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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 421754, 20 pages
http://dx.doi.org/10.1155/2012/421754
Research Article

Uniform Approximate Estimation for Nonlinear Nonhomogenous Stochastic System with Unknown Parameter

School of Information Science and Technology, Donghua University, Shanghai 200051, China

Received 21 June 2012; Accepted 3 August 2012

Academic Editor: Jun Hu

Copyright © 2012 Xiu Kan and Huisheng Shu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

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 [1]. 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, [25]. 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 [6]. Byes estimation (BE), which is a decision rule that minimizes the posterior expected value of a loss function, has been developed in [7]. 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 [8]. 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, [918] 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 [21] for the small noise asymptotic case. In [22], 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 [27] for linear stochastic differential equation. A numerical approximate likelihood method has been developed in [28] based on iterations of the Gaussian transition densities emanating from the Euler scheme. [29] 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. (1) The Itô type approximation for the stochastic integral is introduced to obtain an approximate log-likelihood function. (2) The rate of convergence of the approximation is investigated for Itô type integral. (3) The in probability rate of convergence of the approximate log-likelihood function is established for the nonlinear nonhomogenous stochastic system involving unknown parameter. (4) 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 𝑋𝑡, 𝑡0 on (Ω,,{𝑡}𝑡0,𝐏) satisfying the following stochastic differential equation: 𝑑𝑋𝑡=𝜃𝑓𝑡,𝑋𝑡𝑑𝑡+𝑔𝑡,𝑋𝑡𝑑𝑊𝑡,(2.1) where 𝑊𝑡, 𝑡0 is a standard Wiener process adapted to 𝑡, 𝑡0 such that for 0𝑠<𝑡, 𝑊𝑡𝑊𝑠 is independent of 𝑠, 𝜃Θ open in is the unknown parameter to be estimated. Let 𝜃0 be the true value of the parameter 𝜃.

Throughout this paper 𝐶 is a generic constant, we use following notations: 𝑓𝑥=𝜕𝑓𝜕𝑥,𝑓𝑡=𝜕𝑓𝜕𝑡,𝑓𝑥𝑥=𝜕2𝑓𝜕𝑥2,𝑓𝑡𝑡=𝜕2𝑓𝜕𝑡2,𝑓𝑡𝑥=𝜕2𝑓.𝜕𝑡𝜕𝑥(2.2)

We assume the following condition:(A1)𝑓(,) and 𝑔(,) are Lipschitz continuous in 𝑋𝑡 uniformly in 𝑡+, that is, there exists a constant 𝐾0 such that ||𝑓𝑡,𝑋1𝑓𝑡,𝑋2||2||𝑔𝑡,𝑋1𝑔𝑡,𝑋2||2||𝑋𝐾1𝑋2||2,(2.3)for any 𝑡+ and 𝑋1,𝑋2.(A2)𝑓(,) and 𝑔(,) satisfy linear growth condition, that is, there exists a constant 𝐾0 such that ||||𝑓(𝑡,𝑥)2||||𝑔(𝑡,𝑥)2𝐾1+|𝑥|2,(2.4)for any 𝑡+ and 𝑥.(A3)inf𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡>0.(2.5)(A4)𝑝0,sup𝑡,𝑋𝑡𝔼||𝑓𝑡,𝑋𝑡||𝑝<,sup𝑡,𝑋𝑡𝔼||𝑔𝑡,𝑋𝑡||𝑝<.(2.6)(A5)𝑗𝑓(,) and 𝑔(,) are continuously differentiable with respect to 𝑋𝑡 up to order 𝑗1 and sup0𝑡𝑇𝔼||𝑓𝑥𝑡,𝑋𝑡||8<,sup0𝑡𝑇𝔼||𝑔𝑥𝑡,𝑋𝑡||16<,sup0𝑡𝑇𝔼||𝑓𝑥𝑥𝑡,𝑋𝑡||8<,sup0𝑡𝑇𝔼||𝑔𝑥𝑥𝑡,𝑋𝑡||16<.(2.7)(A6)𝑘𝑓(,) and 𝑔(,) are continuously differentiable with respect to 𝑡 up to order 𝑘1 and sup0𝑡𝑇𝔼||𝑓𝑡𝑡,𝑋𝑡||4<,sup0𝑡𝑇𝔼||𝑔𝑡𝑡,𝑋𝑡||8<,sup0𝑡𝑇𝔼||𝑓𝑡𝑡𝑡,𝑋𝑡||4<,sup0𝑡𝑇𝔼||𝑔𝑡𝑡𝑡,𝑋𝑡||4<.(2.8)(A7)sup0𝑡𝑇𝔼||𝑓𝑡𝑥𝑡,𝑋𝑡||8<,sup0𝑡𝑇𝔼||𝑔𝑡𝑥𝑡,𝑋𝑡||8<.(2.9)(A8)𝔼||𝑋0||8<.(2.10)

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 [30].

Denote 𝑋𝑇0={𝑋𝑡,0𝑡𝑇}. Let 𝑃𝑇𝜃 be the measure generated on the space (𝐶𝑇,𝐵𝑇) of the continuous functions on [0,𝑇] with the associated Borel 𝜎-algebra 𝐵𝑇 generated under the supremum norm by the process 𝑋𝑇0 and 𝑃𝑇0 be the standard Wiener measure. Under assumptions (A3) and (A4), the measure 𝑃𝑇𝜃 and 𝑃𝑇0 are equivalent and the Randon-Nikodym derivative of 𝑃𝑇𝜃 with respect to 𝑃𝑇0 is given by 𝑑𝑃𝑇𝜃𝑑𝑃𝑇0𝜃=exp𝑇0𝑓𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡𝑑𝑋𝑡𝜃22𝑇0𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡𝑑𝑡,(2.11) along the sample path 𝑋𝑇0. Let 𝐿𝑇(𝜃)=log𝑑𝑃𝑇𝜃𝑑𝑃𝑇0=𝜃𝑇0𝑓𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡𝑑𝑋𝑡𝜃22𝑇0𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡𝑑𝑡(2.12) be the log-likelihood function. The maximum likelihood estimate (MLE) of 𝜃 is defined as 𝜃𝑇=argmax𝜃Θ𝐿𝑇(𝜃)𝑇0𝑓𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡𝑑𝑋𝑡𝑇0𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡𝑑𝑡1.(2.13)

Now, we study the approximation of the MLE 𝜃𝑇 when stochastic 𝑋𝑡 is observed at the discrete-time points 0=𝑡0<𝑡1<<𝑡𝑛=𝑇 with 𝑡𝑖𝑖,𝑖=0,1,2,,𝑛 such that 0 as 𝑛. 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: 𝐿𝑛,𝑇(𝜃)=𝜃𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑋𝑡𝑖𝑋𝑡𝑖1𝜃22𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1.(2.14)

The corresponding approximate maximum likelihood estimator (AMLE) is established as follow: 𝜃𝑛,𝑇=𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑋𝑡𝑖𝑋𝑡𝑖1𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖11.(2.15)

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 𝔼|||||𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑖1𝑔2𝑡𝑖1,𝑋𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2𝑇𝐶3𝑛2.(3.1)

Proof. By Itô formula we can derive that for 𝑡[0,𝑇], 𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡𝑓2𝑡𝑖1,𝑋𝑖1𝑔2𝑡𝑖1,𝑋𝑖1=𝑡𝑡𝑖12𝑓𝑢,𝑋𝑢𝑓𝑢𝑢,𝑋𝑢𝑔2𝑢,𝑋𝑢2𝑓2𝑢,𝑋𝑢𝑔𝑢,𝑋𝑢𝑔𝑢𝑢,𝑋𝑢𝑔4𝑢,𝑋𝑢+𝑑𝑢𝑡𝑡𝑖12𝑓𝑢,𝑋𝑢𝑓𝑥𝑢,𝑋𝑢𝑔2𝑢,𝑋𝑢2𝑓2𝑢,𝑋𝑢𝑔𝑢,𝑋𝑢𝑔𝑥𝑢,𝑋𝑢𝑔4𝑢,𝑋𝑢𝜃𝑓𝑢,𝑋𝑢+𝑑𝑢𝑡𝑡𝑖1𝑓2𝑥𝑢,𝑋𝑢𝑔2𝑢,𝑋𝑢+3𝑓2𝑢,𝑋𝑢𝑔2𝑥𝑢,𝑋𝑢4𝑓𝑢,𝑋𝑢𝑔𝑢,𝑋𝑢𝑓𝑥𝑢,𝑋𝑢𝑔𝑥𝑢,𝑋𝑢+𝑑𝑢𝑡𝑡𝑖1𝑓𝑢,𝑋𝑢𝑔2𝑢,𝑋𝑢𝑓𝑥𝑥𝑢,𝑋𝑢𝑓2𝑢,𝑋𝑢𝑔𝑢,𝑋𝑢𝑔𝑥𝑥𝑢,𝑋𝑢+𝑑𝑢𝑡𝑡𝑖12𝑓𝑢,𝑋𝑢𝑓𝑥𝑢,𝑋𝑢𝑔2𝑢,𝑋𝑢2𝑓2𝑢,𝑋𝑢𝑔𝑢,𝑋𝑢𝑔𝑥𝑢,𝑋𝑢𝑔4𝑢,𝑋𝑢𝑑𝑊𝑢𝑡𝑡𝑖1𝐹1𝑢,𝑋𝑢𝑑𝑢+𝑡𝑡𝑖1𝐹2𝑢,𝑋𝑢𝑑𝑊𝑢,(3.2) where 𝐹1𝑢,𝑋𝑢=𝐹11𝑢,𝑋𝑢+𝐹12𝑢,𝑋𝑢+𝐹13𝑢,𝑋𝑢+𝐹14𝑢,𝑋𝑢,𝐹11𝑢,𝑋𝑢=2𝑓𝑢,𝑋𝑢𝑓𝑢𝑢,𝑋𝑢𝑔2𝑢,𝑋𝑢2𝑓2𝑢,𝑋𝑢𝑔𝑢,𝑋𝑢𝑔𝑢𝑢,X𝑢𝑔4𝑢,𝑋𝑢,𝐹12𝑢,𝑋𝑢=2𝑓𝑢,𝑋𝑢𝑓𝑥𝑢,𝑋𝑢𝑔2𝑢,𝑋𝑢2𝑓2𝑢,𝑋𝑢𝑔𝑢,𝑋𝑢𝑔𝑥𝑢,𝑋𝑢𝑔4𝑢,𝑋𝑢𝜃𝑓𝑢,𝑋𝑢,𝐹13𝑢,𝑋𝑢=𝑓2𝑥𝑢,𝑋𝑢𝑔2𝑢,𝑋𝑢+3𝑓2𝑢,𝑋𝑢𝑔2𝑥𝑢,𝑋𝑢4𝑓𝑢,𝑋𝑢𝑔𝑢,𝑋𝑢𝑓𝑥𝑢,𝑋𝑢𝑔𝑥𝑢,𝑋𝑢,𝐹14𝑢,𝑋𝑢=𝑓𝑢,𝑋𝑢𝑔2𝑢,𝑋𝑢𝑓𝑥𝑥𝑢,𝑋𝑢𝑓2𝑢,𝑋𝑢𝑔𝑢,𝑋𝑢𝑔𝑥𝑥𝑢,𝑋𝑢.(3.3)
For 𝐹11(𝑢,𝑋𝑢) and 𝐹13(𝑢,𝑋𝑢), by assumption (A3), (A4), (A5)2, (A6)1, and Hölder’s inequality, one has 𝔼𝑡𝑡𝑖1𝐹11𝑢,𝑋𝑢2=𝔼𝑡𝑡𝑖12𝑓𝑢,𝑋𝑢𝑓𝑢𝑢,𝑋𝑢𝑔2𝑢,𝑋𝑢2𝑓2𝑢,𝑋𝑢𝑔𝑢,𝑋𝑢𝑔𝑢𝑢,𝑋𝑢𝑔4𝑢,𝑋𝑢𝑑𝑢2𝐶𝔼𝑡𝑡𝑖1𝑓2𝑢,𝑋𝑢𝑑𝑢+𝐶𝔼𝑡𝑡𝑖1𝑓2𝑢𝑢,𝑋𝑢𝑑𝑢+𝐶𝔼𝑡𝑡𝑖1𝑔2𝑢𝑢,𝑋𝑢𝔼𝑑𝑢𝐶,𝑡𝑡𝑖1𝐹13𝑢,𝑋𝑢2=𝔼𝑡𝑡𝑖1𝑓2𝑥𝑢,𝑋𝑢𝑔2𝑢,𝑋𝑢+3𝑓2𝑢,𝑋𝑢𝑔2𝑥𝑢,𝑋𝑢4𝑓𝑢,𝑋𝑢𝑔𝑢,𝑋𝑢𝑓𝑥𝑢,𝑋𝑢𝑔𝑥𝑢,𝑋𝑢𝑑𝑢2𝐶𝔼𝑡𝑡𝑖1𝑓4𝑢,𝑋𝑢𝑑𝑢+𝐶𝔼𝑡𝑡𝑖1𝑔4𝑢,𝑋𝑢𝑑𝑢+𝐶𝔼𝑡𝑡i1𝑓4𝑥𝑢,𝑋𝑢𝑑𝑢+𝐶𝔼𝑡𝑡𝑖1𝑔4𝑥𝑢,𝑋𝑢𝑑𝑢𝐶.(3.4)
Similarly, we have 𝔼𝑡𝑡𝑖1𝐹12𝑢,𝑋𝑢2𝐶,𝔼𝑡𝑡𝑖1𝐹14𝑢,𝑋𝑢2𝐶.(3.5)
This means 𝔼𝑡𝑡𝑖1𝐹1𝑢,𝑋𝑢2𝐶.(3.6)
Hence, it follows 𝐶𝑟 inequality that 𝔼|||||𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑖1𝑔2𝑡𝑖1,𝑋𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2|||||=𝔼𝑛𝑖=1𝑡𝑖𝑡𝑖1𝑓2𝑡𝑖1,𝑋𝑖1𝑔2𝑡𝑖1,𝑋𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2|||||=𝔼𝑛𝑖=1𝑡𝑖𝑡𝑖1𝑡𝑡𝑖1𝐹1𝑢,𝑋𝑢𝑑𝑢+𝑡𝑡𝑖1𝐹2𝑢,𝑋𝑢𝑑𝑊𝑢|||||𝑑𝑡2|||||2𝔼𝑛𝑖=1𝑡𝑖𝑡𝑖1𝑡𝑡𝑖1𝐹1𝑢,𝑋𝑢|||||𝑑𝑢𝑑𝑡2|||||+2𝔼𝑛𝑖=1𝑡𝑖𝑡𝑖1𝑡𝑡𝑖1𝐹2𝑢,𝑋𝑢𝑑𝑊𝑢|||||𝑑𝑡22𝐺1+2𝐺2.(3.7)
By assumptions (A3), (A4), (A5)2, and (A6)1, we obtain 𝐺1|||||=𝔼𝑛𝑖=1𝑡𝑖𝑡𝑖1𝑡𝑡𝑖1𝐹1𝑢,𝑋𝑢|||||𝑑𝑢𝑑𝑡2𝔼𝑛𝑖=1𝑡𝑖𝑡𝑖1𝑡𝑡𝑖1𝐹1𝑢,𝑋𝑢𝑑𝑢𝑑𝑡2+𝔼𝑛1=𝑖𝑗𝑡𝑖𝑡𝑖1𝑡𝑡𝑖1𝐹1𝑢,𝑋𝑢𝑑𝑢𝑑𝑡𝑡𝑗𝑡𝑗1𝑡𝑡𝑗1𝐹1𝑢,𝑋𝑢𝑑𝑢𝑑𝑡𝔼𝑛𝑖=1𝑡𝑖𝑡𝑖1𝑡𝑖𝑡𝑖1||||𝑡𝑡𝑖1𝐹1𝑢,𝑋𝑢||||𝑑𝑢𝑑𝑡2+𝑑𝑡𝑛1=𝑖𝑗𝔼𝑡𝑖𝑡𝑖1𝑡𝑡𝑖1𝐹1𝑢,𝑋𝑢𝑑𝑢𝑑𝑡2𝔼𝑡𝑗𝑡𝑗1𝑡𝑡𝑗1𝐹1𝑢,𝑋𝑢𝑑𝑢𝑑𝑡21/2𝑛𝑖=1𝑡𝑖𝑡𝑖1𝑡𝑖𝑡𝑖1𝔼||||𝑡𝑡𝑖1𝐹1𝑢,𝑋𝑢||||𝑑𝑢𝑑𝑡2+𝑑𝑡𝑛1=𝑖𝑗𝑡𝑖𝑡𝑖1𝑡𝑖𝑡𝑖1𝔼𝑡𝑡𝑖1𝐹1𝑢,𝑋𝑢𝑑𝑢2𝑡𝑑𝑡𝑗𝑡𝑗1𝑡𝑗𝑡𝑗1𝔼𝑡𝑡𝑗1𝐹1𝑢,𝑋𝑢𝑑𝑢2𝑑𝑡1/2𝐶𝑛𝑖=1𝑡𝑖𝑡𝑖13+𝐶𝑛1=𝑖𝑗𝑡𝑖𝑡𝑖13𝑡𝑗𝑡𝑗131/2𝑇𝐶3𝑛2.(3.8)
Due to the orthogonality, Itô isomorphism, the Cauchy-Schwarz inequality, assumption (A3), (A4), and (A5)1, we get 𝐺2|||||=𝔼𝑛𝑖=1𝑡𝑖𝑡𝑖1𝑡𝑡𝑖1𝐹2𝑢,𝑋𝑢𝑑𝑊𝑢|||||𝑑𝑡2𝔼𝑛𝑖=1𝑡𝑖𝑡𝑖1𝑡𝑡𝑖1𝐹2𝑢,𝑋𝑢𝑑𝑊𝑢𝑑𝑡2+𝑛1=𝑖𝑗𝔼𝑡𝑖𝑡𝑖1𝑡𝑡𝑖1𝐹2𝑢,𝑋𝑢𝑑𝑊𝑢𝑑𝑡𝑡𝑗𝑡𝑗1𝑡𝑡𝑗1𝐹2𝑢,𝑋𝑢𝑑𝑊𝑢𝑑𝑡𝑛𝑖=1𝑡𝑖𝑡𝑖1𝑡𝑖𝑡𝑖1𝔼||||𝑡𝑡𝑖1𝐹2𝑢,𝑋𝑢𝑑𝑊𝑢||||𝑑𝑡2𝑑𝑡𝑛𝑖=1𝑡𝑖𝑡𝑖1𝑡𝑖𝑡𝑖1𝑡𝑡𝑖1𝔼||𝐹2𝑢,𝑋𝑢||2𝑑𝑢𝑑𝑡𝐶𝑛𝑖=1𝑡𝑖𝑡𝑖1𝑡𝑖𝑡𝑖1𝑡𝑡𝑖1𝑑𝑡𝐶𝑛𝑖=1𝑡𝑖𝑡𝑖13𝑇𝐶3𝑛2.(3.9)
Obviously, it follows from bounds for 𝐺1 and 𝐺2 that 𝔼|||||𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑖1𝑔2𝑡𝑖1,𝑋𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2𝑇𝐶3𝑛2.(3.10)
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 𝔼|||||𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑔𝑡𝑖1,𝑋𝑡𝑖1𝑊𝑡𝑖𝑊𝑡𝑖1𝑇0𝑓𝑡,𝑋𝑡𝑔𝑡,𝑋𝑡𝑑𝑊𝑡|||||2𝑇𝐶3𝑛2.(3.11)

Proof. Let 𝜋𝑛 be the partition 𝜋𝑛=0=𝑡0<𝑡1<<𝑡𝑛=𝑇,𝑡𝑖=𝑖,𝑖=0,1,,𝑛 such that 0. Define 𝑆 and 𝑆𝑛 as 𝑆=𝑇0𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡𝑑𝑊𝑡,𝑆𝑛=𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑔𝑡𝑖1,𝑋𝑡𝑖1𝑊𝑡𝑖𝑊𝑡𝑖1.(3.12)
Let 𝜋𝑛 be a partition which is finer than 𝜋𝑛, obtained by choosing the mid point ̂𝑡𝑖1 from each of the interval 𝑡𝑖1<̂𝑡𝑖1<𝑡𝑖, 𝑖=0,1,,𝑛. Let 0=𝑡0<𝑡1<<𝑡2𝑛=𝑇 be the points of subdivision of the refined partition 𝜋𝑛. Define the approximating sum 𝑆𝜋𝑛 as before. We take two steps to prove the assertion in this lemma.
Step 1. We will first obtain the bounds on 𝐸|𝑆𝜋𝑛𝑆𝜋𝑛|2.
Let ̃𝑡00<̃𝑡1<̃𝑡2𝑇 be three equally space points on [0,𝑇] and let us denote 𝑋̃𝑡𝑖 by 𝑋𝑖 and 𝑊̃𝑡𝑖 by 𝑊𝑖,𝑖=0,1,,𝑛. Define 𝑓̃𝑡𝐻=0,𝑋0𝑔̃𝑡0,𝑋0𝑊2𝑊0𝑓̃𝑡1,𝑋1𝑔̃𝑡1,𝑋1𝑊2𝑊1+𝑓̃𝑡0,𝑋0𝑔̃𝑡0,𝑋0𝑊1𝑊0=𝑊2𝑊1𝑓̃𝑡0,𝑋0𝑔̃𝑡0,𝑋0𝑓̃𝑡1,𝑋1𝑔̃𝑡1,𝑋1.(3.13)
Denote ̃𝑡𝐼=1̃𝑡0𝑓𝑡,𝑋𝑡𝑔𝑡,𝑋𝑡𝑑𝑡.(3.14)
Applying the Taylor expansion, one has 𝑓̃𝑡0,𝑋0𝑔̃𝑡0,𝑋0𝑓̃𝑡1,𝑋1𝑔̃𝑡1,𝑋1=𝑋0𝑋1𝑓𝑥𝑔𝑔𝑥𝑓𝑔2̃𝑡1,𝑋1+̃𝑡0̃𝑡1𝑓𝑡𝑔𝑔𝑡𝑓𝑔2̃𝑡1,𝑋1+12𝑋0𝑋12𝑓𝑥𝑥𝑔2𝑔𝑥𝑥𝑓𝑓𝑔2𝑥𝑔𝑔𝑥𝑓𝑔𝑔𝑥𝑔4𝑡,𝑋+12̃𝑡0̃𝑡12𝑓𝑡𝑡𝑔2𝑔𝑡𝑡𝑓𝑓𝑔2𝑡𝑔𝑔𝑡𝑓𝑔𝑔𝑡𝑔4𝑡,𝑋+̃𝑡0̃𝑡1𝑋0𝑋1𝑓𝑡𝑥𝑔𝑓𝑔𝑡𝑥𝑓𝑥𝑔𝑡𝑓𝑡𝑔𝑥𝑔2+2𝑓𝑔𝑔𝑡𝑔𝑥𝑔4𝑡,𝑋𝑊=1𝑊0𝑓+𝐼𝑥𝑔𝑔𝑥𝑓𝑔2̃𝑡1,𝑋1+̃𝑡0̃𝑡1𝑓𝑡𝑔𝑔𝑡𝑓𝑔2̃𝑡1,𝑋1+12𝑋0𝑋12𝑓𝑥𝑥𝑔2𝑔𝑥𝑥𝑓𝑓𝑔2𝑥𝑔𝑔𝑥𝑓𝑔𝑔𝑥𝑔4𝑡,𝑋+12̃𝑡0̃𝑡12𝑓𝑡𝑡𝑔2𝑔𝑡𝑡𝑓𝑓𝑔2𝑡𝑔𝑔𝑡𝑓𝑔𝑔𝑡𝑔4𝑡,𝑋+̃𝑡0̃𝑡1𝑋0𝑋1𝑓𝑡𝑥𝑔𝑓𝑔𝑡𝑥𝑓𝑥𝑔𝑡𝑓𝑡𝑔𝑥𝑔2+2𝑓𝑔𝑔𝑡𝑔𝑥𝑔4𝑡,𝑋,(3.15) where |𝑋1𝑋|<|𝑋0𝑋1̃𝑡|,|1𝑡̃𝑡|<|0̃𝑡1|.
Relations (3.15) to (3.13) show that 𝑊𝐻=2𝑊1𝐼𝑓𝑥𝑔𝑔𝑥𝑓𝑔2̃𝑡1,𝑋1+𝑊2𝑊1̃𝑡0̃𝑡1𝑓𝑡𝑔𝑔𝑡𝑓𝑔2̃𝑡1,𝑋1+𝑊2𝑊112𝑋0𝑋12𝑓𝑥𝑥𝑔2𝑔𝑥𝑥𝑓𝑓𝑔2𝑥𝑔𝑔𝑥𝑓𝑔𝑔𝑥𝑔4𝑡,𝑋+𝑊2𝑊112̃𝑡0̃𝑡12𝑓𝑡𝑡𝑔2𝑔𝑡𝑡𝑓𝑓𝑔2𝑡𝑔𝑔𝑡𝑓𝑔𝑔𝑡𝑔4𝑡,𝑋+𝑊2𝑊1̃𝑡0̃𝑡1𝑋0𝑋1f𝑡𝑥𝑔𝑓𝑔𝑡𝑥𝑓𝑥𝑔𝑡𝑓𝑡𝑔𝑥𝑔2+2𝑓𝑔𝑔𝑡𝑔𝑥𝑔4𝑡,𝑋.(3.16)
Notice that 𝐻’s corresponding to different subintervals of [0,𝑇]-generated by 𝜋𝑛 form a martingale difference sequence. Observe that 𝔼||𝐻||2𝑊=𝔼2𝑊12𝔼𝑓𝐼𝑥𝑔𝑔𝑥𝑓𝑔2̃𝑡1,𝑋1+̃𝑡0̃𝑡1𝑓𝑡𝑔𝑔𝑡𝑓𝑔2̃𝑡1,𝑋1+12𝑋0𝑋12𝑓𝑥𝑥𝑔2𝑔𝑥𝑥𝑓𝑓𝑔2𝑥𝑔𝑔𝑥𝑓𝑔𝑔𝑥𝑔4𝑡,𝑋+12̃𝑡0̃𝑡12𝑓𝑡𝑡𝑔2𝑔𝑡𝑡𝑓𝑓𝑔2𝑡𝑔𝑔𝑡𝑓𝑔𝑔𝑡𝑔4𝑡,𝑋+̃𝑡0̃𝑡1𝑋0𝑋1(𝒮)𝑔2+2𝑓𝑔𝑔𝑡𝑔𝑥𝑔4𝑡,𝑋2̃𝑡42̃𝑡1𝔼𝐼𝑓𝑥𝑔𝑔𝑥𝑓𝑔2̃𝑡1,𝑋12+̃𝑡0̃𝑡12𝔼𝑓𝑡𝑔𝑔𝑡𝑓𝑔2̃𝑡1,𝑋12+14𝔼𝑋0𝑋14𝑓𝑥𝑥𝑔2𝑔𝑥𝑥𝑓𝑓𝑔2𝑥𝑔𝑔𝑥𝑓𝑔𝑔𝑥𝑔4𝑡,𝑋2+14̃𝑡0̃𝑡14𝔼𝑓𝑡𝑡𝑔2𝑔𝑡𝑡𝑓𝑓𝑔2𝑡𝑔𝑔𝑡𝑓𝑔𝑔𝑡𝑔4𝑡,𝑋2+̃𝑡0̃𝑡12𝔼𝑋0𝑋12(𝒮)𝑔2+2𝑓𝑔𝑔𝑡𝑔𝑥𝑔4𝑡,𝑋2̃𝑡42̃𝑡1𝔼𝐼4𝔼𝑓𝑥𝑔𝑔𝑥𝑓𝑔2̃𝑡1,𝑋141/2+̃𝑡0̃𝑡12𝔼𝑓𝑡𝑔𝑔𝑡𝑓𝑔2̃𝑡1,𝑋12+14𝔼𝑋0𝑋18𝔼𝑓𝑥𝑥𝑔2𝑔𝑥𝑥𝑓𝑓𝑔2𝑥𝑔𝑔𝑥𝑓𝑔𝑔𝑥𝑔4𝑡,𝑋81/2+14̃𝑡0̃𝑡14𝔼𝑓𝑡𝑡𝑔2𝑔𝑡𝑡𝑓𝑓𝑔2𝑡𝑔𝑔𝑡𝑓𝑔𝑔𝑡𝑔4𝑡,𝑋2+̃𝑡0̃𝑡12𝔼𝑋0𝑋14𝔼(𝒮)𝑔2+2𝑓𝑔𝑔𝑡𝑔𝑥𝑔4𝑡,𝑋41/2̃𝑡42̃𝑡1𝔼𝐼41/2𝐶𝔼𝑓8𝑥̃𝑡1,𝑋11/2+𝔼𝑔8𝑥̃𝑡1,𝑋11/2+𝔼𝑓8̃𝑡1,𝑋11/21/2+̃𝑡0̃𝑡12𝐶𝔼𝑓4𝑡̃𝑡1,𝑋11/2+𝔼𝑔4𝑡̃𝑡1,𝑋11/2+𝔼𝑓4̃𝑡1,𝑋11/2+14𝔼𝑋0𝑋181/2𝐶𝔼𝑓8𝑥𝑥𝑡,𝑋1/2+𝔼𝑔8𝑥𝑥𝑡,𝑋1/2+𝔼𝑓8𝑡,𝑋1/2+𝔼𝑓8𝑥𝑡,𝑋1/2+𝔼𝑓𝑥16𝑡,𝑋1/21/2+14̃𝑡0̃𝑡14𝐶𝔼𝑓4𝑡𝑡𝑡,𝑋1/2+𝔼𝑔4𝑡𝑡𝑡,𝑋1/2+𝔼𝑓4𝑡𝑡,𝑋1/2+𝔼𝑔4𝑡𝑡,𝑋1/2+𝔼𝑔8𝑡𝑡,𝑋1/2+̃𝑡0̃𝑡12𝔼𝑋0𝑋141/2𝔼𝑓8𝑡𝑥𝑡,𝑋1/2+𝔼𝑔8𝑡𝑥𝑡,𝑋1/2+𝔼𝑓8𝑥̃𝑡1,𝑋11/2+𝔼𝑔8𝑥̃𝑡1,𝑋11/2+𝔼𝑓8𝑡̃𝑡1,𝑋11/2+𝔼𝑔8𝑡̃𝑡1,𝑋11/21/2.(3.17)where 𝒮 denotes 𝑓𝑡𝑥𝑔𝑓𝑔𝑡𝑥𝑓𝑥𝑔𝑡𝑓𝑡𝑔𝑥.
By Theorem 4 of [31], for any 0𝑠<𝑡𝑇, there exists 𝐶>0 such that 𝔼𝑋𝑡𝑋𝑠2𝑚𝐶𝔼𝑋02𝑚+1(𝑡𝑠)𝑚,𝑚1.(3.18)
Hence 𝔼𝑋𝑡𝑋𝑠8𝐶𝔼𝑋80+1(𝑡𝑠)4,𝔼𝑋𝑡𝑋𝑠4𝐶𝔼𝑋40+1(𝑡𝑠)2.(3.19)
Furthermore by (A2) and (A3), we have 𝔼𝐼4̃𝑡=𝔼1̃𝑡0𝑓𝑡,𝑋𝑡𝑔𝑡,𝑋𝑡𝑑𝑡4̃𝑡𝐶𝔼1̃𝑡0𝑓4𝑡,𝑋𝑡̃𝑡𝑑𝑡𝐶𝔼1̃𝑡0||𝑋1+𝑡||22̃𝑡𝑑𝑡𝐶1̃𝑡04sup0𝑡𝑇𝔼||𝑋1+𝑡||22̃𝑡𝐶1̃𝑡04.(3.20)
Thus 𝔼(𝐻)2̃𝑡𝐶2̃𝑡1̃𝑡1̃𝑡02.(3.21)
Using the property that 𝐻 corresponding to different subintervals forms a martingale difference sequence, it follows that 𝔼||𝑆𝜋𝑛𝑆𝜋𝑛||2𝑇𝐶3𝑛2,(3.22) for some constant 𝐶>0.
Step 2. We will show now the bounds on 𝐸|𝑆𝜋𝑛𝑆|2.
Let 𝜋𝑛(𝑝),𝑝0 be the sequence of partitions such that 𝜋𝑛(𝑖+1) is a refinement of 𝜋𝑛(𝑛) by choosing the midpoint of the subintervals generated by 𝜋𝑛(𝑛). Note that 𝜋𝑛(0)=𝜋𝑛 and 𝜋𝑛(1)=𝜋𝑛. The analysis given above proves that 𝔼||𝑆𝜋𝑛(𝑝)𝑆𝜋𝑛(𝑝+1)||2𝑇𝐶32𝑝𝑛2,𝑝0,(3.23) where 𝑆𝜋𝑛(𝑝) is the approximation corresponding to 𝜋𝑛(𝑝) and 𝑆𝜋𝑛(0)=𝑆𝜋𝑛.
Therefore, applying the Hölder inequality and the Minkovski inequality, one gets 𝔼||𝑆𝜋𝑛(0)𝑆𝜋𝑛(𝑝+1)||2𝔼𝑝𝑘=0𝑆𝜋𝑛(𝑘)𝑆𝜋𝑛(𝑘+1)2𝑝𝑘=0𝔼||𝑆𝜋𝑛(𝑘)𝑆𝜋𝑛(𝑘+1)||21/22𝑝𝑘=0𝐶𝑇32𝑝𝑛21/22𝑇𝐶3𝑛2,(3.24) for all 𝑝0. Let 𝑝. Since the integral 𝑆 exists, 𝑆𝜋𝑛(𝑝+1) converges in 2 to 𝑆 as 𝑝. Note that 𝜋𝑛(𝑝+1),𝑃0 is a sequence of partitions such that the mesh of the partition tends to zero as 𝑝 for any fixed 𝑛.
Thus 𝔼||𝑆𝜋𝑛||𝑆2𝑇𝐶32𝑝𝑛2,𝑝0,(3.25) where 𝑆=lim𝑛𝑆𝜋𝑛=𝑇0𝑓𝑡,𝑋𝑡𝑔𝑡,𝑋𝑡𝑑𝑊𝑡.(3.26)
The proof is now complete.

Theorem 3.3. Under assumptions (A1)–(A4), (A5)2, (A6)2, (A7), and (A8), one has 𝔼||𝐿𝑛,𝑇(𝜃)𝐿𝑇||(𝜃)2𝑇𝐶3𝑛2,𝔼||𝐿𝑛,𝑇(𝜃)𝐿𝑇||(𝜃)2𝑇𝐶3𝑛2.(3.27)

Proof. By the analysis given above, one has ||𝐿𝑛,𝑇(𝜃)𝐿𝑇||(𝜃)2=|||||𝜃𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑋𝑡𝑖𝑋𝑡𝑖1𝑇0𝑓𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡𝑑𝑋𝑡𝜃22𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2𝜃42|||||𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2+2𝜃2|||||𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑊𝑡𝑖𝑊𝑡𝑖1𝑇0𝑓𝑡,𝑋𝑡g2𝑡,𝑋𝑡𝑑𝑊𝑡|||||2.(3.28)
Hence, it follows from Lemmas 3.1 and 3.2 that 𝔼||𝐿𝑛,𝑇(𝜃)𝐿𝑇||(𝜃)2𝜃42𝔼|||||𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2+2𝜃2𝔼|||||𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑊𝑡𝑖𝑊𝑡𝑖1𝑇0𝑓𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡𝑑𝑊𝑡|||||2𝑇𝐶3𝑛2.(3.29)
Next, note that ||𝐿𝑛,𝑇(𝜃)𝐿𝑇(||𝜃)2=|||||𝜃𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑋𝑡𝑖𝑋𝑡𝑖1𝜃𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡𝑑𝑋𝑡𝑇0𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2=(1𝜃)2|||||𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓2𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2+|||||𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑊𝑡𝑖𝑊𝑡𝑖1𝑇0𝑓𝑡,𝑋𝑡𝑔2𝑡,𝑋𝑡𝑑𝑊𝑡|||||2.(3.30)
Similarly, by Lemmas 3.1 and 3.2, we obtain 𝔼||𝐿𝑛,𝑇(𝜃)𝐿𝑇||(𝜃)2𝑇𝐶3𝑛2.(3.31)
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 [27] is the special case for linear stochastic differential equation, furthermore, the conclusions in [9] 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 𝔼||𝜃𝑛,𝑇𝜃𝑇||2𝑇𝐶3𝑛2.(3.32)

Proof. We know 𝜃𝑛,𝑇 and 𝜃𝑇 are the solutions of equations 𝐿𝑛,𝑇(𝜃)=0 and 𝐿𝑇(𝜃)=0, respectively.
Hence, one gets ||𝜃𝑛,𝑇𝜃𝑇||2=|||||𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑋𝑡𝑖𝑋𝑡𝑖1𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓𝑡,𝑋𝑡/𝑔2𝑡,𝑋𝑡𝑑𝑋𝑡𝑇0𝑓2𝑡,𝑋𝑡/𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2=|||||𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑡/𝑔𝑖1,𝑋𝑡𝑖1𝑊𝑡𝑖𝑊𝑡𝑖1𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓𝑡,𝑋𝑡/𝑔𝑡,𝑋𝑡𝑑𝑊𝑡𝑇0𝑓2𝑡,𝑋𝑡/𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2=|||||𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑡/𝑔𝑖1,𝑋𝑡𝑖1𝑊𝑡𝑖𝑊𝑡𝑖1𝑇0𝑓𝑡,𝑋𝑡/𝑔𝑡,𝑋𝑡𝑑𝑊𝑡𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓𝑡,𝑋𝑡/𝑔𝑡,𝑋𝑡𝑑𝑊𝑡𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝒜𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓2𝑡,𝑋𝑡/𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2|||||2𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑡/𝑔𝑖1,𝑋𝑡𝑖1𝑊𝑡𝑖𝑊𝑡𝑖1𝑇0𝑓𝑡,𝑋𝑡/𝑔𝑡,𝑋𝑡𝑑W𝑡𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1|||||2|||||+2𝑇0𝑓𝑡,𝑋𝑡/𝑔𝑡,𝑋𝑡𝑑𝑊𝑡𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝒜𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡𝑖1,𝑋𝑡i1𝑡𝑖𝑡𝑖1𝑇0𝑓2𝑡,𝑋𝑡/𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2.(3.33)
As we know that 𝑛𝑖=1(𝑓2(𝑡𝑖1,𝑋𝑡𝑖1)/𝑔2(𝑡𝑖1,𝑋𝑡𝑖1))(𝑡𝑖𝑡𝑖1)>0, so there exists a constant 𝐶>0 such that 1𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝐶.(3.34)
Therefore, applying Itô isomorphism, the Cauchy-Schwarz inequality, Lemmas 3.1 and 3.2, we obtain 𝔼||𝜃𝑛,𝑇𝜃𝑇||2|||||2𝔼𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑡/𝑔𝑖1,𝑋𝑡𝑖1𝑊𝑡𝑖𝑊𝑡𝑖1𝑇0𝑓𝑡,𝑋𝑡/𝑔𝑡,𝑋𝑡𝑑𝑊𝑡𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1|||||2|||||+2𝔼𝑇0𝑓𝑡,𝑋𝑡/𝑔𝑡,𝑋𝑡𝑑𝑊𝑡𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝒜𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡i1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓2𝑡,𝑋𝑡/𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2|||||𝐶𝔼𝑛𝑖=1𝑓𝑡𝑖1,𝑋𝑡𝑖1𝑡/𝑔𝑖1,𝑋𝑡𝑖1𝑊𝑡𝑖𝑊𝑡𝑖1𝑇0𝑓𝑡,𝑋𝑡/𝑔𝑡,𝑋𝑡𝑑𝑊𝑡|||||2|||||+𝐶𝔼𝑛𝑖=1𝑓2𝑡𝑖1,𝑋𝑡𝑖1/𝑔2𝑡𝑖1,𝑋𝑡𝑖1𝑡𝑖𝑡𝑖1𝑇0𝑓2𝑡,𝑋𝑡/𝑔2𝑡,𝑋𝑡|||||𝑑𝑡2𝑇𝐶3𝑛2,(3.35)where 𝒜 denotes 𝑇0(𝑓2(𝑡,𝑋𝑡)/𝑔2(𝑡,𝑋𝑡))𝑑𝑡.
The proof is now complete.

Remark 3.6. Up to present, the rate of the convergence of the approximate log-likelihood functions and the error bound in probability between the AMLE and the continuous MLE have been obtained for the nonlinear nonhomogenous stochastic system with unknown parameter. As well, the corresponding results gained in [9, 27] are the direct conclusions after applying Chebyshev’s inequality on (3.32).

4. Conclusions

In this paper, we have investigated the error bound in probability between the ALME and the continuous MLE for a class of general nonlinear nonhomogenous stochastic system with unknown parameter. The rates of convergence of the approximations for Itô and ordinary integral have been derived under some regular assumptions. On the basis of these analysis results, we have studied the in probability rate of convergence of the approximate log-likelihood function to the true continuous log-likelihood function 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 has been established. It should be noted that one of the future research topics would be to investigate the asymptotic normality of the ALME for the nonlinear nonhomogenous stochastic system with unknown parameter mentioned in this paper.

Acknowledgment

This work was supported in part by the National Natural Science Foundation of China under Grant 60974030.

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