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We study the asymptotic properties of minimum distance estimator of drift parameter for a class of nonlinear scalar stochastic differential equations driven by mixed fractional Brownian motion. The consistency and limit distribution of this estimator are established as the diffusion coefficient tends to zero under some regularity conditions.

Stochastic differential equations (SDEs) are a natural choice to model the time evolution of dynamic systems which are subject to random influences. For existence and uniqueness of solutions of finite dimensional stochastic differential equations and properties of stochastic integrals, we refer to [

It is natural that a model contains unknown parameters. The parametric estimation problems for diffusion processes satisfying SDEs driven by Brownian motion (hereafter Bm) have been studied earlier. For a more recent comprehensive discussion, we refer to [

The mixed fractional Brownian motion (hereafter mfBm) was introduced by Patrick [

There are several heuristic methods available for use in case of SDEs driven by mfBm, such as MLE, LSE, and sequential estimation. In the continuous case, since the MLE has desirable asymptotic properties of consistency, normality, and efficiency under broad conditions, perhaps the most direct method is the MLE. However, MLE has some shortcomings; MLE’s calculation is often cumbersome as the expressions for MLE involve stochastic integrals which need good approximations for computational purposes. Moreover, the generally good asymptotic properties are not always satisfied in the discrete case. Paper [

Though mfBm has stationary self-similar increments, it does not have stationary increments and is not a Markov process. So, state-space models and Kalman filter estimators cannot be applied to the parameters of this process. Under the circumstances, in order to overcome those difficulties, the minimum distance approach is proposed.

The interest for this method of parametric estimation is that the minimum distance estimation (hereafter MDE) method has several features. It makes MDE an attractive method. From one part it is sometimes easy to calculate. On the other side this estimator is known to be consistent (see [

For the SDEs driven by Brownian motions, Kutoyants [

However, it appears that there are few works studying the estimators of mfBm. Zili [

In present paper, our aim is to obtain the MDE of the drift parameter for a class of nonlinear scalar SDEs driven by mfBm and study the asymptotic properties of this estimator.

The remainder of this paper proceeds as follows. Section

Let

A fractional Brownian motion

From (

The notation

A mixed fractional Brownian motion of parameters

From Zili [

By using the self-similarity of mfBm, we obtain the following lemma.

Let

The value of (

For any stopping time

B-D-G have a long history and we cite only some works in this area. Maybe the first general results were due to Novikov (

Let

Let

Now we consider the parameter estimation problem for a class of nonlinear scalar mixed SDE in the following framework:

Denote

Assume that the trend functional of the above equation has the following form:

The function

The

We also need the following additional conditions.

Denote

In the paper, we will use

If the above condition

Condition

As a consequence of the above theorem, we obtain that

Under conditions

Let

Introduce the set

In fact, according to (

Then, by Theorem

We have

Introduce the functions

The authors declare that there is no conflict of interests regarding the publication of this paper.

_{1}-norm estimate of the parameter of the Ornstein-Uhlenbeck process