Stochastic modeling by processes driven by a fractional Brownian motion has been used for phenomena with long range dependence. Statistical inference for stochastic processes satisfying stochastic differential equations driven by a fractional Brownian motion has been studied earlier and a comprehensive survey of various methods is given in Mishura  and in Prakasa Rao . However it was observed that modeling of the financial markets by processes driven by fractional Brownian motion lead to arbitrage opportunities which is contrary to the rational market behaviour. Cheridito  proposed modeling by processes driven by a mixed fractional Brownian motion to avoid this problem. There has been a recent interest to study problems of statistical inference for stochastic processes driven by a mixed fractional Brownian motion (mfBm). Existence and uniqueness for solutions of stochastic differential equations driven by a mfBm are investigated in Mishura and Shevchenko , Shevchenko , Guerra and Nualart  and more recently by Luis da Silva et al.  among others . Maximum likelihood estimation for estimation of drift parameter in a linear stochastic differential equations driven by a mfBm is investigated in Prakasa Rao . The method of instrumental variable estimation for such parametric models is investigated in Prakasa Rao . Some applications of such models in finance are presented in Prakasa Rao [10,11]. Stochastic differential equation models with random effects are used in the biomedical field for the study of repeated measurements collected on a series of individuals/subjects. For instance, see Antic et al. , Delattre et al. , Ditlevsen and De Gaetano , Nie and Yang , Nie [16,17], Picchini et al.  and Picchini and Ditlevsen  among others. Parametric inference for linear stochastic differential equations driven by a mixed fractional Brownian motion with random effects based on discrete observations has been studied in Prakasa Rao . Nonparametric estimation for fractional diffusion processes with random effects has been investigated in El Omari et al. . They study the properties of kernel and histogram estimators for estimation of the density of random effects. We discussed nonparametric estimation for models governed by stochastic differential equations with random effects driven by a mixed fractional Brownian motion (mfBm) with Hurst indexin Prakasa Rao . For parametric inference for processes driven by mfBm, see Marushkevych , Rudomino-Dusyatska , Song and Liu , Mishra and Prakasa Rao , Prakasa Rao  and Miao  among others. El Omari et al.  studied estimation of parameters and when the random effects are Gaussian with mean
and variancebased on discrete observations on the process. Maximum likelihood estimation of stochastic differential equations with random effects driven by fractional brownian motion is investigated in Dai et al. . Our aim in this paper is to study similar problem for processes driven by mfBm.
2 Mixed fractional Brownian motion
We will now summarize some properties of stochastic processes which are solutions of stochastic differential equations driven by a mixed fractional Brownian motion . We assume that sufficient conditions hold to ensure existence and uniqueness of the solution. Let be a stochastic basis satisfying the usual conditions. The natural filtration of a stochastic process is understood as the -completion of the filtration generated by this process. Let be a standard Wiener process and be an independent normalized fractional Brownian motion with Hurst parameter , that is, a Gaussian process with continuous sample paths such that and
The process is called the mixed fractional Brownian motion with Hurst index We assume here after that Hurst index is known. Following the results in Cheridito , it is known that the process is a semimartingale in its own filtration if and only if either or Let us consider a stochastic process defined by the stochastic integral equation
where the process is an -adapted process. For convenience, we write the above integral equation in the form of a stochastic differential equation
driven by the mixed fractional Brownian motion Following the recent works by Cai et al.  and Chigansky and Kleptsyna , one can construct an integral transformation that transforms the mixed fractional Brownian motion into a martingale Let be the solution of the integro-differential equation
Cai et al.  proved that the process
is a Gaussian martingale with quadratic variation
For convenience, we denote the function by Furthermore the natural filtration of the martingale coincides with that of the mixed fractional Brownian motion Suppose that, for the martingale defined by the equation (2.5), the sample paths of the process are smooth enough in the sense that the process
is well defined. Define the process
As a consequence of the results in Cai et al. , it follows that the process is a fundamental semimartingale associated with the process in the following sense. Theorem 2.1: Let be the solution of the equation (2.4). Define the process as given in the equation (2.8). Then the following relations hold.
(i) The process is a semimartingale with the decomposition
where is the martingale defined by the equation (2.5).
(ii) The process admits the representation
(iii) The natural filtrations and of the processes and respectively coincide.
Applying Corollary 2.9 in Cai et al. , it follows that the probability measuresand generated by the processes and on an interval are absolutely continuous with respect to each other and the Radon-Nikodym derivative is given by
which is also the likelihood function based on the observation Since the filtrations generated by the processes and are the same, the information contained in the families of -algebras and is the same and hence the problem of the estimation of the parameters involved based on the observation and are equivalent.
Let us consider a system of stochastic differential equations
where the processes are independent mixed fractional Brownian motions with common Hurst index Suppose the random effects are independent and identically distributed as that of the distribution of
We assume that the random variablesare independent of the random processes and sufficient conditions hold for the existence and uniqueness of the solutions for the system (3.1). See Mishura and Shevchenko , Shevchenko , Guerra and Nualart  and Luis da Silva  for sufficient conditions on the function for the existence and uniqueness of the solution for the system (3.1). The problem is to estimate the unknown parameters based on the set of observations and study the asymptotic properties of the estimators as tends to infinity. Here after we assume that the random effect with respect to a -finite measure on and it is known but for the parameter which is unknown. The problem is to estimate the parameter based on the observations Let denote the probability measure generated by the process over the interval when is the observe value of the random effect. Let be another value of From the results in Section 2, it follows that, given the log-likelihood ratio, based on the observation of the process over the interval is given by
where the process is given by
as defined by the equation (2.7). Without loss of generality, suppose that the parameter satisfies the condition Then the log-likelihood function, given can be written in the form
Since the parameter has the probability density with respect to the -finite measure it follows that the likelihood function based on the observation of the process over the interval is given by
and the likelihood function based on the processes is given by
Suppose there exists an estimator defined by the relation
The estimator is called a maximum likelihood estimator (MLE) of the parameter Existence and uniqueness of the MLE can be ensured under some conditions on the function and the parameter space (cf. Prakasa Rao ). Under standard regularity conditions on the function , it can be shown that the MLE based on i.i.d. observations is consistent and asymptotically normal as in the sense that
as where is the Fisher information matrix defined by
See Theorems 16.2 and Theorem 16.3, DasGupta  among others for details about the asymptotic theory of maximum likelihood estimators based on independent and identically distributed observations. We omit the details.
4 Linear multiplier case
Suppose that the function where and the function is known. We assume that the function satisfies sufficient conditions so that the system (3.1) has a unique solution. Further suppose that
Observe that the processes are independent and identically distributed. From the computations given in the Sections 2 and 3, it is easy to check that the likelihood function based on these observed data is given by
for Maximizing the likelihood function, we obtain the maximum likelihood estimator of the parameter Special case: Suppose that the functions and the -finite measure are such that is the Gaussian probability density with mean and variance Here Following the computations similar to those given in Dai et al. (2020), it can be shown that, if the parameter is known, then the MLE of is given by
then, by the Strong law of large numbers (SLLN) for independent and identically distributed random variables (i.i.d.), it follows that
Applying the SLLN for i.i.d. random variables, it follows that
Applying the standard central limit theorem for independent and identically distributed random variables, it can be checked that
in distribution as If both the parameters and are unknown, then the MLEs and of and are given by the system of equations
but the explicit computation of the estimators and and study of their asymptotic properties as tends to infinity is cumbersome. Acknowledgment: This work was supported under the scheme “INSA Senior Scientist” by the Indian National Science Academy at the CR Rao Advanced Institute for Mathematics, Statistics and Computer Science, Hyderabad 500046, India. References
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