Parameter estimation for Ornstein–Uhlenbeck processes driven by fractional Lévy process

Abstract

We study the minimum Skorohod distance estimation ${\theta }_{\epsilon }^{\ast }$ and minimum $L_1$-norm estimation $\widetilde{{\theta }_{\epsilon }}$ of the drift parameter θ of a stochastic differential equation $d{X}_t=\theta X_{t}dt+\epsilon d{L}_t^d$, $X_0=x_0$, where $\{L_t^d,0\le t\le T\}$ is a fractional Lévy process, $\epsilon \in (0,1]$. We obtain their consistency and limit distribution for fixed T, when $\epsilon \to 0$. Moreover, we also study the asymptotic laws of their limit distributions for $T\to \mathrm{\infty }$.

Introduction

Statistical inference for stochastic equations is a main research direction in probability theory and its applications. The asymptotic theory of parametric estimation for diffusion processes with small noise is well developed. Genon-Catalot [8] and Laredo [17] considered the efficient estimation for drift parameters of small diffusions from discrete observations as $ϵ\to 0$ and $n\to \mathrm{\infty }$. Using martingale estimating function, Sørensen [27] obtained consistency and asymptotic normality of the estimators of drift and diffusion coefficient parameters as $ϵ\to 0$ and n is fixed. Using a contrast function under suitable conditions on ϵ and n, Sørensen and Uchida [28] and Gloter and Sørensen [9] considered the efficient estimation for unknown parameters in both drift and diffusion coefficient functions. Long [20], Ma [21] studied parameter estimation for Ornstein–Uhlenbeck processes driven by small Lévy noises for discrete observations when $ϵ\to 0$ and $n\to \mathrm{\infty }$ simultaneously. Shen and Yu [26] obtained consistency and the asymptotic distribution of the estimator for Ornstein–Uhlenbeck processes with small fractional Lévy noises.

Recently, Diop and Yode [4] obtained the minimum Skorohod distance estimate for the parameter θ of a stochastic differential equation with a centered Lévy processes $\{Z_t,0\le t\le T\}$, $ϵ\in (0,1]$,

$$d{X}_t=\theta X_{t}dt+ϵd{Z}_t,X_0=x_0.$$

When $\{Z_t,0\le t\le T\}$ is a Brownian motion, Millar [24] obtained the asymptotic behavior of the estimator of the parameter θ. The minimum uniform metric estimate of parameters of diffusion-type processes was considered in Kutoyants and Pilibossian [14, 15]. Hénaff [10] considered the asymptotics of a minimum distance estimator of the parameter of the Ornstein–Uhlenbeck process. Prakasa Rao [25] studied the minimum $L_1$-norm estimates of the drift parameter of Ornstein–Uhlenbeck process driven by fractional Brownian motion and investigated the asymptotic properties following Kutoyants and Pilibossian [14, 15]. Some surveys on the parameter estimates of fractional Ornstein–Uhlenbeck process can be found in Hu and Nualart [11], El Onsy, Es-Sebaiy and Ndiaye [5], Xiao, Zhang and Xu [29], Jiang and Dong [12], Liu and Song [19].

Motivated by the above results, in this paper we consider the minimum Skorohod distance estimation ${\theta }_{\epsilon }^{\ast }$ and minimum $L_1$-norm estimation $\widetilde{{\theta }_{\epsilon }}$ of the drift parameter θ for Ornstein–Uhlenbeck processes driven by the fractional Lévy process $\{L_t^d,0\le t\le T\}$ which satisfies the following stochastic differential equation:

$$d{X}_t=\theta X_{t}dt+ϵd{L}_t^d,X_0=x_0,$$ 1

where the shift parameter $\theta \in \Theta =({\theta }_1,{\theta }_2)\subseteq R$ is unknown, $\epsilon \in (0,1]$. Denote by ${\theta }_0$ the true value of the unknown parameter θ. Note that

$$X_t(\theta )=x_t(\theta )+\epsilon e^{\theta t}{\int }_0^t{e}^{-\theta s}d{L}_s^d,$$

where $x_t(\theta )=x_0{e}^{\theta t}$ is a solution of (1) with $\epsilon =0$.

Recall that fractional Lévy processes is a natural generalization of the integral representation of fractional Brownian motion. Analogously to Mandelbrot and Van Ness [22] for fractional Brownian motion we introduce the following definition.

Definition 1.1

(Marquardt [23])

Let $L=(L(t),t\in R)$ be a zero-mean two-sided Lévy process with $E[L{(1)}^2]<\mathrm{\infty }$ and without a Brownian component. For $d\in (0,\frac{1}{2})$, a stochastic process

$$L_t^d:=\frac{1}{\Gamma (d+1)}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}[{(t-s)}_{+}^d-{(-s)}_{+}^d]L(ds),t\in R,$$ 2

is called a fractional Lévy process (fLp), where

$$L(t)=L_1(t),t\ge 0,L(t)=-L_2(-t_{-}),t<0.$$ 3

$\{L_1(t),t\ge 0\}$ and $\{L_2(t),t\ge 0\}$ are two independent copies of a one-side Lévy process.

Lemma 1.1

(Marquardt [23])

Let $g\in H$, H is the completion of $L^1(R)\cap L^2(R)$ with respect to the norm ${\parallel g\parallel }_H^2=E[L{(1)}^2]{\int }_R{(I_{-}^{d}g)}^2(u)du$, then

$${\int }_{R}g(s)d{L}_s^d={\int }_R\left(I_{-}^{d}g\right)(u)dL(u),$$ 4

where the equality holds in the $L^2$ sense and $I_{-}^{d}g$ denotes the Riemann–Liouville fractional integral defined by

$$\left(I_{-}^{d}g\right)(x)=\frac{1}{\Gamma (d)}{\int }_x^{\mathrm{\infty }}g(t){(t-x)}^{d-1}dt.$$

Lemma 1.2

(Marquardt [23])

Let $|f|$, $|g|\in H$. Then

$$E[{\int }_{R}f(s)d{L}_s^d{\int }_{R}g(s)d{L}_s^d]=\frac{\Gamma (1-2d)E[L{(1)}^2]}{\Gamma (d)\Gamma (1-d)}{\int }_R{\int }_{R}f(t)g(s)|t-s{|}^{2d-1}dsdt.$$ 5

Lemma 1.3

(Bender et al. [2])

Let $L_t^d$ be a fLp. Then for every $p\ge 2$ and $\delta >0$ such that $d+\delta <\frac{1}{2}$ there exists a constant $C_{p,\delta ,d}$ independent of the driving Lévy process L such that for every $T\ge 1$

$$E\left(\underset{0\le t\le T}{sup}\right|L_t^d{|}^p)\le C_{p,\delta ,d}E\left(\right|L(1){|}^p)T^{p(d+1/2+\delta )}.$$

For the study of fLp see Bender et al. [3], Fink and Klüppelberg [7], Lin and Cheng [18], Benassi et al. [1], Lacaux [16], Engelke [6] and the references therein.

The rest of this paper is organized as follows. In Sect. 2, we consider the minimum Skorohod distance estimation ${\theta }_{\epsilon }^{\ast }$ of the drift parameter θ, its consistency and limit distribution are studied for fixed T, when $\epsilon \to 0$. Moreover, the asymptotic law of its limit distribution are also studied for $T\to \mathrm{\infty }$. The similar problems for minimum $L_1$-norm estimation $\widetilde{{\theta }_{\epsilon }}$ of the drift parameter θ were studied in Sect. 3.

Minimum Skorohod distance estimation

In this section, we consider the minimum Skorohod distance estimation which defined by

$${\theta }_{\epsilon }^{\ast }=arg\underset{\theta \in \Theta }{min}\rho (X,x(\theta )),$$ 6

where

$$\rho (x,y)=\underset{\mu \in \Lambda ([0,T])}{inf}(H(\mu )+sup|x(\mu (t))-y(t)\left|\right)$$ 7

on the Skorohod space $D([0,T],R)$ consists of càdlàg functions on $[0,T]$, $\Lambda ([0,T])$ is the set of functions μ defined on $[0,T]$ with values in $[0,T]$, continuous, strictly increasing such that $\mu (0)=0$ and $\mu (T)=T$, and

$$H(\mu )=\underset{s,t\in [0,T],s\ne t}{sup}|log\left(\frac{\mu (s)-\mu (t)}{s-t}\right)|<\mathrm{\infty }.$$

Let

$${\eta }_T=arg\underset{u\in R}{min}\rho (Y({\theta }_0),u\dot{x}({\theta }_0)),$$ 8

where $\dot{x}({\theta }_0)=x_{0}t{e}^{{\theta }_{0}t}$ is the derivative of $x_t({\theta }_0)$ with respect to ${\theta }_0$ and

$$Y_t({\theta }_0)=e^{{\theta }_{0}t}{\int }_0^t{e}^{{\theta }_{0}s}d{L}_s^d.$$ 9

Let

$$f(\kappa )=\underset{|\theta -{\theta }_0|>\kappa }{inf}|X_t-x({\theta }_0){|}_{\mathrm{\infty }}=\underset{|\theta -{\theta }_0|>\kappa }{inf}\underset{0\le t\le T}{sup}|X_t-x({\theta }_0)|,\kappa >0$$ 10

and $P_{{\theta }_0}^{(\epsilon )}$ denotes the probability measure induced by the process $X_t$ for fixed ε.

Theorem 2.1

(Consistency)

For every $p\ge 2$ and $\kappa >0$ such that, for every $T\ge 1$, we have

$$P_{{\theta }_0}^{(\epsilon )}\left(\right|{\theta }_{\epsilon }^{\ast }-{\theta }_0|>\kappa )\le C_{p,\kappa ,d}E\left(\right|L(1){|}^p)T^{p(d+1/2+\kappa )}{\left(\frac{2\epsilon e^{|{\theta }_0|T}}{f(\kappa )}\right)}^p=O\left({(f(\kappa ))}^{-p}{\epsilon }^p\right),$$ 11

where constant $C_{p,\kappa ,d}$ is only dependent on p, κ, d.

Proof

Fixed $\kappa >0$ and let

$${\mathcal{I}}_0=\{\omega :\underset{|\theta -{\theta }_0|<\kappa }{inf}\rho (X,x(\theta ))>\underset{|\theta -{\theta }_0|>\kappa }{inf}\rho (X,x(\theta ))\}.$$

Then we can obtain ${\mathcal{I}}_0=\{|{\theta }_{\epsilon }^{\ast }-{\theta }_0|>\kappa \}$. In fact, for $\omega \in {\mathcal{I}}_0$, we have

$$\underset{|\theta -{\theta }_0|<\kappa }{inf}\rho (X(\omega ),x(\theta ))\ge \underset{\theta \in \Theta }{inf}\rho (X(\omega ),x(\theta ))=\rho (X(\omega ),x\left({\theta }_{\epsilon }^{\ast }\right)),$$

thus, $|{\theta }_{\epsilon }^{\ast }(\omega )-{\theta }_0|>\kappa$. On the other hand, assume that $|{\theta }_{\epsilon }^{\ast }(\omega )-{\theta }_0|>\kappa$,

$$\rho (X(\omega ),x\left({\theta }_{\epsilon }^{\ast }\right))=\underset{|\theta -{\theta }_0|>\kappa }{inf}\rho (X(\omega ),x(\theta ))<\underset{|\theta -{\theta }_0|<\kappa }{inf}\rho (X(\omega ),x(\theta )).$$

For any $\kappa >0$, we have

$$\begin{array}{rcl}{P}_{{\theta }_0}^{(\epsilon )}({\mathcal{I}}_0)& =& P_{{\theta }_0}^{(\epsilon )}(\underset{|\theta -{\theta }_0|<\kappa }{inf}\rho (X,x(\theta ))>\underset{|\theta -{\theta }_0|>\kappa }{inf}\rho (X,x(\theta )))\\ & \le & P_{{\theta }_0}^{(\epsilon )}(\underset{|\theta -{\theta }_0|<\kappa }{inf}\rho (X,x(\theta ))>\underset{|\theta -{\theta }_0|>\kappa }{inf}|\rho (X,x(\theta ))-\rho (x({\theta }_0),x(\theta ))\left|\right)\\ & \le & P_{{\theta }_0}^{(\epsilon )}(\underset{|\theta -{\theta }_0|<\kappa }{inf}\rho (X,x(\theta ))>\underset{|\theta -{\theta }_0|>\kappa }{inf}\rho (x({\theta }_0),x(\theta ))-\rho (X,x({\theta }_0)))\\ & \le & P_{{\theta }_0}^{(\epsilon )}(\underset{|\theta -{\theta }_0|<\kappa }{inf}\rho (x(\theta ),x({\theta }_0))+2\rho (X,x({\theta }_0))>\underset{|\theta -{\theta }_0|>\kappa }{inf}\rho (x({\theta }_0),x(\theta )))\\ & \le & P_{{\theta }_0}^{(\epsilon )}({\parallel X-x({\theta }_0)\parallel }_{\mathrm{\infty }}>\frac{f(\kappa )}{2}).\end{array}$$

Besides, since the process $X_t$ satisfies the stochastic differential Eqs. (1), it follows that

$$X_t-x_t({\theta }_0)=x_0+{\theta }_0{\int }_0^t{X}_{s}ds+\epsilon L_t^d-x_t({\theta }_0)={\theta }_0{\int }_0^t(X_s-x_s({\theta }_0))ds+\epsilon L_t^d.$$ 12

Then

$$|X_t-x_t({\theta }_0)|=|{\theta }_0{\int }_0^t(X_s-x_s({\theta }_0))ds+\epsilon L_t^d|\le |{\theta }_0|{\int }_0^t|X_s-x_s({\theta }_0)|ds+\epsilon \left|L_t^d\right|.$$ 13

Hence, we have

$${\parallel X-x({\theta }_0)\parallel }_{\mathrm{\infty }}=\underset{0\le t\le T}{sup}|X_t-x_t({\theta }_0)|\le \epsilon e^{|{\theta }_{0}T|}\underset{0\le t\le T}{sup}\left|L_t^d\right|$$ 14

because of the Gronwall–Bellman lemma. Thus,

$$P_{{\theta }_0}^{(\epsilon )}({\parallel X-x({\theta }_0)\parallel }_{\mathrm{\infty }}>\frac{f(\kappa )}{2})\le P\left(\underset{0\le t\le T}{sup}\right|L_t^d|\ge \frac{f(\kappa )}{2\epsilon e^{|{\theta }_{0}T|}}).$$ 15

According to Lemma 1.3 and Chebyshev’s inequality, for all $p\ge 2$, we get

$$\begin{array}{rl}{P}_{{\theta }_0}^{(\epsilon )}\left(\right|{\theta }_{\epsilon }^{\ast }-{\theta }_0|>\kappa )& \le E{\left(\underset{0\le t\le T}{sup}\right|L_t^d\left|\right)}^p{\left(\frac{2\epsilon e^{|{\theta }_{0}T|}}{f(\kappa )}\right)}^p\\ & \le C_{p,\kappa ,d}E\left(\right|L(1){|}^p)T^{p(d+1/2+\kappa )}{2}^p{e}^{|{\theta }_{0}T|p}{(f(\kappa ))}^{-p}{\epsilon }^p\\ & =O\left({(f(\kappa ))}^{-p}{\epsilon }^p\right).\end{array}$$ 16

This completes the proof. □

Remark 2.1

As a consequence of the above theorem, we obtain the result that ${\theta }_{\epsilon }^{\ast }$ converges in probability to ${\theta }_0$ under $P_{{\theta }_0}^{(\epsilon )}$-measure as $\epsilon \to 0$. Furthermore, the rate of convergence is of order $O({\epsilon }^p)$ for every $p\ge 2$.

Theorem 2.2

(Limit distribution)

For any $h\in D([0,T],R)$ satisfying $h(0)=0$, ${\varphi }_h^{\alpha }=\rho (h,u\cdot a)$, $a(t)=t{e}^{\alpha t}$, $\alpha \in R$, $u\in R$ admits a unique minimum at u. Then we have, as $\epsilon \to 0$, ${\epsilon }^{-1}({\theta }_{\epsilon }^{\ast }-{\theta }_0)\stackrel{d}{\to }{\zeta }_T$, where the notation “$\stackrel{d}{\to }$” denotes “convergence in distribution”.

Remark 2.2

${\varphi }_h^{\alpha }$ is a convex function and ${\varphi }_h^{\alpha }\to +\mathrm{\infty }$ when $|u|\to +\mathrm{\infty }$, so ${\varphi }_h^{\alpha }$ admits a minimum.

The following lemma due to Diop and Yode [4] which is vital for our proof of Theorem 2.2.

Lemma 2.1

Let ${\{K_{\epsilon }\}}_{\epsilon >0}$ be a sequence of continuous functions on R and $K_0$ be a convex function which admits a unique minimum η on R. Let ${\{L_{\epsilon }\}}_{\epsilon >0}$ be a sequence of positive numbers such that $L_{\epsilon }\to +\mathrm{\infty }$ as $\epsilon \to 0$. We suppose that

$$\underset{\epsilon \to 0}{lim}\underset{|u|\le L_{\epsilon }}{sup}|K_{\epsilon }(u)-K_0(u)|=0.$$

Then

$$\underset{\epsilon \to 0}{lim}arg\underset{|u|\le L_{\epsilon }}{min}{K}_{\epsilon }(u)=\eta ,$$

where if there are several minima of $K_{\epsilon }$, we choose one of them arbitrarily.

Proof of Theorem 2.2

We introduce the following notations:

$$\begin{array}{rl}& K_{\epsilon }(u)=\rho (Y,\frac{1}{\epsilon }(x({\theta }_0+\epsilon u)-x({\theta }_0))),\\ & K_0(u)=\rho (Y,u\dot{x}({\theta }_0)).\end{array}$$

Since

$$\begin{array}{rl}|K_{\epsilon }(u)-K_0(u)|& =|\underset{\mu \in \Lambda ([0,T])}{inf}(H(\mu )+{\parallel Y_{\mu }-\frac{1}{\epsilon }(x({\theta }_0+\epsilon u)-x({\theta }_0))\parallel }_{\mathrm{\infty }})\\ & -\underset{\mu \in \Lambda ([0,T])}{inf}(H(\mu )+{\parallel Y_{\mu }-u\dot{x}({\theta }_0)\parallel }_{\mathrm{\infty }})|\\ & =|\underset{\mu \in \Lambda ([0,T])}{inf}(H(\mu )+\parallel Y_{\mu }-u\dot{x}({\theta }_0)-\frac{1}{2}\epsilon u^2\ddot{x}(\tilde{\theta })){\parallel }_{\mathrm{\infty }})\\ & -\underset{\mu \in \Lambda ([0,T])}{inf}(H(\mu )+{\parallel Y_{\mu }-u\dot{x}({\theta }_0)\parallel }_{\mathrm{\infty }})|\end{array}$$

with $\tilde{\theta }={\tilde{\theta }}_{\epsilon ,u,t}\in ({\theta }_0,{\theta }_0+\epsilon u)$, where the second equality is because of the Taylor expansion. If we take $L_{\epsilon }={\epsilon }^{\delta -1}$ with $\delta \in (1/2,1)$, we get

$$\begin{array}{rcl}\underset{|u|\le L_{\epsilon }}{sup}|K_{\epsilon }(u)-K_0(u)|& =& |\underset{\mu \in \Lambda ([0,T])}{inf}(H(\mu )+{\parallel Y_{\mu }-u\dot{x}({\theta }_0)-\frac{1}{2}\epsilon u^2\ddot{x}(\tilde{\theta })\parallel }_{\mathrm{\infty }})\\ & & -\underset{\mu \in \Lambda ([0,T])}{inf}(H(\mu )+{\parallel Y_{\mu }-u\dot{x}({\theta }_0)\parallel }_{\mathrm{\infty }})|\\ & \le & \underset{|u|\le L_{\epsilon }}{sup}[\frac{1}{2}\epsilon u^2\underset{0\le t\le T}{sup}\ddot{x}(\tilde{\theta })]\le \frac{\epsilon L_{\epsilon }^2}{2}|x_0|T^2{e}^{(|{\theta }_0|+\epsilon L_{\epsilon })T}\\ & =& \frac{{\epsilon }^{2\delta -1}}{2}|x_0|T^2{e}^{(|{\theta }_0|+\epsilon L_{\epsilon })T}\to 0(\epsilon \to 0).\end{array}$$

Therefore, we get the desired results by Lemma 2.1. □

In the following, we will consider the limiting behavior of ${\eta }_T$ for $T\to +\mathrm{\infty }$. Let us introduce the following notations:

$$\begin{array}{rl}& A_t={\int }_t^{+\mathrm{\infty }}{e}^{-{\theta }_{0}s}d{L}_s^d,\\ & B_t={\int }_0^t{e}^{-{\theta }_{0}s}d{L}_s^d.\end{array}$$

From Theorem 3.6.6 of Jurek and Mason [13] and Lemma 4 of Diop and Yode [4], we can get the logarithmic moment condition is necessary and sufficient for the existence of the improper integral $A_0$.

Lemma 2.2

Suppose that $E(log(1+|L_1|))<+\mathrm{\infty }$. Then

$$A_t\stackrel{d}{=}{e}^{-{\theta }_{0}s}{A}_0$$ 17

where “$\stackrel{d}{=}$” denotes “identical distribution”.

Proof

It is not hard to see,

$$\begin{array}{rcl}{A}_t& =& {\int }_t^{+\mathrm{\infty }}{e}^{-{\theta }_{0}s}d{L}_s^d={\int }_t^{+\mathrm{\infty }}\left(I_{-}^d{e}^{-{\theta }_{0}u}\right)(s)dL(s)\\ & =& {\int }_t^{+\mathrm{\infty }}\left(\frac{1}{\Gamma (d)}{\int }_s^{\mathrm{\infty }}{e}^{-{\theta }_{0}u}{(u-s)}^{d-1}du\right)dL(s)\\ & =& {\int }_t^{+\mathrm{\infty }}\left(\frac{1}{\Gamma (d)}{\int }_0^{\mathrm{\infty }}{e}^{-{\theta }_0(s+x)}{x}^{d-1}dx\right)dL(s)\\ & =& {\int }_t^{+\mathrm{\infty }}(\frac{1}{\Gamma (d)}{e}^{-{\theta }_{0}s}{\theta }_0^{-d}{\int }_s^{\mathrm{\infty }}{e}^{-{\theta }_{0}x}{({\theta }_{0}x)}^{d-1}d({\theta }_{0}x))dL(s)\\ & =& {\theta }_0^{-d}{\int }_t^{+\mathrm{\infty }}{e}^{-{\theta }_{0}s}dL(s).\end{array}$$

In a similar way,

$$A_0={\theta }_0^{-d}{\int }_0^{+\mathrm{\infty }}{e}^{-{\theta }_{0}s}dL(s).$$

From Lemma 4 of Diop and Yode [4], we have immediately

$$A_t\stackrel{d}{=}{e}^{-{\theta }_{0}s}{A}_0.$$

 □

The next theorem gives the asymptotic behavior of the limit distribution ${\eta }_T$ for large T.

Theorem 2.3

Suppose that ${\theta }_0>0$ and $E(log(1+|L_1|))<+\mathrm{\infty }$. Then ${\xi }_T=x_{0}T{\eta }_T$ converges in distribution to $A_0$ as $T\to +\mathrm{\infty }$.

Proof

Recall that

$${\eta }_T=arg\underset{u\in R}{min}\rho (Y({\theta }_0),u\dot{x}({\theta }_0)).$$

By changing variable, we have

$${\xi }_T=arg\underset{\omega \in R}{min}\rho (Y({\theta }_0),M_t(\omega )):=arg\underset{\omega \in R}{min}N(\omega ),$$ 18

where $M_t(\omega )=\frac{\omega t{e}^{{\theta }_{0}t}}{T}$ and $N(\cdot )=\rho (Y({\theta }_0),M(\cdot ))$.

We want to show that, for every $\mathrm{\Delta }>0$,

$$\underset{T\to +\mathrm{\infty }}{lim}{P}_{{\theta }_0}\{|{\xi }_T-A_0|>\mathrm{\Delta }\}=0.$$ 19

Therefore, let us consider the set

$$V_{\mathrm{\Delta }}=\{\omega :|\omega -A_0|>\mathrm{\Delta }\},$$

where $P_{{\theta }_0}$ is the probability measure induced by the process $X_t$ when ${\theta }_0$ is the true parameter and $\epsilon \to 0$. We can get

$$\begin{array}{rcl}N(A_0)& =& \rho (Y({\theta }_0),M(A_0))\le {\parallel Y({\theta }_0)-M(A_0)\parallel }_{\mathrm{\infty }}\\ & =& {\parallel e^{{\theta }_{0}t}({\int }_0^t{e}^{-{\theta }_{0}s}d{L}_s^d-\frac{A_{0}t}{T}-A_0+A_0)\parallel }_{\mathrm{\infty }}\\ & =& {\parallel e^{{\theta }_{0}t}({\int }_0^t{e}^{-{\theta }_{0}s}d{L}_s^d-A_0+(1-\frac{t}{T})A_0)\parallel }_{\mathrm{\infty }}\\ & =& {\parallel e^{{\theta }_{0}t}({\int }_0^t{e}^{-{\theta }_{0}s}d{L}_s^d-A_0)\parallel }_{\mathrm{\infty }}+|A_{0}t|{\parallel (1-\frac{t}{T})e^{{\theta }_{0}t}\parallel }_{\mathrm{\infty }}.\end{array}$$

On the other hand, for $\omega \in V_{\mathrm{\Delta }}$, we have

$$\begin{array}{rcl}N(\omega )& =& \rho (Y({\theta }_0),M(\omega ))\\ & \ge & \rho (M(A_0),M(\omega ))-\rho (Y({\theta }_0),M(A_0))\\ & =& {\parallel M(\omega )-M(A_0)\parallel }_{\mathrm{\infty }}-N(A_0)\\ & =& |\omega -A_0|{\parallel \frac{t{e}^{{\theta }_{0}t}}{T}\parallel }_{\mathrm{\infty }}-N(A_0)\\ & \ge & \mathrm{\Delta }{\parallel \frac{t{e}^{{\theta }_{0}t}}{T}\parallel }_{\mathrm{\infty }}-N(A_0).\end{array}$$

Hence, we have

$$\begin{array}{c}\frac{N(\omega )}{N(A_0)}\ge \frac{\mathrm{\Delta }{\parallel \frac{t{e}^{{\theta }_{0}t}}{T}\parallel }_{\mathrm{\infty }}}{N(A_0)}-1,\hfill \\ \begin{array}{rl}\frac{{inf}_{\omega \in V_{\mathrm{\Delta }}}N(\omega )}{N(A_0)}& \ge \mathrm{\Delta }{[\frac{T{\parallel e^{{\theta }_{0}t}({\int }_0^t{e}^{-{\theta }_{0}s}d{L}_s^d-A_0)\parallel }_{\mathrm{\infty }}}{{\parallel t{e}^{{\theta }_{0}t}\parallel }_{\mathrm{\infty }}}+\frac{|A_0|{\parallel (T-t)e^{{\theta }_{0}t}\parallel }_{\mathrm{\infty }}}{{\parallel t{e}^{{\theta }_{0}t}\parallel }_{\mathrm{\infty }}}]}^{-1}-1\\ & =\mathrm{\Delta }{[\frac{T{\parallel e^{{\theta }_{0}t}(B_t-A_0)\parallel }_{\mathrm{\infty }}}{{\parallel t{e}^{{\theta }_{0}t}\parallel }_{\mathrm{\infty }}}+\frac{|R_0|{\parallel (T-t)e^{{\theta }_{0}t}\parallel }_{\mathrm{\infty }}}{{\parallel t{e}^{{\theta }_{0}t}\parallel }_{\mathrm{\infty }}}]}^{-1}-1\\ & ={[e^{-{\theta }_{0}T}{\parallel e^{{\theta }_{0}t}{A}_t\parallel }_{\mathrm{\infty }}+\frac{|A_0|}{T{\theta }_{0}e}]}^{-1}-1,\end{array}\hfill \end{array}$$

where we get the maximum value of the function $(T-t)e^{{\theta }_{0}t}$ by taking the derivative.

We obtain

$$\frac{|A_0|}{T{\theta }_{0}e}\to 0\text{a.s. as }T\to +\mathrm{\infty }.$$ 20

Using Lemma 2.2 we have

$$P_{{\theta }_0}(e^{-{\theta }_{0}T}{\parallel e^{{\theta }_{0}t}{A}_t\parallel }_{\mathrm{\infty }}>\mathrm{\Delta })=P_{{\theta }_0}(|A_0|>e^{{\theta }_{0}T}\mathrm{\Delta })\le e^{-{\theta }_{0}T}\frac{E_{{\theta }_0}(|A_0|)}{\mathrm{\Delta }}\to 0,T\to +\mathrm{\infty }.$$ 21

By (20) and (21), we obtain

$$\frac{{inf}_{\omega \in V_{\mathrm{\Delta }}}N(\omega )}{N(A_0)}\stackrel{P}{⟶}+\mathrm{\infty },T\to +\mathrm{\infty }.$$ 22

In addition, using (18), ${\xi }_T\in V_{\mathrm{\Delta }}$, we have

$$N({\xi }_T)=\underset{\omega \in V_{\mathrm{\Delta }}}{inf}N(\omega )\le N(A_0).$$ 23

We can get the desired result (19) by Eqs. (22) and (23). □

Minimum $L_1$-norm estimation

In this section, we will study the minimum $L_1$-norm estimation $\widetilde{{\theta }_{\epsilon }}$ of the drift parameter θ. Let

$$D_T(\theta )={\int }_0^T|X_t-x_t(\theta )|dt.$$ 24

It is well known that $\widetilde{{\theta }_{\epsilon }}$ is the minimum $L_1$-norm estimator if there exists a measurable selection $\widetilde{{\theta }_{\epsilon }}$ such that

$$D_T(\widetilde{{\theta }_{\epsilon }})=\underset{\theta \in \Theta }{inf}{D}_T(\theta ).$$ 25

Suppose that there exists a measurable selection $\widetilde{{\theta }_{\epsilon }}$ satisfying the above equation. We can also define the estimator $\widetilde{{\theta }_{\epsilon }}$ by the relation

$$\widetilde{{\theta }_{\epsilon }}=arg\underset{\theta \in \Theta }{inf}{\int }_0^T|X_t-x_t(\theta )|dt.$$ 26

For any $\kappa >0$, we define

$$\tilde{f}(\kappa )=\underset{|\theta -{\theta }_0|>\kappa }{inf}{\int }_0^T|X_t(\theta )-x_t({\theta }_0)|dt>0,\text{for any }\kappa >0.$$ 27

Theorem 3.1

(Consistency)

For any $p\ge 2$, there exists a constant $C_{p,\kappa ,d}$ (only depending the p, κ, d), such that, for every $\kappa >0$, we have

$$P_{{\theta }_0}^{(\epsilon )}(|\widetilde{{\theta }_{\epsilon }}-{\theta }_0|>\kappa )\le C_{p,\kappa ,d}E\left(\right|L(1){|}^p)T^{p(d+1/2+\kappa )}{\left(\frac{2\epsilon e^{|{\theta }_0|T}}{f(\kappa )}\right)}^p=O\left({(\tilde{f}(\kappa ))}^{-p}{\epsilon }^p\right).$$ 28

Proof

Set $\parallel \cdot \parallel$ denotes the $L_1$-norm, then we have

$$\begin{array}{rcl}{P}_{{\theta }_0}^{(\epsilon )}(|\widetilde{{\theta }_{\epsilon }}-{\theta }_0|>\kappa )& =& P_{{\theta }_0}^{(\epsilon )}\{\underset{|\theta -{\theta }_0|\le \kappa }{inf}\parallel X-x(\theta )\parallel >\underset{|\theta -{\theta }_0|>\kappa }{inf}\parallel X-x(\theta )\parallel \}\\ & \le & P_{{\theta }_0}^{(\epsilon )}\{\underset{|\theta -{\theta }_0|\le \kappa }{inf}(\parallel X-x({\theta }_0)\parallel +\parallel x(\theta )-x({\theta }_0)\parallel )\\ & & >\underset{|\theta -{\theta }_0|>\kappa }{inf}(\parallel x(\theta )-x({\theta }_0)\parallel -\parallel X-x({\theta }_0)\parallel )\}\\ & =& P_{{\theta }_0}^{(\epsilon )}\{2\parallel X-x(\theta )\parallel >\underset{|\theta -{\theta }_0|>\kappa }{inf}\parallel x(\theta )-x({\theta }_0)\parallel \}\\ & =& P_{{\theta }_0}^{(\epsilon )}\{\parallel X-x(\theta )\parallel >\frac{1}{2}\tilde{f}(\kappa )\}.\end{array}$$

Since the process $X_t$ satisfies the stochastic differential equation (1), it follows that

$$X_t-x_t({\theta }_0)=x_0+{\theta }_0{\int }_0^t{X}_{s}ds+\epsilon L_t^d-x_t({\theta }_0)={\theta }_0{\int }_0^t(X_s-x_s({\theta }_0))ds+\epsilon L_t^d,$$ 29

where $x_t(\theta )=x_0{e}^{\theta t}$.

Similar to the proof of Theorem 2.1, we have

$$\underset{0\le t\le T}{sup}|X_t-x_t({\theta }_0)|\le \epsilon e^{|{\theta }_{0}T|}\underset{0\le t\le T}{sup}\left|L_t^d\right|.$$ 30

Thus,

$$P_{{\theta }_0}^{(\epsilon )}\{\parallel X-x(\theta )\parallel >\frac{1}{2}\tilde{f}(\kappa )\}\le P\left(\underset{0\le t\le T}{sup}\right|L_t^d|\ge \frac{\tilde{f}(\kappa )}{2\epsilon e^{|{\theta }_{0}T|}}).$$ 31

Applying Lemma 1.3 to the estimate obtained above, we have

$$\begin{array}{rcl}{P}_{{\theta }_0}^{(\epsilon )}(|{\tilde{\theta }}_{\epsilon }-{\theta }_0|>\kappa )& \le & E{\left(\underset{0\le t\le T}{sup}\right|L_t^d\left|\right)}^p{\left(\frac{2\epsilon e^{|{\theta }_{0}T|}}{\tilde{f}(\kappa )}\right)}^p\\ & \le & C_{p,\kappa ,d}E\left(\right|L(1){|}^p)T^{p(d+1/2+\kappa )}{2}^p{e}^{|{\theta }_{0}T|p}{(\tilde{f}(\kappa ))}^{-p}{\epsilon }^p\\ & =& O\left({(\tilde{f}(\kappa ))}^{-p}{\epsilon }^p\right).\end{array}$$

This completes the proof. □

Remark 3.1

It follows from Theorem 3.1 that we have ${\tilde{\theta }}_{\epsilon }$ converges in probability to ${\theta }_0$ under $P_{{\theta }_0}^{(\epsilon )}$-measure as $\epsilon \to 0$. Furthermore, the rate of convergence is of order $O({\epsilon }^p)$ for every $p\ge 2$.

Theorem 3.2

(Limit distribution)

As $\epsilon \to 0$, ${\epsilon }^{-1}({\tilde{\theta }}_{\epsilon }-{\theta }_0)\stackrel{d}{\to }\xi$, ξ has the same probability distribution as η̃ under $P_{{\theta }_0}^{(\epsilon )}$

$$\tilde{\eta }=arg\underset{-\mathrm{\infty }<u<+\mathrm{\infty }}{inf}{\int }_0^T|Y_t(\theta )-ut{x}_0{e}^{{\theta }_{0}t}|dt.$$ 32

Proof

Let

$$Z_{\epsilon }(u)=\parallel Y-{\epsilon }^{-1}(x({\theta }_0+\epsilon u)-x({\theta }_0))\parallel$$ 33

and

$$Z_0(u)=\parallel Y-u\dot{x}({\theta }_0)\parallel .$$ 34

Furthermore, let

$$A_{\epsilon }=\{\omega :|{\tilde{\theta }}_{\epsilon }-{\theta }_0|<{\delta }_{\epsilon }\},{\delta }_{\epsilon }={\epsilon }^{\tau }\tau ,\tau \in (\frac{1}{2},1),L_{\epsilon }={\epsilon }^{\tau -1}.$$ 35

It is easy to see that the random variable ${\tilde{u}}_{\epsilon }={\epsilon }^{-1}({\tilde{\theta }}_{\epsilon }-{\theta }_0)$ satisfies the equation

$$Z_{\epsilon }({\tilde{u}}_{\epsilon })=\underset{|u|<L_{\epsilon }}{inf}{Z}_{\epsilon }(u),\omega \in A_{\epsilon }.$$ 36

Define

$${\tilde{\eta }}_{\epsilon }=arg\underset{|u|<L_{\epsilon }}{inf}{Z}_0(u).$$ 37

Observe that, with probability one,

$$\begin{array}{rl}\underset{|u|<L_{\epsilon }}{sup}|Z_{\epsilon }(u)-Z_0(u)|& =|\parallel Y-u\dot{x}({\theta }_0)-1/2\epsilon u^2\ddot{x}(\tilde{\theta })\parallel -\parallel Y-u\dot{x}({\theta }_0)\parallel |\\ & \le \frac{\epsilon }{2}{L}_{\epsilon }^2\underset{|\theta -{\theta }_0|<{\delta }_{\epsilon }}{sup}{\int }_0^T|\ddot{x}(\theta )|dt\le C{\epsilon }^{2\tau -1}\to 0,\epsilon \to 0,\end{array}$$ 38

where $\tilde{\theta }={\theta }_0+\alpha (\theta -{\theta }_0)$ for some $\alpha \in (0,1]$. Note that the last term in the above inequality tends to zero as $\epsilon \to 0$. This follows from the arguments given in Theorem 2 of Kutoyants and Pilibossian [14, 15]. In addition, we can choose the interval $[-L,L]$ such that

$$P_{{\theta }_0}^{(\epsilon )}\{u_{\epsilon }^{\ast }\in (-L,L)\}\ge 1-\beta \tilde{f}{(L)}^{-p}$$ 39

and

$$P\{u^{\ast }\in (-L,L)\}\ge 1-\beta \tilde{f}{(L)}^{-p},\beta >0.$$ 40

Note that $\tilde{f}(L)$ increases as L increases. The process $\{Z_{\epsilon }(u),u\in [-L,L]\}$ and $\{Z_0(u),u\in [-L,L]\}$ satisfy the Lipschitz conditions and $Z_{\epsilon }(u)$ converges uniformly to $Z_0(u)$ over $u\in [-L,L]$. Hence the minimizer of $Z_{\epsilon }(\cdot )$ converges to the minimizer of $Z_0(u)$. This completes the proof. □

Although the distribution of η̃ is not clear, we can consider its limiting behaviors as $T\to +\mathrm{\infty }$.

Theorem 3.3

(Asymptotic law)

Suppose that ${\theta }_0>0$ and $E(log(1+|L_1|))<+\mathrm{\infty }$. Then

$${\tilde{\xi }}_T=x_{0}T{\tilde{\eta }}_T\stackrel{d}{\to }{A}_0,T\to +\mathrm{\infty },$$

where $L_1$, $A_0$ and other notations in the following are the same as Theorem 2.3.

Proof

Recall that

$${\tilde{\eta }}_T=arg\underset{u\in R}{inf}{\int }_0^T|Y_t({\theta }_0)-ut{x}_0{e}^{{\theta }_{0}t}|dt.$$

Let $\parallel \cdot \parallel$ denote the $L_1$-norm. By changing variable, we have the following:

$${\tilde{\xi }}_T=arg\underset{\omega \in R}{inf}\parallel Y-\tilde{M}(\omega )\parallel :=arg\underset{\omega \in R}{inf}\tilde{N}(\omega ),$$ 41

where ${\tilde{M}}_t(\omega )=\frac{\omega t{e}^{{\theta }_{0}t}}{T}$ and $\tilde{N}(\cdot )=\parallel Y-\tilde{M}(\cdot )\parallel$.

We want to show that, for every $\mathrm{\Delta }>0$,

$$\underset{T\to +\mathrm{\infty }}{lim}{P}_{{\theta }_0}\{|{\tilde{\xi }}_T-A_0|>\mathrm{\Delta }\}=0.$$ 42

Therefore, we consider the set

$$V_{\mathrm{\Delta }}=\{\omega :|\omega -A_0|>\mathrm{\Delta }\},$$

where $P_{{\theta }_0}$ is the probability measure induced by the process $X_t$ when ${\theta }_0$ is the true parameter and $\epsilon \to 0$.

Besides, we have

$$\begin{array}{rcl}\tilde{N}(A_0)& =& \parallel Y-\tilde{N}(A_0)\parallel \\ & =& \parallel e^{{\theta }_{0}t}({\int }_0^t{e}^{-{\theta }_{0}s}d{L}_s^d-\frac{A_{0}t}{T}-A_0+A_0)\parallel \\ & =& \parallel e^{{\theta }_{0}t}({\int }_0^t{e}^{-{\theta }_{0}s}d{L}_s^d-A_0+(1-\frac{t}{T})A_0)\parallel \\ & =& \parallel e^{{\theta }_{0}t}({\int }_0^t{e}^{-{\theta }_{0}s}d{L}_s^d-A_0)\parallel +|A_{0}t|\parallel (1-\frac{t}{T})e^{{\theta }_{0}t}\parallel .\end{array}$$

On the other hand, for $\omega \in V_{\mathrm{\Delta }}$, we can get

$$\begin{array}{rcl}\tilde{N}(\omega )& =& \parallel Y-\tilde{M}(\omega )\parallel \\ & \ge & \parallel \tilde{M}(A_0)-\tilde{M}(\omega )\parallel -\parallel Y-\tilde{M}(A_0)\parallel \\ & =& \parallel \tilde{M}(\omega )-\tilde{M}(A_0)\parallel -\tilde{N}(A_0)\\ & =& |\omega -R_0|\parallel \frac{t{e}^{{\theta }_{0}t}}{T}\parallel -\tilde{N}(A_0)\\ & \ge & \mathrm{\Delta }\parallel \frac{t{e}^{{\theta }_{0}t}}{T}\parallel -\tilde{N}(A_0).\end{array}$$

Obviously, we have

$$\begin{array}{rl}& \frac{\tilde{N}(\omega )}{\tilde{N}(A_0)}\ge \frac{\mathrm{\Delta }\parallel \frac{t{e}^{{\theta }_{0}t}}{T}\parallel }{\tilde{N}(A_0)}-1,\\ & \begin{array}{rl}\frac{{inf}_{\omega \in V_{\mathrm{\Delta }}}\tilde{N}(\omega )}{\tilde{N}(A_0)}& \ge \mathrm{\Delta }{[\frac{T\parallel e^{{\theta }_{0}t}({\int }_0^t{e}^{-{\theta }_{0}s}d{M}_s^d-A_0)\parallel }{\parallel t{e}^{{\theta }_{0}t}\parallel }+\frac{|A_0|\parallel (T-t)e^{{\theta }_{0}t}\parallel }{\parallel t{e}^{{\theta }_{0}t}\parallel }]}^{-1}-1\\ & =\mathrm{\Delta }{[\frac{T\parallel e^{{\theta }_{0}t}(B_t-A_0)\parallel }{\parallel t{e}^{{\theta }_{0}t}\parallel }+\frac{|A_0|\parallel (T-t)e^{{\theta }_{0}t}\parallel }{\parallel t{e}^{{\theta }_{0}t}\parallel }]}^{-1}-1\\ & =\mathrm{\Delta }{(I_1+I_2)}^{-1}-1,\end{array}\end{array}$$

with

$$\begin{array}{rl}& I_1=\frac{T\parallel e^{{\theta }_{0}t}(B_t-A_0)\parallel }{\parallel t{e}^{{\theta }_{0}t}\parallel }=\frac{T\parallel e^{{\theta }_{0}t}{A}_t\parallel }{{\int }_0^{T}t{e}^{{\theta }_{0}t}dt}=\frac{T\parallel e^{{\theta }_{0}t}{A}_t\parallel }{{\theta }_0^{-1}T{e}^{{\theta }_{0}T}-{\theta }_0^{-2}{e}^{{\theta }_{0}T}+{\theta }_0^{-2}},\\ & I_2=\frac{|A_0|\parallel (T-t)e^{{\theta }_{0}t}\parallel }{\parallel t{e}^{{\theta }_{0}t}\parallel }=\frac{|A_0|{\int }_0^T{e}^{{\theta }_{0}t}dt}{{\int }_0^{T}t{e}^{{\theta }_{0}t}dt}=\frac{|A_0|({\theta }_0^{-2}{e}^{{\theta }_{0}T}-{\theta }_{0}T-{\theta }_0^{-2})}{{\theta }_0^{-1}T{e}^{{\theta }_{0}T}-{\theta }_0^{-2}{e}^{{\theta }_{0}T}+{\theta }_0^{-2}}.\end{array}$$

We obtain with probability one

$$\begin{array}{rl}\underset{T\to +\mathrm{\infty }}{lim}{I}_2& =\underset{T\to +\mathrm{\infty }}{lim}\frac{|A_0|({\theta }_0^{-2}{e}^{{\theta }_{0}T}-{\theta }_{0}T-{\theta }_0^{-2})}{{\theta }_0^{-1}T{e}^{{\theta }_{0}T}-{\theta }_0^{-2}{e}^{{\theta }_{0}T}+{\theta }_0^{-2}}\\ & =\underset{T\to +\mathrm{\infty }}{lim}\frac{|A_0|{\theta }_0^{-2}{e}^{{\theta }_{0}T}}{{\theta }_0^{-1}T{e}^{{\theta }_{0}T}}=\underset{T\to +\mathrm{\infty }}{lim}\frac{|A_0|}{{\theta }_{0}T}=0.\end{array}$$ 43

Moreover, using Lemma 2.2 we obtain

$$\begin{array}{rl}\underset{T\to +\mathrm{\infty }}{lim}{P}_{{\theta }_0}(I_1>\mathrm{\Delta })& =\underset{T\to +\mathrm{\infty }}{lim}{P}_{{\theta }_0}(\frac{T\parallel e^{{\theta }_{0}t}{R}_t\parallel }{{\theta }_0^{-1}T{e}^{{\theta }_{0}T}-{\theta }_0^{-2}{e}^{{\theta }_{0}T}+{\theta }_0^{-2}}>\mathrm{\Delta })\\ & =\underset{T\to +\mathrm{\infty }}{lim}{P}_{{\theta }_0}(\frac{T\parallel e^{{\theta }_{0}t}{R}_t\parallel }{{\theta }_0^{-1}T{e}^{{\theta }_{0}T}}>\mathrm{\Delta })\\ & =\underset{T\to +\mathrm{\infty }}{lim}{P}_{{\theta }_0}(|R_0|>{\theta }_0{e}^{{\theta }_{0}T}\mathrm{\Delta })\le \underset{T\to +\mathrm{\infty }}{lim}{\theta }_0^{-1}{e}^{-{\theta }_{0}T}\frac{E_{{\theta }_0}(|R_0|)}{\mathrm{\Delta }}=0.\end{array}$$ 44

By (43) and (44), we obtain as $T\to +\mathrm{\infty }$

$$\frac{{inf}_{\omega \in V_{\mathrm{\Delta }}}\tilde{N}(\omega )}{\tilde{N}(A_0)}\stackrel{P}{\to }+\mathrm{\infty }.$$ 45

Using (41), ${\tilde{\xi }}_T\in V_{\mathrm{\Delta }}$ implies

$$\tilde{N}({\xi }_T)=\underset{\omega \in V_{\mathrm{\Delta }}}{inf}\tilde{N}(\omega )\le \tilde{N}(A_0).$$ 46

Therefore, from Eqs. (45) and (46), we have the result (42). □

Remark 3.2

If $L_t^d$ is a Brownian motion, then ${\tilde{\xi }}_T$ is asymptotically Gaussian, this is treated by Kutoyants and Pilibossian [14, 15].

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Funding

This research is supported by the Distinguished Young Scholars Foundation of Anhui Province (1608085J06), the National Natural Science Foundation of China (11271020, 11601260), the Top Talent Project of University Discipline (Speciality) (gxbjZD03), the Natural Science Foundation of Chuzhou University (2016QD13), and Natural Science Foundation of Shandong Province (ZR2016AB01).

Competing interests

The authors declare that they have no competing interests.