A problem of drift parameter estimation is studied for a nonergodic weighted fractional Vasicek model defined as

The statistical inference of non-ergodic Itô-type diffusions has a long history. For motivation and further references, we refer the reader to Basawa and Scott [

Let

The purpose of this paper is to estimate jointly the drift parameters of the weighted fractional Vasicek (also called weighted fractional mean-reverting Ornstein–Uhlenbeck) process

In recent years, several researchers have been interested in studying statistical estimation problems for Gaussian Ornstein–Uhlenbeck processes. Estimation of the drift parameters in fractional-noise-driven Ornstein–Uhlenbeck processes is a problem that is both well-motivated by practical needs and theoretically challenging. In the finance context, our practical motivation to study this estimation problem is to provide tools to understand volatility modeling in finance. Indeed, any mean-reverting model in discrete or continuous time can be taken as a model for stochastic volatility. Let us mention some important results in this field where the volatility exhibits long-memory, which means that the volatility today is correlated to past volatility values with a dependence that decays very slowly. The authors of [

An example of interesting problem related to (

Let us mention that similar drift statistical problems for nonergodic Vasicek models were recently studied. The work [

Let us also describe what is known about the parameter estimation for the model (

Now we recall several approaches to estimate the parameters of (

A least squares approach has been also considered by several researchers to study statistical estimation problems for (

The paper is organized as follows. In Section

Throughout the paper, we shall use notation

This section is devoted to study pathwise properties of the nonergodic weighted fractional Vasicek model (

Because (

Let us introduce the following processes, for every

Since the process

We will make use of the following two technical lemmas.

Let us prove (

Notice that the convergence (

On the other hand, using (

Using similar arguments as in (

Using (

Now we prove (

In this section we will prove the strong consistency of the estimators

Combining (

Now we prove (

Using the formula (

On the other hand, taking

Consequently, the convergence (

In this section we analyze the joint asymptotic distribution of the LSEs

Both convergences (

Consequently, combining (

In order to investigate the asymptotic behavior in distribution of the estimators

Using similar arguments as in [

The convergence (

Recall that if

First we prove (

Combining all these facts together with (

Using (

Further, according to (

Let us now prove (

In this section, we briefly recall some basic elements of the Young integral (see [

The Young integral obeys the following formula. Let

We thank the two anonymous reviewers for their very careful reading and suggestions, which have led to significant improvements in the presentation of our results.