Asymptotic normality of discretized maximum likelihood estimator for drift parameter in homogeneous diffusion model

We prove the asymptotic normality of the discretized maximum likelihood estimator for the drift parameter in the homogeneous ergodic diffusion model.


Introduction
The statistical inference for diffusion models has been thoroughly studied by now; see the books [4,[6][7][8]10] and references therein.
In this paper, we consider the homogeneous diffusion process given by the stochastic differential equation where W t is a standard Wiener process, and θ is an unknown parameter.
The standard maximum likelihood estimator for the parameter θ constructed by the observations of X on the interval [0, T ] has the form K. Ralchenkô see, for instance, [7,Example 1.37] and [9]. If the equation has a weak solution, the coefficient a is not identically zero, and the functions 1 b 2 , a 2 b 2 , a 2 b 4 are locally integrable, then this estimator is strongly consistent [9,Thm. 3.3]. Moreover, if the model is ergodic, then this estimator is asymptotically normal [7,Ex. 1.37]. Note that in the nonergodic case the maximum likelihood estimatorθ T may have different limit distributions; some examples can be found in [7,Sect. 3.5].
If the data are the observations of the trajectory {X t , t ≥ 0} at discrete time moments t 1 , t 2 , . . . , we obtain the discrete-time version of the model. Parameter estimation in such models has been studied since the mid-1980s; see [2,3,11]. A review of this problem and many references can be found in [5] and [13]. For recent results, see [6,9,12].
In this paper, we are interested in the scheme of observations that is called "rapidly increasing experimental design." The process X is observed at time moments t i = i∆ n , i = 0, . . . , n, such that ∆ n → 0 and n∆ n → ∞ as n → ∞. One of possible approaches to parameter estimation is to consider a discretized version of the continuous-time MLEθ T . The most general results in this direction were obtained by Yoshida [14]. He proved the consistency and asymptotic normality of the discretized MLE in the model, where the process was multidimensional, the drift coefficient depended on θ nonlinearly, and the diffusion coefficient also contained an unknown parameter.
Assume that we observe the process X at discrete time moments t n k = k/n, 0 ≤ k ≤ n 1+α , where 0 < α < 1 2 . In this scheme, Mishura [9] proposed the following discretized version of the maximum likelihood estimator: She proved its strong consistency in the case where the coefficients a and b are bounded. The aim of this paper is to establish the asymptotic normality of this estimator. Additionally, we assume the ergodicity of the model, but the boundedness of the coefficients is not required. In comparison with general results of Yoshida [14], our assumptions are less restrictive. We assume the polynomial growth of the function 1/b instead of the condition inf x b(x) 2 > 0. Also, we do not assume the smoothness of the coefficients and the polynomial growth of their derivatives; any Lipschitz continuous a(x) and b(x) are possible. The paper is organized as follows. In Section 2, we describe the model and formulate the results. In Section 3, some simulation experiments are considered. The proof of the main theorem is given in Section 4.

Model description and main result
Let (Ω, F) be a measurable space. Assume that θ ∈ R is fixed but unknown. Consider a probability measure P θ such that F is P θ -complete.
Let X solve the equation where x 0 ∈ R, a, b : R → R are measurable functions, and {W t , t ≥ 0} is a standard Wiener process on (Ω, F, P θ ).
Assume that the following conditions hold.
(A1) For some L > 0 and for any x, y ∈ R, It is well known that under assumption (A1) the stochastic differential equation (1) has a unique strong solution. This assumption also yields that the functions a(x) and b(x) satisfy the linear growth condition, that is, for some M > 0 and for all x ∈ R. Assume additionally that (A4) There exist K > 0 and p ≥ 0 such that Then, for some M 2 > 0 and for any x ∈ R, Under assumptions (A2)-(A3), the diffusion process X is positive recurrent; see, for example, [7,Prop. 1.15]. In this case, it has ergodic properties with the invariant density given by Let ξ θ denote a random variable with density µ θ (x). Then, for any measurable func- Assume that the invariant distribution satisfies the condition Let 0 < α < 1. Suppose that we observe the process X at discrete time moments t n k = k/n, 0 ≤ k ≤ n 1+α . Consider the estimator Then E θ d(ξ θ ) > 0. Note also that by (3) and (A5), E θ d(ξ θ ) < ∞. Now we are ready to formulate the main result.
The proof is given in Section 4. The following result gives sufficient conditions for consistency and asymptotic normality in the case where the parameter θ is positive. Then Proof. Note that condition (6), together with (A4), implies that assumptions (A2)-(A3) are satisfied and, moreover, all polynomial moments of the invariant density are finite; see [7, p. 3]. Hence, the result follows directly from Theorem 2.1.
If the coefficients are bounded, then the consistency and asymptotic normality of θ n can be obtained without assumption (A5).
Sketch of proof. This result can be proved similarly to Theorem 2.1 using the boundedness of a(x), b(x), c(x), d(x) instead of the growth conditions (2), (3), and (A4) to-gether with the boundedness of moments of the invariant density. In this case, (8) for all m ∈ N and t ∈ [ k−1 n , k n ], k = 1, 2, . . . n α . This estimate is used in the proof instead of Lemmas 4.1-4.2.

Some simulation results
In this section, we illustrate quality of the estimator by simulation experiments. We consider the diffusion process (1) with drift parameter θ = 2 and initial value x 0 = 1 in three following cases: Using the Milstein method, we simulate 100 sample paths of each process and find the estimateθ n for different values of n and α. The average values ofθ n and the corresponding standard deviations are presented in Tables 1-3.  In this section, we prove the main theorem and some auxiliary lemmas. In what follows, C, C 1 , C 2 , . . . are positive generic constants that may vary from line to line. If they depend on some arguments, we will write C(θ), C(m, θ), and so on.
Proof. By (8), Using assumption (A1) and (2), we get The same estimate holds for E θ t k−1 n b(X s ) 2m ds. Therefore, and the result follows from the Gronwall lemma.
Proof. Applying the Hölder inequality and Lemma 4.1, we get By Jensen's inequality we have By (5) the expression in brackets is bounded. This completes the proof. (ii) if, additionally, (A4) holds, then Proof. (i) In the case m = 0, the result is trivial. Let m ≥ 1. By (5) we have Hence, it suffices to prove that converges to zero as n → ∞. By the inequality |x| ≤ |x − y| + |y|, Therefore, and, by Lemma 4.3, (ii) For arbitrary x and y, By (A1), (A4), and (2), The rest of the proof can be done similarly to part (i) using estimate (10) instead of (9).
Proof. (i) Let us prove the convergence in L 2 . We have Then by (5) we have a(x) − a(y) .