We investigate large deviation properties of the maximum likelihood drift parameter estimator for Ornstein–Uhlenbeck process driven by mixed fractional Brownian motion.

Our purpose is to establish large deviations principle for the maximum likelihood estimator of drift parameter of the Ornstein–Uhlenbeck process driven by a mixed fractional Brownian motion:

It is important to notice that the parameter

Two following chapters contain information about maximum likelihood estimation procedure for the mixed fractional Ornstein–Uhlenbeck process and description of basic concepts of large deviations theory.

The formulation of our main results and their proofs are given in Section

The interest to mixed fractional Brownian motion was triggered by Cheridito [

An interesting change in properties of a mixed fractional Brownian motion

The main contribution of paper [

In fact, there is an integral transformation that changes the mixed fractional Brownian motion to a martingale. In particular (see [

Moreover, the natural filtration of the martingale

Further, to what has just been mentioned concerning the mixed fractional Brownian motion, an auxiliary semimartingale, appropriate for the purposes of statistical analysis, can be also associated to the corresponding Ornstein–Uhlenbeck process

One of the most important results of [

In addition, it was shown by Chigansky and Kleptsyna [

The specific structure of the process

We will develop this result by proving the large deviation principle for the maximum-likelihood estimator (

The large deviations principle characterizes the limiting behavior of a family of random variables (or corresponding probability measures) in terms of a rate function.

A rate function

We say that a family of real random variables

We shall prove the large deviation principle for a family of maximum likelihood estimators (

In order to prove the large deviations principle for the drift parameter estimator of mixed fractional Ornstein–Uhlenbeck process (

Note that, for some

As it was mentioned in the previous section, in order to establish the large deviation principle for

For arbitrary

Note that

Observe that, due to (

Under the new probability measure

For

Observe that the rate function

We can observe that the following lemma plays a key role in the proof of Theorem

We shall prove the lemma using an approach similar to that in [

The Gaussian vector

We shall search solution of (

My deepest gratitude is to my advisor Marina Kleptsyna, who proposed me to consider the problem presented in this paper and whose advices and help were very important. I also would like to thank the reviewers whose remarks and advices allowed me to significantly improve this article.