**1. Introduction**

22 Will-be-set-by-IN-TECH

[24] Wang, Z., Yang, F., Ho, D. & Liu, X. [2006]. Robust *H*∞ filtering for stochastic time-delay systems with missing measurements, *IEEE Transactions on Signal Processing*

54(No. 7): 2579–2587.

Assume (Ω, F,(F(*t*), *t* ≥ 0), *P*) is a given filtered probability space and *W* = (*W*(*t*), *t* ≥ 0), *V* = (*V*(*t*), *t* ≥ 0) are real-valued standard Wiener processes on (Ω, F,(F(*t*), *t* ≥ 0), *P*), adapted to (F(*t*)) and mutually independent. Further assume that *X*<sup>0</sup> = (*X*0(*t*), *t* ∈ [−1, 0]) and *Y*<sup>0</sup> are a real-valued cadlag process and a real-valued random variable on (Ω, F,(F(*t*), *t* ≥ 0), *P*) respectively with

$$E \int\_{-1}^{0} X\_0^2(s) ds < \infty \text{ and } EY\_0^2 < \infty.$$

Assume *Y*<sup>0</sup> and *X*0(*s*) are F0−measurable, *s* ∈ [−1, 0] and the quantities *W*, *V*, *X*<sup>0</sup> and *Y*<sup>0</sup> are mutually independent.

Consider a two–dimensional random process (*X*,*Y*)=(*X*(*t*),*Y*(*t*), *t* ≥ 0) described by the system of stochastic differential equations

$$dX(t) = aX(t)dt + bX(t-1)dt + dW(t),\tag{1}$$

$$dY(t) = X(t)dt + dV(t), \; t \ge 0 \tag{2}$$

with the initial conditions *X*(*t*) = *X*0(*t*), *t* ∈ [−1, 0], and *Y*(0) = *Y*0. The process *X* is supposed to be hidden, i.e., unobservable, and the process *Y* is observed. Such models are used in applied problems connected with control, filtering and prediction of stochastic processes (see, for example, [1, 4, 17–20] among others).

The parameter *ϑ* = (*a*, *b*)� ∈ Θ is assumed to be unknown and shall be estimated based on continuous observation of *<sup>Y</sup>*, <sup>Θ</sup> is a subset of <sup>R</sup><sup>2</sup> ((*a*, *<sup>b</sup>*)� denotes the transposed (*a*, *<sup>b</sup>*)). Equations (1) and (2) together with the initial values *X*0(·) and *Y*<sup>0</sup> respectively have uniquely solutions *X*(·) and *Y*(·), for details see [19].

©2012 Küchler and Vasiliev, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Küchler and Vasiliev, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 24 Stochastic Modeling and Control On Guaranteed Parameter Estimation of Stochastic Delay Differential Equations by Noisy Observations <sup>3</sup>

Equation (1) is a very special case of stochastic differential equations with time delay, see [5, 6] and [20] for example.

*X*(*t*) = *x*0(*t*)*X*0(*t*) + *b*

The solution *X* has the property *E*

representation (*r*(*ξ*),*s*(*ξ*)) in <sup>R</sup><sup>2</sup> :

equation of (4):

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

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.

behaviour of *x*0(*t*) as *t* → ∞ (see [3] for details).

decomposed into eleven subsets. Here we use another notation.

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

*x*0(*t*) = 1 +

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

which equals (*<sup>λ</sup>* <sup>−</sup> *<sup>a</sup>* <sup>−</sup> *<sup>b</sup>*e−*λ*)−1, *<sup>λ</sup>* any complex number.

 0 −1

> *t*

0

*T*

*x*0(*t* − *s* − 1)*X*0(*s*)*ds* +

On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 25

<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,

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

*r*(*ξ*) = *ξ* cot *ξ*, *s*(*ξ*) = −*ξ*/ sin *ξ*

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>

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

*<sup>λ</sup>* <sup>−</sup> *<sup>a</sup>* <sup>−</sup> *<sup>b</sup>*e−*<sup>λ</sup>* <sup>=</sup> 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

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

Θ = Θ<sup>1</sup> ∪ Θ<sup>2</sup> ∪ Θ<sup>3</sup> ∪ Θ4,

<sup>Θ</sup><sup>11</sup> <sup>=</sup> {*<sup>ϑ</sup>* ∈ R2<sup>|</sup> *<sup>v</sup>*0(*ϑ*) <sup>&</sup>lt; <sup>0</sup>},

<sup>Θ</sup><sup>12</sup> <sup>=</sup> {*<sup>ϑ</sup>* ∈ R2<sup>|</sup> *<sup>v</sup>*0(*ϑ*) <sup>&</sup>gt; 0 and *<sup>v</sup>*0(*ϑ*) �∈ <sup>Λ</sup>},

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

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

 *t* 0

(*ax*0(*s*) + *bx*0(*s* − 1))*ds*, *t* ≥ 0, (4)

*x*0(*t* − *s*)*dW*(*s*). (3)

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

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

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

A related result in such problem statement was published first for estimators of an another structure and without proofs in [15].

A similar problem for partially observed stochastic dynamic systems without time-delay was solved in [22, 23].

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.
