Parameter estimation for non-stationary reflected Ornstein–Uhlenbeck processes driven by α -stable noises

Statistics and Probability Letters 156 (2020) 108617

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Parameter estimation for non-stationary reflected Ornstein–Uhlenbeck processes driven by α -stable noises Xuekang Zhang a , Haoran Yi a , Huisheng Shu b , a b

∗

School of Information Science and Technology, Donghua University, Shanghai, China School of Science, Donghua University, Shanghai, China

article

info

Article history: Received 23 April 2019 Received in revised form 14 July 2019 Accepted 6 September 2019 Available online 12 September 2019 Keywords: Non-stationary reflected Ornstein–Uhlenbeck processes α -stable processes Trajectory fitting method Consistency Stable distribution

a b s t r a c t This paper is concerned with the parameter estimation problem for non-stationary reflected Ornstein–Uhlenbeck processes driven by stable noises by using the trajectory fitting method combined with the weighted least squares technique. Under some regularity conditions, we obtain the consistency and the rate of convergence of the proposed estimator of the drift rate. The asymptotic stability property of the estimator in our setting is also proved. © 2019 Elsevier B.V. All rights reserved.

1. Introduction During the last decades, reflected stochastic dynamics have received considerable attention in many fields for example in queueing system (see, e.g., Borovkov, 1984; Harrison, 1986; Ward and Glynn, 2003a,b), financial engineering (see, e.g., Goldstein and Keirstead, 1997; Hanson et al., 1999; Bo et al., 2011a,b), mathematical biology (see Ricciardi and Sacerdote, 1987). In practice, some important fields of performance of a queueing system (see, e.g., customers’ waiting times, traffic intensities) may not be directly observable and therefore such performance measures and their related model parameters need to be statistically inferred from the available data. Statistical inference for reflected stochastic differential equations attracts more and more attention of statisticians. Bo et al. (2011c) are first to study the asymptotics behaviour of the maximum likelihood estimator (MLE) for the ergodic reflected Ornstein–Uhlenbeck (ROU) processes, and then Lee et al. (2012) focus on discuss the properties of the sequential MLE for ROU processes. In particular, Zang and Zhang (2016) study the strong consistency and asymptotic normality of MLE for ROU processes with the general drift coefficient. Zang and Zhu (2016) and Zang and Zhang (2019) discuss the asymptotics behaviour of estimators of the parameters in the non-stationary ROU process

⎧ ⎨dXt = θ Xt dt + dBt + dLt , X ≥ bL for all t ≥ 0, ⎩ t X0 = x0 ,

(1.1)

where bL > 0, x0 ≥ bL , θ ≥ 0, and {Bt , t ≥ 0} is a one-dimensional standard Wiener process defined on a complete probability space (Ω , F , P) equipped with a right continuous and increasing family of σ -algebras {Ft , t ≥ 0}. The process ∗ Corresponding author. E-mail addresses: [email protected] (X. Zhang), [email protected] (H. Yi), [email protected] (H. Shu). https://doi.org/10.1016/j.spl.2019.108617 0167-7152/© 2019 Elsevier B.V. All rights reserved.

2

X. Zhang, H. Yi and H. Shu / Statistics and Probability Letters 156 (2020) 108617

L = {Lt ; t ≥ 0} is the minimal nondecreasing and nonnegative process, which makes Xt ≥ bL for all t ≥ 0 with L0 = 0. The process L increases only when X hits the boundary bL , so that ∞

∫

I{Xt >bL } dLt = 0.

0

There also exist some works which study the parameter estimate problem in reflected stochastic processes with Lévy noises. Bo and Yang (2012) investigate a sequential maximum likelihood estimator of the unknown drift parameter for a class of reflected generalized Ornstein–Uhlenbeck processes driven by spectrally positive Lévy processes. Zhao and Zhang (2018) study the maximum likelihood estimation for the ROU processes with Lévy noises. On the other hand, it is well known that many natural phenomena also perform a power-law distribution. Some examples of these phenomena are finance and economics (see e.g., Bouchaud, 2001; Gabaix, 2009), biology (see Jeong et al., 2000). By the general central limit theorem (see Nolan, 2009), it follows that the sum of a sequence of independently and identically distributed (for short i.i.d.) random variables with power-law tail distributions will tend to an α -stable distribution. Hence, it is more reasonable to replace the driving Brownian motion by the α -stable process for such phenomena. Therefore, in this paper, it is interesting to study the parametric estimation for non-stationary reflected Ornstein– Uhlenbeck processes with α -stable noises

⎧ ⎨dXt = θ Xt dt + dZt + dLt , X ≥ bL for all t ≥ 0, ⎩ t X0 = x0 ,

(1.2)

where {Zt , t ≥ 0} is a standard α -stable Lévy motion, with Z1 ∼ Sα (1, β, 0) and β ∈ [−1, 1] is a skewness parameter. The works of Bo and Yang (2012) and Zhao and Zhang (2018) cannot cover α -stable noises, since the α -stable process has infinite variance. To the best of our knowledge, Eq. (1.2) is new. In addition, statistical inference problems of stochastic differential equations driven by α -stable processes have been studied by Hu and Long (2007, 2009a,b), Zhang and Zhang (2013), Pan and Yan (2018), Zhang et al. (2019a,b). In particular, due to the infinite variance property of α -stable processes, Hu and Long (2007) study the strong consistency and asymptotic distributions of the estimator for non-ergodic OU processes driven by α -stable processes by using the trajectory fitting method combined with the weighted least squares technique. The trajectory fitting method was first proposed by Kutoyants (1991) as a numerically attractive alternative to the well-developed MLEs for continuous diffusion processes (see, e.g., Dietz and Kutoyants, 1997, 2003; Dietz, 2001; Kutoyants, 2004). To obtain our estimator we introduce t

∫

Xs ds,

At =

t ≥ 0.

0

Eq. (1.2) can be written as Xt = x0 + θ At + Zt + Lt . Let ωt be a deterministic positive (weight) function. Multiply the above equation by ωt we have

ωt Xt = ωt x0 + θ ωt At + ωt Zt + ωt Lt . The weighted trajectory fitting estimate (TFE) of θ is to minimize T

∫

|ωt Xt − (ωt x0 + θωt At )|2 dt , 0

where T is the time instant. It is easy to see that the minimum is attained when θ is given by

∫T ˆ θT =

0

ωt2 (Xt − x0 )At dt . ∫T 2 2 ωt At dt 0

(1.3)

Since the observed process {Xt } is the solution of (1.2), we find

∫T 2 ∫T 2 ωt At Zt dt ωt At Lt dt 0 ˆ θT = θ + ∫ T + ∫0 T . 2 2 ωt At dt ωt2 A2t dt 0 0

(1.4)

In this paper, we consider the asymptotics of the weighted TFE ˆ θT of Eq. (1.2). We obtain some new asymptotic distributions for the weighted TFE in our setting, which are stable distributions.

X. Zhang, H. Yi and H. Shu / Statistics and Probability Letters 156 (2020) 108617

3

2. Preliminaries In this section, we give the precise definition for the solution to Eq. (1.2) and present some preliminaries on α -stable distributions, α -stable Lévy processes, and moment inequalities for stable stochastic integrals. Throughout the paper, let D(R+ , R+ ) denotes the space of càdlàg functions from R+ to R+ equipped with Skorohod topology. We denote also by D0 (R+ , R+ ) the set of the functions f ∈ D(R+ , R+ ), such that f (0) = 0. Let we use notation ‘‘ →p ’’ to denote ‘‘convergence d

in probability’’ and notation ‘‘ ⇒’’ to denote ‘‘convergence in distribution’’. We write ‘‘ =’’ for equality in distribution. Similar to Doney and Zhang (2005) and Zhang (2017), here, we will give the precise definition for the solution to Eq. (1.2). Definition 2.1. A pair of one-dimensional Ft -adapted process (Xt , Lt , t ≥ 0) is called a solution of Eq. (1.2) if, (1) X ∈ D(R+ , R+ ) and X0 = x0 ≥ bL ; (2) L∫ is non-decreasing, L ∈ D0 (R+ , R+ ); t (3) 0 I{Xs =bL } dLs = Lt ; (4) for every t ≥ 0, X t = x0 + θ

t

∫

Xs ds + Zt + Lt ,

a.s.

0

Recall that a random variable η is said to follow a stable distribution, denote by η ∼ Sα (σ , β, µ), if it has characteristic function of the following form:

} { ( ⎧ απ ) α α ⎪ ⎨ exp −σ |u| 1 − iβ sgn(u) tan 2 + iµu , if α ̸= 1 ) } { ( φη (u) = E exp{iuη} = 2 ⎪ ⎩ exp −σ |u| 1 + iβ sgn(u) log |u| + iµu , if α = 1, π where α ∈ (0, 2], σ ∈ (0, ∞), β ∈ [−1, 1], and µ ∈ (−∞, ∞) are the index of stability, the scale, skewness, and location parameters, respectively. We denote η ∼ Sα (σ , β, µ). When µ = 0, we say η is strictly α -stable. If in addition β = 0, we call η symmetric α -stable. For more detailed discussion on stable distributions, we refer to Janicki and Weron (1994), Sato (1999), and Applebaum (2009). d

1

Note that α -stable Lévy motion has the scaling (or self-similarity) property Zat = a α Zt with a > 0. It is well known that every α -stable Lévy process has a unique modification which is càdlàg (right continuous with left limits) and which is also an α -Lévy process. Therefore we assume that our α -stable Lévy processes are càdlàg. We refer to Sato (1999) for more details on α -stable Lévy processes. Now we give the moment inequalities for stable stochastic integrals in the following lemma, which comes from Long (2010) and will play an important role in the proof of our main results. Lemma 2.1. Let φ (t) be a predictable process satisfying 0 |φ (t)|α dt < ∞ almost surely for T < ∞, and F : [0, ∞) → [0, ∞) be a continuous function. We assume that (i) either φ is nonnegative or Z is symmetric, and (ii) there exist positive constants λ0 , C and α0 < α such that F (λν ) ≤ C λα0 F (ν ) for all ν > 0 and all λ ≥ λ0 . Then there exist positive constants C1 and C2 depending only on α, α0 , β, C , and λ0 such that for each T > 0

∫T

⎡ ⎛

T

(∫

C1 E ⎣F ⎝

|φ (t)|α dt

0

) α1

⎞⎤

⎡ ⎛ ⏐∫ t ⏐)] (∫ ⏐ ⏐ ⎠⎦ ≤ E F sup ⏐ φ (s)dZs ⏐ ≤ C2 E ⎣F ⎝ ⏐ ⏐ t ≤T

T

[ (

0

|φ (t)|α dt

) α1

⎞⎤ ⎠⎦ .

0

A important lemma is the following well-known integral version of the Toeplitz lemma, which comes from Dietz and Kutoyants (1997). Lemma 2.2 (Toeplitz Lemma). If ϕT is a probability measure defined on [0, ∞) such that ϕT ([0, T ]) = 1 and ϕT ([0, K ]) → 0 as T → ∞ for each K > 0, then

∫

T

lim

T →∞

ft ϕT (dt) = f∞

0

for every bounded and measure function f : [0, ∞) → R for which the limit f∞ := limt →∞ ft exists. In the paper, for convenience, we let T

∫

2 2θ t dt te

ω

h1 (T ) = 0

,

T

∫ h2 (T ) = 0

ωt2 eθ t dt .

4

X. Zhang, H. Yi and H. Shu / Statistics and Probability Letters 156 (2020) 108617

We will make use of the following assumptions: (A1) The index of stability α of the state process Z satisfies α ∈ (1, 2) h (M) (A2) The positive weight function ωt satisfies that hi (T ) → ∞ and hi (T ) → 0 as T → ∞ for each M > 0 and i = 1, 2. i (A3) The positive weight function ωt satisfies the following condition:

ωt1 t2 = ωt1 ωt2 ,

t1 , t2 ≥ 0.

(A4) There exist constants C0 > 0, b < 0 such that when T is large enough,

ωt2 eθ t h2 (T )

≤ C0 eb(T −t) ,

t ∈ [0, T ].

3. Consistency of estimator ˆ θT In the section, we give some results about the consistency of the weighted TFE ˆ θT . Before we state our main results, we first establish a preliminary lemma. Lemma 3.1.

If X ∈ D+ (R+ , R+ ) and X0 = x0 ≥ bL , then there exists a unique solution (Xt , Lt , t ≥ 0) to Eq. (1.2).

Proof. We will use the Picard iteration method. Let Xt0 = x0 ≥ bL , t ≥ 0. By Theorem 3.2 of Zhang (2017), it follows that (Xtn+1 , Ltn+1 , t ≥ 0) is the solution of the equation Xtn+1 = x0 + θ

t

∫

Xsn ds + Zt + Lnt +1 ,

n ∈ N.

(3.1)

0

We begin with the case n = 0. First, by the reflection principle, it follows that L1t

{

} = max 0, max (−x0 − Zs ) . 0≤s≤t

Thus, we have

|Xt1 | ≤ |x0 + Zt | + sup |x0 + Zs | ≤ 2 sup |x0 + Zs | , s∈[0,t ]

s∈[0,t ]

and this implies that sup |Xs1 | ≤ 2 sup |x0 + Zs | .

s∈[0,t ]

s∈[0,t ]

By Lemma 2.1, it follows that

[

Xs1

sup |

E

s∈[0,t ]

] [ ] 1 | ≤ E 2x0 + 2 sup |Zs | ≤ 2x0 + 2C2 t α . s∈[0,t ]

(3.2)

We now consider the cases for general n ∈ N. By (3.1) and the reflection principle, we have

{

(

Lnt +1 = max 0, max − x0 + θ

s

∫

0≤s≤t

Xrn dr + Zs

)}

.

(3.3)

0

Combining (3.1) with (3.3) gives that

|Xtn+1 − Xtn | ≤ θ

t

∫

|Xsn − Xsn−1 |ds + |Lnt +1 − Lnt | ∫ t ∫ s n n−1 ≤θ |Xs − Xs |ds + θ sup |Xrn − Xrn−1 |dr . 0

s∈[0,t ]

0

0

This implies that t

∫

sup |Xsn+1 − Xsn | ≤ 2θ

s∈[0,t ]

sup |Xrn − Xrn−1 |dr .

0 r ∈[0,s]

(3.4)

In the rest of the proof, we will use C to denote a generic constant which may change from line to line. By (3.2) and (3.4), we conclude that for fixed T > 0,

E sup |Xsn+1 − Xsn | ≤ s∈[0,T ]

C

(3.5)

n!

where n ∈ N. By Markov inequality, it follows that

[ P

sup |Xsn+1 − Xsn | ≥

s∈[0,T ]

1 2n

] ≤

C n!

.

X. Zhang, H. Yi and H. Shu / Statistics and Probability Letters 156 (2020) 108617

5

By the Borel–Cantelli’s lemma, we have

[

P lim sup sup |Xsn+1 − Xsn | ≥ n→∞

s∈[0,T ]

1

]

2n

= 0.

Therefore, we deduce that Xtn converges uniformly to a càdlàg, adapted process Xt on [0, T ] almost surely. Define

= x0 + θ

Ytn

t

∫

Xsn ds

+ Zt ,

n ∈ N,

Y t = x0 + θ

t

∫

Xs ds + Zt ,

Lt = Xt − Yt .

(3.6)

0

0

Now, we verify that (Xt , Lt , t ≥ 0) satisfies Eq. (1.2). On one hand, by Xtn → Xt uniformly on [0, T ] a.s., it follows that Ytn → Yt ,

[0, T ] a.s.

n → ∞ uniformly on

(3.7)

This implies that Lnt → Lt ,

n → ∞ uniformly on

[0, T ] a.s.

(3.8)

By (3.6)–(3.8) and the fact that T is arbitrary, we have X t = x0 + θ

t

∫

Xs ds + Zt + Lt ,

t≥0

a.s.

0

On the other hand, to show that (Xt , Lt , t ≥ 0) is a solution to Eq. (1.2), we need to show that t

∫

I{Xs =bL } dLs = Lt .

0

By the fact that (Xtn , Lnt , t ≥ 0) is a solution of Eq. (3.1), we have t

∫

I{Xsn =bL } dLns = Lnt ,

0

∀ n ∈ N,

t ≥ 0.

Therefore, for any bounded continuous function ϕ : R+ → R with compact support, we find t

∫

ϕ (Xsn )dLns = 0. 0

It then follows from Proposition 2.9 of Jakubowski et al. (1989) that t

∫

ϕ (Xs )dLs = lim

n→∞

0

t

∫

ϕ (Xsn )dLns = 0, 0 (1)

(1)

(2)

(2)

and the desired result is obtained. Next we show the uniqueness of the solution of Eq. (1.2). Let (Xt , Lt ) and (Xt , Lt ) be two distinct solutions to Eq. (1.2). Similar to the discussion of Eq. (3.4), we obtain

[ E

]

sup |Xs(1) − Xs(2) | ≤ C E

[∫

s∈[0,t ]

t

]

sup |Xs(1) − Xs(2) |du .

0 s∈[0,u]

Thus, by Gronwall’s inequality, it follows that

[ E

]

sup |Xs(1) − Xs(2) | = 0.

s∈[0,t ]

(1) Xt

= Xt(2) a.s. for all t ≥ 0. By the right continuity of the sample path, we have { } (1) (2) P Xt = Xt , ∀ t ≥ 0 = 1.

Hence,

This implies that

{

(1)

P Lt

= L(2) t ,

} ∀ t ≥ 0 = 1.

Thus, (Xt , Lt , t ≥ 0) is a unique solution to Eq. (1.2). This completes the proof.

□

Theorem 3.1. (i) Under conditions (A1)–(A2) and θ > 0, we have lim ˆ θT = θ ,

T →∞

a.s.,

i.e., ˆ θ is strongly consistent. (ii) If conditions (A1), (A3) and θ = 0, then ˆ θT →p θ , as T → ∞.

(3.9)

6

X. Zhang, H. Yi and H. Shu / Statistics and Probability Letters 156 (2020) 108617

Proof. (i) If θ > 0, then the process X of Eq. (1.2) is not recurrent. Applying Ito’s formula to the function e−θ t Xt , it follows that de−θ t Xt = −θ e−θ t Xt dt + e−θ t (θ Xt dt + dZt + dLt )

= e−θ t dZt + e−θ t dLt . Integrating both sides from 0 to t, we have e

−θ t

t

∫

e

X t = x0 +

−θ s

t

∫

e−θ s dLs =: ηt .

dZs +

(3.10)

0

0

∫t

Because the process L increase only when X hits the boundary zero, 0 e−θ s dLs is a continuous nondecreasing process which makes ηt ≥ 0 and increase only when ηt = 0. By virtue of Lemma 3.1 and the reflection principle, we immediately conclude that t

∫

[

{ (

e−θ s dLs = max 0, max − x0 + 0≤s≤t

0

s

∫

e−θ u dZu

)}]

.

(3.11)

0

∫t

For 0 e−θ u dZu , by the self-similarity property for the α -stable Lévy motion, we find that there exists another stable process ∫t {Zˆ , t ≥ 0} such that − 0 e−θ u dZu = Zˆ 1−e−αθ t . Hence, we conclude that αθ

[

t

∫

e

−θ s

]

dLs = max 0, −x0 +

0

max

−αθ t 0≤s≤ 1−eαθ

Zˆs .

(3.12)

Then, it is easy to see that e

lim

t →∞

]

[

t

∫

−θ s

dLs = max 0, −x0 + max Zˆs .

(3.13)

1 0≤s≤ αθ

0

It is obvious to see that

∫t 0

e−θ s dZs is a Lp -bounded càdlàg Ft -martingale (1 < p < α ). Moreover, 1

∫t 0

e−θ s dZs is an α -stable

random variable with distribution Sα (τt α , β, 0), where

τt =

t

∫

α

|e−θ s | ds =

1 − e−αθ t

αθ ∫t

0

. 1

Letting t → ∞, we find that 0 e−θ s dZs converges to an α -stable random variable with distribution Sα ((αθ ) α , β, 0). Therefore, by (3.10), (3.13) and martingales convergence theorem, it follows that lim e

t →∞

−θ t

[

∞

∫ X t = x0 +

e

−θ s

]

dZs + max 0, −x0 +

0

max

−αθ t 0≤s≤ 1−eαθ

Zˆs =: η∞ .

(3.14)

Now, we are in a position to study the rate of convergence of Lt as t → ∞. By Lemma 3.1 and the reflection principle, we find that

{

Lt = max 0, max

(

0≤s≤t

−x0 − θ

s

∫

)} Xu du − Zs

.

(3.15)

0

By the self-similarity property for the α -stable Lévy motion and (3.15), we have

{

Lt ≤ max 0, max −x0 − θ bL s − Zs

(

} )

0≤s≤t

( )} 1 L α = max 0, max −x0 − θ b ut − Zu t . d

{

(3.16)

0≤u≤1

By the fact that |Zu |, u ∈ [0, 1] is almost surely finite, it follows that lim max

(

t →∞ 0
1

−x0 − θ bL ut − Zu t α

)

( ) 1 = − lim min x0 + θ bL ut + Zu t α = −∞. t →∞ 0
) ( 1 Combining maxu=0 −x0 − θ bL ut − Zu t α = −x0 with (3.17) gives that ( ) 1 lim max −x0 − θ bL ut − Zu t α = −x0 . t →∞ 0≤u≤1

(3.17)

(3.18)

By (3.16) and (3.18), we have lim Lt = 0.

t →∞

(3.19)

X. Zhang, H. Yi and H. Shu / Statistics and Probability Letters 156 (2020) 108617

7

By the strong law of large numbers, we have lim

t →∞

Zt

= 0,

eθ t

a.s.

(3.20)

By the Toeplitz Lemma 2.2, it follows that lim

t →∞

∫t

At

0

∫t

eθ s ds

0 = lim t →∞ eθ t ∫ ∫ t t θs e ds λt eθ s = θt (e−θ s Xs − η∞ ) ds + η∞ 0 θ t e λt e 0 η∞ = , a.s., θ

eθ t

where λt = conclude that

(e−θ s Xs − η∞ )eθ s ds + η∞

∫t 0

(3.21)

eθ s ds. Then, under conditions (A1)–(A2), by combining (3.19)–(3.21) and the Toeplitz Lemma 2.2, we

∫T lim (ˆ θT − θ ) = lim

T →∞

ωt2 At Zt dt

0

∫T

T →∞

ωt2 A2t

0

∫T T →∞

2 2θ t Z t ωt e

∫T

At dt 0 eθ t eθ t h1 (T ) 2 2θ t T At 2 ω t e dt 0 h1 (T ) eθ t

= lim ∫ ( T →∞ = 0,

ωt2 At Lt dt ∫T 2 2 ωt At 0 ∫ T At L t

0

+ lim

)

ωt2 e2θ t dt 0 eθ t eθ t h1 (T ) 2 2θ t T At 2 ωt e dt 0 h1 (T ) eθ t

+ lim ∫ ( T →∞

)

a.s.

(3.22)

This completes the desired proof. (ii) If θ = 0, we have X t = x 0 + Z t + Lt . By conditions (A1) and (A3), we have T

∫

ω

2 t At (Zt

+ Lt )dt = T

1 1+ α

1

∫ 0

0 2

ω ∫ 1

2 sT

= T 2+ α ωT2

0

ZsT + LsT

(∫

sT

1

0 Tα (∫ 2 ZsT + LsT

ωs

(Zν + Lν + x0 )dν s

ZuT + LuT + x0

1

1

Tα

) ds

) du ds

Tα

0

and T

∫

ωt2 A2t dt = T

1

∫

2 ωsT

(Zν + Lν + x0 )dν

)2 ds

0

0

0

sT

(∫

2

= T 3+ α ωT2

1

∫

ωs2 0

s

(∫

ZuT + LuT + x0 1

0

)2 du

Tα

ds.

Combining Lemma 3.1 with the reflection principle gives that

{

}

Lt = max 0, max (−Zs ) . 0≤s≤t

By the scaling property of α -stable process, we have

{ ( { } ( )}) 1 d {(Zt , Lt ); t ≥ 0} = T α Z˜ t , max 0, max −Z˜ Ts ;t ≥ 0 T

0≤s≤t

d

where = denotes equal in distribution, {Z˜t , t ≥ 0} denote a fixed stable process, independent of x0 (on an enlarged d

1

probability space, if necessary) and {Zs , s ≥ 0} = {T α Z˜ s , s ≥ 0} for T > 0. Therefore, in view of the continuous mapping T theorem, we have

∫T 0

ω ∫

2 Lt )dt t At (Zt T 2 2 t At dt 0

+

ω

∫1 d

=

1

0

T

( ) (∫ ( ) ) s ˜u + L˜ u + x01 du ds ωs2 Z˜s + L˜ s Z 0 Tα ) )2 ∫ 1 (∫ s ( x ωs2 0 Z˜u + L˜ u + 01 du ds 0 Tα

→p 0,

(3.23)

8

X. Zhang, H. Yi and H. Shu / Statistics and Probability Letters 156 (2020) 108617

{

(

as T → ∞, where L˜ s = max 0, max0≤r ≤s −Z˜r

)}

. Hence, by (1.4) and (3.23), we conclude that ˆ θT →p θ , as T → ∞. By

the reflection principle, it follows that

{

(

s

∫

Lt = max 0, max − x0 + θ

)} xr dr + Zs

0≤s≤t

.

0

This completes the proof. □ 4. Asymptotics distribution of estimator ˆ θT In the section, we will study the asymptotic distribution of the weighted TFE ˆ θT . Theorem 4.1. Under conditions (A1)–(A2), (A4) and θ > 0, we have h1 (T ) 1

h2 (T )T α

U

(ˆ θT − θ ) ⇒ θ

η∞

,

(4.1)

as T → ∞, where η∞ is given in (3.14) and U is a random variable with α -stable distribution Sα (1, β, 0) independent of η∞ . Proof. Note that 1

h1 (T ) 1

h2 (T )T α

(ˆ θT − θ ) =

=

∫T

1 −α h− 2 (T )T 1 h− 1 (T )

1 h− 1 (T )

+

∫T

ηT2 ∫T 0

0

1

ωt2 At Zt dt

0

+

ωt2 A2t dt (

ωt2 At Lt dt

1 h− 1 (T ) 1

ωt2 At dt

∫T 0

0

ωt2 A2t dt

ηT2

ωt2 A2t dt ∫ −1 T α

0

∫T

1 −α ZT h− 2 (T )T

1 h− 2 (T )T

∫T

1 −α h− 2 (T )T

ωt2 At (ZT − Zt )dt

0

ηT2

+

ηT2

∫T 0

ωt2 At Lt dt

)

1

h2 (T )T α

:= H1 (T )(H2 (T ) + H3 (T ) + H4 (T )).

(4.2)

We shall study the asymptotic behaviour of Hi (T ), i = 1, . . . , 4, respectively. It follows from conditions (A2) and the Toeplitz Lemma 2.2 that T

∫

−1

ω

2 2 t At dt

lim h1 (T )

T →∞

T

∫

(

= lim

T →∞

0

At

)2

ωt2 e2θ t

eθ t

0

h1 (T )

dt =

2 η∞ , θ2

a.s.

Thus, we immediately conclude that lim H1 (T ) = θ 2 ,

a.s.

T →∞

(4.3)

Now let us consider H2 (T ). Note that H2 (T ) =

1 h− 2 (T )

∫T

1

ωt2 At dt T − α ZT . ηT ηT 0

(4.4)

By the Toeplitz Lemma 2.2 again, it follows that −1

T

∫

ω

2 t At dt

lim h2 (T )

T →∞

T

∫

(

= lim

T →∞

0

At

)

eθ t

0

ωt2 eθ t h2 (T )

dt =

η∞ , θ

a.s.

Consequently, lim

1 h− 2 (T )

∫T 0

ωt2 At dt

ηT

T →∞

=

1

θ

,

a.s.

(4.5)

For the second factor in H2 (T ), we obtain 1

T − α (ZT − Z

1

T − α ZT

ηT

=

η

1

Tα

1

1

Tα

) + T−α Z

+ (η T − η

1

Tα

)

1

Tα

.

We have the following claims. (1) Under condition (A1), the random variable

1 1

Tα

(ZT − Z

1

1

Tα

1

) has an α -stable distribution Sα ((1 − T α −1 ) α , β, 0), which

converges weakly to a random variable U with stable distribution Sα (1, β, 0) as T → ∞.

X. Zhang, H. Yi and H. Shu / Statistics and Probability Letters 156 (2020) 108617

9

(2) By strong law of large numbers, it follows that Z

1

Tα

= 0, a.s. Tα (3) It is easy to see that lim

1

T →∞

lim η

= η∞ ,

1

Tα

T →∞

P − a.s.

1

(4) T − α (ZT − Z 1 ) and η 1 are independent. Tα Tα (5) We have that ηT − η 1 converges to zero in probability as T → ∞. Tα

Proof of (5). By the definition of ηt , we find that

ηT − η

T

∫ =

1

Tα

1

Tα

T

∫

e−θ s dLs +

e−θ s dZs .

1

Tα

(4.6)

It follows that

|ηT − η

T

⏐∫ ⏐ ⏐ | ≤ ⏐ α 1

T 1

Tα

⏐ ⏐

⏐∫ ⏐

e−θ s dLs ⏐⏐ + ⏐⏐

T

⏐ ⏐

e−θ s dZs ⏐⏐ .

1

Tα

Similar to the discussion of (3.11), we can conclude that

[

T

∫

−θ s

1

e

1

Tα

For

∫T

1

Tα

( ∫ − dLs = max 0, max T α ≤s≤T

−θ s

e

1

dZs

Tα

.

e−θ s dZs , by the self-similarity property for the α -stable Lévy motion, we find that there exists another stable

process {Zˆt , t ≥ 0} such that −

∫T

1

Tα

e−θ s dZs = Zˆ 1−e−αθ T − Zˆ αθ

⎡ ∫

)]

s

1

Tα

αθ

(

T

e−θ s dLs = max ⎣0,

1

1−e−αθ T α

) Zˆs − Zˆ

max

⎢

−αθ T

⎤ ⎥ ⎦.

1 1−e−αθ T α

(4.7)

αθ

≤s≤ 1−eαθ

αθ

. Hence, we get that

1 1−e−αθ T α

By the Markov inequality, Lemma 2.1, and (4.7), we find that for given δ > 0,

(⏐∫ ⏐ P ⏐⏐

T

−θ s

1

e

Tα

⏐∫ ⏐ ) ⏐ ⏐ −1 ⏐ ⏐ dLs ⏐ > δ ≤ δ E ⏐

T

−θ s

1

e

Tα

⏐ ⏐ dLs ⏐⏐

⎡

(

⎢ = δ −1 E max ⎣0,

1−e−αθ T α

⎛ ∫ ⎢ −1 ⎝ ≤ δ ⎣0, C2 ⏐∫ ⏐ T

which tends to zero as T → ∞. For ⏐ given δ > 0

{⏐∫ ⏐ P ⏐⏐

T

−θ s

1

e

Tα

1

Tα

Zˆs − Zˆ

max 1

αθ

⎡

)

−αθ T

≤s≤ 1−eαθ

1 1−e−αθ T α

⎤ ⎥ ⎦

αθ

⎞1 ⎤ α

1−e−αθ T

αθ

α

⎥ 1 ds⎠ ⎦ ,

1

1−e−αθ T α

αθ

(4.8)

⏐ ⏐

e−θ s dZs ⏐, by the Markov inequality and Lemma 2.1 again, we find that for any

⏐∫ T ⏐ ⏐ } ⏐ ⏐ ⏐ −1 ⏐ −θ s ⏐ dZs ⏐ > δ ≤ δ E ⏐ 1 e dZs ⏐⏐ Tα (∫ T ) α1 −αθ s ≤ δ −1 C2 e ds 1 Tα

⎛ ≤ δ −1 C2 ⎝

e

−αθ T

1

−αθ T α

−e αθ

⎞ α1 ⎠ ,

(4.9)

which tends to zero as T → ∞ for any given δ > 0. By combining (4.6)–(4.9), we conclude that (5) holds. Therefore, for all the claims (1)–(5), we conclude that 1

T − α ZT

ηT

⇒

U

η∞

,

(4.10)

10

X. Zhang, H. Yi and H. Shu / Statistics and Probability Letters 156 (2020) 108617

where U and η∞ are independent. Combining (4.4) with (4.10) gives that H2 (T ) ⇒

U

,

θ η∞

(4.11)

as T → ∞. Next, we shall prove that H3 (T ) → 0 in probability as T → ∞. We have

⏐ ∫ ⏐ − 1 −1 ⏐T α h (T ) 2 ⏐

T

ω

2 t At (ZT

0

⏐ ⏐∫ T ⏐ ∫ T ⏐ ⏐ ⏐ − α1 −1 2⏐ ⏐ − Zt )dt ⏐ ≤ T h2 (T ) ωt ⏐ Xs ds⏐⏐ |ZT − Zt |dt 0 0 ) ∫ T (∫ T − α1 −1 |e−θ s Xs |eθ s ds |ZT − Zt |ωt2 dt ≤ T h2 (T ) 0 0 ∫ T 1 1 1 ≤ sup |e−θ t Xt |T − α h− (T ) |ZT − Zt |ωt2 eθ t dt . 2 θ t ≥0 0 1

1 It is not difficult to see that supt ≥0 |e−θ t Xt | is almost surely finite. We will prove that the last factor T − α h− 2 (T )

|ZT − Zt |ωt2 eθ t dt in the above inequality converges to zero in probability. It is easy to see that, by Assuming (A4), 0 we have for T large enough

∫T

[

1

1 E T − α h− 2 (T )

T

∫

] ∫ 1 |ZT − Zt |ωt2 eθ t ≤ T − α

0

T

E[|ZT − Zt |]

0 1

≤ C0 T − α

ωt2 eθ t

T

∫

dt

h2 (T ) 1

C (1, α )(T − t) α eb(T −t) dt 0

1

= C0 T − α

T

∫

1

C (1, α )u α ebu du 0 1

≤ C (1, α )C0 T − α

∞

∫

1

u α ebu du 0

≤ C (1, α )C0 T which tends to zero as T → ∞, where C (1, α ) =

− α1

Γ

1 ) 4Γ (− α √ α π Γ (− 12 )

( 1+

1

α

)

1

|b|−(1+ α ) , 1

1 (see Zolotarev, 1986). This implies that T − α h− 2 (T )

∫T 0

ωt2 At (ZT −

Zt )dt converges to zero in probability as T → ∞. That is to say, H3 (T ) → 0

(4.12)

in probability as T → ∞. Finally, for H4 (T ), by the Toeplitz Lemma 2.2, (3.19) and (3.21), we have

ηT2

∫T

At 0 eθ t

Lt

ωt2 eθ t h2 (T )

dt

= 0, a.s. Tα Therefore, from (4.2), (4.3), and (4.11)–(4.13), we conclude that (4.1) holds. This completes the proof. lim H4 (T ) = lim

T →∞

(4.13)

1

T →∞

□

Remark 4.1. We shall specify some conditions on ωt so that the condition (A4) is satisfied. We consider two special classes of weight functions. (i) Let ωt = t p , p ≥ 0. Some basic calculation yields T

∫ h2 (T ) =

ωt2 eθ t dt =

T

∫

0

t 2p eθ t dt ≥ K1 T 2p eθ T ,

0

for each T ≥ T0 with some T0 > 0 and some K1 > 0. It follows that for T large enough

ωt2 eθ t h2 (T )

≤

t 2p eθ t K1 T 2p eθ T

≤

1 K1

e−θ (T −t) ,

t ∈ [0, T ].

This implies that (A4) is satisfied with b = −θ . (ii) Let ωt2 = ert . Then T

∫ h2 (T ) =

ert eθ t dt =

0

e(r +θ )T − 1 r +θ

.

It follows that for T large enough

ωt2 eθ t h2 (T )

≤ K2 e(−θ −r)(T −t) ,

and b = −θ − r < 0. Thus, if r > −θ , then the condition (A4) is satisfied.

X. Zhang, H. Yi and H. Shu / Statistics and Probability Letters 156 (2020) 108617

11

We have the following result with regard to the limiting distribution of ˆ θT under the case θ = 0. Theorem 4.2. If θ = 0, (A1), and (A3), we have

∫1 Tˆ θT ⇒

0

(∫ ) s ωs2 (Z˜s + L˜ s ) 0 (Z˜u + L˜ u )du ds , )2 ∫ 1 (∫ 1 ωs2 0 (Z˜u + L˜ u )du ds 0

(4.14)

{

(

where Z˜u is another standard stable process and L˜ s = max 0, max0≤r ≤s −Z˜r

)}

.

Proof. By (1.4) and (3.23), we can conclude that (4.14) holds. This completes the proof. □ Remark 4.2. Let ωt = t p , p ≥ 0. It is easy to see that the condition (A3) is satisfied. Acknowledgements The authors are very grateful to the referees and the editor for their insightful and valuable comments which have greatly improved the presentation of the paper. This work was supported in part by the National Natural Science Foundation of China (61673103) and the Fundamental Research Funds for the Central Universities, China and Graduate Student Innovation Fund of Donghua University, China (CUSF-DH-D-2019086). References Applebaum, D., 2009. Lévy Processes and Stochastic Calculus, second ed. Cambridge University Press. Bo, L., Tang, D., Wang, Y., Yang, X., 2011a. On the conditional default probability in a regulated market: a structural approach. Quant. Finance 11, 1695–1702. Bo, L., Wang, Y., Yang, X., 2011b. Some integral functionals of reflected SDEs and their applications in finance. 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