# Parameter maximum likelihood estimation problem for time periodic modulated drift Ornstein Uhlenbeck processes

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* Corresponding author 1 IRMAR - Institut de Recherche Mathématique de Rennes

Abstract : In this paper we investigate the large-sample behaviour of the maximum likelihood estimate MLE of the unknown parameter $\theta$ for processes following the model $d\xi t = \theta ft\xi t dt + dB t$, where $f : R ightarrow R$ is a continuous function with period, say $P > 0$. Here the periodic function $f\cdot$ is assumed known. We establish the consistency of the MLE and we point out its minimax optimality. These results comply with the well-established case of an Ornstein Uhlenbek process when the function $f\cdot$ is constant. However the case when $\int^P 0 ft dt = 0$ and $f\cdot$ is not identically null presents some special features. For instance in this case whatever is the value of $\theta$, the rate of convergence of the MLE is T as in the case when $\theta = 0$ and $\int^P 0 ftdt eq 0$.

Keywords : Langevin stochastic differential equation periodicity local asymptotic minimax bound maximum likelihood estimator Ornstein Uhlenbeck process Time-inhomogeneous diffusion process

Author: ** Dominique Dehay - **

Source: https://hal.archives-ouvertes.fr/