The statistical inference for diffusion models has been thoroughly studied by now; see the books [4, 6–8, 10] and references therein.

In this paper, we consider the homogeneous diffusion process given by the stochastic differential equation dXt=θa(Xt)dt+b(Xt)dWt, where Wt 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 θˆT=∫0Ta(Xt)b(Xt)2dXt∫0Ta(Xt)2b(Xt)2dt; 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 1b2 , a2b2 , a2b4 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 estimator θˆT may have different limit distributions; some examples can be found in [7, Sect. 3.5].

If the data are the observations of the trajectory {Xt,t≥0} at discrete time moments t1,t2,… , 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 ti=iΔn , i=0,…,n , such that Δn→0 and nΔn→∞ as n→∞ . One of possible approaches to parameter estimation is to consider a discretized version of the continuous-time MLE θˆT . 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 tkn=k/n , 0≤k≤n1+α , where 0<α<12 . In this scheme, Mishura [9] proposed the following discretized version of the maximum likelihood estimator: θˆn=∑k=0n1+αa(Xkn)(Xk+1n−Xkn)/b(Xkn)2n−1∑k=0n1+αa(Xkn)2/b(Xkn)2. 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 infxb(x)2>0 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.

Let (Ω,F) be a measurable space. Assume that θ∈R is fixed but unknown. Consider a probability measure Pθ such that F is Pθ -complete.

Let X solve the equation (1) Xt=x0+θ∫0ta(Xs)ds+∫0tb(Xs)dWs, where x0∈R , a,b:R→R are measurable functions, and {Wt,t≥0} is a standard Wiener process on (Ω,F,Pθ) .

Denote c(x)=a(x)b(x)2 , d(x)=a(x)2b(x)2 , φθ(x)=exp{−2θ∫0xc(y)dy} , and Φθ(x)=∫0xφθ(y)dy .

Assume that the following conditions hold.

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, (2) |a(x)|+|b(x)|≤M1(1+|x|) for some M>0 0$]]> and for all x∈R .

Assume additionally that (A4)

There exist K>0 0$]]> and p≥0 such that |b(x)|−1≤K(1+|x|p).

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 μθ(x)=1Gθb(x)2φθ(x),x∈R. Let ξθ denote a random variable with density μθ(x) . Then, for any measurable function h such that Eθ|h(ξθ)|<∞ , (4) 1T∫0Th(Xt)dt→∫−∞+∞h(x)μθ(x)dx≡Eθh(ξθ)a.s. asT→∞, see [7, Thm. 1.16]. Moreover, according to [1, Sect. II.37], the convergence (4) holds also in L1 , that is, (5) 1TEθ∫0Th(Xt)dt→Eθh(ξθ). Assume that the invariant distribution satisfies the condition

Let 0<α<1 . Suppose that we observe the process X at discrete time moments tkn=k/n , 0≤k≤n1+α . Consider the estimator θˆn=∑k=1n1+αc(Xk−1n)ΔXknn−1∑k=1n1+αd(Xk−1n), where ΔXkn=Xkn−Xk−1n .

Assume also that

Then Eθd(ξθ)>0 0$]]>. Note also that by (3) and (A5), Eθd(ξθ)<∞ . 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.

If the coefficients are bounded, then the consistency and asymptotic normality of θˆn can be obtained without assumption (A5). Corollary 2.3.

Assume that conditions (A1)–(A3) are satisfied, the coefficients a(x) and b(x) are bounded, and infx∈R|b(x)|>0 0$]]>. Then (i)

θˆn→Pθθ as n→∞ ,

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 x0=1 in three following cases: (1)

a(x)=1−x , b(x)=2+sinx ,

In this section, we prove the main theorem and some auxiliary lemmas. In what follows, C,C1,C2,… 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.

By (1), ΔXkn=θ∫k−1nkna(Xt)dt+∫k−1nknb(Xt)dWt=θa(Xk−1n)1n+θ∫k−1nkn(a(Xt)−a(Xk−1n))dt+b(Xk−1n)ΔWkn+∫k−1nkn(b(Xt)−b(Xk−1n))dWt. Therefore, θˆn=θ+ ∑k=1n1+α(c(Xk−1n)θ∫k−1nkn(a(Xt)−a(Xk−1n))dt+a(Xk−1n)b(Xk−1n)ΔWkn+c(Xk−1n)∫k−1nkn(b(Xt)−b(Xk−1n))dWt)/(1n ∑k=1n1+αd(Xk−1n)). Then (7) θˆn−θ=n−α(An+Bn+En)Dn, where Dn=n−1−α ∑k=1n1+αd(Xk−1n),An= ∑k=1n1+αc(Xk−1n)θ∫k−1nkn(a(Xt)−a(Xk−1n))dt,Bn= ∑k=1n1+αa(Xk−1n)b(Xk−1n)ΔWkn,En= ∑k=1n1+αc(Xk−1n)∫k−1nkn(b(Xt)−b(Xk−1n))dWt.