Statistical Inference for Stochastic Differential Equations with Small Noises

Abstract

This paper proposes the least squares method to estimate the drift parameter for the stochastic differential equations driven by small noises, which is more general than pure jump 𝛼-stable noises. The asymptotic property of this least squares estimator is studied under some regularity conditions. The asymptotic distribution of the estimator is shown to be the convolution of a stable distribution and a normal distribution, which is completely different from the classical cases.

1. Introduction

Stochastic differential equations (SDEs) are being extensively used as a model to describe some phenomena which are subject to random influences; it has found many applications in biology [1], medicine [2], econometrics [3, 4], finance [5], geophysics [6], and oceanography [7]. Then, statistical inference for these differential equations was of great interest and became a challenging theoretical problem. For a more recent comprehensive discussion, we refer to [8, 9].

The asymptotic theory of parametric estimation for diffusion processes with small white noise based on continuous time observations is well developed and it has been studied by many authors (see, e.g., [10–14]). There have been many applications of small noise in mathematical finance; see, for example, [15–18].

In parametric inference, due to the impossibility of observing diffusions continuously throughout a time interval, it is more practical and interesting to consider asymptotic estimation for diffusion processes with small noise based on discrete observations. There are many approaches to drift estimation for discretely observed diffusions (see, e.g., [19–23]). Long [24] has started the study on parameter estimation for a class of stochastic differential equations driven by small stable noise {𝑍_𝑡,𝑡≥0}. However, there has been no study on parametric inference for stochastic processes with small Lévy noises yet.

In this paper, we are interested in the study of parameter estimation for the following stochastic differential equations driven by more general Lévy noise {𝐿_𝑡,𝑡≥0} based on discrete observations. We will employ the least squares method to obtain an asymptotically consistent estimator.

Let (Ω,ℱ,{ℱ}_𝑡≥0,ℙ) be a basic complete filtered probability space satisfying the usual conditions; that is, the filtration is continuous on the right and ℱ₀ contains all ℙ-null sets. In this paper, we consider a class of stochastic differential equations as follows: 𝑑𝑋_𝑡=𝜃𝑓𝑋_𝑡𝑑𝑡+𝜀𝑔𝑋_𝑡−𝑑𝐿_𝑡[],𝐿,𝑡∈0,1_𝑡=𝑎𝐵_𝑡+𝑏𝑍_𝑡,𝑋(0)=𝑥₀, (1) where 𝑓∶ℝ→ℝ and 𝑔∶ℝ→ℝ are known functions and 𝑎, 𝑏 are known constants. Let {𝐵_𝑡,𝑡≥0} be a standard Brownian motion and let {𝑍_𝑡,𝑡≥0} be a standard 𝛼-stable Lévy motion independent of {𝐵_𝑡,𝑡≥0}, with 𝑍₁∼𝑆_𝛼(1,𝛽,0) for 𝛽∈[0,1], 1<𝛼<2.

Let 𝑋={𝑋_𝑡,𝑡≥0} be a real-valued, stationary process satisfying the stochastic differential equation (1) and we assume that this process is observed at regularly spaced time points {𝑡_𝑖=𝑖/𝑛,𝑖=1,2,…,𝑛}. Assume 𝑋⁰_𝑡 is the solution of the underlying ordinary differential equation (ODE) with the true value of the drift parameter 𝜃₀: 𝑑𝑋⁰_𝑡=𝜃₀𝑓𝑋⁰_𝑡𝑑𝑡,𝑋⁰₀=𝑥₀. (2) Then, we get 𝑋_𝑡𝑖−𝑋_𝑡𝑖−1=^𝑡𝑖_𝑡𝑖−1𝜃₀𝑓𝑋_𝑠𝑑𝑠+𝜀^𝑡𝑖_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐿_𝑠. (3)

2. Preliminaries

In this paper, we denote 𝐶 as a generic constant whose value may vary from place to place.

The following regularity conditions are assumed to hold:(𝒜₁)The functions 𝑓(𝑥) and 𝑔(𝑥) satisfy the Lipschitz conditions; that is, there exists a constant 𝐿>0 such that 𝑓(𝑥)−𝑓𝑦+𝑔(𝑥)−𝑔𝑦≤𝐿𝑥−𝑦,𝑥,𝑦∈ℝ. (4) (𝒜₂) There exist constants 𝑀>0 and 𝑟≥0 satisfying the growth condition 𝑔⁻²(𝑥)≤𝑀1+|𝑥|^𝑟,𝑥∈ℝ. (5) (𝒜₃) There exists a positive constant 𝑁>0 such that 0<|𝑔(𝑥)|≤𝑁<∞. (𝒜₄) For 𝐶_𝑟=2^𝑟−1∨1,𝑟>0, 𝑋_𝑡𝑖−1^𝑟≤𝐶_𝑟𝑋⁰_𝑡𝑖−1^𝑟+𝑋_𝑡𝑖−1−𝑋⁰_𝑡𝑖−1^𝑟. (6)The LSE of 𝜃_𝑛,𝜀 is defined as 𝜃_𝑛,𝜀∶=argmin_𝜃𝜌_𝑛,𝜀(𝜃), (7) where the contrast function 𝜌_𝑛,𝜀(𝜃)=^𝑛_𝑖=1𝑋_𝑡𝑖−𝑋_𝑡𝑖−1−𝜃𝑓𝑋_𝑡𝑖−1Δ𝑡_𝑖−1𝜀𝑔𝑋_𝑡𝑖−1². (8) Then the 𝜃_𝑛,𝜀 can be represented explicitly as follows: 𝜃_𝑛,𝜀=∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓𝑋_𝑡𝑖−1𝑋_𝑡𝑖−𝑋_𝑡𝑖−1𝑛⁻¹∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓²𝑋_𝑡𝑖−1. (9) Based on (3) and (9), there is a special decomposition for 𝜃_𝑛,𝜀𝜃_𝑛,𝜀=𝜃₀∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓𝑋_𝑡𝑖−1∫^𝑡𝑖_𝑡𝑖−1𝑓𝑋_𝑠𝑑𝑠𝑛⁻¹∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓²𝑋_𝑡𝑖−1+𝜀∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓𝑋_𝑡𝑖−1∫^𝑡𝑖_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐿_𝑠𝑛⁻¹∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓²𝑋_𝑡𝑖−1=𝜃₀+𝜃₀∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓𝑋_𝑡𝑖−1∫^𝑡𝑖_𝑡𝑖−1𝑓𝑋_𝑠−𝑓𝑋_𝑡𝑖−1𝑑𝑠𝑛⁻¹∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓²𝑋_𝑡𝑖−1+𝜀∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓𝑋_𝑡𝑖−1∫^𝑡𝑖_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐿_𝑠𝑛⁻¹∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓²𝑋_𝑡𝑖−1=𝜃₀+𝜃₀∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓𝑋_𝑡𝑖−1∫^𝑡𝑖_𝑡𝑖−1𝑓𝑋_𝑠−𝑓𝑋_𝑡𝑖−1𝑑𝑠𝑛⁻¹∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓²𝑋_𝑡𝑖−1+𝑏𝜀∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓𝑋_𝑡𝑖−1∫^𝑡𝑖_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝑍_𝑠𝑛⁻¹∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓²𝑋_𝑡𝑖−1+𝑎𝜀∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓𝑋_𝑡𝑖−1∫^𝑡𝑖_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐵_𝑠𝑛⁻¹∑^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓²𝑋_𝑡𝑖−1∶=𝜃₀+Φ₂(𝑛,𝜀)Φ₁()+Φ𝑛,𝜀³(𝑛,𝜀)Φ₁()+Φ𝑛,𝜀⁴(𝑛,𝜀)Φ₁().𝑛,𝜀 (10) Now we give an explicit expression for 𝜀⁻¹(𝜃_𝑛,𝜀−𝜃₀). By using (10), we have 𝜀⁻¹𝜃_𝑛,𝜀−𝜃₀=𝜀⁻¹Φ₂(𝑛,𝜀)Φ₁()+𝜀𝑛,𝜀⁻¹Φ₃(𝑛,𝜀)Φ₁()+𝜀𝑛,𝜀⁻¹Φ₄(𝑛,𝜀)Φ₁()Ψ𝑛,𝜀∶=²(𝑛,𝜀)Φ₁()+Ψ𝑛,𝜀³(𝑛,𝜀)Φ₁()+Ψ𝑛,𝜀⁴(𝑛,𝜀)Φ₁().𝑛,𝜀 (11)

One of the important tools we will employ is the underlying lemma (see (3.5) in the Lemma 3.2 of [24]).

Lemma 1. Under conditions (𝒜₁)-(𝒜₂), one has 𝑋_𝑡−𝑋⁰_𝑡≤𝜀𝑒^{𝐿|𝜃0|𝑡𝑡}₀𝑔𝑋_𝑠−𝑑𝑍_𝑠,sup_0≤𝑡≤1𝑋_𝑡−𝑋⁰_𝑡⟶_𝑃0,as𝜀⟶0. (12)(13)

3. Asymptotic Property of the Least Squares Estimator

Theorem 2. Under the conditions (𝒜₁)--(𝒜₄), as 𝑛→∞,  𝜀→0, 𝑛𝜀→∞, and 𝑛𝜀^{𝛼/(𝛼−1)}→∞, one has 𝜀⁻¹𝜃_𝑛,𝜀−𝜃₀∫⟹𝑎¹⁰𝑔⁻²𝑋⁰_𝑠𝑓²𝑋⁰_𝑠𝑑𝑠^1/2∫¹₀𝑔⁻²𝑋⁰_𝑠𝑓²𝑋⁰_𝑠𝑁𝑑𝑠+𝑏¹₀𝑔𝑋⁰_𝑠^−2𝛼𝑓𝑋⁰_𝑠𝑔𝑋⁰_𝑠^𝛼₊𝑑𝑠^1/𝛼𝑈₁−¹₀𝑔𝑋⁰_𝑠^−2𝛼𝑓𝑋⁰_𝑠𝑔𝑋⁰_𝑠^𝛼₋𝑑𝑠^1/𝛼𝑈₂×¹₀𝑔⁻²𝑋⁰_𝑠𝑓²𝑋⁰_𝑠𝑑𝑠⁻¹, (14) where 𝑈₁ and 𝑈₂ are independent random variables with 𝛼-stable distribution 𝑆_𝛼(1,𝛽,0) and  𝑁 is an independent random variable with standard normal distribution.

The theorem will be proved by establishing several propositions. We will consider the asymptotic behaviors of Φ₁(𝑛,𝜀), Ψ_𝑖(𝑛,𝜀), 𝑖=2,3,4, respectively.

Proposition 3. Under conditions (𝒜₁)--(𝒜₄), and 𝑛→∞, 𝜀→0, one has Φ₁(𝑛,𝜀)⟶_𝑃¹₀𝑔⁻²𝑋⁰_𝑠𝑓²𝑋⁰_𝑠𝑑𝑠. (15)

Proof. Under conditions (𝒜₁)--(𝒜₃), Proposition 3 can be proved by using condition (𝒜₄) (see the proof of Proposition 3.3 in [24]).

Proposition 4. Under conditions (𝒜₁)--(𝒜₄), as 𝑛→∞, 𝜀→0 and 𝑛𝜀→∞, one has Ψ₂(𝑛,𝜀)⟶_𝑃0. (16)

Proof. For 𝑡_𝑖−1≤𝑡≤𝑡_𝑖, 𝑖=1,2,…,𝑛, 𝑋_𝑡=𝑋_𝑡𝑖−1+^𝑡_𝑡𝑖−1𝜃₀𝑓𝑋_𝑠𝑑𝑠+𝜀^𝑡_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐿_𝑠. (17) It follows that 𝑋_𝑡−𝑋_𝑡𝑖−1≤^𝑡_𝑡𝑖−1𝜃₀𝑓𝑋_𝑠−𝑓𝑋_𝑡𝑖−1+𝑓𝑋_𝑡𝑖−1𝑑𝑠+𝜀^𝑡_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐿_𝑠≤𝜃₀𝑀^𝑡_𝑡𝑖−1𝑓𝑋_𝑠−𝑓𝑋_𝑡𝑖−1+𝑛⁻¹𝜃₀𝑓𝑋_𝑡𝑖−1+𝑎𝜀sup_{𝑡𝑖−1≤𝑡≤𝑡𝑖}^𝑡_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐵_𝑠+𝑏𝜀sup_{𝑡𝑖−1≤𝑡≤𝑡𝑖}^𝑡_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝑍_𝑠. (18) Using Gronwall inequality, we get 𝑋_𝑡−𝑋_𝑡𝑖−1≤𝑒^{|𝜃0|𝑀(𝑡−𝑡𝑖−1)}𝜃₀𝑓𝑋_𝑡𝑖−1𝑛+𝑎𝜀sup_{𝑡𝑖−1≤𝑡≤𝑡𝑖}^𝑡_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐵_𝑠+𝑏𝜀sup_{𝑡𝑖−1≤𝑡≤𝑡𝑖}^𝑡_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝑍_𝑠, (19) which yields sup_{𝑡𝑖−1≤𝑡≤𝑡𝑖}𝑋_𝑡−𝑋_𝑡𝑖−1≤𝑒^{|𝜃0|𝑀/𝑛}𝜃₀𝑓𝑋_𝑡𝑖−1𝑛+𝑎𝜀sup_{𝑡𝑖−1≤𝑡≤𝑡𝑖}^𝑡_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐵_𝑠+𝑏𝜀sup_{𝑡𝑖−1≤𝑡≤𝑡𝑖}^𝑡_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝑍_𝑠, (20) thus, under conditions (𝒜₁) and (𝒜₃), Φ₂(≤𝜃𝑛,𝜀)_0𝑛𝑖=1𝑀1+𝑋_𝑡𝑖−1^𝑟𝑓𝑋_𝑡𝑖−1×^𝑡_𝑡𝑖−1𝑓𝑋_𝑠−𝑓𝑋_𝑡𝑖−1≤𝜃𝑑𝑠𝑀𝐾₀𝑛^𝑛_𝑖=11+𝑋_𝑡𝑖−1^𝑟𝑓𝑋_𝑡𝑖−1×sup_{𝑡𝑖−1≤𝑡≤𝑡𝑖}𝑋_𝑡−𝑋_𝑡𝑖−1≤𝑀𝐾𝜃₀²𝑒^{|𝜃0|𝑀/𝑛}𝑛^2𝑛_𝑖=11+𝑋_𝑡𝑖−1^𝑟𝑓𝑋_𝑡𝑖−1²+𝑀𝐾𝜃₀²𝑒^{(|𝜃0|𝑀)/𝑛}𝑛𝑏𝜀^𝑛_𝑖=11+𝑋_𝑡𝑖−1^𝑟𝑓𝑋_𝑡𝑖−1×sup_{𝑡𝑖−1≤𝑡≤𝑡𝑖}^𝑡_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝑍_𝑠+𝑀𝐾𝜃₀²𝑒^{|𝜃0|𝑀/𝑛}𝑛𝑎𝜀^𝑛_𝑖=11+𝑋_𝑡𝑖−1^𝑟𝑓𝑋_𝑡𝑖−1×sup_{𝑡𝑖−1≤𝑡≤𝑡𝑖}^𝑡_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐵_𝑠∶=Φ_2,1(𝑛,𝜀)+Φ_2,2(𝑛,𝜀)+Φ_2,3(𝑛,𝜀). (21) Then, Ψ₂(𝑛,𝜀)≤𝜀⁻¹Φ_2,1(𝑛,𝜀)+𝜀⁻¹Φ_2,2(𝑛,𝜀)+𝜀⁻¹Φ_2,3(𝑛,𝜀)∶=Ψ_2,1(𝑛,𝜀)+Ψ_2,2(𝑛,𝜀)+Ψ_2,3(𝑛,𝜀). (22) Using (13) in Lemma 1, conditions (𝒜₁) and (𝒜₄), we get Ψ_2,1(𝑛,𝜀)→_𝑃0 as 𝑛→∞, 𝜀→0 and 𝑛𝜀→∞ (see (3.26) in [24]). By using the same techniques, under condition(𝒜₂), we can prove that Ψ_2,𝑗(𝑛,𝜀)→_𝑃0,  𝑗=2,3, as 𝑛→∞, 𝜀→0, respectively.

Proposition 5. Under conditions (𝒜₁)--(𝒜₄), as 𝑛→∞, 𝜀→0 and 𝑛𝜀^{𝛼/(𝛼−1)}→∞, one has Ψ₃(𝑛,𝜀)⟹𝑏¹₀𝑔𝑋⁰_𝑠^−2𝛼𝑓𝑋⁰_𝑠𝑔𝑋⁰_𝑠^𝛼₊𝑑𝑠^1/𝛼𝑈₁−𝑏¹₀𝑔𝑋⁰_𝑠^−2𝛼𝑓𝑋⁰_𝑠𝑔𝑋⁰_𝑠^𝛼₋𝑑𝑠^1/𝛼𝑈₂. (23)

Proof. Under conditions (𝒜₁)--(𝒜₃), Proposition 5 can be proved by using condition (𝒜₄) (see the proof of Proposition 4.4 in [24]).

Proposition 6. Under conditions (𝒜₁)--(𝒜₄), as 𝑛→∞, 𝜀→0, one has Ψ₄(𝑛,𝜀)⟹𝑎¹₀𝑔⁻²𝑋⁰_𝑠𝑓²𝑋⁰_𝑠𝑑𝑠^1/2𝑁. (24)

Proof. Note that Ψ₄(𝑛,𝜀)=𝑎^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1𝑓𝑋_𝑡𝑖−1^𝑡𝑖_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐵_𝑠=𝑎^𝑛_𝑖=1𝑔⁻²𝑋⁰_𝑡𝑖−1𝑓𝑋⁰_𝑡𝑖−1^𝑡𝑖_𝑡𝑖−1𝑔𝑋⁰_𝑠−𝑑𝐵_𝑠+𝑎^𝑛_𝑖=1𝑔⁻²𝑋⁰_𝑡𝑖−1𝑓𝑋⁰_𝑡𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑔𝑋_𝑠−−𝑔𝑋⁰_𝑠−𝑑𝐵_𝑠+𝑎^𝑛_𝑖=1𝑔⁻²𝑋⁰_𝑡𝑖−1𝑓𝑋_𝑡𝑖−1−𝑓𝑋⁰_𝑡𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐵_𝑠+𝑎^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1−𝑔⁻²𝑋⁰_𝑡𝑖−1𝑓𝑋⁰_𝑡𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐵_𝑠+𝑎^𝑛_𝑖=1𝑔⁻²𝑋_𝑡𝑖−1−𝑔⁻²𝑋⁰_𝑡𝑖−1×𝑓𝑋_𝑡𝑖−1−𝑓𝑋⁰_𝑡𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑔𝑋_𝑠−𝑑𝐵_𝑠∶=⁵_𝑗=1Ψ_4,𝑗(𝑛,𝜀). (25) For Ψ_4,1(𝑛,𝜀), let 𝑌_𝑖=∫^𝑡𝑖_𝑡𝑖−1𝑔(𝑋_𝑠−)𝑑𝐵_𝑠, 𝑗=1,…,𝑛. Then it is easy to see that 𝑌_𝑖∼𝑁(0,∫^𝑡𝑖_𝑡𝑖−1𝑔²(𝑋_𝑠−)𝑑𝑠) and 𝑌₁,…,𝑌_𝑛 are independent normal random variables.
It follows that Ψ_4,1(𝑛,𝜀)=𝑎^𝑛_𝑖=1𝑔⁻²𝑋⁰_𝑡𝑖−1𝑓𝑋⁰_𝑡𝑖−1𝑌_𝑖∼𝑁0,𝑎^2𝑛_𝑖=1𝑔⁻⁴𝑋⁰_𝑡𝑖−1𝑓²𝑋⁰_𝑡𝑖−1^𝑡𝑖_𝑡𝑖−1𝑔²𝑋_𝑠−𝑑𝑠⟹𝑎¹₀𝑔⁻²𝑋⁰_𝑠𝑓²𝑋⁰_𝑠𝑑𝑠^1/2𝑁 (26) as 𝑛→∞, 𝜀→0.
For Ψ_4,2(𝑛,𝜀), using Markov inequality and Ito’s isometry property, for any given 𝜂>0, Ψ_4,2()≤1𝑛,𝜀𝜂𝔼𝑎^𝑛_𝑖=1𝑔⁻²𝑋⁰_𝑡𝑖−1𝑓𝑋⁰_𝑡𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑔𝑋_𝑠−−𝑔𝑋⁰_𝑠−𝑑𝐵_𝑠≤𝑎𝜂^𝑛_𝑖=1𝑔⁻²𝑋⁰_𝑡𝑖−1𝑓𝑋⁰_𝑡𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑔𝑋_𝑠−−𝑔𝑋⁰_𝑠−²𝑑𝑠^1/2≤𝐿𝑎𝜂^𝑛_𝑖=1𝑔⁻²𝑋⁰_𝑡𝑖−1𝑓𝑋⁰_𝑡𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑋_𝑠−−𝑋⁰_𝑠−²𝑑𝑠^1/2≤𝐿𝑎𝜂^𝑛_𝑖=1𝑔⁻²𝑋⁰_𝑡𝑖−1𝑓𝑋⁰_𝑡𝑖−1×sup_{𝑡𝑖−1≤𝑡≤𝑡𝑖}𝑋_𝑠−−𝑋⁰_𝑠−𝑛^−1/2. (27) By using (13), Ψ_4,2(𝑛,𝜀)→0, as 𝑛→∞,𝜀→0.
Applying similar techniques to Ψ_4,𝑗(𝑛,𝜀), 𝑗=3,4,5, we get Ψ_4,𝑗(𝑛,𝜀)→0, 𝑗=3,4,5, as 𝑛→∞, 𝜀→0.

Now we can prove Theorem 2.

Proof. By using Propositions 3, 4, 5, 6 and Slutsky’s theorem, we can get the conclusion.

4. Example

We consider the following nonlinear SDE driven by general Lévy noises: 𝑑𝑋_𝑡=𝜃𝑋_𝑡𝜀𝑑𝑡+1+𝑋²_𝑡−𝑑𝐿_𝑡,𝑡∈[0,1];𝑋₀=𝑥₀, (28) where 𝑓(𝑥)=𝑥, 𝑔(𝑥)=1/(1+𝑥²),  𝑥₀ and 𝜀 are known constants, and 𝜃≠0 is an unknown parameter.

For simplicity, let 𝑥₀>0, 𝜀=0; we get the ODE: 𝑑𝑋⁰_𝑡=𝜃₀𝑋⁰_𝑡𝑑𝑡,𝑡∈[0,1];𝑋⁰₀=𝑥₀ (29) and the solution 𝑋⁰_𝑡=𝑥₀𝑒^𝜃0𝑡. (30) Then, the asymptotic distribution is 𝑎¹₀1+𝑥²₀𝑒^2𝜃0𝑠2𝑥²₀𝑒^2𝜃0𝑠𝑑𝑠^−1/2𝑁∫+𝑏¹⁰1+𝑥²₀𝑒^{2𝜃0𝑠𝛼}𝑥^𝛼₀𝑒^{𝛼𝜃0𝑠}𝑑𝑠^1/𝛼∫¹₀1+𝑥²₀𝑒^2𝜃0𝑠2𝑥²₀𝑒^2𝜃0𝑠𝑆𝑑𝑠^𝛼1,𝛽,0. (31)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.