**2. Preliminaries**

To construct sequential plans for estimation of the parameter *ϑ* we need some preparation. At first we shall summarize some known facts about the equation (1). For details the reader is referred to [3]. Together with the mentioned initial condition the equation (1) has a uniquely determined solution *X* which can be represented for *t* ≥ 0 as follows:

$$X(t) = \mathbf{x}\_0(t)X\_0(t) + b \int\_{-1}^0 \mathbf{x}\_0(t-s-1)X\_0(s)ds + \int\_0^t \mathbf{x}\_0(t-s)dW(s). \tag{3}$$

Here *x*<sup>0</sup> = (*x*0(*t*), *t* ≥ −1) denotes the fundamental solution of the deterministic equation

$$\mathbf{x}\_0(t) = 1 + \int\_0^t (a\mathbf{x}\_0(s) + b\mathbf{x}\_0(s-1)) ds, \quad t \ge 0,\tag{4}$$

corresponding to (1) with *x*0(*t*) = 0, *t* ∈ [−1, 0), *x*0(0) = 1.

The solution *X* has the property *E T* <sup>0</sup> *<sup>X</sup>*2(*s*)*ds* <sup>&</sup>lt; <sup>∞</sup> for every *<sup>T</sup>* <sup>&</sup>gt; 0.

From (3) it is clear, that the limit behaviour for *t* → ∞ of *X* very depends on the limit behaviour of *x*0(·). The asymptotic properties of *x*0(·) can be studied by the Laplace-transform of *x*0, which equals (*<sup>λ</sup>* <sup>−</sup> *<sup>a</sup>* <sup>−</sup> *<sup>b</sup>*e−*λ*)−1, *<sup>λ</sup>* any complex number.

Let *<sup>s</sup>* <sup>=</sup> *<sup>u</sup>*(*r*) (*<sup>r</sup>* <sup>&</sup>lt; <sup>1</sup>) and *<sup>s</sup>* <sup>=</sup> *<sup>w</sup>*(*r*) (*<sup>r</sup>* ∈ R1) be the functions given by the following parametric representation (*r*(*ξ*),*s*(*ξ*)) in <sup>R</sup><sup>2</sup> :

$$r(\xi) = \xi \cot \xi \text{ s} \\ \text{(\(\xi\) = -\(\xi / \sin \xi\))}$$

with *ξ* ∈ (0, *π*) and *ξ* ∈ (*π*, 2*π*) respectively.

2 Will-be-set-by-IN-TECH

Equation (1) is a very special case of stochastic differential equations with time delay, see [5, 6]

To estimate the true parameter *ϑ* with a prescribed least square accuracy *ε* we shall construct a sequential plan (*T*∗(*ε*), *ϑ*∗(*ε*)) working for all *ϑ* ∈ Θ. Here *T*∗(*ε*) is the duration of observations which is a special chosen stopping time and *ϑ*∗(*ε*) is an estimator of *ϑ*. The set Θ is defined to be the intersection of the set <sup>Θ</sup> with an arbitrary but fixed ball <sup>B</sup>0,*<sup>R</sup>* ⊂ R2. Sequential estimation problem has been solved for sets Θ of a different structure in [7]-[9], [11, 13, 14, 16]

In this chapter the set <sup>Θ</sup> of parameters consists of all (*a*, *<sup>b</sup>*)� from <sup>R</sup><sup>2</sup> which do not belong to

This sequential plan is a composition of several different plans which follow the regions to which the unknown true parameter *ϑ* = (*a*, *b*)� may belong to. Each individual plan is based on a weighted correlation estimator, where the weight matrices are chosen in such a way that this estimator has an appropriate asymptotic behaviour being typical for the corresponding region to which *ϑ* belongs to. Due to the fact that this behaviour is very connected with the asymptotic properties of the so-called fundamental solution *x*0(·) of the deterministic delay differential equation corresponding to (1) (see Section 2 for details), we have to treat different regions of <sup>Θ</sup> <sup>=</sup> <sup>R</sup><sup>2</sup> \ *<sup>L</sup>*, *<sup>L</sup>* <sup>=</sup> *<sup>L</sup>*<sup>1</sup> <sup>∪</sup> *<sup>L</sup>*2, separately. If the true parameter *<sup>ϑ</sup>* belongs to *L*, the weighted correlation estimator under consideration converges weakly only, and thus the assertions of Theorem 3.1 below cannot be derived by means of such estimators. In general, the exception of the set *L* does not disturb applications of the results below in adaptive filtration, control theory and other applications because of its Lebesgue zero measure.

In the papers [10, 12] the problem described above was solved for the two special sets of parameters Θ*<sup>I</sup>* (a straight line) and Θ*I I* (where *X*(·) satisfies (1) is stable or periodic (unstable)) respectively. The general sequential estimation problem for all *<sup>ϑ</sup>* = (*a*, *<sup>b</sup>*)� from <sup>R</sup><sup>2</sup> except of two lines was solved in [13, 14, 16] for the equation (1) based on the observations of *X*(·). In this chapter the sequential estimation method developed in [10, 12] for the system (1), (2) is extended to the case, considered by [13, 14, 16] for the equation (1) (as already mentioned, for

A related result in such problem statement was published first for estimators of an another

A similar problem for partially observed stochastic dynamic systems without time-delay was

The organization of this chapter is as follows. Section 2 presents some preliminary facts needed for the further studies about we have spoken. In Section 3 we shall present the main result, mentioned above. In Section 4 all proofs are given. Section 5 includes conclusions.

To construct sequential plans for estimation of the parameter *ϑ* we need some preparation. At first we shall summarize some known facts about the equation (1). For details the reader is referred to [3]. Together with the mentioned initial condition the equation (1) has a uniquely

by observations of the process (1) and in [10, 12, 15] – by noisy observations (2).

lines *L*<sup>1</sup> or *L*<sup>2</sup> defined in Section 2 below and having Lebesgue measure zero.

all *<sup>ϑ</sup>* from <sup>R</sup><sup>2</sup> except of two lines for the observations without noises).

determined solution *X* which can be represented for *t* ≥ 0 as follows:

structure and without proofs in [15].

solved in [22, 23].

**2. Preliminaries**

and [20] for example.

Now we define the parameter set <sup>Θ</sup> to be the plane <sup>R</sup><sup>2</sup> without the lines *<sup>L</sup>*<sup>1</sup> = (*a*, *<sup>u</sup>*(*a*))*a*≤<sup>1</sup> and *<sup>L</sup>*<sup>2</sup> = (*a*, *<sup>w</sup>*(*a*))*a*∈R<sup>1</sup> such that <sup>R</sup><sup>2</sup> <sup>=</sup> <sup>Θ</sup> <sup>∪</sup> *<sup>L</sup>*<sup>1</sup> <sup>∪</sup> *<sup>L</sup>*2.

It seems not to be possible to construct a general simple sequential procedure which has the desired properties under *P<sup>ϑ</sup>* for all *ϑ* ∈ Θ. Therefore we are going to divide the set Θ into some appropriate smaller regions where it is possible to do. This decomposition is very connected with the structure of the set Λ of all (real or complex) roots of the so-called characteristic equation of (4):

$$
\lambda - a - b \mathbf{e}^{-\lambda} = 0.
$$

Put *v*<sup>0</sup> = *v*0(*ϑ*) = max{*Reλ*|*λ* ∈ Λ}, *v*<sup>1</sup> = *v*1(*ϑ*) = max{*Reλ*|*λ* ∈ Λ, *Reλ* < *v*0}. Beside of the case *b* = 0 it holds −∞ < *v*<sup>1</sup> < *v*<sup>0</sup> < ∞. By *m*(*λ*) we denote the multiplicity of the solution *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>. Note that *<sup>m</sup>*(*λ*) = 1 for all *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> beside of (*a*, *<sup>b</sup>*) ∈ R<sup>2</sup> with *<sup>b</sup>* <sup>=</sup> <sup>−</sup>*ea*. In this cases we have *λ* = *a* − 1 ∈ Λ and *m*(*a* − 1) = 2. The values *v*0(*ϑ*) and *v*1(*ϑ*) determine the asymptotic behaviour of *x*0(*t*) as *t* → ∞ (see [3] for details).

Now we are able to divide Θ into some appropriate for our purposes regions. Note, that this decomposition is very related to the classification used in [3]. There the plane <sup>R</sup><sup>2</sup> was decomposed into eleven subsets. Here we use another notation.

*Definition* (Θ). The set Θ of parameters is decomposed as

$$
\Theta = \Theta\_1 \cup \Theta\_2 \cup \Theta\_3 \cup \Theta\_4.
$$

where Θ<sup>1</sup> = Θ<sup>11</sup> ∪ Θ<sup>12</sup> ∪ Θ13, Θ<sup>2</sup> = Θ<sup>21</sup> ∪ Θ22, Θ<sup>3</sup> = Θ31, Θ<sup>4</sup> = Θ<sup>41</sup> ∪ Θ<sup>42</sup> with

$$\Theta\_{11} = \{ \theta \in \mathcal{R}^2 \, | \, v\_0(\theta) < 0 \},$$

$$\Theta\_{12} = \{ \theta \in \mathcal{R}^2 \, | \, v\_0(\theta) > 0 \text{ and } v\_0(\theta) \notin \Lambda \},$$

$$\Theta\_{13} = \{\theta \in \mathfrak{R}^2 \, | \, v\_0(\theta) > 0; \, v\_0(\theta) \in \Lambda, \, m(v\_0) = 2\},$$

$$\Theta\_{21} = \{\theta \in \mathfrak{R}^2 \, | \, v\_0(\theta) > 0, v\_0(\theta) \in \Lambda, \, m(v\_0) = 1, \, v\_1(\theta) > 0 \text{ and } v\_1(\theta) \in \Lambda\},$$

$$\Theta\_{22} = \{\theta \in \mathfrak{R}^2 \, | \, v\_0(\theta) > 0, v\_0(\theta) \in \Lambda, \, m(v\_0) = 1, \, v\_1(\theta) > 0 \text{ and } v\_1(\theta) \notin \Lambda\},$$

$$\Theta\_{31} = \{\theta \in \mathfrak{R}^2 \, | \, v\_0(\theta) > 0, \, v\_0(\theta) \in \Lambda, \, m(v\_0) = 1 \text{ and } v\_1(\theta) < 0\},$$

$$\Theta\_{41} = \{\theta \in \mathfrak{R}^2 \, | \, v\_0(\theta) = 0, \, v\_0(\theta) \in \Lambda, \, m(v\_0) = 1\},$$

$$\Theta\_{42} = \{\theta \in \mathfrak{R}^2 \, | \, v\_0(\theta) > 0, \, v\_0(\theta) \in \Lambda, \, m(v\_0) = 1, \, v\_1(\theta) = 0 \text{ and } v\_1(\theta) \in \Lambda\}.$$

Using this operator and (5) we obtain the following equation:

**3. Construction of sequential estimation plans**

process (Δ˜ *<sup>Y</sup>*(*t*), *<sup>t</sup>* <sup>≥</sup> <sup>2</sup>) depending on the unknown parameters *<sup>a</sup>* and *<sup>b</sup>*.

with the initial condition <sup>Δ</sup>˜ *<sup>Y</sup>*(1) = *<sup>Y</sup>*(1) <sup>−</sup> *<sup>Y</sup>*0.

case if we observe (*X*(·)) instead of (*Y*(·)).

known continuous distribution function.

**3.1. Sequential estimation procedure for** *ϑ* ∈ Θ<sup>1</sup>

observations.

very similar.

the following condition:

is referred for details.

– the sequence of stopping times

*<sup>τ</sup>*1(*n*,*ε*) = *<sup>h</sup>*<sup>1</sup> inf{*<sup>k</sup>* <sup>≥</sup> 1 : *kh*<sup>1</sup>

– the functions

*<sup>d</sup>*Δ˜ *<sup>Y</sup>*(*t*) = *<sup>a</sup>*Δ˜ *<sup>Y</sup>*(*t*)*dt* <sup>+</sup> *<sup>b</sup>*Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>)*dt* <sup>+</sup> <sup>Δ</sup>˜ *<sup>ξ</sup>*(*t*)*dt* <sup>+</sup> *<sup>d</sup>*Δ˜ *<sup>V</sup>*(*t*) (6)

On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 27

Thus we have reduced the system (1), (2) to a single differential equation for the observed

In this section we shall construct the sequential estimation procedure for each of the cases Θ<sup>1</sup> ..., Θ<sup>4</sup> separately. Then we shall define, similar to [11, 13, 14, 16], the final sequential estimation plan, which works in Θ as a sequential plan with the smallest duration of

We shall construct the sequential estimation procedure of the parameter *ϑ* on the basis of the correlation method in the cases Θ1, Θ<sup>4</sup> (similar to [12, 14, 15]) and on the basis of correlation estimators with weights in the cases Θ<sup>2</sup> ∪ Θ3. The last cases and Θ<sup>13</sup> are new. It should be noted, that the sequential plan, constructed e.g. in [2] does not work for Θ<sup>3</sup> here, even in the

Consider the problem of estimating *ϑ* ∈ Θ1. We will use some modification of the estimation procedure from [12], constructed for the Case II thereon. It can be easily shown, that Proposition 3.1 below can be proved for the cases Θ<sup>11</sup> ∪ Θ<sup>12</sup> similarly to [12]. Presented below modified procedure is oriented, similar to [16] on all parameter sets Θ11, Θ12, Θ13. Thus we will prove Proposition 3.1 in detail for the case Θ<sup>13</sup> only. The proofs for cases Θ<sup>11</sup> ∪ Θ<sup>12</sup> are

For the construction of the estimation procedure we assume *h*<sup>10</sup> is a real number in (0, 1/5) and *h*<sup>1</sup> is a random variable with values in [*h*10, 1/5] only, F(0)-measurable and having a

Assume (*cn*)*n*≥<sup>1</sup> is a given unboundedly increasing sequence of positive numbers satisfying

1 *cn*

This construction follows principally the line of [14, 16] (see [12] as well), for which the reader


<sup>Ψ</sup>*s*(*t*) = (Δ˜ *<sup>Y</sup>*(*t*), <sup>Δ</sup>˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*))� for *<sup>t</sup>* <sup>≥</sup> <sup>1</sup> <sup>+</sup> *<sup>s</sup>*, (0, 0)� for *t* < 1 + *s*;

< ∞. (7)

<sup>−</sup>1*cn*} for *<sup>n</sup>* <sup>≥</sup> 1;

∑ *n*≥1

We introduce for every *ε* > 0 and every *s* ≥ 0 several quantities:

0

It should be noted, that the cases (Q2 ∪ Q3) and (Q5) considered in [3] correspond to our exceptional lines *L*<sup>1</sup> and *L*<sup>2</sup> respectively.

Here are some comments concerning the Θ subsets.

The unions Θ1,..., Θ<sup>4</sup> are marked out, because the Fisher information matrix and related design matrices which will be considered below, have similar asymptotic properties for all *ϑ* throughout every Θ*<sup>i</sup>* (*i* = 1, . . . , 4).

Obviously, all sets <sup>Θ</sup>11,..., <sup>Θ</sup><sup>42</sup> are pairwise disjoint, the closure of <sup>Θ</sup> equals to <sup>R</sup><sup>2</sup> and the exceptional set *L*<sup>1</sup> ∪ *L*<sup>2</sup> has Lebesgue measure zero.

The set Θ<sup>11</sup> is the set of parameters *ϑ* for which there exists a stationary solution of (1).

Note that the one-parametric set Θ<sup>4</sup> is a part of the boundaries of the following regions: Θ11, Θ12, Θ21, Θ3. In this case *b* = −*a* holds and (1) can be written as a differential equation with only one parameter and being linear in the parameter.

We shall use a truncation of all the introduced sets. First chose an arbitrary but fixed positive R. Define the set Θ = {*ϑ* ∈ Θ| ||*ϑ*|| ≤ R} and in a similar way the subsets Θ11,..., Θ42.

Sequential estimators of *ϑ* with a prescribed least square accuracy we have already constructed in [10, 12]. But in these articles the set of possible parameters *ϑ* were restricted to Θ<sup>11</sup> ∪ Θ<sup>12</sup> ∪ {Θ<sup>41</sup> \ {(0, 0)}} ∪ Θ42.

To construct a sequential plan for estimating *ϑ* based on the observation of *Y*(·) we follow the line of [10, 12]. We shall use a single equation for *Y* of the form:

$$dY(t) = \theta^\prime A(t)dt + \xi(t)dt + dV(t),\tag{5}$$

where *A*(*t*)=(*Y*(*t*),*Y*(*t* − 1))� ,

$$
\xi(t) = X(0) - aY(0) - bY(0) + b \int\_{-1}^{0} X\_0(s) ds - aV(t) - bV(t-1) + W(t).
$$

The random variables *A*(*t*) and *ξ*(*t*) are F(*t*)-measurable for every fixed *t* ≥ 1 and a short calculation shows that all conditions of type (7) in [12], consisting of

$$E\int\_{1}^{T} (|Y(t)| + |\tilde{\xi}(t)|) dt < \infty \text{ for all } T > 1,$$

$$E[\tilde{\Delta}\tilde{\xi}(t)|\mathcal{F}(t-2)] = 0, \; E[(\tilde{\Delta}\tilde{\xi}(t))^2|\mathcal{F}(t-2)] \le 1 + R^2$$

hold in our case. Here <sup>Δ</sup>˜ denotes the difference operator defined by <sup>Δ</sup>˜ *<sup>f</sup>*(*t*) = *<sup>f</sup>*(*t*) <sup>−</sup> *<sup>f</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>).

Using this operator and (5) we obtain the following equation:

$$d\tilde{\Delta}Y(t) = a\tilde{\Delta}Y(t)dt + b\tilde{\Delta}Y(t-1)dt + \tilde{\Delta}\tilde{\xi}(t)dt + d\tilde{\Delta}V(t) \tag{6}$$

with the initial condition <sup>Δ</sup>˜ *<sup>Y</sup>*(1) = *<sup>Y</sup>*(1) <sup>−</sup> *<sup>Y</sup>*0.

4 Will-be-set-by-IN-TECH

<sup>Θ</sup><sup>13</sup> <sup>=</sup> {*<sup>ϑ</sup>* ∈ R2<sup>|</sup> *<sup>v</sup>*0(*ϑ*) <sup>&</sup>gt; 0; *<sup>v</sup>*0(*ϑ*) <sup>∈</sup> <sup>Λ</sup>, *<sup>m</sup>*(*v*0) = <sup>2</sup>}, <sup>Θ</sup><sup>21</sup> <sup>=</sup> {*<sup>ϑ</sup>* ∈ R2<sup>|</sup> *<sup>v</sup>*0(*ϑ*) <sup>&</sup>gt; 0, *<sup>v</sup>*0(*ϑ*) <sup>∈</sup> <sup>Λ</sup>, *<sup>m</sup>*(*v*0) = 1, *<sup>v</sup>*1(*ϑ*) <sup>&</sup>gt; 0 and *<sup>v</sup>*1(*ϑ*) <sup>∈</sup> <sup>Λ</sup>}, <sup>Θ</sup><sup>22</sup> <sup>=</sup> {*<sup>ϑ</sup>* ∈ R2<sup>|</sup> *<sup>v</sup>*0(*ϑ*) <sup>&</sup>gt; 0, *<sup>v</sup>*0(*ϑ*) <sup>∈</sup> <sup>Λ</sup>, *<sup>m</sup>*(*v*0) = 1, *<sup>v</sup>*1(*ϑ*) <sup>&</sup>gt; 0 and *<sup>v</sup>*1(*ϑ*) �∈ <sup>Λ</sup>}, <sup>Θ</sup><sup>31</sup> <sup>=</sup> {*<sup>ϑ</sup>* ∈ R2<sup>|</sup> *<sup>v</sup>*0(*ϑ*) <sup>&</sup>gt; 0, *<sup>v</sup>*0(*ϑ*) <sup>∈</sup> <sup>Λ</sup>, *<sup>m</sup>*(*v*0) = 1 and *<sup>v</sup>*1(*ϑ*) <sup>&</sup>lt; <sup>0</sup>}, <sup>Θ</sup><sup>41</sup> <sup>=</sup> {*<sup>ϑ</sup>* ∈ R2<sup>|</sup> *<sup>v</sup>*0(*ϑ*) = 0, *<sup>v</sup>*0(*ϑ*) <sup>∈</sup> <sup>Λ</sup>, *<sup>m</sup>*(*v*0) = <sup>1</sup>}, <sup>Θ</sup><sup>42</sup> <sup>=</sup> {*<sup>ϑ</sup>* ∈ R2<sup>|</sup> *<sup>v</sup>*0(*ϑ*) <sup>&</sup>gt; 0, *<sup>v</sup>*0(*ϑ*) <sup>∈</sup> <sup>Λ</sup>, *<sup>m</sup>*(*v*0) = 1, *<sup>v</sup>*1(*ϑ*) = 0 and *<sup>v</sup>*1(*ϑ*) <sup>∈</sup> <sup>Λ</sup>}. It should be noted, that the cases (Q2 ∪ Q3) and (Q5) considered in [3] correspond to our

The unions Θ1,..., Θ<sup>4</sup> are marked out, because the Fisher information matrix and related design matrices which will be considered below, have similar asymptotic properties for all *ϑ*

Obviously, all sets <sup>Θ</sup>11,..., <sup>Θ</sup><sup>42</sup> are pairwise disjoint, the closure of <sup>Θ</sup> equals to <sup>R</sup><sup>2</sup> and the

Note that the one-parametric set Θ<sup>4</sup> is a part of the boundaries of the following regions: Θ11, Θ12, Θ21, Θ3. In this case *b* = −*a* holds and (1) can be written as a differential equation with

We shall use a truncation of all the introduced sets. First chose an arbitrary but fixed positive R. Define the set Θ = {*ϑ* ∈ Θ| ||*ϑ*|| ≤ R} and in a similar way the subsets Θ11,..., Θ42.

Sequential estimators of *ϑ* with a prescribed least square accuracy we have already constructed in [10, 12]. But in these articles the set of possible parameters *ϑ* were restricted to

To construct a sequential plan for estimating *ϑ* based on the observation of *Y*(·) we follow the

 0 −1

The random variables *A*(*t*) and *ξ*(*t*) are F(*t*)-measurable for every fixed *t* ≥ 1 and a short

*<sup>E</sup>*[Δ˜ *<sup>ξ</sup>*(*t*)|F(*<sup>t</sup>* <sup>−</sup> <sup>2</sup>)] = 0, *<sup>E</sup>*[(Δ˜ *<sup>ξ</sup>*(*t*))2|F(*<sup>t</sup>* <sup>−</sup> <sup>2</sup>)] <sup>≤</sup> <sup>1</sup> <sup>+</sup> *<sup>R</sup>*<sup>2</sup> hold in our case. Here <sup>Δ</sup>˜ denotes the difference operator defined by <sup>Δ</sup>˜ *<sup>f</sup>*(*t*) = *<sup>f</sup>*(*t*) <sup>−</sup> *<sup>f</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>).

(|*Y*(*t*)| + |*ξ*(*t*)|)*dt* < ∞ for all *T* > 1,

*A*(*t*)*dt* + *ξ*(*t*)*dt* + *dV*(*t*), (5)

*X*0(*s*)*ds* − *aV*(*t*) − *bV*(*t* − 1) + *W*(*t*).

The set Θ<sup>11</sup> is the set of parameters *ϑ* for which there exists a stationary solution of (1).

exceptional lines *L*<sup>1</sup> and *L*<sup>2</sup> respectively.

throughout every Θ*<sup>i</sup>* (*i* = 1, . . . , 4).

Θ<sup>11</sup> ∪ Θ<sup>12</sup> ∪ {Θ<sup>41</sup> \ {(0, 0)}} ∪ Θ42.

where *A*(*t*)=(*Y*(*t*),*Y*(*t* − 1))�

Here are some comments concerning the Θ subsets.

exceptional set *L*<sup>1</sup> ∪ *L*<sup>2</sup> has Lebesgue measure zero.

only one parameter and being linear in the parameter.

line of [10, 12]. We shall use a single equation for *Y* of the form:

,

*ξ*(*t*) = *X*(0) − *aY*(0) − *bY*(0) + *b*

*E T* 1

*dY*(*t*) = *ϑ*�

calculation shows that all conditions of type (7) in [12], consisting of

Thus we have reduced the system (1), (2) to a single differential equation for the observed process (Δ˜ *<sup>Y</sup>*(*t*), *<sup>t</sup>* <sup>≥</sup> <sup>2</sup>) depending on the unknown parameters *<sup>a</sup>* and *<sup>b</sup>*.

## **3. Construction of sequential estimation plans**

In this section we shall construct the sequential estimation procedure for each of the cases Θ<sup>1</sup> ..., Θ<sup>4</sup> separately. Then we shall define, similar to [11, 13, 14, 16], the final sequential estimation plan, which works in Θ as a sequential plan with the smallest duration of observations.

We shall construct the sequential estimation procedure of the parameter *ϑ* on the basis of the correlation method in the cases Θ1, Θ<sup>4</sup> (similar to [12, 14, 15]) and on the basis of correlation estimators with weights in the cases Θ<sup>2</sup> ∪ Θ3. The last cases and Θ<sup>13</sup> are new. It should be noted, that the sequential plan, constructed e.g. in [2] does not work for Θ<sup>3</sup> here, even in the case if we observe (*X*(·)) instead of (*Y*(·)).
