Asymptotic Properties of Estimators for Ornstein-Uhlenbeck Processes with Small Symmetric α-Stable Motions

PANYurong，JIAChaoyong，LIUXiaoyan

1 School of Science， Bengbu University， Bengbu 233030， China2 College of Arts and Sciences， University of La Verne， La Verne 91750， USA

Abstract： The asymptotic behaviors for estimators of the drift parameters in the Ornstein-Uhlenbeck process driven by small symmetric α-stable motion are studied in this paper. Based on the discrete observations， the conditional least squares estimators(CLSEs) of all the parameters involved in the Ornstein-Uhlenbeck process are proposed. We establish the consistency and the asymptotic distributions of our estimators as ε goes to 0 and n goes to ∞ simultaneously.

Key words： Ornstein-Uhlenbeck process； symmetric α-stable motion； conditional least squares estimator(CLSE)； consistency； asymptotic distribution

Introduction

It is well-known that the Ornstein-Uhlenbeck process is defined as the unique strong solution to the following stochastic differential equation

dXt=(-θXt+γ)dt+εdBt，

(1)

withθ，γ∈，ε∈+.Btis a standard Brownian motion. This differential equation is also called Vasicek model， which is a popular short-term interest rate model in the financial world. Parameter estimation problem about the Vasicek model is an important issue in application， and it has been well studied. Kutoyants[1]proved that the maximum likelihood estimators of the parametersγandθwere consistent and had asymptotic normality whenθ>0. Jiang and Dong[2]discussed the asymptotic properties of estimators of two parametersγ，θwhenθ<0 orθ=0. Many surveys and literatures discussed the problem of unknown parameter estimation for more general stochastic differential equations[3-5]. Manynancial processes exhibited discontinuous sample paths and heavy tailed distributions. So， some scholars started to study the parameter estimation for Ornstein-Uhlenbeck processes driven by α-stable processes[6-11]. On the other hand， small diffusion asymptotic theory has

been extensively studied with wide applications to real world problems. There exist many meaningful works about statistical estimation for stochastic differential equations with small white noises[12-16]. However， in the case of small α-stable Lévy noises， the related literatures are very limited. The main reason is that α-stable process has the infinite variance. Long[17]studied the least squares estimator(LSE) for discretely observed Ornstein-Uhlenbeck processes with small Lévy noises， the consistency and the rate of convergence of the LSE were established. In a similar framework， Longetal.[18-19]extended Ref. [17] to more complicated cases. Recently， Shen and Yu[20]dealt with the problem of the least squares estimation for Ornstein-Uhlenbeck processes with small fractional Lévy noises.

In the present paper， we mainly consider the Ornstein-UhlenbeckX=(Xt，t≥0) which is the solution of the following stochastic differential equation with small symmetric α-stable Lévy motion

dXt=(-θXt+γ)dt+εdZt，t∈[0，1]，X0=x0，(2)

1 Construction of CLSEs

1.1 Preliminaries

The following Lemma 1 gives a moment equality forα-stable integrals， which is a very critical tool of our research. We may refer to Ref. [17] for its detailed proof.

Lemma 1Letf(·)：[0， 1]→+be a deterministic function satisfyingfα(s)ds<∞. Then， for any 0

where

1.2 CLSEs

(3)

Then it is easy to obtain the following stochastic regressive equation.

(4)

The CLSEs of (γ，θ) can be given by minimizing the sum of squares.

(5)

By basic calculation， we easily obtain

(6)

and

(7)

2 Consistency of CLSEs

This section discusses the consistency of the proposed estimators. To get the main result， we need the following assumption.

Our main object is to expound and to prove Theorem 1 which gives the consistency of the CLSEs.

Theorem 1If the condition(A1) is satisfied， then the two estimators are consistent，i.e.，

Before proving Theorem 1， we need to establish several preliminary propositions.

As we know， the drift coefficient and diffusion coefficient of Eq.(2) satisfy the Lipschitz condition and linear growth condition， so this differential equation has a unique strong solution， which can be expressed as

(8)

According to Eqs.(4) and(6)， we find

then

(9)

Therefore， we have

-log1+eθ0nΦ2(n, ε)nΦ1(n, ε)é?êêù?úúnΦ1(n, ε)eθ0nΦ2(n, ε)·eθ0nΦ2(n, ε)Φ1(n, ε).

(10)

Next， we start to discuss the asymptotic behavior of Φ1(n，ε) and Φ2(n，ε)， respectively.

ProofBy Eq.(8)， we have the following decomposition.

Φ1，1(n，ε)+Φ1，2(n，ε)+Φ1，3(n，ε)+Φ1，4(n，ε)+Φ1，5(n，ε)+Φ1，6(n，ε).

(11)

According to the basic calculation， whenn→∞， we can get

(12)

For Φ1， 2(n，ε)， using the Markov inequality and Lemma 1， we find that for any givenδ>0，

(13)

which tends to zero asn→∞，ε→0.

By the Markov inequality and Lemma 1， we find that for 1≤p<α,

P(|Φ1， 3(n，ε)|>δ)≤

O(εpn1-p/2)，

(14)

Using the same proof technique as Φ1， 3(n，ε)， we find that the following result holds：

(15)

Now we estimate Φ1， 6(n，ε).

Φ1， 6， 1(n，ε)+Φ1， 6， 2(n，ε).

(16)

(17)

In additional， for Φ1， 6， 2(n，ε)， using the Markov inequality， Lemma 1 and the independence of two stochastic integrals， we find that for any givenδ>0,which tends to zero asn→∞，ε→0.

(18)

Finally， combining Eqs.(11)-(18)， we conclude that Proposition 1 holds.

ProofAn elementary calculation shows that

Φ2，1(n，ε)+Φ2，2(n，ε)+Φ2，3(n，ε)+Φ2，4(n，ε).

(19)

For Φ2， 1(n，ε)， applying the Markov inequality and Lemma 1， it is clear that for any givenδ>0

(20)

which tends to zero whenn→∞ andε→0.

For Φ2，2(n，ε)， by the Markov inequality， Lemma 1 and the independence of two stochastic integrals， we have for any givenδ>0

P(|Φ2， 2(n，ε)|>δ)≤

O(ε2n1-1/α).

(21)

1+eθ0nΦ2(n, ε)nΦ1(n, ε)é?êêù?úúnΦ1(n, ε)eθ0nΦ2(n, ε)P→e.

Finally， we complete the proof of Theorem 1.

3 Asymptotic Distributions of CLSEs

In this section， we will be devoted to studying the

asymptotic distribution of the CLSEs. To obtain the result， we need the condition below.

Theorem 2Suppose that the Condition(A2) is fulfilled， then the convergence results hold.

whereUdenotes a random variable with standard α-stable distributionSα(1，0，0)， and

ProofBy Eqs.(4) and(6)， we obtain the following explicit expression

then

(22)

Applying Eq.(8) to Φ3(n，ε)， we easily have

Φ3，1(n，ε)+Φ3，2(n，ε)+Φ3，3(n，ε)+Φ3，4(n，ε).

(23)

(24)

and

(25)

Using the Markov inequality， Lemma 1 and the independence of two stochastic integrals， the following convergence results hold

(26)

(27)

Thus， combining Eqs.(22)-(27) and Proposition 1， under Condition(A2)， we deduce that the following result holds.

Φ4， 1(n，ε)+Φ4，2(n，ε)+Φ4，3(n，ε)+Φ4， 4(n，ε).

In conclusion， we finish the proof of Theorem 2.

4 Conclusions

In this paper， we mainly study drift estimation for discretely observed stochastic differential equations driven by small symmetric α-stable motions. We conduct the CLSEs of two unknown parameters. Under certain regularity conditions， we prove that our estimators are consistent and give the rate of convergence of the CLSEs when ε→0 andn→∞ simultaneously.