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Parameter Estimation for Stochastic Differential Equations Driven by Mixed Fractional Brownian Motion
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 [1–3] and the references therein.
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 [4, 5] and the references therein. In case of statistical inference for diffusion processes satisfying SDEs driven by a fractional Brownian motion (hereafter fBm), substantial progress has been made in this direction; we refer to [6–9] for more details.
The mixed fractional Brownian motion (hereafter mfBm) was introduced by Patrick  to present a stochastic model of the discounted stock price in some arbitrage-free and complete financial markets. As a result, in order to take into account long memory and exclude arbitrage, it is natural to use mfBm to replace the standard Brownian motion. Consequently, there has been a growing interest in parameter estimation for stochastic processes driven by mfBm.
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  showed that the approach of estimating parameters of an Itô process by applying MLE to a discretization of the SDE does not yield consistent estimators. LSE is asymptotically equivalent to the MLE. It is well known that the sequential estimation methods might lead to equally efficient estimators from the process observed possibly over a shorter expected period of observation time. Although there exists a wide range of estimation techniques developed for the problem of parameters estimation for SDEs driven by mfBm, we should choose a suitable estimation method.
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 ) under some general conditions. Millar  studied a general framework for (MDE) of Hilbertian type and showed that the MDE is efficient and asymptotically normal in some situations. Furthermore, the MDE is a class of estimators that is automatically robust in the same sense (for more details see ), which is generically optimal according to some quantitative measure of robustness.
For the SDEs driven by Brownian motions, Kutoyants  and Kutoyants and Pilibossian  proved that converge in probability to the random variable with or supremum norm and he also proved that is asymptotically normal when and . Hénaff  established the same results in the general case of a norm in some Banach space of functions on . For the SDEs driven by fBm, Prakasa Rao  studied the minimum -norm estimator of the drift parameter of a fractional Ornstein-Uhlenbeck type process and proved that converges in probability under to a random variable . Kouame et al.  studied asymptotic properties of minimum distance estimator of the parameter of stochastic process driven by a fBm as the diffusion coefficient tends to zero.
However, it appears that there are few works studying the estimators of mfBm. Zili  obtained some general stochastic properties of the mfBm and treated the Hölder continuity of the sample paths and -differentiability of the trajectories of mfBm. Miao  obtained the asymptotic properties of the minimum -norm estimator of the drift parameter for a linear SDE driven by an mfBm. Xiao et al.  studied the problem of estimating the parameters for the mfBm from discrete observations based on the MLE.
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 2 starts with a short description of definition of mfBm and then provides some basic lemmas that will be used in the forthcoming sections. And we obtain the MDE of the drift parameter for a class of nonlinear scalar SDEs driven by mfBm. In Section 3, we study the consistency of the above estimator. In Section 4, the limit distribution of this estimator is established as the diffusion coefficient tends to zero under some regularity conditions.
2. Notation and Preliminaries
Let be a fractional Brownian motion defined on the probability space and is a filtration of -algebra of , where the usual conditions are satisfied; that is, is a complete probability space, contains all -null sets of , and, for each , .
A fractional Brownian motion of Hurst parameter is a continuous and centered Gaussian process; that is, for all , with covariance function
From (1), it is easy to obtain that , , for all . Moreover, the fBm reduces to a standard Brownian motion denoted by for .
The notation means that the and have the same law. Denote by the supremum process A standard fBm has the following properties (for more details see , Page 5, Definition 1.1.1).(1) has homogeneous increments; that is, for .(2) has continuous trajectories.Let us take and which are two real constants such that . By Patrick , we introduce the following.
Definition 1. A mixed fractional Brownian motion of parameters , and is a process , defined on the probability space by
Remark 2. From Zili , we know that mfBm is a mixed-self-similar process: where is a constant. Furthermore it follows for the supremum process that
By using the self-similarity of mfBm, we obtain the following lemma.
Lemma 3. Let be a constant and an mfBm with parameter ; then for every ,
The value of (6) is not known. However it is fortunate that we have the following two lemmas which give the bounds for the standard Bm and fBm, respectively.
Lemma 4 (Burkholder-Davis-Gundy inequalities). For any stopping time with respect to the filtration generated by the Bm , one has where the constants and depend only upon the parameter .
Lemma 5 (see ). Let be a stopping time with respect to the filtration generated by the fBm . Then, for , one has and, for , one has where the constants and depend only upon the parameters .
Lemma 6 (see [16, page 120]). Let , , , be a sequence of continuous functions and a convex function which admits a unique minimum . Let , , be a sequence of positive numbers such that as . We suppose that Then where if there are several minima of , we choose the arbitrary one.
Now we consider the parameter estimation problem for a class of nonlinear scalar mixed SDE in the following framework: where is a known measurable functional, the unknown drift parameter , and .
Denote by the true parameter of . Let be the probability measure induced by the process and the solution of the differential equation (12) with .
Assume that the trend functional of the above equation has the following form: where and are two measurable functions.
The function is measurable with respect to and, for any , denote Note that for any .
The or supremum norm can be denoted by and the MDE (see ) is defined by
We also need the following additional conditions.
The functional is measurable and nonanticipative and satisfies the following inequalities: for all and , where are positive constants.
The measurable functions and are continuously twice differentiable in and .
Suppose that and define by the random variable ,
Denote by the derivatives of with respect to and introduce a Gaussian process which satisfies the equation where and are the derivatives of and with respect to and is a derivative with probability one of with respect to as .
In the paper, we will use to denote a generic constant which may vary from place to place.
Theorem 7. If the above condition is satisfied, then, for any , there exist constants , such that, for every ,
Remark 8. As a consequence of the above theorem, we obtain that converges in probability to under -measure as . Furthermore, the rate of convergence is of order for every .
4. Asymptotic Distribution
Theorem 9. Under conditions and , we have that the random variable converges in probability to a random variable whose probability distribution is the same as that of under .
Proof. Let and as .
Introduce the set For any , we have .
In fact, according to (17) of condition and Taylor formula, we obtain where , , , and .
Then, by Theorem 7, we have Therefore, we just need to consider the behavior of the norm , for .
We have where Then using (12), (13), (19), and (21), with some constants . From (19), condition , and Lemma 4.13 (see ), we obtain then, By using Lemma 4.13 once more, we get Now consider ; using the Taylor formula, we obtain where , .
Introduce the functions By (33) and (34), Therefore, if we choose such that when , then, with probability one, we have Using Lemma 6, we obtain the result. The proof completes.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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