On Parameter Estimation of Hidden Ergodic Ornstein-Uhlenbeck Process

TL;DR

This paper develops a two-step maximum likelihood estimator for unknown parameters in a partially observed Ornstein-Uhlenbeck process, demonstrating its consistency and asymptotic normality using Kalman-Bucy filtering.

Contribution

It introduces a novel two-step MLE process for joint estimation of the unobserved process and parameters in linear stochastic differential equations.

Findings

Estimator is consistent and asymptotically normal.

Recurrent estimators are constructed using Kalman-Bucy filtering.

Theoretical properties are established for large samples.

Abstract

We consider the problem of parameter estimation for the partially observed linear stochastic differential equation. We assume that the unobserved Ornstein-Uhlenbeck process depends on some unknown parameter and estimate the unobserved process and the unknown parameter simultaneously. We construct the two-step MLE-process for the estimator of the parameter and describe its large sample asymptotic properties, including consistency and asymptotic normality. Using the Kalman-Bucy filtering equations we construct recurrent estimators of the state and the parameter.

Equations

$$\mathrm{d}{X}_t$$

$$\mathrm{d}{Y}_t$$

$$X_t^2$$

$${\hat{\sigma }}_t^2=t^{-1}{X}_t^2-2t^{-1}{\int }_0^t{X}_s\mathrm{d}{X}_s={\sigma }^2$$

$$X_t=ab{\int }_0^t{\int }_0^s{e}^{-f\left(s-r\right)}\mathrm{d}{V}_r\mathrm{d}s+\sigma W_t+o\left(1\right).$$

$$L\left(\vartheta ,X^T\right)$$

$$L({\hat{\vartheta }}_T,X^T)=\underset{\vartheta \in \Theta }{\sup }L\left(\vartheta ,X^T\right).$$

$$\mathrm{d}m\left(\vartheta ,t\right)$$

$$=-\left[\vartheta +\frac{\gamma \left(\vartheta ,t\right)a^2}{{\sigma }^2}\right]m\left(\vartheta ,t\right)\mathrm{d}t+\frac{\gamma \left(\vartheta ,t\right)a}{{\sigma }^2}\mathrm{d}{X}_t,$$

$$\frac{\partial \gamma \left(\vartheta ,t\right)}{\partial t}=-2\vartheta \gamma \left(\vartheta ,t\right)-\frac{\gamma {\left(\vartheta ,t\right)}^2{a}^2}{{\sigma }^2}+b^2,\gamma \left(\vartheta ,0\right)=d^2.$$

$$\frac{{\hat{\vartheta }}_{\epsilon }-\vartheta }{\epsilon }⟹\mathcal{N}\left(0,\mathrm{I}{\left(\vartheta \right)}^{-1}\right).$$

$$\sqrt{T}\left({\hat{\vartheta }}_T-\vartheta \right)⟹\mathcal{N}\left(0,\mathrm{I}{\left(\vartheta \right)}^{-1}\right).$$

$$\frac{{\hat{\vartheta }}_{\epsilon }-\vartheta }{\sqrt{\epsilon }}⟹\mathcal{N}\left(0,\mathrm{I}{\left(\vartheta \right)}^{-1}\right).$$

$${\text{SS}}_K$$

$${\Phi }_{\ast }\left(\vartheta \right)=\frac{a^2{\vartheta }^2}{f^3}\left[e^{-f}-1+f\right]+{\sigma }^2,\hat{\Phi }\left(\vartheta \right)=\frac{{\vartheta }^2{b}^2}{f^3}\left[e^{-f}-1+f\right]+{\sigma }^2$$

$$\mathrm{d}{X}_t$$

$$\mathrm{d}{Y}_t$$

$${\bar{\vartheta }}_K={\vartheta }_K^{\ast }\text{mbox}{1I}_{\left\{{\mathcal{A}}_K\right\}}+\alpha \text{mbox}{1I}_{\left\{{\mathcal{A}}_K^{-}\right\}}+\beta \text{mbox}{1I}_{\left\{{\mathcal{A}}_K^{+}\right\}}.$$

$${\mathcal{A}}_K$$

$${\mathcal{A}}_K^{+}$$

$$\underset{{\vartheta }_0\in \Theta }{\sup }{\mathbf{E}}_{{\vartheta }_0}{\left|{\bar{\vartheta }}_K-{\vartheta }_0\right|}^2\le \frac{C}{K}$$

$${\mathbf{E}}_{{\vartheta }_0}{\left[{\bar{\vartheta }}_K-{\vartheta }_0\right]}^2$$

$$+{\left(\beta -{\vartheta }_0\right)}^2{\mathbf{P}}_{{\vartheta }_0}\left({\mathcal{A}}_K^{+}\right).$$

$${\mathbf{P}}_{{\vartheta }_0}\left({\mathcal{A}}_K^{-}\right)$$

$${\mathbf{P}}_{{\vartheta }_0}\left({\mathcal{A}}_K^{+}\right)$$

$${\text{SS}}_K$$

$$=\frac{a^2}{K}\sum _{k=1}^K{\eta }_k^2+\frac{2a\sigma }{K}\sum _{k=1}^K{\eta }_k\left[W_k-W_{k-1}\right]+\frac{{\sigma }^2}{K}\sum _{k=1}^K{\left[W_k-W_{k-1}\right]}^2,$$

$${\eta }_k={\int }_{k-1}^k{Y}_t\mathrm{d}t.$$

$${\mathbf{E}}_{{\vartheta }_0}{\text{SS}}_K$$

$$Y_t=\xi e^{-{\vartheta }_{0}t}+b{\int }_0^t{e}^{-{\vartheta }_0\left(t-r\right)}\mathrm{d}{V}_r.$$

$${\mathbf{E}}_{{\vartheta }_0}{Y}_t{Y}_s$$

$$=\left[d^2-\frac{b^2}{2{\vartheta }_0}\right]e^{-{\vartheta }_0\left(t+s\right)}+\frac{b^2}{2{\vartheta }_0}{e}^{-{\vartheta }_0|t-s|}$$

$${\mathbf{E}}_{{\vartheta }_0}{\eta }_k^2$$

$$=\left[d^2-\frac{b^2}{2{\vartheta }_0}\right]{\left({\int }_{k-1}^k{e}^{-{\vartheta }_{0}t}\mathrm{d}t\right)}^2+\frac{b^2}{2{\vartheta }_0}{\int }_{k-1}^k{\int }_{k-1}^k{e}^{-{\vartheta }_0|t-s|}\mathrm{d}s\mathrm{d}t$$

$$=\left[\frac{d^2}{{\vartheta }_0^2}-\frac{b^2}{2{\vartheta }_0^3}\right]{\left[e^{{\vartheta }_0}-1\right]}^2{e}^{-2{\vartheta }_{0}k}+\frac{b^2}{{{\vartheta }_0}^3}\left[e^{-{\vartheta }_0}-1+{\vartheta }_0\right].$$

Full text

MSC 2000 Classification: 62M02, 62G10, 62G20.

Key words: Partially observed linear system, parameter estimation, hidden process, ergodic process, One-step MLE-process.

1 Introduction

We are given a partially observed linear system, defined by the equations

[TABLE]

where $a\not=0,\sigma\not=0,b\not=0$ and $f>0$ are constants, $W^{T}=\left(W_{t},0\leq t\leq T\right)$ and $V^{T}=\left(V_{t},0\leq t\leq T\right)$ are two independent Wiener processes. The random variable $\xi\sim{\cal N}\left(0,d^{2}\right)$ is independent of $W^{T}$ and $V^{T}$.

The system (1)-(2) is defined by the four parameters $a,f,b,\sigma^{2}$. Recall that the parameter $\sigma^{2}$ can be estimated without error by continuous time observations $X^{T}$ as follows. By the Itô formula we can write

[TABLE]

Hence, for any $t\in(0,T]$, we have the estimator

[TABLE]

and this estimator equals the true value. Therefore we consider only the estimation of the three other parameters $f,b$ and $a$. Note that the consistent estimation of the three-dimensional parameter $\vartheta=\left(a,b,f\right)$ is impossible because the observed process can be written as follows

[TABLE]

This means that the parameters $a$ and $b$ appear as product $ab$. We can have consistent estimation of two-dimensional parameters $\left(f,a\right)$ and $\left(f,b\right)$. This possibility we discuss in the last section.

The observations are $X^{T}=\left(X_{t},0\leq t\leq T\right)$ and the Ornstein-Uhlenbeck process $Y^{T}$ is unobservable (hidden), i.e., we have partially observed linear model of observations.

We consider estimation of the one-dimensional parameters $f$, $b$ and $a$ separately given the continuous time observations $X^{T}$. The unknown parameter will be denoted by $\vartheta$ and we will assume that $\vartheta\in\Theta=\left(\alpha,\beta\right)$ for some constants $\alpha<\beta$. In all the cases the set $\Theta$ does not contain 0. Thus we are faced with three different problems: $\vartheta=f$, $\vartheta=b$ and $\vartheta=a$. In each problem we propose a two-step construction of asymptotically efficient estimator-process of recurrent nature. First we propose a preliminary consistent estimator $\vartheta_{T^{\delta}}$ based on the observations $X^{T^{\delta}}=\left(X_{t},0\leq t\leq T^{\delta}\right)$ with $\delta\in\left(1/2,1\right)$. Then this estimator is used for construction of One-step MLE-process, which has recurrent structure. In the last section we discuss the possibilities of the joint estimation of two dimensional parameters $\vartheta=\left(f,b\right)$ and $\vartheta=\left(f,a\right)$.

Equations (1)-(2) is a prototypical model in the Kalman-Bucy filtering theory, which provides a closed form system of equations for the conditional expectation $m\left(t\right)=\mathbf{E}\left(Y_{t}|X_{s},0\leq s\leq t\right)$ ([1], [9],[18]). The statistical problems for discretely observed hidden Markov processes were studied by many authors (see [2], [3], [6], [7] and the references therein). However, the literature on continuous time models is limited. For the results in continuous time setup, we refer the interested reader to [13] (linear and non linear partially observed systems with small noise), [6] (continuous-time hidden Markov models estimation), [4] and [11] (hidden telegraph process observed in the white Gaussian noise).

In the present paper we are particularly interested in the asymptotic behavior of the maximum likelihood estimator (MLE) $\hat{\vartheta}_{T}$ in the large sample asymptotic regime, i.e., when $T\rightarrow\infty$. The statistical problems for such observation models have been widely studied, motivated by the importance of the Kalman-Bucy filtering in engineering applications.

Let us now recall the definitions of the MLE in the case $\vartheta=f$, when the other two parameters $a$ and $b$ are known. As the parameters of the model take finite values and $\sigma^{2}>0$, the measures $\left\{\mathbf{P}_{\vartheta}^{\left(T\right)},\vartheta\in\Theta\right\}$ induced by the observations (1) on the space of continuous functions on $\left[0,T\right]$ are equivalent. The likelihood ratio function ([18]) is given by the expression

[TABLE]

Then the MLE $\hat{\vartheta}_{T}$ is defined by the equation

[TABLE]

This means that to calculate $\hat{\vartheta}_{T}$ we need the values of the family of stochastic processes $\left(m\left(\vartheta,t\right),0\leq t\leq T\right),\vartheta\in\Theta$. The random process $m\left(\vartheta,\cdot\right)$ is solution of the Kalman-Bucy filtering equations (see [1], [9], [18])

[TABLE]

where $m\left(\vartheta,0\right)=\mathbf{E}_{\vartheta}\left(\xi|X_{0}\right)=0$. The function $\gamma\left(\vartheta,t\right)=\mathbf{E}_{\vartheta}\left(m\left(\vartheta,t\right)-Y_{t}\right)^{2}$ is the solution of the Ricatti equation

[TABLE]

Due to importance of this model in many applied problems, much engineering literature is concerned with identification of this model.

The behavior of the MLE was studied at least in three asymptotics:

• •

*Small noise in both equations * $\sigma=b=\varepsilon\rightarrow 0$ ($T$ is fixed) [12], [13]

[TABLE]

• •

Large sample $T\rightarrow\infty$ ($\sigma$ and $b$ are fixed) [14]

[TABLE]

• •

Small noise in observation only, $\sigma\rightarrow 0$, ($T$ and $b$ are fixed) [16]

[TABLE]

In all three cases ${\rm I}\left(\vartheta\right)$ the Fisher information is different. It was also shown that the polynomial moments of the scaled estimation error converge and the MLE is asymptotically efficient.

It is evident that the numerical calculation of the MLE $\hat{\vartheta}_{T}$ according to (3)-(6) is quite a difficult problem. The goal of this work is to suggest the new estimator, called One-step MLE-process $\vartheta_{t}^{\star},\tau\leq t\leq T$, which has two advantages. First, its numerical calculation is much more simple than that of the MLE and, second, this estimator has a recurrent structure and can be used for the joint estimation of the hidden process $Y_{t}$ and the parameter $\vartheta$. Similar One-step MLE’s and Multi-step MLE-processes, introduced in [15], have been applied in the problem of parameter estimation of the hidden telegraph process [11], parameter estimation in diffusion processes by the discrete time observations [10], in the problem of frequency estimation [8], intensity parameter estimation for inhomogeneous Poisson processes [5], parameter estimation for the Markov sequences [17].

2 Preliminary estimator.

Following [11] One-step MLE process will be constructed in two steps. First we introduce a consistent and asymptotically normal preliminary estimator and then this estimator is used to define One-step MLE-process. Preliminary estimator is constructed using an asymptotically negligible amount of the observations $X^{K}=\left(X_{t},0\leq t\leq K\right)$, where $K=T^{\delta},\delta\in\left(1/2,1\right)$.

Suppose that $\vartheta=f$ and introduce the statistic $\SS_{K}$ and the function $\Phi\left(\vartheta\right),\vartheta\in\Theta$:

[TABLE]

In the cases $\vartheta=b$ and $\vartheta=a$ the counterparts of the latter function are

[TABLE]

respectively.

In this section we consider the case $\vartheta=f$ only. Therefore

[TABLE]

Note that the function $\Phi\left(\vartheta\right),\alpha<\vartheta<\beta$ is strictly decreasing. Define the preliminary estimator $\bar{\vartheta}_{K}$, base the observations $X^{K}$:

[TABLE]

Here $\vartheta_{K}^{*}$ is the root of equation $\Phi\left(\vartheta_{K}^{*}\right)=\SS_{K}$ and ${\cal A}_{K},{\cal A}_{K}^{-},{\cal A}_{K}^{+}$ are the sets

[TABLE]

The asymptotic behavior of $\bar{\vartheta}_{K}$ as $K\rightarrow\infty$ is described in the following proposition.

Proposition 1

*The estimator $\bar{\vartheta}_{K}$ is consistent, uniformly on compacts

$\left[\bar{\alpha},\bar{\beta}\right]$ $\subset\Theta$, and*

[TABLE]

with some constant $C>0$.

Proof. We have

[TABLE]

For the probabilities we have the estimates

[TABLE]

Therefore we have to study the asymptotics of the statistic $\SS_{K}$ as $K\rightarrow\infty$:

[TABLE]

where

[TABLE]

We have

[TABLE]

because $Y^{T}=\left(Y_{t},0\leq t\leq T\right)$ and $W^{T}=\left(W_{t},0\leq t\leq T\right)$ are independent.

The process $Y^{T}$ can be written as

[TABLE]

Hence

[TABLE]

and

[TABLE]

Therefore

[TABLE]

Using similar calculations we obtain the estimate

[TABLE]

which allows us to prove the law of large numbers: for $K\rightarrow\infty$ we have convergence in mean square

[TABLE]

and

[TABLE]

Hence

[TABLE]

The function $\Phi\left(\vartheta\right),\alpha<\vartheta<\beta$ is strictly decreasing. If we denote its inverse function as $\Psi\left(\phi\right)=\Phi^{-1}\left(\phi\right),\Phi\left(\beta\right)<\phi<\Phi\left(\alpha\right)$, then we have

[TABLE]

and

[TABLE]

We can write

[TABLE]

as $K\rightarrow\infty$.

If we put $K=T^{\delta}$, then

[TABLE]

3 One-Step MLE-process. Case $\vartheta=f$.

Suppose that the unknown parameter is $\vartheta=f$ and we have the model (7)-(8), where the process $X^{T}$ is observable and the Ornstein-Uhlenbeck process $Y^{T}$ is “hidden”. We realize the asymptotically efficient estimation of the parameter $\vartheta\in\Theta$ in two steps. First we calculate the preliminary estimator $\bar{\vartheta}_{T^{\delta}}$ and then using this estimator we construct the One-step MLE-process.

Recall that the equation (6) has explicit solution

[TABLE]

Here $\gamma_{0}=d^{2}$,

[TABLE]

Therefore we have exponential convergence of $\gamma\left(\vartheta,t\right)$ to the stationary solution $\gamma\left(\vartheta\right)$

[TABLE]

To simplify the exposition we suppose that $d^{2}=\gamma\left(\vartheta\right)$; then we have $\gamma\left(\vartheta,t\right)=\gamma\left(\vartheta\right)$. The case with an arbitrary $d^{2}$ requires cumbersome calculations, but the main results remain intact.

The equation for $m\left(\vartheta,t\right)$ in this case is

[TABLE]

Denote $m_{t}=m\left(\vartheta_{0},t\right)$ and $\gamma_{*}\left(\vartheta_{0}\right)=\gamma_{*}$, where $\vartheta_{0}$ is the true value. Then for the process $m_{t},0\leq t\leq T$ we obtain the equation

[TABLE]

Here we used the innovation theorem (see [18], ??)

[TABLE]

The innovation Wiener process $\bar{W}_{t}$ is defined by this equation and $m_{0}$ is independent on $\bar{W}_{t},0\leq t\leq T$. With probability 1, the random process $m\left(\vartheta,t\right)$ has continuous derivatives w.r.t. $\vartheta$ and derivative processes $\dot{m}\left(\vartheta,t\right),\ddot{m}\left(\vartheta,t\right)$ satisfy the equations

[TABLE]

The Fisher information for this model of observations is

[TABLE]

Note that ${\rm I}\left(\vartheta\right)$ has continuous bounded derivatives and is uniformly in $\vartheta\in\Theta$ separated from zero.

According to [15] the One-step MLE-process $\vartheta_{t}^{\star},T^{\delta}<t\leq T$ is introduced as follows

[TABLE]

Let us change the variables $t=\tau T$ and denote $\vartheta_{\tau T}^{\star}=\vartheta_{T}^{\star}\left(\tau\right),T^{\delta-1}<\tau\leq 1$.

Theorem 1

One-step MLE-process $\vartheta_{T}^{\star}\left(\tau\right),T^{\delta-1}<\tau\leq 1$ with $\delta\in\left(1/2,1\right)$ is consistent: for any $\nu>0$ and any $\tau\in(0,1]$

[TABLE]

and asymptotically normal

[TABLE]

Proof. Consider the difference

[TABLE]

Note that as it follows from the equations (3)-(3), the Gaussian processes $\dot{m}\left(\vartheta,t\right)$ and $\ddot{m}\left(\vartheta_{0},t\right)$ have bounded variances and therefore for any $p>1$ we have

[TABLE]

where the constants do not depend on $t$. We can write

[TABLE]

because

[TABLE]

Here $\left|\tilde{\vartheta}-\bar{\vartheta}_{T^{\delta}}\right|\leq\left|\vartheta_{0}-\bar{\vartheta}_{T^{\delta}}\right|$.

Further, for the Fisher information we have

[TABLE]

This allows us to write

[TABLE]

By the law of large numbers

[TABLE]

and therefore by the central limit theorem

[TABLE]

The similar arguments allow us to write

[TABLE]

Recall that as we have stationary regime $\mathbf{E}_{\vartheta_{0}}\dot{m}(\vartheta_{0},s)^{2}=\sigma^{2}a^{-2}{\rm I}\left(\vartheta_{0}\right)$. Therefore

[TABLE]

where the integral (see, e.g., Proposition 1.23 in [14])

[TABLE]

Hence we obtained the representation

[TABLE]

Substitution of this relation into the initial representation (3) yields the final expression

[TABLE]

since $\delta\in(1/2,1)$.

Note that the process $\vartheta_{t}^{\star},T^{\delta}<t\leq T$ can be written in recurrent form

[TABLE]

and we can introduce the adaptive filtering equations as follows

[TABLE]

with the initial value $m_{T^{\delta}}=m\left(\bar{\vartheta}_{T^{\delta}},T^{\delta}\right)$. Here

[TABLE]

It will be interesting to see the behavior of the system (16)-(18) using numerical simulations.

Recall that if we put $\tau=1$, then $\vartheta_{T}^{\star}$ is One-step MLE with

[TABLE]

studied for ergodic diffusion processes in the Section 2.5 [14]. Therefore the estimator $\vartheta_{T}^{\star}$ is asymptotically equivalent to the asymptotically efficient MLE $\hat{\vartheta}_{T}$ defined by the equation (4). There is essential computational difference between these two estimators. The calculation of $\hat{\vartheta}_{T}$ using (3)-(6) requires solving the differential equations (1)-(6) for numerous values of $\vartheta\in\Theta$, which is computationally inefficient. To construct One-step MLE-process $\vartheta_{T}^{\star}$ we have to calculate a simple preliminary estimator $\bar{\vartheta}_{T^{\delta}}$ and then to solve the system (1)-(6) for just one value $\vartheta=\bar{\vartheta}_{T^{\delta}}$. The difference between these two approaches becomes even more significant in the case of multidimensional $\vartheta$.

4 One-Step MLE-process. Case $\vartheta=b$.

Suppose that the volatility $b=\vartheta$ is the unknown parameter and we have the equations

[TABLE]

As before all parameters $a,\sigma,\vartheta$ do not vanish and $f>0$. The volatility $\vartheta\in\left(\alpha,\beta\right)$ with $\alpha>0$ and the function

[TABLE]

is strictly increasing.

The statistic $\SS_{K}$, with the new notations, converges to this function

[TABLE]

Therefore we have the explicit expression for the preliminary estimator

[TABLE]

where

[TABLE]

Here the sets ${\cal B}^{\pm}$ are defined by the similar relations

[TABLE]

As before, we have the consistency

[TABLE]

and

[TABLE]

We need the equation for $\dot{m}\left(\vartheta,t\right)$ and expression for Fisher information

[TABLE]

in this case. The filtering equations in the stationary regime are

[TABLE]

Therefore

[TABLE]

For $\vartheta=\vartheta_{0}$

[TABLE]

where

[TABLE]

Since

[TABLE]

we obtain

[TABLE]

Therefore the Fisher information is

[TABLE]

Now we can write the One-step MLE-process $\vartheta_{t}^{\star},T^{\delta}<t\leq T$ as follows

[TABLE]

If we change the variables $t=\tau T$ and denote $\vartheta_{\tau T}^{\star}=\vartheta_{T}^{\star}\left(\tau\right),T^{\delta-1}<\tau\leq 1$, then we obtain the same assertions as in the Theorem 1:

Proposition 2

One-step MLE-process $\vartheta_{T}^{\star}=\left(\vartheta_{T}^{\star}\left(\tau\right),T^{\delta-1}<\tau\leq 1\right)$ with $\delta\in\left(1/2,1\right)$ is consistent: for any $\nu>0$ and any $\tau\in(0,1]$

[TABLE]

and asymptotically normal

[TABLE]

Proof. Similarly to (16), we have exactly the same representation for the estimator $\vartheta_{t}^{\star}$ as in (14), with the only difference in the forms of $\dot{m}\left(\vartheta,t\right)$ and ${\rm I}\left(\vartheta\right)$. Thus the previous proof works in this case as well.

It is possible to write the system of recurrent equations as in (16)-(18).

5 One-Step MLE-process. Case $\vartheta=a$.

It is clear that the suggested estimation approach also works for the partially observed system

[TABLE]

where the unknown parameter is the drift $\vartheta=a$.

The function

[TABLE]

is strictly increasing and the corresponding preliminary estimator $\bar{\vartheta}_{K}$ admits the same asymptotic properties as in the preceding section.

The filtering equations are

[TABLE]

Therefore

[TABLE]

To calculate Fisher information ${\rm I}\left(\vartheta_{0}\right)=\sigma^{-2}\mathbf{E}_{\vartheta_{0}}\left[m\left(\vartheta_{0},t\right)+\vartheta_{0}\dot{m}\left(\vartheta_{0},t\right)\right]^{2}$ we write the representations

[TABLE]

In the last integral we change the order of integration

[TABLE]

Introduce notations

[TABLE]

Then $\dot{m}\left(\vartheta_{0},t\right)$ we can write as follows

[TABLE]

Hence

[TABLE]

Therefore the Fisher information in this problem is the function

[TABLE]

Having the preliminary estimator $\bar{\vartheta}_{T^{\delta}}$, expression for Fisher information ${\rm I}\left(\vartheta_{0}\right)$ and the equation for $\dot{m}\left(\vartheta,t\right)$ we can construct the One-step MLE-process $\vartheta_{t}^{\star},T^{\delta}<t\leq T$ of the same form as in (14), with $a$ replaced by $\bar{\vartheta}_{T^{\delta}}$.

This estimator has the same asymptotic properties: it is consistent and asymptotically normal

[TABLE]

The proof follows the same pattern as in the previous cases.

6 Discussion

The results, presented above, can be developed in several directions by means of already known approaches.

It is interesting to find preliminary estimator in the cases of unknown parameters $\vartheta=\left(f,b,a\right)$. Of course, with one statistic $\SS_{T^{\delta}}$ it is impossible and we need at least three different statistics.

Consider the case of two-dimensional parameter $\vartheta=\left(f,b\right)$ or $\vartheta=\left(f,a\right)$ and two statistics

[TABLE]

The limits are

[TABLE]

Therefore

[TABLE]

The function

[TABLE]

is strictly increasing and $\lim_{x\rightarrow 0}\phi\left(x\right)=1$, $\lim_{x\rightarrow\infty}\phi\left(x\right)=\infty$. Therefore, the parameter $f$ can be estimated with the help of the statistic $\mathbb{Q}_{K}$:

[TABLE]

Having this estimator the second parameter, say, $a$ or $b$ can be obtained as solution of one of these equations

[TABLE]

with obvious notation. As soon as we have a consistent preliminary estimator, say, $\bar{\vartheta}_{T^{\delta}}=\left(f_{T^{\delta}}^{*},b_{T^{\delta}}^{*}\right)$ and explicit expression for the information matrix ${\rm I}\left(\vartheta\right)$, then

[TABLE]

Recall that such processes were studied in [15]. 2. 2.

The One-step MLE-process has learning interval $\left[0,T^{\delta}\right]$ with $\delta\in(\frac{1}{2},1]$. It can be interesting to have such process with shorter learning. This can be done with the help of another construction called Two-step MLE-process introduced in [15]. Let us recall this construction using the model of observation (7)-(8). The first preliminary estimator $\bar{\vartheta}_{T^{\delta}}$ is constructed using the observations $X^{T^{\delta}}=\left(X_{t},0\leq t\leq T^{\delta}\right)$ with $\delta\in(1/3,1/2]$ (shorter learning interval). The second preliminary estimator-process $\vartheta_{t}^{*},T^{\delta}<t\leq T$ is

[TABLE]

The Two-step MLE-process is

[TABLE]

Following the same arguments as in the proof of Theorem 2 in [15] it can be shown that

[TABLE]

where $\vartheta_{T}^{\star\star}\left(\tau\right)=\vartheta_{\tau T}^{\star\star}$.

The learning interval $\left[0,T^{\delta}\right]$ can be made even shorter if $\delta\in(1/4,1/3]$. In this case we use Three-step MLE-process (see details in [15]). 3. 3.

Consider the model (7)-(8) and the estimator-process $\vartheta_{T}^{\star}\left(\tau\right),\left[\kappa,1\right]$, where $\kappa>0$. Let us denote by ${\cal P}_{T}$ the measure induced by the process

[TABLE]

in the measurable space $\left({\mathscr{C}}\left[\kappa,1\right],{\mathscr{B}}\right)$ of continuous on $\left[\kappa,1\right]$ functions. It is possible to verify the weak convergence

[TABLE]

where ${\cal P}$ corresponds to the Gaussian process $\zeta\left(\tau\right),\left[\kappa,1\right]$ with

[TABLE]

i.e. $\zeta\left(\cdot\right)$ is a Wiener process on the interval $\left[\kappa,1\right]$.

The proof in similar situation can be found in [15], Theorem 1. It consists of proving convergence of the finite-dimensional distributions

[TABLE]

and the estimate

[TABLE]

where the constant $C>0$ does not depend on $T$. The approach applied in the present work allows us the direct verification these two conditions.

Acknowledgment. I would like to thank P. Chigansky for useful comments.