THE LEAST-SQUARES ESTIMATOR OF SINUSOIDAL SIGNAL OF DIFFUSION PROCESS FOR DISCRETE OBSERVATIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. There is notably paucity of studies on least-squares estimator of diffusion process for discrete observations. This paper discusses sufficient conditions of the least-squares estimator of diffusion process for discrete observations in order to gain an estimator that is strongly consistent of [1]. We assume that the process Y is arranged by a function such as sinusoidal signal a(θ ,Yt) = sin(2πtθ),θ ∈ [ 0, 2 ] and function b(σ ,Yt). For a given a sample (Y0,Yh, . . . ,Ynh), h→ 0, we demonstrate an asymptotic theory of least-squares estimator θ̂n. The results of the study show that the least-squares estimator is strongly consistent and asymptotic normal, assuming that nh→∞ and n3h4→∞; θ that represents the frequency of sinusoidal signal of the unity of time which has a rate of convergence, namely √ n3h4.


INTRODUCTION
The stochastic differential equation application has been widely used in the field of industry, economics and environment as can be seen in [2], [3], [4], [5], [6], [7]. A part from these examples, studies on nonlinear problems have been investigated by numerous researchers, examples of which can be seen in [1] and [8]. [9] exploited the nonlinear model as follows: (1) y t = cos(2πtθ 0 ) + e t , where t ∈ N, {e t } are i.i.d normal random variable with mean zero and finite positive variance The stochastic differential equation (SDE) can be viewed as problems of nonlinear regression model. Few published studies have examined parameter estimation of diffusion process using the least-squares method for discrete observations (see for examples, [10] dan [11]). Hence, this study aimed to investigate the least-squares estimator (LSE) of diffusion process for discrete observations.
Consider Y = (Y t ) t∈R + is the solution to the SDE where • a(θ ,Y t ) = sin(2πθt), θ ∈ 0, 1 2 , a is a measurable function, a : R → R; • θ is unknown parameter and will be estimated; • Y > 0 and Y 0 is an initial value of Y when t = 0; • w is a one-dimensional Wiener process.
This study inspired the nonlinear models of [1], [12], [9] and [13]. These studies discussed nonlinear model in general, i.e., y t = f t (α) + e t with {e t } t∈N which are independent random variables with mean zero and finite variance. In this paper we develop schemes, namely: first we discretize (2), from this discretization model, we define a target function using the leastsquare approach by minimizing errors of the squares between the process Y and function of a(.). Based on the target function, a verification of the almost sure convergence of the estimator will be performed as suggested by [1] whether it fulfills for discrete observations. With a strong consistency of the estimator θ , we will then discuss how to determine an asymptotic normality for θ and verify an assumption required for asymptotic normal of the estimator.
The paper is structured as follows. The model is presented in Section 2, followed by our theoretical results, namely asymptotic theory of the estimator in Section 3. The study of numeric aiming to simulate the least-squares estimator is set out in Section 4.

PRELIMINARIES
defined on an underlying complete filtered probability space (Ω, F , (F t ) t∈R + , P) where • the true parameter value is denoted by θ 0 ∈ Θ which does exist with the P 0 assumed as the true image measure; • θ ∈ Θ ⊂ R, where Θ is supposed to be bounded convex domains, and the closure of Θ is denoted by Θ which satisfies Θ ⊂ θ ∈ 0, 1 2 ; • P θ for the image measure of a solution process Y associated with θ .
We assume that Y is observed at discrete sample points (Y t 0 ,Y t 1 , . . . ,Y t n ) with 0 ≡ t 0 < t 1 < · · · < t n , where t n i = t i = ih, i ≤ n and h > 0 is a non-random sampling discrete of step size such that for n → ∞ satisfies (4) h → 0, Several notations which will be used in this paper are: for a function f (θ , .), ∂ j θ f (θ , .) stands for jth derivative of f with respect to θ , j = 1, 2; symbol L − → indicates the convergence in law under P 0 .

MAIN RESULTS
Discretization of Y (3) by deploying approximations of Euler-Maruyama we can provide: We also assume From here and the next section, we define the increased process ρ i as Next, we define a target function of (2) namely The definition of target function of diffusion process for discrete observations can be seen also in [10] and [11].
From the target function (9), we define the LSE of θ asθ n namely a measurable function which satisfies and based on the function (11), we then shall discuss the consistency of LSEθ n .
Proof. We shall verify a condition of almost sure convergence of θ from Lemma 1 of [1] whether it can be fulfilled for LSE of diffusion process for discrete observations. Verification can be done by showing: for C > 0, inequality By re-calling (9) and (8), we have Let us observe the first part of the right hand-side of an equation (13), Since we apply the following results, then we have Next, we need to know that the last part of the right hand-side of an equation (13). Since Therefore, {σ ∆ i w} i≤n can be seen as {ε t i } i≤n that fulfill Assumption 3.1 and 3.2 of [13], hence, we can state the following Lemma.   [13] in Lemma 4.2 in view of discrete observations. Therefore, using Lemma 3.2 we may know that (12) is fulfilled because and since we have (14) and (16), the verification has been completed.
Next, we shall discuss the asymptotic normality ofθ n . (17) Proof. Note that, from the equation (9), we have the relation To solve the equation (18), we need the following Lemma.
Lemma 3.5. Under (5) and (6), we have Proof. Re-calling v and q, we get that n 1 2 h → 0 for LSE of diffusion process, meanwhile in the maximum likelihood method, the assumption of nh 3 → 0 given by [14], whereas [15] assumes nh p , p ∈ N.
By applying Lemma 3.5, we can rewrite (18) as follows or we can write it as By applying Central Limit Theorem (CLT), we shall determine Σ 1 such that (9), we can obtain: Use (8) and (19) for expectation and variance of − 1 √ n 3 h 4 ∂ H n (θ 0 ) . Note that,

by using Corrolary 3.3 it is obvious that
by using the result of trigonometry of identity: we obtain Σ 1 = 8π 2 σ 2 3 + o(1). So we can make a claim that Next, using Law of Large Number (LLN), we find Σ 2 such that Note that and we get By using Corollary 3.3 and the result of trigonometry identity, we obtain Σ 2 = 4π 2 3 . Now let us say 1

SIMULATION
The simulation is done through the use of software R. In the first simulation, we choose

CONCLUSION
Based on the above discussion, we can conclude that: the LSEθ n of diffusion process (2) at discrete observations is strongly consistent according to Wu [1] under (4), (5) and (6) with rate of convergence is √ n 3 h 4 .