The error bound in probability between the approximate maximum likelihood estimator (AMLE) and the continuous maximum likelihood estimator (MLE) is investigated for nonlinear nonhomogenous stochastic system with unknown parameter. The rates of convergence of the approximations for Itô and ordinary integral are introduced under some regular assumptions. Based on these results, the in probability rate of convergence of the approximate log-likelihood function to the true continuous log-likelihood function is studied for the nonlinear nonhomogenous stochastic system involving unknown parameter. Finally, the main result which gives the error bound in probability between the ALME and the continuous MLE is established.
It is now well known that the parameter estimation is one of the foundational problems in stochastic differential equations which are used to model practical systems that with random influences. Since 1962, Arato et al. who first applied parameter estimation to a geophysical problem in . Various parameter estimation methods have been developed for many advanced models with an increasing number of application to physical, biological and financial systems. Over the past few decades, a lot of effective approaches have proposed in this research area, see for example, [2–5]. In particular, maximum likelihood estimation (MLE) gives a unified approach to estimation, which is well defined in the case of the normal distribution and many other statistical models. Therefore the MLE technique has been widely used for the parameter estimation problem of stochastic systems . Byes estimation (BE), which is a decision rule that minimizes the posterior expected value of a loss function, has been developed in . Since some inconvenience is encountered in the real-time application that location and scale parameters are not uniquely determined, M-estimator has been studied toward the theory of robust estimation . Other widely used parameter estimation methods can be generally categorized as least squares estimation (LSE), maximum probability estimation (MPE), minimum distance estimation (MDE), minimum contrast estimation (MCE), and filtering method for parameter estimation, see for example, [9–18] and the references therein.
In reality, nonhomogenous stochastic differential equations are useful for modeling term structure of interest rates in finance and other fields. A large number of results have been published in the literature on a variety of research topics including strong or weak consistency and asymptotic efficiency as well as asymptotic normality on various parameter estimators of nonhomogenous stochastic systems [19, 20]. On the other hand, recognizing that nonlinearity is commonly encountered in engineering practice, the parameter estimation problem for nonlinear nonhomogenous stochastic systems deserves more research attention from both the theoretical and practical viewpoints and, accordingly, some promising results have been reported. For example, weak consistency, asymptotic normality, and convergence of moments of MLE and BE of the drift parameter in the nonlinear nonhomogenous Itô stochastic differential equations having nonstationary solutions have been studied in  for the small noise asymptotic case. In , the martingale approach but under some stronger regularity conditions has been used to study strong consistency and asymptotic normality for nonlinear nonhomogenous stochastic system in the large sample case. It should be pointed out that, so far, many parameter estimation methods and corresponding probability properties have been widely investigated for nonlinear nonhomogenous Itô stochastic differential equation with constant diffusion. Unfortunately, the parameter problem of general nonlinear nonhomogenous system has gained much less research attention despite its potential in practical application.
The stochastic processes which can be observed continuously over a specified time period are first used to model real system for the most part [23, 24]. In practice, it is obviously impossible to observe a process continuously over any given time period, due to the limitations on the precision of the measuring instrument or to unavailability of observations at every time point, and so forth. In other words, stochastic inference based on discrete observations is of major importance in dealing with practical problems. Hence, parameter estimation problem based on discrete observations has naturally become a hot topic in recent years [25, 26]. An approximation method has been proposed based on the discretization of the continuous time likelihood function in  for linear stochastic differential equation. A numerical approximate likelihood method has been developed in  based on iterations of the Gaussian transition densities emanating from the Euler scheme.  has used a specific transformation of the diffusion to obtain accurate theoretical approximations based on the Hermite function expansions and studied the asymptotic behavior of the approximate MLE. Up to now, although some parameter estimation problems have been established based on discretization scheme, how close are the discrete parameter estimator to the true continuous one for general nonlinear nonhomogenous stochastic system has not been fully studied due probably to the mathematical complexity, and this situation motivates our present paper.
Summarizing the above discussions, in this paper, we are motivated to study the rate of convergence of the approximate maximum likelihood estimator (AMLE) to the true continuous MLE for a class of general nonlinear nonhomogenous stochastic system with unknown parameter. The main contributions of this paper lie in the following aspects. The Itô type approximation for the stochastic integral is introduced to obtain an approximate log-likelihood function. The rate of convergence of the approximation is investigated for Itô type integral. The in probability rate of convergence of the approximate log-likelihood function is established for the nonlinear nonhomogenous stochastic system involving unknown parameter. The error bound in probability of the ALME and the LME is studied for the nonlinear nonhomogenous stochastic system. The rest of this paper is outlined as follows. In Section 2, the approximate log-likelihood function is proposed and the problem under consideration is formulated. In Section 3, several lemmas are given to analyze the rates of convergence of the approximations for Itô and ordinary integral; furthermore, the main results are discussed to analyze the rate of convergence of the approximate log-likelihood function and the error bound of the ALME and the LME. Finally, we conclude the paper in Section 4.
2. Problem Formulation and Preliminaries
Consider the real valued diffusion process , on satisfying the following stochastic differential equation:
where , is a standard Wiener process adapted to , such that for , is independent of , open in is the unknown parameter to be estimated. Let be the true value of the parameter .
Throughout this paper is a generic constant, we use following notations:
We assume the following condition:(A1) and are Lipschitz continuous in uniformly in , that is, there exists a constant such that
for any and .(A2) and satisfy linear growth condition, that is, there exists a constant such that
for any and .(A3)(A4) and are continuously differentiable with respect to up to order and
and are continuously differentiable with respect to up to order and
Remark 2.1. As (A1) and (A2) are established, it is well known that stochastic differential equation (2.1) has a unique solution. Please see the details in .
Denote . Let be the measure generated on the space of the continuous functions on with the associated Borel -algebra generated under the supremum norm by the process and be the standard Wiener measure. Under assumptions (A3) and (A4), the measure and are equivalent and the Randon-Nikodym derivative of with respect to is given by
along the sample path . Let
be the log-likelihood function. The maximum likelihood estimate (MLE) of is defined as
Now, we study the approximation of the MLE when stochastic is observed at the discrete-time points with such that as . Itô approximation of the stochastic integral and rectangular approximation of the ordinary integral in the log-likelihood (2.12) yields the approximate log-likelihood function:
The corresponding approximate maximum likelihood estimator (AMLE) is established as follow:
The main purpose of this paper is to study the rate of the convergence of the approximate log-likelihood functions and furthermore analyze the error bound in probability between the AMLE and the continuous MLE.
3. Main Results
Firstly, let us give the following lemmas which will be used in the proof of our main results.
Lemma 3.1. Under the assumptions (A1)–(A4), (A5)2, and (A6)1, one has
Proof. By Itô formula we can derive that for ,
For and , by assumption (A3), (A4), (A5)2, (A6)1, and Hölder’s inequality, one has
Similarly, we have
Hence, it follows inequality that
By assumptions (A3), (A4), (A5)2, and (A6)1, we obtain
Due to the orthogonality, Itô isomorphism, the Cauchy-Schwarz inequality, assumption (A3), (A4), and (A5)1, we get
Obviously, it follows from bounds for and that
The proof is now complete.
Next, we will go on to analyze the rate of convergence of the approximations for Itô integral whose result will be used in the following theorems.
Lemma 3.2. Under the assumptions (A1)–(A4), (A5)2, (A6)2, (A7), and (A8), one has
Proof. Let be the partition such that . Define and as
Let be a partition which is finer than , obtained by choosing the mid point from each of the interval , . Let be the points of subdivision of the refined partition . Define the approximating sum as before. We take two steps to prove the assertion in this lemma. Step 1. We will first obtain the bounds on . Let be three equally space points on and let us denote by and by . Define
Applying the Taylor expansion, one has
where . Relations (3.15) to (3.13) show that
Notice that ’s corresponding to different subintervals of -generated by form a martingale difference sequence. Observe that
where denotes . By Theorem 4 of , for any , there exists such that
Furthermore by (A2) and (A3), we have
Using the property that corresponding to different subintervals forms a martingale difference sequence, it follows that
for some constant . Step 2. We will show now the bounds on . Let be the sequence of partitions such that is a refinement of by choosing the midpoint of the subintervals generated by . Note that and . The analysis given above proves that
where is the approximation corresponding to and . Therefore, applying the Hölder inequality and the Minkovski inequality, one gets
for all . Let . Since the integral exists, converges in to as . Note that is a sequence of partitions such that the mesh of the partition tends to zero as for any fixed . Thus
The proof is now complete.
Theorem 3.3. Under assumptions (A1)–(A4), (A5)2, (A6)2, (A7), and (A8), one has
Proof. By the analysis given above, one has
Hence, it follows from Lemmas 3.1 and 3.2 that
Next, note that
Similarly, by Lemmas 3.1 and 3.2, we obtain
The proof is now complete.
Remark 3.4. The rate of convergence of the approximations for Itô and ordinary integral have been investigated in Lemmas 3.1 and 3.2. Based on these analysis results, the rate of convergence of the approximate log-likelihood function for nonlinear nonhomogenous stochastic system with unknown parameter has been established in Theorem 3.3. It should be pointed out that the corresponding approximate result gained in  is the special case for linear stochastic differential equation, furthermore, the conclusions in  also can be regarded as a special example under the result in Theorem 3.3 for nonlinear nonhomogenous stochastic system with constant diffusion.
Finally, we will study the error bound in probability between the AMLE and the continuous MLE for nonlinear nonhomogenous stochastic system with unknown parameter.
Theorem 3.5. Under assumption (A1)–(A4), (A5)2, (A6)2, (A7), and (A8), one has
Proof. We know and are the solutions of equations and , respectively.
Hence, one gets
As we know that , so there exists a constant such that
Therefore, applying Itô isomorphism, the Cauchy-Schwarz inequality, Lemmas 3.1 and 3.2, we obtain