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

Dominique Dehay (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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 $d\xi_t = \theta f(t)\xi_t dt + dB_t$, where $f : R \rightarrow R$ 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 $\int^P_0 f(t) dt = 0$ 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 $\int^P_0 f(t)dt \neq 0$.

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

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Dominique Dehay. Parameter maximum likelihood estimation problem for time periodic modulated drift Ornstein Uhlenbeck processes. Statistical Inference for Stochastic Processes, 2015, 18 (1), pp.69-98. ⟨10.1007/s11203-014-9104-7⟩. ⟨hal-00794615v2⟩