Uniform Approximate Estimation for Nonlinear Nonhomogenous Stochastic System with Unknown Parameter

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, [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 [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, [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 [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 𝑋_𝑡, 𝑡≥0 on (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝐏) satisfying the following stochastic differential equation: 𝑑𝑋_𝑡=𝜃𝑓𝑡,𝑋_𝑡𝑑𝑡+𝑔𝑡,𝑋_𝑡𝑑𝑊_𝑡,(2.1) where 𝑊_𝑡, 𝑡≥0 is a standard Wiener process adapted to ℱ_𝑡, 𝑡≥0 such that for 0≤𝑠<𝑡, 𝑊_𝑡−𝑊_𝑠 is independent of ℱ_𝑠, 𝜃∈Θ open in ℝ 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: 𝑓_𝑥=𝜕𝑓𝜕𝑥,𝑓^𝑡=𝜕𝑓𝜕𝑡,𝑓^𝑥𝑥=𝜕²𝑓𝜕𝑥²,𝑓_𝑡𝑡=𝜕²𝑓𝜕𝑡²,𝑓_𝑡𝑥=𝜕²𝑓𝜕𝑡𝜕𝑥.(2.2)

We assume the following condition:(A1)𝑓(⋅,⋅) and 𝑔(⋅,⋅) are Lipschitz continuous in 𝑋_𝑡∈ℝ uniformly in 𝑡∈ℝ₊, that is, there exists a constant 𝐾≥0 such that ||𝑓𝑡,𝑋₁−𝑓𝑡,𝑋₂||²∨||𝑔𝑡,𝑋₁−𝑔𝑡,𝑋₂||²≤𝐾||𝑋₁−𝑋₂||²,(2.3)for any 𝑡∈ℝ₊ and 𝑋₁,𝑋₂∈ℝ.(A2)𝑓(⋅,⋅) and 𝑔(⋅,⋅) satisfy linear growth condition, that is, there exists a constant 𝐾≥0 such that ||𝑓(𝑡,𝑥)||²∨||𝑔(𝑡,𝑥)||²≤𝐾1+|𝑥|²,(2.4)for any 𝑡∈ℝ₊ and 𝑥∈ℝ.(A3)inf_{𝑡,𝑋𝑡}𝑔²𝑡,𝑋_𝑡>0.(2.5)(A4)∀𝑝≥0,sup_{𝑡,𝑋𝑡}𝔼||𝑓𝑡,𝑋_𝑡||^𝑝<∞,sup_{𝑡,𝑋𝑡}𝔼||𝑔𝑡,𝑋_𝑡||^𝑝<∞.(2.6)(A5)_𝑗𝑓(⋅,⋅) and 𝑔(⋅,⋅) are continuously differentiable with respect to 𝑋_𝑡 up to order 𝑗≥1 and ssuupp_{0≤𝑡≤𝑇0≤𝑡≤𝑇}𝔼||𝑓_𝑥𝑡,𝑋_𝑡||⁸<∞,sup_{0≤𝑡≤𝑇}𝔼||𝑔_𝑥𝑡,𝑋_𝑡||¹⁶<∞,𝔼||𝑓_𝑥𝑥𝑡,𝑋_𝑡||⁸<∞,sup_{0≤𝑡≤𝑇}𝔼||𝑔_𝑥𝑥𝑡,𝑋_𝑡||¹⁶<∞.(2.7)(A6)_𝑘𝑓(⋅,⋅) and 𝑔(⋅,⋅) are continuously differentiable with respect to 𝑡 up to order 𝑘≥1 and ssuupp_{0≤𝑡≤𝑇0≤𝑡≤𝑇}𝔼||𝑓_𝑡𝑡,𝑋_𝑡||⁴<∞,sup_{0≤𝑡≤𝑇}𝔼||𝑔_𝑡𝑡,𝑋_𝑡||⁸<∞,𝔼||𝑓_𝑡𝑡𝑡,𝑋_𝑡||⁴<∞,sup_{0≤𝑡≤𝑇}𝔼||𝑔_𝑡𝑡𝑡,𝑋_𝑡||⁴<∞.(2.8)(A7)sup_{0≤𝑡≤𝑇}𝔼||𝑓_𝑡𝑥𝑡,𝑋_𝑡||⁸<∞,sup_{0≤𝑡≤𝑇}𝔼||𝑔_𝑡𝑥𝑡,𝑋_𝑡||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 𝑋^𝑇0={𝑋_𝑡,0≤𝑡≤𝑇}. Let 𝑃^𝑇𝜃 be the measure generated on the space (𝐶_𝑇,𝐵_𝑇) of the continuous functions on [0,𝑇] with the associated Borel 𝜎-algebra 𝐵_𝑇 generated under the supremum norm by the process 𝑋^𝑇0 and 𝑃^𝑇0 be the standard Wiener measure. Under assumptions (A3) and (A4), the measure 𝑃^𝑇𝜃 and 𝑃^𝑇0 are equivalent and the Randon-Nikodym derivative of 𝑃^𝑇𝜃 with respect to 𝑃^𝑇0 is given by 𝑑𝑃^𝑇𝜃𝑑𝑃^𝑇0=exp𝜃^𝑇0𝑓𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑋^𝑡−𝜃²2^𝑇0𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑡,(2.11) along the sample path 𝑋^𝑇0. Let 𝐿_𝑇(𝜃)=log𝑑𝑃^𝑇𝜃𝑑𝑃^𝑇0=𝜃^𝑇0𝑓𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑋^𝑡−𝜃²2^𝑇0𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑡(2.12) be the log-likelihood function. The maximum likelihood estimate (MLE) of 𝜃 is defined as 𝜃_𝑇=argmax_𝜃∈Θ𝐿_𝑇(𝜃)^𝑇0𝑓𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑋^𝑡^𝑇0𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑡⁻¹.(2.13)

Now, we study the approximation of the MLE 𝜃_𝑇 when stochastic 𝑋_𝑡 is observed at the discrete-time points 0=𝑡₀<𝑡₁<⋯<𝑡_𝑛=𝑇 with 𝑡_𝑖−𝑖ℎ,𝑖=0,1,2,…,𝑛 such that ℎ→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: 𝐿_𝑛,𝑇(𝜃)=𝜃^𝑛_𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑋^𝑡𝑖−𝑋_𝑡𝑖−1−𝜃²2^𝑛_𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡^𝑖−𝑡_𝑖−1.(2.14)

The corresponding approximate maximum likelihood estimator (AMLE) is established as follow: 𝜃_𝑛,𝑇=^𝑛_𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑋^𝑡𝑖−𝑋_𝑡𝑖−1^𝑛_𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡^𝑖−𝑡_𝑖−1⁻¹.(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|||||^𝑛_𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑖−1𝑡^𝑖−𝑡_𝑖−1−^𝑇0𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²≤𝐶𝑇³𝑛².(3.1)

Proof. By Itô formula we can derive that for 𝑡∈[0,𝑇], 𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡−𝑓²𝑡_𝑖−1,𝑋_𝑖−1=_{𝑡𝑡𝑖−1}𝑔²𝑡_𝑖−1,𝑋_𝑖−12𝑓𝑢,𝑋_𝑢𝑓_𝑢𝑢,𝑋_𝑢𝑔²𝑢,𝑋_𝑢−2𝑓²𝑢,𝑋_𝑢𝑔𝑢,𝑋_𝑢𝑔_𝑢𝑢,𝑋_𝑢+𝑔⁴𝑢,𝑋_𝑢𝑑𝑢_{𝑡𝑡𝑖−1}2𝑓𝑢,𝑋_𝑢𝑓_𝑥𝑢,𝑋_𝑢𝑔²𝑢,𝑋_𝑢−2𝑓²𝑢,𝑋_𝑢𝑔𝑢,𝑋_𝑢𝑔_𝑥𝑢,𝑋_𝑢𝑔⁴𝑢,𝑋_𝑢𝜃𝑓𝑢,𝑋^𝑢+𝑑𝑢_{𝑡𝑡𝑖−1}+𝑓^2𝑥𝑢,𝑋_𝑢𝑔²𝑢,𝑋_𝑢+3𝑓²𝑢,𝑋_𝑢𝑔^2𝑥𝑢,𝑋_𝑢−4𝑓𝑢,𝑋_𝑢𝑔𝑢,𝑋_𝑢𝑓_𝑥𝑢,𝑋_𝑢𝑔_𝑥𝑢,𝑋_𝑢𝑑𝑢_{𝑡𝑡𝑖−1}+𝑓𝑢,𝑋_𝑢𝑔²𝑢,𝑋_𝑢𝑓_𝑥𝑥𝑢,𝑋_𝑢−𝑓²𝑢,𝑋_𝑢𝑔𝑢,𝑋_𝑢𝑔_𝑥𝑥𝑢,𝑋_𝑢𝑑𝑢_{𝑡𝑡𝑖−1}2𝑓𝑢,𝑋_𝑢𝑓_𝑥𝑢,𝑋_𝑢𝑔²𝑢,𝑋_𝑢−2𝑓²𝑢,𝑋_𝑢𝑔𝑢,𝑋_𝑢𝑔_𝑥𝑢,𝑋_𝑢𝑔⁴𝑢,𝑋_𝑢𝑑𝑊^𝑢≜^𝑡𝑡_𝑖−1𝐹₁𝑢,𝑋_𝑢𝑑𝑢+^𝑡𝑡_𝑖−1𝐹₂𝑢,𝑋_𝑢𝑑𝑊_𝑢,(3.2) where 𝐹𝐹₁₁₁𝑢,𝑋_𝑢=𝐹₁₁𝑢,𝑋_𝑢+𝐹₁₂𝑢,𝑋_𝑢+𝐹₁₃𝑢,𝑋_𝑢+𝐹₁₄𝑢,𝑋_𝑢,𝑢,𝑋_𝑢=2𝑓𝑢,𝑋^𝑢𝑓_𝑢𝑢,𝑋_𝑢𝑔²𝑢,𝑋_𝑢−2𝑓²𝑢,𝑋_𝑢𝑔𝑢,𝑋_𝑢𝑔_𝑢𝑢,X_𝑢𝐹₁₂𝑔⁴𝑢,𝑋_𝑢,𝑢,𝑋_𝑢=2𝑓𝑢,𝑋^𝑢𝑓_𝑥𝑢,𝑋_𝑢𝑔²𝑢,𝑋_𝑢−2𝑓²𝑢,𝑋_𝑢𝑔𝑢,𝑋_𝑢𝑔_𝑥𝑢,𝑋_𝑢𝑔⁴𝑢,𝑋_𝑢𝜃𝑓𝑢,𝑋^𝑢𝐹₁₃,𝐹₁₄𝑢,𝑋_𝑢=𝑓^2𝑥𝑢,𝑋_𝑢𝑔²𝑢,𝑋_𝑢+3𝑓²𝑢,𝑋_𝑢𝑔^2𝑥𝑢,𝑋_𝑢−4𝑓𝑢,𝑋_𝑢𝑔𝑢,𝑋_𝑢𝑓_𝑥𝑢,𝑋_𝑢𝑔_𝑥𝑢,𝑋_𝑢,𝑢,𝑋_𝑢=𝑓𝑢,𝑋_𝑢𝑔²𝑢,𝑋_𝑢𝑓_𝑥𝑥𝑢,𝑋_𝑢−𝑓²𝑢,𝑋_𝑢𝑔𝑢,𝑋_𝑢𝑔_𝑥𝑥𝑢,𝑋_𝑢.(3.3)
For 𝐹₁₁(𝑢,𝑋_𝑢) and 𝐹₁₃(𝑢,𝑋_𝑢), by assumption (A3), (A4), (A5)₂, (A6)₁, and Hölder’s inequality, one has 𝔼^𝑡𝑡_𝑖−1𝐹₁₁𝑢,𝑋_𝑢²=𝔼^𝑡𝑡_𝑖−12𝑓𝑢,𝑋_𝑢𝑓_𝑢𝑢,𝑋_𝑢𝑔²𝑢,𝑋_𝑢−2𝑓²𝑢,𝑋_𝑢𝑔𝑢,𝑋_𝑢𝑔_𝑢𝑢,𝑋_𝑢𝑔⁴𝑢,𝑋_𝑢𝑑𝑢²𝔼≤≤𝐶𝐶𝔼,_{𝑡𝑡𝑖−1}^𝑡𝑡_𝑖−1𝑓²𝑢,𝑋_𝑢𝑑𝑢+𝐶𝔼^𝑡𝑡_𝑖−1𝑓^2𝑢𝑢,𝑋_𝑢𝑑𝑢+𝐶𝔼^𝑡𝑡_𝑖−1𝑔^2𝑢𝑢,𝑋_𝑢𝑑𝑢𝐹₁₃𝑢,𝑋_𝑢²=𝔼−4^𝑡𝑡_𝑖−1𝑓𝑓^2𝑥𝑢,𝑢𝑋,_𝑢𝑋_𝑢𝑔²𝑢,𝑋_𝑢+3𝑓²𝑢,𝑋_𝑢𝑔^2𝑥𝑢,𝑋_𝑢𝑔𝑢,𝑋_𝑢𝑓_𝑥𝑢,𝑋_𝑢𝑔_𝑥𝑢,𝑋_𝑢𝑑𝑢²≤≤𝐶𝐶𝔼.^𝑡𝑡_𝑖−1𝑓⁴𝑢,𝑋_𝑢𝑑𝑢+𝐶𝔼^𝑡𝑡_𝑖−1𝑔⁴𝑢,𝑋_𝑢𝑑𝑢+𝐶𝔼^𝑡𝑡_i−1𝑓^4𝑥𝑢,𝑋_𝑢𝑑𝑢+𝐶𝔼^𝑡𝑡_𝑖−1𝑔^4𝑥𝑢,𝑋_𝑢𝑑𝑢(3.4)
Similarly, we have 𝔼^𝑡𝑡_𝑖−1𝐹₁₂𝑢,𝑋_𝑢²≤𝐶,𝔼^𝑡𝑡_𝑖−1𝐹₁₄𝑢,𝑋_𝑢²≤𝐶.(3.5)
This means 𝔼^𝑡𝑡_𝑖−1𝐹1𝑢,𝑋_𝑢²≤𝐶.(3.6)
Hence, it follows 𝐶_𝑟 inequality that 𝔼|||||^𝑛_𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑖−1𝑡^𝑖−𝑡_𝑖−1−^𝑇0𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²=𝔼|||||^𝑛_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑓²𝑡_𝑖−1,𝑋_𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑖−1𝑡^𝑖−𝑡_𝑖−1−^𝑇0𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²=𝔼|||||^𝑛_𝑖=1^𝑡𝑖_𝑡𝑖−1^𝑡𝑡_𝑖−1𝐹₁𝑢,𝑋_𝑢𝑑𝑢+^𝑡𝑡_𝑖−1𝐹₂𝑢,𝑋_𝑢𝑑𝑊_𝑢𝑑𝑡|||||²≤2𝔼|||||^𝑛_𝑖=1^𝑡𝑖_𝑡𝑖−1^𝑡𝑡_𝑖−1𝐹₁𝑢,𝑋_𝑢𝑑𝑢𝑑𝑡|||||²+2𝔼|||||^𝑛_𝑖=1^𝑡𝑖_𝑡𝑖−1^𝑡𝑡_𝑖−1𝐹₂𝑢,𝑋_𝑢𝑑𝑊_𝑢𝑑𝑡|||||²≜2𝐺₁+2𝐺₂.(3.7)
By assumptions (A3), (A4), (A5)₂, and (A6)₁, we obtain 𝐺₁=𝔼|||||^𝑛_𝑖=1^𝑡𝑖_𝑡𝑖−1^𝑡𝑡_𝑖−1𝐹₁𝑢,𝑋_𝑢𝑑𝑢𝑑𝑡|||||²≤𝔼^𝑛_𝑖=1^𝑡𝑖_𝑡𝑖−1^𝑡𝑡_𝑖−1𝐹₁𝑢,𝑋_𝑢𝑑𝑢𝑑𝑡²+𝔼^𝑛₁₌_𝑖≠𝑗^𝑡𝑖_𝑡𝑖−1^𝑡𝑡_𝑖−1𝐹₁𝑢,𝑋_𝑢𝑑𝑢𝑑𝑡^𝑡𝑗_𝑡𝑗−1^𝑡𝑡_𝑗−1≤𝔼_𝑛𝐹₁𝑢,𝑋_𝑢𝑑𝑢𝑑𝑡_𝑖=1𝑡_𝑖−𝑡_𝑖−1^𝑡𝑖_𝑡𝑖−1||||^𝑡𝑡_𝑖−1+𝐹₁𝑢,𝑋_𝑢𝑑𝑢𝑑𝑡||||²𝑑𝑡_𝑛1=_𝑖≠𝑗⎧⎪⎨⎪⎩𝔼^𝑡_{𝑖𝑡𝑖−1}^𝑡𝑡_𝑖−1𝐹₁𝑢,𝑋_𝑢𝑑𝑢𝑑𝑡²𝔼^𝑡𝑗_𝑡𝑗−1^𝑡𝑡_𝑗−1𝐹₁𝑢,𝑋_𝑢𝑑𝑢𝑑𝑡²⎫⎪⎬⎪⎭^1/2≤^𝑛_𝑖=1𝑡_𝑖+−𝑡_𝑖−1^𝑡𝑖_𝑡𝑖−1𝔼||||^𝑡𝑡_𝑖−1𝐹₁𝑢,𝑋_𝑢𝑑𝑢𝑑𝑡||||²𝑑𝑡_𝑛1=_𝑖≠𝑗⎧⎪⎨⎪⎩𝑡_𝑖−𝑡_𝑖−1^𝑡𝑖_𝑡𝑖−1𝔼^𝑡𝑡_𝑖−1𝐹₁𝑢,𝑋_𝑢𝑑𝑢²𝑑𝑡𝑡_𝑗−𝑡_𝑗−1^𝑡𝑗_𝑡𝑗−1𝔼^𝑡𝑡_𝑗−1𝐹₁𝑢,𝑋_𝑢𝑑𝑢²𝑑𝑡⎫⎪⎬⎪⎭^1/2≤𝐶^𝑛_𝑖=1𝑡_𝑖−𝑡_𝑖−1³+𝐶^𝑛₁₌_𝑖≠𝑗𝑡^𝑖−𝑡_𝑖−1³𝑡_𝑗−𝑡_𝑗−1³^1/2≤𝐶𝑇³𝑛².(3.8)
Due to the orthogonality, Itô isomorphism, the Cauchy-Schwarz inequality, assumption (A3), (A4), and (A5)₁, we get 𝐺₂=𝔼|||||^𝑛_𝑖=1^𝑡𝑖_𝑡𝑖−1^𝑡𝑡_𝑖−1𝐹₂𝑢,𝑋_𝑢𝑑𝑊_𝑢𝑑𝑡|||||²≤𝔼^𝑛_𝑖=1^𝑡𝑖_𝑡𝑖−1^𝑡𝑡_𝑖−1𝐹₂𝑢,𝑋_𝑢𝑑𝑊_𝑢𝑑𝑡²+^𝑛₁₌_𝑖≠𝑗𝔼^𝑡𝑖_𝑡𝑖−1^𝑡𝑡_𝑖−1𝐹₂𝑢,𝑋_𝑢𝑑𝑊_𝑢𝑑𝑡^𝑡𝑗_𝑡𝑗−1^𝑡𝑡_𝑗−1≤𝐹₂𝑢,𝑋_𝑢𝑑𝑊_𝑢𝑑𝑡_𝑛_𝑖=1𝑡_𝑖≤−𝑡_𝑖−1^𝑡𝑖_𝑡𝑖−1𝔼||||^𝑡𝑡_𝑖−1𝐹₂𝑢,𝑋_𝑢𝑑𝑊_𝑢𝑑𝑡||||²𝑑𝑡_𝑛_𝑖=1𝑡_𝑖−𝑡_𝑖−1^𝑡𝑖_𝑡𝑖−1^𝑡𝑡_𝑖−1≤𝐶_𝑛𝔼||𝐹₂𝑢,𝑋_𝑢||²𝑑𝑢𝑑𝑡_𝑖=1𝑡_𝑖≤𝐶_𝑛−𝑡_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡−𝑡_𝑖−1𝑑𝑡_𝑖=1𝑡_𝑖−𝑡_𝑖−1³≤𝐶𝑇³𝑛².(3.9)
Obviously, it follows from bounds for 𝐺₁ and 𝐺₂ that 𝔼|||||^𝑛_𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑖−1𝑡^𝑖−𝑡_𝑖−1−^𝑇0𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²≤𝐶𝑇³𝑛².(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|||||^𝑛_𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑊^𝑡𝑖−𝑊_𝑡𝑖−1−^𝑇0𝑓𝑡,𝑋_𝑡𝑔𝑡,𝑋_𝑡𝑑𝑊^𝑡|||||²≤𝐶𝑇³𝑛².(3.11)

Proof. Let 𝜋_𝑛 be the partition 𝜋_𝑛=0=𝑡₀<𝑡₁<⋯<𝑡_𝑛=𝑇,𝑡_𝑖=𝑖ℎ,𝑖=0,1,…,𝑛 such that ℎ→0. Define 𝑆 and 𝑆_𝑛 as 𝑆=^𝑇0𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑊^𝑡𝑆_𝑛,=^𝑛_𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑊^𝑡𝑖−𝑊_𝑡𝑖−1.(3.12)
Let 𝜋^𝑛 be a partition which is finer than 𝜋_𝑛, obtained by choosing the mid point 𝑡̂_𝑖−1 from each of the interval 𝑡_𝑖−1<𝑡̂_𝑖−1<𝑡_𝑖, 𝑖=0,1,…,𝑛. Let 0=𝑡^0<𝑡^1<⋯<𝑡^2𝑛=𝑇 be the points of subdivision of the refined partition 𝜋^𝑛. Define the approximating sum 𝑆_𝜋^′𝑛 as before. We take two steps to prove the assertion in this lemma.
Step 1. We will first obtain the bounds on 𝐸|𝑆_𝜋𝑛−𝑆_𝜋^′𝑛|².
Let 0≤𝑡₀̃<𝑡₁̃<𝑡₂̃≤𝑇 be three equally space points on [0,𝑇] and let us denote 𝑋𝑡_𝑖̃ by 𝑋_𝑖 and 𝑊𝑡_𝑖̃ by 𝑊_𝑖,𝑖=0,1,…,𝑛. Define 𝐻=𝑓𝑡⁰̃,𝑋₀𝑔𝑡₀̃,𝑋₀𝑊²−𝑊₀−𝑓𝑡¹̃,𝑋₁𝑔𝑡₁̃,𝑋₁𝑊²−𝑊₁+𝑓𝑡⁰̃,𝑋₀𝑔𝑡₀̃,𝑋₀𝑊¹=𝑊₂−𝑊₀−𝑊₁𝑓𝑡⁰̃,𝑋₀𝑔𝑡₀̃,𝑋₀−𝑓𝑡¹̃,𝑋₁𝑔𝑡₁̃,𝑋₁.(3.13)
Denote 𝐼=𝑡¹̃𝑡₀̃𝑓𝑡,𝑋_𝑡𝑔𝑡,𝑋_𝑡𝑑𝑡.(3.14)
Applying the Taylor expansion, one has 𝑓𝑡₀̃,𝑋₀𝑔𝑡₀̃,𝑋₀−𝑓𝑡¹̃,𝑋₁=𝑋₀𝑔𝑡₁̃,𝑋₁−𝑋₁𝑓^𝑥𝑔−𝑔_𝑥𝑓𝑔²𝑡₁̃,𝑋₁+𝑡₀̃−𝑡₁̃𝑓^𝑡𝑔−𝑔_𝑡𝑓𝑔²+1𝑡₁̃,𝑋₁2𝑋⁰−𝑋₁²𝑓_𝑥𝑥𝑔²−𝑔_𝑥𝑥𝑓𝑔−2𝑓_𝑥𝑔−𝑔_𝑥𝑓𝑔𝑔_𝑥𝑔⁴+1𝑡^∗,𝑋^∗2𝑡⁰̃−𝑡₁̃²𝑓_𝑡𝑡𝑔²−𝑔_𝑡𝑡𝑓𝑔−2𝑓_𝑡𝑔−𝑔_𝑡𝑓𝑔𝑔_𝑡𝑔⁴+𝑡₀̃𝑡^∗,𝑋^∗−𝑡₁̃𝑋₀−𝑋₁𝑓^𝑡𝑥𝑔−𝑓𝑔𝑡𝑥−𝑓_𝑥𝑔_𝑡−𝑓_𝑡𝑔_𝑥𝑔²+2𝑓𝑔𝑔_𝑡𝑔_𝑥𝑔⁴=−𝑊₁𝑡^∗,𝑋^∗−𝑊₀+𝐼𝑓^𝑥𝑔−𝑔_𝑥𝑓𝑔²𝑡₁̃,𝑋₁+𝑡₀̃−𝑡₁̃𝑓^𝑡𝑔−𝑔_𝑡𝑓𝑔²+1𝑡₁̃,𝑋₁2𝑋⁰−𝑋₁²𝑓_𝑥𝑥𝑔²−𝑔_𝑥𝑥𝑓𝑔−2𝑓_𝑥𝑔−𝑔_𝑥𝑓𝑔𝑔_𝑥𝑔⁴+1𝑡^∗,𝑋^∗2𝑡⁰̃−𝑡₁̃²𝑓_𝑡𝑡𝑔²−𝑔_𝑡𝑡𝑓𝑔−2𝑓_𝑡𝑔−𝑔_𝑡𝑓𝑔𝑔_𝑡𝑔⁴+𝑡₀̃𝑡^∗,𝑋^∗−𝑡₁̃𝑋₀−𝑋₁𝑓^𝑡𝑥𝑔−𝑓𝑔𝑡𝑥−𝑓_𝑥𝑔_𝑡−𝑓_𝑡𝑔_𝑥𝑔²+2𝑓𝑔𝑔_𝑡𝑔_𝑥𝑔⁴𝑡^∗,𝑋^∗,(3.15) where |𝑋₁−𝑋^∗|<|𝑋₀−𝑋₁|,|𝑡₁̃−𝑡^∗|<|𝑡₀̃−𝑡₁̃|.
Relations (3.15) to (3.13) show that 𝐻=−𝑊₂−𝑊₁𝐼𝑓^𝑥𝑔−𝑔_𝑥𝑓𝑔²𝑡₁̃,𝑋₁+𝑊₂−𝑊₁𝑡₀̃−𝑡₁̃𝑓^𝑡𝑔−𝑔_𝑡𝑓𝑔²+𝑊₂𝑡₁̃,𝑋₁−𝑊₁12𝑋⁰−𝑋₁²𝑓_𝑥𝑥𝑔²−𝑔_𝑥𝑥𝑓𝑔−2𝑓_𝑥𝑔−𝑔_𝑥𝑓𝑔𝑔_𝑥𝑔⁴+𝑊₂𝑡^∗,𝑋^∗−𝑊₁12𝑡⁰̃−𝑡₁̃²𝑓_𝑡𝑡𝑔²−𝑔_𝑡𝑡𝑓𝑔−2𝑓_𝑡𝑔−𝑔_𝑡𝑓𝑔𝑔_𝑡𝑔⁴+𝑊₂𝑡^∗,𝑋^∗−𝑊₁𝑡₀̃−𝑡₁̃𝑋₀−𝑋₁f^𝑡𝑥𝑔−𝑓𝑔_𝑡𝑥−𝑓_𝑥𝑔_𝑡−𝑓_𝑡𝑔_𝑥𝑔²+2𝑓𝑔𝑔_𝑡𝑔_𝑥𝑔⁴𝑡^∗,𝑋^∗.(3.16)
Notice that 𝐻’s corresponding to different subintervals of [0,𝑇]-generated by 𝜋_𝑛 form a martingale difference sequence. Observe that 𝔼||𝐻||²=𝔼𝑊₂−𝑊₁²𝔼−𝐼𝑓^𝑥𝑔−𝑔_𝑥𝑓𝑔²𝑡₁̃,𝑋₁+𝑡₀̃−𝑡₁̃𝑓^𝑡𝑔−𝑔_𝑡𝑓𝑔²+1𝑡₁̃,𝑋₁2𝑋⁰−𝑋₁²𝑓_𝑥𝑥𝑔²−𝑔_𝑥𝑥𝑓𝑔−2𝑓_𝑥𝑔−𝑔_𝑥𝑓𝑔𝑔_𝑥𝑔⁴+1𝑡^∗,𝑋^∗2𝑡⁰̃−𝑡₁̃²𝑓_𝑡𝑡𝑔²−𝑔_𝑡𝑡𝑓𝑔−2𝑓_𝑡𝑔−𝑔_𝑡𝑓𝑔𝑔_𝑡𝑔⁴+𝑡₀̃𝑡^∗,𝑋^∗−𝑡₁̃𝑋₀−𝑋₁(𝒮)𝑔²+2𝑓𝑔𝑔_𝑡𝑔_𝑥𝑔⁴𝑡^∗,𝑋^∗²≤4𝑡₂̃−𝑡₁̃⎧⎪⎨⎪⎩𝔼𝐼𝑓^𝑥𝑔−𝑔_𝑥𝑓𝑔²𝑡₁̃,𝑋₁²+𝑡₀̃−𝑡₁̃²𝔼𝑓^𝑡𝑔−𝑔_𝑡𝑓𝑔²𝑡₁̃,𝑋₁²+14𝔼⎡⎢⎢⎣𝑋⁰−𝑋₁⁴𝑓^𝑥𝑥𝑔²−𝑔_𝑥𝑥𝑓𝑔−2𝑓_𝑥𝑔−𝑔_𝑥𝑓𝑔𝑔_𝑥𝑔⁴𝑡^∗,𝑋^∗²+1⎤⎥⎥⎦4𝑡⁰̃−𝑡₁̃⁴𝔼𝑓^𝑡𝑡𝑔²−𝑔_𝑡𝑡𝑓𝑔−2𝑓_𝑡𝑔−𝑔_𝑡𝑓𝑔𝑔_𝑡𝑔⁴𝑡^∗,𝑋^∗²+𝑡₀̃−𝑡₁̃²𝔼𝑋₀−𝑋₁²(𝒮)𝑔²+2𝑓𝑔𝑔_𝑡𝑔_𝑥𝑔⁴𝑡^∗,𝑋^∗²≤4𝑡₂̃−𝑡₁̃⎧⎪⎨⎪⎩𝔼𝐼⁴𝔼𝑓^𝑥𝑔−𝑔_𝑥𝑓𝑔²𝑡₁̃,𝑋₁⁴^1/2+𝑡₀̃−𝑡₁̃²𝔼𝑓^𝑡𝑔−𝑔_𝑡𝑓𝑔²𝑡₁̃,𝑋₁²+14⎡⎢⎢⎣𝔼𝑋⁰−𝑋₁⁸𝔼𝑓^𝑥𝑥𝑔²−𝑔_𝑥𝑥𝑓𝑔−2𝑓_𝑥𝑔−𝑔_𝑥𝑓𝑔𝑔_𝑥𝑔⁴𝑡^∗,𝑋^∗⁸⎤⎥⎥⎦^1/2+14𝑡⁰̃−𝑡₁̃⁴𝔼𝑓^𝑡𝑡𝑔²−𝑔_𝑡𝑡𝑓𝑔−2𝑓_𝑡𝑔−𝑔_𝑡𝑓𝑔𝑔_𝑡𝑔⁴𝑡^∗,𝑋^∗²+𝑡₀̃−𝑡₁̃²𝔼𝑋⁰−𝑋₁⁴𝔼(𝒮)𝑔²+2𝑓𝑔𝑔_𝑡𝑔_𝑥𝑔⁴𝑡^∗,𝑋^∗⁴^1/2≤4𝑡₂̃⎫⎪⎬⎪⎭−𝑡₁̃𝔼𝐼⁴^1/2𝐶𝔼𝑓^8𝑥𝑡₁̃,𝑋₁^1/2+𝔼𝑔^8𝑥𝑡₁̃,𝑋₁^1/2+𝔼𝑓⁸𝑡₁̃,𝑋₁^1/2^1/2++1𝑡₀̃−𝑡₁̃²𝐶𝔼𝑓^4𝑡𝑡₁̃,𝑋₁^1/2+𝔼𝑔^4𝑡𝑡₁̃,𝑋₁^1/2+𝔼𝑓⁴𝑡₁̃,𝑋₁^1/24𝔼𝑋⁰−𝑋₁⁸^1/2𝐶𝔼𝑓^8𝑥𝑥𝑡^∗,𝑋^∗^1/2+𝔼𝑔^8𝑥𝑥𝑡^∗,𝑋^∗^1/2+𝔼𝑓⁸𝑡^∗,𝑋^∗^1/2+𝔼𝑓^8𝑥𝑡^∗,𝑋^∗^1/2+𝔼𝑓^𝑥16𝑡^∗,𝑋^∗^1/2^1/2+14𝑡⁰̃−𝑡₁̃⁴𝐶𝔼𝑓^4𝑡𝑡𝑡^∗,𝑋^∗^1/2+𝔼𝑔^4𝑡𝑡𝑡^∗,𝑋^∗^1/2+𝔼𝑓^4𝑡𝑡^∗,𝑋^∗^1/2+𝑡₀̃+𝔼𝑔^4𝑡𝑡^∗,𝑋^∗^1/2+𝔼𝑔^8𝑡𝑡^∗,𝑋^∗^1/2−𝑡₁̃²𝔼𝑋⁰−𝑋₁⁴^1/2𝔼𝑓^8𝑡𝑥𝑡^∗,𝑋^∗^1/2+𝔼𝑔^8𝑡𝑥𝑡^∗,𝑋^∗^1/2+𝔼𝑓^8𝑥𝑡₁̃,𝑋₁^1/2+𝔼𝑔^8𝑥𝑡₁̃,𝑋₁^1/2+𝔼𝑓^8𝑡𝑡₁̃,𝑋₁^1/2+𝔼𝑔^8𝑡𝑡₁̃,𝑋₁^1/2^1/2.(3.17)where 𝒮 denotes 𝑓_𝑡𝑥𝑔−𝑓𝑔_𝑡𝑥−𝑓_𝑥𝑔_𝑡−𝑓_𝑡𝑔_𝑥.
By Theorem 4 of [31], for any 0≤𝑠<𝑡≤𝑇, there exists 𝐶>0 such that 𝔼𝑋_𝑡−𝑋_𝑠^2𝑚≤𝐶𝔼𝑋^02𝑚+1(𝑡−𝑠)^𝑚,𝑚≥1.(3.18)
Hence 𝔼𝔼𝑋𝑋_𝑡𝑡−𝑋_𝑠⁸≤𝐶𝔼𝑋⁸⁰+1(𝑡−𝑠)⁴,−𝑋_𝑠⁴≤𝐶𝔼𝑋⁴⁰+1(𝑡−𝑠)².(3.19)
Furthermore by (A2) and (A3), we have 𝔼𝐼⁴=𝔼𝑡¹̃𝑡₀̃𝑓𝑡,𝑋_𝑡𝑔𝑡,𝑋_𝑡𝑑𝑡⁴≤𝐶𝔼𝑡¹̃𝑡₀̃≤𝐶𝔼𝑡₁̃𝑓⁴𝑡,𝑋_𝑡𝑑𝑡𝑡₀̃1+||𝑋^𝑡||²≤𝐶𝑡₁̃²𝑑𝑡−𝑡₀̃⁴sup_{0≤𝑡≤𝑇}𝔼1+||𝑋_𝑡||²²≤𝐶𝑡₁̃−𝑡₀̃⁴.(3.20)
Thus 𝔼(𝐻)²≤𝐶𝑡₂̃−𝑡₁̃𝑡₁̃−𝑡₀̃².(3.21)
Using the property that 𝐻 corresponding to different subintervals forms a martingale difference sequence, it follows that 𝔼||𝑆_𝜋𝑛−𝑆_𝜋^′𝑛||²≤𝐶𝑇³𝑛²,(3.22) for some constant 𝐶>0.
Step 2. We will show now the bounds on 𝐸|𝑆_𝜋^′𝑛−𝑆|².
Let 𝜋^𝑛(𝑝),𝑝≥0 be the sequence of partitions such that 𝜋^𝑛(𝑖+1) is a refinement of 𝜋^𝑛(𝑛) by choosing the midpoint of the subintervals generated by 𝜋^𝑛(𝑛). Note that 𝜋^𝑛(0)=𝜋_𝑛 and 𝜋^𝑛(1)=𝜋^𝑛. The analysis given above proves that 𝔼||𝑆_𝜋𝑛(𝑝)−𝑆_{𝜋𝑛(𝑝+1)}||²≤𝐶𝑇³2^𝑝𝑛²,𝑝≥0,(3.23) where 𝑆_𝜋𝑛(𝑝) is the approximation corresponding to 𝜋^𝑛(𝑝) and 𝑆_𝜋𝑛(0)=𝑆_𝜋𝑛.
Therefore, applying the Hölder inequality and the Minkovski inequality, one gets 𝔼||𝑆_𝜋𝑛(0)−𝑆_{𝜋𝑛(𝑝+1)}||²≤𝔼^𝑝_𝑘=0𝑆_𝜋𝑛(𝑘)−𝑆_{𝜋𝑛(𝑘+1)}²≤^𝑝_𝑘=0𝔼||𝑆^𝜋_𝑛(𝑘)−𝑆_{𝜋𝑛(𝑘+1)}||²^1/2²≤^𝑝_𝑘=0𝐶2^𝑝𝑇𝑛³²^1/2²≤𝐶𝑇³𝑛²,(3.24) for all 𝑝≥0. Let 𝑝→∞. Since the integral 𝑆 exists, 𝑆_𝜋𝑛(𝑝+1) converges in ℒ₂ to 𝑆 as 𝑝→∞. Note that 𝜋^𝑛(𝑝+1),𝑃≥0 is a sequence of partitions such that the mesh of the partition tends to zero as 𝑝→∞ for any fixed 𝑛.
Thus 𝔼||𝑆_𝜋𝑛−𝑆||²≤𝐶𝑇³2^𝑝𝑛²,𝑝≥0,(3.25) where 𝑆=lim_𝑛→∞𝑆_𝜋𝑛=^𝑇0𝑓𝑡,𝑋_𝑡𝑔𝑡,𝑋_𝑡𝑑𝑊^𝑡.(3.26)
The proof is now complete.

Theorem 3.3. Under assumptions (A1)–(A4), (A5)₂, (A6)₂, (A7), and (A8), one has 𝔼||𝐿_𝑛,𝑇(𝜃)−𝐿_𝑇(𝜃)||²≤𝐶𝑇³𝑛²𝔼||𝐿_𝑛,𝑇,(𝜃)−𝐿^𝑇(𝜃)||²≤𝐶𝑇³𝑛².(3.27)

Proof. By the analysis given above, one has ||𝐿_𝑛,𝑇(𝜃)−𝐿_𝑇(𝜃)||²=|||||𝜃^𝑛_𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑋^𝑡𝑖−𝑋_𝑡𝑖−1−^𝑇0𝑓𝑡,𝑋_𝑡−𝜃₂𝑔²𝑡,𝑋_𝑡𝑑𝑋^𝑡2^𝑛_𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡^𝑖−𝑡_𝑖−1−^𝑇0𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²≤𝜃⁴2|||||^𝑛_𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡^𝑖−𝑡_𝑖−1−^𝑇0𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²+2𝜃²|||||_𝑛_𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑊^𝑡𝑖−𝑊_𝑡𝑖−1−^𝑇0𝑓𝑡,𝑋_𝑡g²𝑡,𝑋_𝑡𝑑𝑊^𝑡|||||².(3.28)
Hence, it follows from Lemmas 3.1 and 3.2 that 𝔼||𝐿_𝑛,𝑇(𝜃)−𝐿_𝑇(𝜃)||²≤𝜃⁴2𝔼|||||^𝑛_𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡^𝑖−𝑡_𝑖−1−^𝑇0𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²+2𝜃²𝔼|||||^𝑛_𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑊^𝑡𝑖−𝑊_𝑡𝑖−1−^𝑇0𝑓𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑊^𝑡|||||²≤𝐶𝑇³𝑛².(3.29)
Next, note that ||𝐿^𝑛,𝑇(𝜃)−𝐿^𝑇(𝜃)||²=|||||𝜃^𝑛_𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑋^𝑡𝑖−𝜃_𝑛−𝑋_𝑡𝑖−1_𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡^𝑖−𝑡_𝑖−1−^𝑇0𝑓𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑋^𝑡−^𝑇0𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²=(1−𝜃)²|||||_𝑛_𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡^𝑖−𝑡_𝑖−1−^𝑇0𝑓²𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²+|||||^𝑛_𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑊^𝑡𝑖−𝑊_𝑡𝑖−1−^𝑇0𝑓𝑡,𝑋_𝑡𝑔²𝑡,𝑋_𝑡𝑑𝑊^𝑡|||||².(3.30)
Similarly, by Lemmas 3.1 and 3.2, we obtain 𝔼||𝐿^𝑛,𝑇(𝜃)−𝐿^𝑇(𝜃)||²≤𝐶𝑇³𝑛².(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||𝜃_𝑛,𝑇−𝜃_𝑇||²≤𝐶𝑇³𝑛².(3.32)

Proof. We know 𝜃_𝑛,𝑇 and 𝜃_𝑇 are the solutions of equations 𝐿^𝑛,𝑇(𝜃)=0 and 𝐿^𝑇(𝜃)=0, respectively.
Hence, one gets ||𝜃_𝑛,𝑇−𝜃_𝑇||²=|||||∑^𝑛𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑋_𝑡𝑖−𝑋_𝑡𝑖−1∑^𝑛𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡_𝑖−𝑡_𝑖−1−∫^𝑇0𝑓𝑡,𝑋_𝑡/𝑔²𝑡,𝑋_𝑡𝑑𝑋_𝑡∫^𝑇0𝑓²𝑡,𝑋_𝑡/𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²=|||||∑^𝑛𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑊_𝑡𝑖−𝑊_𝑡𝑖−1∑^𝑛𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡_𝑖−𝑡_𝑖−1−∫^𝑇0𝑓𝑡,𝑋_𝑡/𝑔𝑡,𝑋_𝑡𝑑𝑊_𝑡∫^𝑇0𝑓²𝑡,𝑋_𝑡/𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²=|||||∑^𝑛𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑊_𝑡𝑖−𝑊_𝑡𝑖−1−∫^𝑇0𝑓𝑡,𝑋_𝑡/𝑔𝑡,𝑋_𝑡𝑑𝑊_𝑡−∫_𝑇0∑^𝑛𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡_𝑖−𝑡_𝑖−1𝑓𝑡,𝑋_𝑡/𝑔𝑡,𝑋_𝑡𝑑𝑊_𝑡∑^𝑛𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡_𝑖−𝑡_𝑖−1−𝒜∑^𝑛𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡_𝑖−𝑡_𝑖−1∫^𝑇0𝑓²𝑡,𝑋_𝑡/𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²≤2|||||∑^𝑛𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑊_𝑡𝑖−𝑊_𝑡𝑖−1−∫^𝑇0𝑓𝑡,𝑋_𝑡/𝑔𝑡,𝑋_𝑡𝑑W_𝑡∑^𝑛𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡_𝑖−𝑡_𝑖−1|||||²+2|||||∫^𝑇0𝑓𝑡,𝑋_𝑡/𝑔𝑡,𝑋_𝑡𝑑𝑊_𝑡∑^𝑛𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡_𝑖−𝑡_𝑖−1−𝒜∑^𝑛𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_𝑖−1,𝑋_𝑡i−1𝑡_𝑖−𝑡_𝑖−1∫^𝑇0𝑓²𝑡,𝑋_𝑡/𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||².(3.33)
As we know that ∑^𝑛𝑖=1(𝑓²(𝑡_𝑖−1,𝑋_𝑡𝑖−1)/𝑔²(𝑡_𝑖−1,𝑋_𝑡𝑖−1))(𝑡_𝑖−𝑡_𝑖−1)>0, so there exists a constant 𝐶>0 such that 1∑^𝑛𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡_𝑖−𝑡_𝑖−1≤𝐶.(3.34)
Therefore, applying Itô isomorphism, the Cauchy-Schwarz inequality, Lemmas 3.1 and 3.2, we obtain 𝔼||𝜃_𝑛,𝑇−𝜃_𝑇||²≤2𝔼|||||∑^𝑛𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑊_𝑡𝑖−𝑊_𝑡𝑖−1−∫^𝑇0𝑓𝑡,𝑋_𝑡/𝑔𝑡,𝑋_𝑡𝑑𝑊_𝑡∑^𝑛𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡_𝑖−𝑡_𝑖−1|||||²+2𝔼|||||∫^𝑇0𝑓𝑡,𝑋_𝑡/𝑔𝑡,𝑋_𝑡𝑑𝑊_𝑡∑^𝑛𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡_𝑖−𝑡_𝑖−1−𝒜∑^𝑛𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_i−1,𝑋_𝑡𝑖−1𝑡_𝑖−𝑡_𝑖−1∫^𝑇0𝑓²𝑡,𝑋_𝑡/𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²≤𝐶𝔼|||||^𝑛_𝑖=1𝑓𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑊_𝑡𝑖−𝑊_𝑡𝑖−1−^𝑇0𝑓𝑡,𝑋_𝑡/𝑔𝑡,𝑋_𝑡𝑑𝑊_𝑡|||||²+𝐶𝔼|||||^𝑛_𝑖=1𝑓²𝑡_𝑖−1,𝑋_𝑡𝑖−1/𝑔²𝑡_𝑖−1,𝑋_𝑡𝑖−1𝑡_𝑖−𝑡_𝑖−1−^𝑇0𝑓²𝑡,𝑋_𝑡/𝑔²𝑡,𝑋_𝑡𝑑𝑡|||||²≤𝐶𝑇³𝑛²,(3.35)where 𝒜 denotes ∫^𝑇0(𝑓²(𝑡,𝑋_𝑡)/𝑔²(𝑡,𝑋_𝑡))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.