On the multi-step MLE-process for ergodic diffusion

Abstract

We propose a new method of the construction of the asymptotically efficient estimator-processes asymptotically equivalent to the MLE and the same time much more easy to calculate. We suppose that the observed process is ergodic diffusion and that there is a learning time interval of the length negligeable with respect to the whole time of observations. The preliminary estimator obtained after the learning time is then used in the construction of one-step and twostep MLE processes. We discuss the possibility of the applications of the proposed estimation procedure to several other observations models.

On approximation of BSDE and multi-step MLE-processes

“…In the work Kutoyants (2015) we aleady studied the multi-step MLE-process for ergodic diffusion process, and the structure of estimator-process proposed in the present work is quite similar.…”

“…The uniform consistency can be shown following the proof of such uniform consistency presented in Kutoyants (2015), Theorem 1.…”

“…The main contribution of the present work is due to a new class of estimatorprocesses called multi-step MLE-process introduced in Kutoyants (2015). These estimator-processes allow us to construct the approximations of the solutions of BSDEs for three above mentioned models with vanishing learning intervals (models (1) and (2)) or negligible with respect to the whole volume of observations learning interval (model (3)).…”

“…The one-step MLE-process, introduced in Kutoyants (2015), for this model of observations can be written as follows. Let us denoteθ N the premilinary estimator constructed by the first N = n δ observations X N = (X 1 , .…”

“…That is why we propose to use the so-called multi-step MLE-process (Kutoyants 2015), which is introduced as follows. First we construct a preliminary estimatorθ τ ε by the observations X τ ε = (X s , 0 ≤ s ≤ τ ε ) on some learning interval [0, τ ε ], where τ ε = ε δ with 0 < δ < 1 and then we propose an estimator-process ϑ t,ε , τ ε ≤ t ≤ T based on this preliminary estimator.…”