance estimator
• deviation in norm p
• central limit theorem

AMS Subject Classification Codes :

• 60F05
• 60F25
• 60J60
• 60H05
• 62M02
• 62M05

Acknowledgements

The authors want to thank two anonymous referees whose remarks have improved the first versions of the present article. The work of the second author was financed by the LOCTI-Total project of Venezuela‘Transporte de contaminantes en el lago de Valencia’.