**3.4. Sequential estimation procedure for** *ϑ* ∈ Θ<sup>4</sup>

In this case *b* = −*a* and (6) is the differential equation of the first order:

$$d\tilde{\Delta}Y(t) = aZ^\*(t)dt + \tilde{\Delta}\xi(t)dt + dV(t) - dV(t-1), \; t \ge 2,$$

where

$$Z^\*(t) = \begin{cases} \tilde{\Delta}Y(t) - \tilde{\Delta}Y(t-1) & \text{for} \quad t \ge 2, \\ 0 & \text{for} \quad t < 2. \end{cases}$$

#### 12 Will-be-set-by-IN-TECH 34 Stochastic Modeling and Control On Guaranteed Parameter Estimation of Stochastic Delay Differential Equations by Noisy Observations <sup>13</sup>

We shall construct sequential plan (*T*4(*ε*), *ϑ*4(*ε*)) for estimation of the vector parameter *ϑ* = *a*(1, −1)� with the (*δ*4*ε*)-accuracy in the sense of the *L*2-norm for every *ε* > 0 and fixed chosen *δ*<sup>4</sup> ∈ (0, 1).

**Proposition 3.4.** *Assume that the sequence* (*cn*) *defined above satisfy the condition* (7). *Then we*

*I. For any ε* > 0 *and every ϑ* ∈ Θ<sup>4</sup> *the sequential plan* (*T*4(*ε*), *ϑ*4(*ε*)) *defined by* (22) *is closed and has*

*ε*→0

lim*n*∨*<sup>ε</sup> <sup>ϑ</sup>*4(*n*,*ε*) = *<sup>ϑ</sup> <sup>P</sup><sup>ϑ</sup>* <sup>−</sup> *a.s.*

In this paragraph we construct the sequential estimation procedure for the parameters *a* and

SEP�(*ε*)=(*T*�(*ε*), *<sup>ϑ</sup>*∗(*ε*)), *<sup>T</sup>*�(*ε*) = *Tj*� (*ε*), *<sup>ϑ</sup>*∗(*ε*) = *<sup>ϑ</sup>j*� (*ε*).

**Theorem 3.1.** *Assume that the underlying processes* (*X*(*t*)) *and* (*Y*(*t*)) *satisfy the equations* (1)*,* (2)*, the parameter ϑ to be estimated belongs to the region* Θ *and for the numbers δ*1,..., *δ*<sup>4</sup> *in the definitions*

*<sup>T</sup>*�(*ε*) <sup>&</sup>lt; <sup>∞</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> *a.s.;*

*<sup>E</sup>ϑ*�*ϑ*�(*ε*) <sup>−</sup> *<sup>ϑ</sup>*�<sup>2</sup> <sup>≤</sup> *<sup>ε</sup>*;

*ε* · *T*∗(*ε*) < ∞;

*<sup>E</sup>ϑ*||*ϑ*4(*ε*) <sup>−</sup> *<sup>ϑ</sup>*||<sup>2</sup> <sup>≤</sup> *<sup>δ</sup>*4*ε*;

On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 35

[*T*4(*ε*) <sup>−</sup> <sup>1</sup>

*ε* · *T*4(*ε*) < ∞ *P<sup>ϑ</sup>* − *a.s.*,

2*v*<sup>0</sup> ln *ε*

*Tj*(*ε*). We define the sequential plan (*T*�(*ε*), *<sup>ϑ</sup>*�(*ε*)) of estimation *<sup>ϑ</sup>* <sup>∈</sup> <sup>Θ</sup>

4 ∑ *j*=1

*δ<sup>j</sup>* = 1 *is fulfilled.*

<sup>−</sup>1] <sup>&</sup>lt; <sup>∞</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> *a.s.;*

1◦. sup *ϑ*∈Θ<sup>4</sup>

*ε* · *T*4(*ε*) ≤ lim

<sup>−</sup>1] <sup>≤</sup> lim *ε*→0

**3.5. General sequential estimation procedure of the time-delayed process**

*b* of the process (1) on the bases of the estimators, presented in subsections 3.1-3.4.

*Then the sequential estimation plan* (*T*�(*ε*), *ϑ*�(*ε*)) *possess the following properties:*

sup *ϑ*∈Θ

> lim *ε*→0

0 < lim *ε*→0

[*T*4(*ε*) <sup>−</sup> <sup>1</sup>

2*v*<sup>0</sup> ln *ε*

*II. For every ϑ* ∈ Θ<sup>4</sup> *the estimator ϑ*4(*n*,*ε*) *is strongly consistent:*

on the bases of all constructed above estimators by the formulae

(10), (15), (20) *and* (22) *of sequential plans the condition*

3◦. *the following relations hold with P<sup>ϑ</sup> – probability one:*

*obtain the following result:*

*the following properties:*

*– if ϑ* ∈ Θ<sup>41</sup> *then*

*– if ϑ* ∈ Θ<sup>42</sup> *then*

Denote *j*

2◦. *the following relations hold:*

0 < lim *ε*→0

� <sup>=</sup> arg min *<sup>j</sup>*=1,4

The following theorem is valid.

1◦. *for any ε* > 0 *and for every ϑ* ∈ Θ

2◦. *for any ε* > 0

*– for ϑ* ∈ Θ<sup>11</sup> ∪ Θ<sup>3</sup> ∪ Θ<sup>41</sup>

First define the sequential estimation plans for the scalar parameter *a* on the bases of correlation estimators which are generalized least squares estimators:

$$\begin{aligned} a\_4(T) &= G\_4^{-1}(T)\Phi\_4(T), \\ G\_4(T) &= \int\_0^T Z^\*(t-2)Z^\*(t)dt, \\ \Phi\_4(T) &= \int\_0^T Z^\*(t-2)d\tilde{\Delta}Y(t), \; T>0. \end{aligned}$$

Let (*cn*, *n* ≥ 1) be an unboundedly increasing sequence of positive numbers, satisfying the condition (7).

We shall define

– the sequence of stopping times (*τ*4(*n*,*ε*), *n* ≥ 1) as

$$\pi\_4(n,\varepsilon) = \inf\{T > 2 \, : \, \int\_0^T (Z^\*(t-2))^2 dt = \varepsilon^{-1} c\_n\}, \ n \ge 1;$$

– the sequence of estimators

$$a\_4(n, \varepsilon) = a\_4(\pi\_4(n, \varepsilon)) = G\_4^{-1}(\pi\_4(n, \varepsilon)) \Phi\_4(\pi\_4(n, \varepsilon));$$

– the stopping time

$$\sigma\_4(\varepsilon) = \inf \{ n \ge 1 \, : \, \mathcal{S}\_4(\mathcal{N}) > (\rho\_4 \delta\_4^{-1})^{1/2} \},\tag{21}$$

where *S*4(*N*) = *N* ∑ *n*=1 *G*˜ <sup>−</sup><sup>2</sup> <sup>4</sup> (*n*,*ε*), *<sup>ρ</sup>*<sup>4</sup> <sup>=</sup> *<sup>ρ</sup>*3, *<sup>G</sup>*˜ <sup>4</sup>(*n*,*ε*)=(*ε*−1*cn*)−1*G*4(*τ*4(*n*,*ε*)). The deviation of *a*4(*n*,*ε*) has the form

$$a\_4(n, \varepsilon) - a = (\varepsilon^{-1} \varepsilon\_n)^{-1/2} \tilde{G}\_4^{-1}(n, \varepsilon) \tilde{\zeta}\_4(n, \varepsilon), \ n \ge 1, \nu$$

where

$$\tilde{\zeta}\_4(n,\varepsilon) = (\varepsilon^{-1}c\_n)^{-1/2} \int\_0^{\pi\_4(n,\varepsilon)} Z^\*(t-2)(\tilde{\Delta}\_\varepsilon^\pi(t)dt + dV(t) - dV(t-1))$$

and we have

$$E\_{\theta} ||\tilde{\zeta}\_{4}(n, \varepsilon)||^{2} \le \mathfrak{Z}(\mathfrak{Z} + \mathbb{R}^{2}), \ n \ge 1, \ \varepsilon > 0.$$

We define the sequential plan (*T*4(*ε*), *ϑ*4(*ε*)) for the estimation of *ϑ* as

$$T\_4(\varepsilon) = \tau\_4(\sigma\_4(\varepsilon), \varepsilon), \ \theta\_4(\varepsilon) = a\_4(\sigma\_4(\varepsilon), \varepsilon)(1, -1)^\prime. \tag{22}$$

The following proposition presents the conditions under which *T*4(*ε*) and *ϑ*4(*ε*) are well-defined and have the desired property of preassigned mean square accuracy.

**Proposition 3.4.** *Assume that the sequence* (*cn*) *defined above satisfy the condition* (7). *Then we obtain the following result:*

*I. For any ε* > 0 *and every ϑ* ∈ Θ<sup>4</sup> *the sequential plan* (*T*4(*ε*), *ϑ*4(*ε*)) *defined by* (22) *is closed and has the following properties:*

$$1^\diamond . \sup\_{\vartheta \in \overline{\Theta}\_4} \sup\_{\vartheta} E\_{\vartheta} ||\vartheta\_4(\varepsilon) - \vartheta||^2 \le \delta\_4 \varepsilon;$$

2◦. *the following relations hold:*

*– if ϑ* ∈ Θ<sup>41</sup> *then*

12 Will-be-set-by-IN-TECH

We shall construct sequential plan (*T*4(*ε*), *ϑ*4(*ε*)) for estimation of the vector parameter *ϑ* = *a*(1, −1)� with the (*δ*4*ε*)-accuracy in the sense of the *L*2-norm for every *ε* > 0 and fixed chosen

First define the sequential estimation plans for the scalar parameter *a* on the bases of

<sup>4</sup> (*T*)Φ4(*T*),

Let (*cn*, *n* ≥ 1) be an unboundedly increasing sequence of positive numbers, satisfying the

*Z*∗(*t* − 2)*Z*∗(*t*)*dt*,

*<sup>Z</sup>*∗(*<sup>t</sup>* <sup>−</sup> <sup>2</sup>)*d*Δ˜ *<sup>Y</sup>*(*t*), *<sup>T</sup>* <sup>&</sup>gt; 0.

(*Z*∗(*<sup>t</sup>* <sup>−</sup> <sup>2</sup>))2*dt* <sup>=</sup> *<sup>ε</sup>*

<sup>4</sup> (*n*,*ε*) ˜

<sup>4</sup> (*τ*4(*n*,*ε*))Φ4(*τ*4(*n*,*ε*));

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

<sup>4</sup>(*n*,*ε*)=(*ε*−1*cn*)−1*G*4(*τ*4(*n*,*ε*)). The deviation of

*ζ*4(*n*,*ε*), *n* ≥ 1,

*<sup>Z</sup>*∗(*<sup>t</sup>* <sup>−</sup> <sup>2</sup>)(Δ˜ *<sup>ξ</sup>*(*t*)*dt* <sup>+</sup> *dV*(*t*) <sup>−</sup> *dV*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>))

<sup>4</sup> )1/2}, (21)

. (22)

correlation estimators which are generalized least squares estimators:

*a*4(*T*) = *G*−<sup>1</sup>

 *T* 0

 *T* 0

> *T* 0

*<sup>σ</sup>*4(*ε*) = inf{*<sup>n</sup>* <sup>≥</sup> 1 : *<sup>S</sup>*4(*N*) <sup>&</sup>gt; (*ρ*4*δ*−<sup>1</sup>

<sup>−</sup>1*cn*)−1/2*G*˜ <sup>−</sup><sup>1</sup>

*<sup>ζ</sup>*4(*n*,*ε*)||<sup>2</sup> <sup>≤</sup> <sup>3</sup>(<sup>3</sup> <sup>+</sup> <sup>R</sup>2), *<sup>n</sup>* <sup>≥</sup> 1, *<sup>ε</sup>* <sup>&</sup>gt; 0.

*T*4(*ε*) = *τ*4(*σ*4(*ε*),*ε*), *ϑ*4(*ε*) = *a*4(*σ*4(*ε*),*ε*)(1, −1)�

The following proposition presents the conditions under which *T*4(*ε*) and *ϑ*4(*ε*) are

well-defined and have the desired property of preassigned mean square accuracy.

*τ*4(*n*,*ε*) 

0

We define the sequential plan (*T*4(*ε*), *ϑ*4(*ε*)) for the estimation of *ϑ* as

*a*4(*n*,*ε*) = *a*4(*τ*4(*n*,*ε*)) = *G*−<sup>1</sup>

<sup>4</sup> (*n*,*ε*), *<sup>ρ</sup>*<sup>4</sup> <sup>=</sup> *<sup>ρ</sup>*3, *<sup>G</sup>*˜

*a*4(*n*,*ε*) − *a* = (*ε*

<sup>−</sup>1*cn*)−1/2

*<sup>E</sup>ϑ*|| ˜

*G*4(*T*) =

Φ4(*T*) =

– the sequence of stopping times (*τ*4(*n*,*ε*), *n* ≥ 1) as

*τ*4(*n*,*ε*) = inf{*T* > 2 :

*δ*<sup>4</sup> ∈ (0, 1).

condition (7). We shall define

– the sequence of estimators

*N* ∑ *n*=1

*ζ*4(*n*,*ε*)=(*ε*

*G*˜ <sup>−</sup><sup>2</sup>

– the stopping time

*a*4(*n*,*ε*) has the form

˜

where *S*4(*N*) =

where

and we have

$$0 < \varliminf\_{\varepsilon \to 0} \varepsilon \cdot T\_4(\varepsilon) \le \overline{\lim}\_{\varepsilon \to 0} \varepsilon \cdot T\_4(\varepsilon) < \infty \ P\_\vartheta - a.s., \iota$$

*– if ϑ* ∈ Θ<sup>42</sup> *then*

$$0 < \varliminf\_{\varepsilon \to 0} \left[ T\_4(\varepsilon) - \frac{1}{2v\_0} \ln \varepsilon^{-1} \right] \le \varlimsup\_{\varepsilon \to 0} \left[ T\_4(\varepsilon) - \frac{1}{2v\_0} \ln \varepsilon^{-1} \right] < \infty \ P\_{\vartheta} - a.s.;$$

*II. For every ϑ* ∈ Θ<sup>4</sup> *the estimator ϑ*4(*n*,*ε*) *is strongly consistent:*

$$\lim\_{n \vee \varepsilon} \vartheta\_4(n, \varepsilon) = \vartheta\_1 P\_{\vartheta} - a.s.$$

#### **3.5. General sequential estimation procedure of the time-delayed process**

In this paragraph we construct the sequential estimation procedure for the parameters *a* and *b* of the process (1) on the bases of the estimators, presented in subsections 3.1-3.4.

Denote *j* � <sup>=</sup> arg min *<sup>j</sup>*=1,4 *Tj*(*ε*). We define the sequential plan (*T*�(*ε*), *<sup>ϑ</sup>*�(*ε*)) of estimation *<sup>ϑ</sup>* <sup>∈</sup> <sup>Θ</sup> on the bases of all constructed above estimators by the formulae

$$\text{SEP}^\star(\varepsilon) = (T^\star(\varepsilon), \theta^\*(\varepsilon)), \quad T^\star(\varepsilon) = T\_{\mathfrak{f}^\*}(\varepsilon), \quad \theta^\*(\varepsilon) = \vartheta\_{\mathfrak{f}^\*}(\varepsilon).$$

The following theorem is valid.

**Theorem 3.1.** *Assume that the underlying processes* (*X*(*t*)) *and* (*Y*(*t*)) *satisfy the equations* (1)*,* (2)*, the parameter ϑ to be estimated belongs to the region* Θ *and for the numbers δ*1,..., *δ*<sup>4</sup> *in the definitions*

(10), (15), (20) *and* (22) *of sequential plans the condition* 4 ∑ *j*=1 *δ<sup>j</sup>* = 1 *is fulfilled.*

*Then the sequential estimation plan* (*T*�(*ε*), *ϑ*�(*ε*)) *possess the following properties:*

1◦. *for any ε* > 0 *and for every ϑ* ∈ Θ

$$T^\star(\varepsilon) < \infty \quad P\_\theta-a.s.;$$

2◦. *for any ε* > 0

$$\sup\_{\vartheta \in \overline{\Theta}} E\_{\theta} \left\| \theta^{\star}(\varepsilon) - \theta \right\|^{2} \le \varepsilon;$$

3◦. *the following relations hold with P<sup>ϑ</sup> – probability one: – for ϑ* ∈ Θ<sup>11</sup> ∪ Θ<sup>3</sup> ∪ Θ<sup>41</sup>

$$\overline{\lim\_{\varepsilon \to 0}} \varepsilon \cdot T^\*(\varepsilon) \ll \infty;$$

*– for ϑ* ∈ Θ<sup>12</sup> ∪ Θ<sup>42</sup> lim *ε*→0 [*T*∗(*ε*) <sup>−</sup> <sup>1</sup> 2*v*<sup>0</sup> ln *ε* <sup>−</sup>1] < ∞;

*– for ϑ* ∈ Θ<sup>13</sup>

$$\varlimsup\_{\varepsilon \to 0} \left[ T^\*(\varepsilon) + \frac{1}{v\_0} \ln T\_1(\varepsilon) - \Psi\_{13}^{\prime\prime}(\varepsilon) \right] < \infty$$

where

and *A*˜ *i*, *B*˜ *φ*˜*i*(*t*) = *A*˜

*<sup>i</sup>*, *ξ<sup>i</sup>* are some constants (see [10, 12]).

<sup>−</sup>1e−*v*0*<sup>t</sup>*

 *T*

1

 *T*

1

*<sup>T</sup>*2e2*<sup>v</sup>*0*<sup>T</sup> <sup>G</sup>*1(*T*,*s*) = *<sup>G</sup>*13(*s*), lim

*X* 2*v*<sup>0</sup>

lim

<sup>13</sup>(*s*) = <sup>2</sup>*v*0e(3+11*s*)*v*<sup>0</sup> *sC*˜ *XC*˜ *Z*

lim*n*∨*ε*

*<sup>T</sup>*→<sup>∞</sup> *<sup>T</sup>*<sup>−</sup>1e2*<sup>v</sup>*0*TG*−<sup>1</sup>

From (23) and by the definition of the stopping times *τ*1(*n*,*ε*) we have

*τ*2

*<sup>G</sup>*13(*s*) = *<sup>C</sup>*˜2

*G*˜

is a non-random matrix function.

in the case Θ<sup>13</sup> the following limits:

*<sup>T</sup>*→<sup>∞</sup> <sup>|</sup> *<sup>T</sup>*<sup>−</sup>2e−2*v*0*<sup>T</sup>*

*<sup>T</sup>*→<sup>∞</sup> <sup>|</sup> *<sup>T</sup>*<sup>−</sup>1e−2*v*0*<sup>T</sup>*

and, as follows, for *u* ≥ 0

lim

lim

where *C*˜*x*, *CY* and *C*˜

lim *T*→∞

(24) we obtain the limits:

1

and, as follows, we can find

lim *<sup>t</sup>*→<sup>∞</sup> *<sup>t</sup>*

lim *<sup>t</sup>*→<sup>∞</sup> <sup>e</sup>−*v*0*<sup>t</sup>*

the processes *<sup>X</sup>*˜(*t*), *<sup>Y</sup>*˜(*t*) = *<sup>X</sup>*˜(*t*) <sup>−</sup> *<sup>λ</sup>X*˜(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>), *<sup>λ</sup>* <sup>=</sup> <sup>e</sup>*v*<sup>0</sup> , <sup>Δ</sup>˜ *<sup>Y</sup>*(*t*) and

*<sup>i</sup>* cos *ξit* + *B*˜

On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 37

The processes *X*˜(*t*) and Δ˜ *V*(*t*) are mutually independent and the process *X*˜(*t*) has the representation similar to (3). Then, after some algebra similar to those in [10, 12] we get for

*<sup>Z</sup>*(*t*) = <sup>Δ</sup>˜ *<sup>Y</sup>*(*t*) <sup>−</sup> *<sup>λ</sup>*Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>) for *<sup>t</sup>* <sup>≥</sup> 2,

*<sup>t</sup>*→<sup>∞</sup> *<sup>t</sup>*

<sup>−</sup>1e−*v*0*<sup>t</sup>*

*<sup>t</sup>*→<sup>∞</sup> <sup>e</sup>−*v*0*<sup>t</sup>*

*X* 2*v*<sup>0</sup> <sup>1</sup> <sup>−</sup> *<sup>u</sup> T* 

*XC*˜ *Z* 2*v*<sup>0</sup>

 <sup>1</sup> <sup>−</sup> *<sup>u</sup> T* 

*<sup>Z</sup>* are some nonzero constants, which can be found from [10, 12]. From

*<sup>T</sup>*→<sup>∞</sup> *<sup>T</sup>*<sup>−</sup>1e−4*v*0*T*|*G*1(*T*,*s*)<sup>|</sup> <sup>=</sup> *<sup>G</sup>*13e−(3+11*s*)*v*<sup>0</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> *<sup>a</sup>*.*s*.,

<sup>13</sup>(*s*) *P<sup>ϑ</sup>* − *a*.*s*.,

<sup>e</sup>−(1+6*s*)*v*<sup>0</sup> <sup>−</sup>e−(1+5*s*)*v*<sup>0</sup> <sup>−</sup>e−2(1+3*s*)*v*<sup>0</sup> <sup>e</sup>−(2+5*s*)*v*<sup>0</sup>

, *<sup>G</sup>*<sup>13</sup> <sup>=</sup> *sC*˜3

0 for *t* < 2

Δ˜ *Y*(*t*) = lim

*Y*˜(*t*) = *CY*, lim

<sup>Δ</sup>˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>u</sup>*)Δ˜ *<sup>Y</sup>*(*t*)*dt*<sup>−</sup> *<sup>C</sup>*˜2

<sup>Δ</sup>˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>u</sup>*)*Z*(*t*)*dt*−*C*˜

e−(2+5*s*)*v*<sup>0</sup> e−(1+5*s*)*v*<sup>0</sup> e−2(1+3*s*)*v*<sup>0</sup> e−(1+6*s*)*v*<sup>0</sup>

<sup>1</sup> (*n*,*ε*)e2*τ*1(*n*,*ε*)*v*<sup>0</sup> *ε*−1*cn*

<sup>1</sup> (*T*,*s*) = *<sup>G</sup>*˜

= *g*∗

*<sup>i</sup>* sin *ξit*

*X*˜(*t*) = *C*˜

*Z*(*t*) = *C*˜

*<sup>X</sup> P<sup>ϑ</sup>* − a.s., (23)

<sup>e</sup>−*uv*<sup>0</sup> <sup>|</sup> <sup>=</sup> <sup>0</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s., (24)

<sup>e</sup>−*uv*<sup>0</sup> <sup>|</sup> <sup>=</sup> <sup>0</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.,

*XC*˜ *Z* 4*v*<sup>2</sup> 0

<sup>13</sup> *P<sup>ϑ</sup>* − a.s., (25)

*<sup>Z</sup> P<sup>ϑ</sup>* − a.s.,

*the function* Ψ�� <sup>13</sup>(*ε*) *is defined in (30);*

*– for ϑ* ∈ Θ<sup>2</sup>

$$\overline{\lim\_{\varepsilon \to 0}} \left[ T^\*(\varepsilon) - \frac{1}{2v\_1} \ln \varepsilon^{-1} \right] < \infty.$$

### **4. Proofs**

**Proof of Proposition 3.1.** The closeness of the sequential estimation plan, as well as assertions I.2 and II of Proposition 3.1 for the cases Θ<sup>11</sup> ∪ Θ<sup>12</sup> can be easily verified similar to [10, 12, 14, 16]. Now we verify the finiteness of the stopping time *T*1(*ε*) in the new case Θ13.

By the definition of Δ˜ *Y*(*t*) we have:

$$
\tilde{\Delta}Y(t) = \tilde{X}(t) + \tilde{\Delta}V(t), \quad t \ge 1,
$$

where

$$\tilde{X}(t) = \int\_{t-1}^{t} X(t)dt.$$

It is easy to show that the process (*X*˜(·)) has the following representation:

$$\tilde{X}(t) = \mathfrak{x}\_0(t)X\_0(0) + b \int\_{-1}^0 \mathfrak{x}\_0(t-s-1)X\_0(s)ds + \int\_0^t \mathfrak{x}\_0(t-s)dW(s)$$

for *<sup>t</sup>* <sup>≥</sup> 1, *<sup>X</sup>*˜(*t*) = � <sup>0</sup> *<sup>t</sup>*−<sup>1</sup> *<sup>X</sup>*0(*s*)*ds* <sup>+</sup> � *<sup>t</sup>* <sup>0</sup> *<sup>X</sup>*(*s*)*ds* for *<sup>t</sup>* <sup>∈</sup> [0, 1) and *<sup>X</sup>*˜(*t*) = 0 for *<sup>t</sup>* <sup>∈</sup> [−1, 0). Based on the representation above for the function *x*0(·), the subsequent properties of *x*0(*t*) the function *<sup>x</sup>*˜0(*t*) = � *<sup>t</sup> <sup>t</sup>*−<sup>1</sup> *<sup>x</sup>*0(*s*)*ds* can be easily shown to fulfill *<sup>x</sup>*˜0(*t*) = 0, *<sup>t</sup>* <sup>∈</sup> [−1, 0] and as *<sup>t</sup>* <sup>→</sup> <sup>∞</sup>

*x*˜0(*t*) = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *o*(*eγ<sup>t</sup>* ), *γ* < 0, *ϑ* ∈ Θ11, *φ*˜0(*t*)*ev*0*<sup>t</sup>* + *o*(*eγ*0*<sup>t</sup>* ), *<sup>γ</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>v</sup>*0, *<sup>ϑ</sup>* <sup>∈</sup> <sup>Θ</sup>12, <sup>2</sup> *<sup>v</sup>*<sup>0</sup> [(<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*v*<sup>0</sup> )*<sup>t</sup>* <sup>+</sup> *<sup>e</sup>*−*v*<sup>0</sup> <sup>−</sup> <sup>1</sup>−*e*−*v*<sup>0</sup> *<sup>v</sup>*<sup>0</sup> ]*ev*0*<sup>t</sup>* <sup>+</sup> *<sup>o</sup>*(*eγ*0*<sup>t</sup>* ), *γ*<sup>0</sup> < *v*0, *ϑ* ∈ Θ13, <sup>1</sup>−*e*−*v*<sup>0</sup> *<sup>v</sup>*0(*v*0−*a*+1)*ev*0*<sup>t</sup>* <sup>+</sup> <sup>1</sup>−*e*−*v*<sup>1</sup> *<sup>v</sup>*1(*a*−*v*1−1)*ev*1*<sup>t</sup>* <sup>+</sup> *<sup>o</sup>*(*eγ*1*<sup>t</sup>* ), *γ*<sup>1</sup> < *v*1, *ϑ* ∈ Θ21, <sup>1</sup>−*e*−*v*<sup>0</sup> *<sup>v</sup>*0(*v*0−*a*+1)*ev*0*<sup>t</sup>* <sup>+</sup> *<sup>φ</sup>*˜1(*t*)*ev*1*<sup>t</sup>* <sup>+</sup> *<sup>o</sup>*(*eγ*1*<sup>t</sup>* ), *γ*<sup>1</sup> < *v*1, *ϑ* ∈ Θ22, <sup>1</sup>−*e*−*v*<sup>0</sup> *<sup>v</sup>*0(*v*0−*a*+1)*ev*0*<sup>t</sup>* <sup>+</sup> *<sup>o</sup>*(*eγ<sup>t</sup>* ), *γ* < 0, *ϑ* ∈ Θ3, 1 <sup>1</sup>−*<sup>a</sup>* <sup>+</sup> *<sup>o</sup>*(*eγ<sup>t</sup>* ), *γ* < 0, *ϑ* ∈ Θ41, <sup>1</sup>−*e*−*v*<sup>0</sup> *<sup>v</sup>*0(*v*0−*a*+1)*ev*0*<sup>t</sup>* <sup>−</sup> <sup>1</sup> *<sup>a</sup>*−<sup>1</sup> <sup>+</sup> *<sup>o</sup>*(*eγ<sup>t</sup>* ), *γ* < 0, *ϑ* ∈ Θ42,

where

14 Will-be-set-by-IN-TECH

2*v*<sup>0</sup> ln *ε*

ln *T*1(*ε*) − Ψ��

<sup>−</sup>1] < ∞;

<sup>−</sup>1] < ∞.

<sup>13</sup>(*ε*)] < ∞,

[*T*∗(*ε*) <sup>−</sup> <sup>1</sup>

*v*0

[*T*∗(*ε*) <sup>−</sup> <sup>1</sup>

16]. Now we verify the finiteness of the stopping time *T*1(*ε*) in the new case Θ13.

*X*˜(*t*) =

It is easy to show that the process (*X*˜(·)) has the following representation:

� 0

−1

*<sup>v</sup>*<sup>0</sup> [(<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*v*<sup>0</sup> )*<sup>t</sup>* <sup>+</sup> *<sup>e</sup>*−*v*<sup>0</sup> <sup>−</sup> <sup>1</sup>−*e*−*v*<sup>0</sup>

*<sup>v</sup>*0(*v*0−*a*+1)*ev*0*<sup>t</sup>* <sup>+</sup> *<sup>φ</sup>*˜1(*t*)*ev*1*<sup>t</sup>* <sup>+</sup> *<sup>o</sup>*(*eγ*1*<sup>t</sup>*

*<sup>a</sup>*−<sup>1</sup> <sup>+</sup> *<sup>o</sup>*(*eγ<sup>t</sup>*

*<sup>v</sup>*0(*v*0−*a*+1)*ev*0*<sup>t</sup>* <sup>+</sup> <sup>1</sup>−*e*−*v*<sup>1</sup>

*<sup>v</sup>*0(*v*0−*a*+1)*ev*0*<sup>t</sup>* <sup>+</sup> *<sup>o</sup>*(*eγ<sup>t</sup>*

2*v*<sup>1</sup> ln *ε*

**Proof of Proposition 3.1.** The closeness of the sequential estimation plan, as well as assertions I.2 and II of Proposition 3.1 for the cases Θ<sup>11</sup> ∪ Θ<sup>12</sup> can be easily verified similar to [10, 12, 14,

<sup>Δ</sup>˜ *<sup>Y</sup>*(*t*) = *<sup>X</sup>*˜(*t*) + <sup>Δ</sup>˜ *<sup>V</sup>*(*t*), *<sup>t</sup>* <sup>≥</sup> 1,

� *t*

*X*(*t*)*dt*.

*x*˜0(*t* − *s* − 1)*X*0(*s*)*ds* +

� *t* 0

<sup>0</sup> *<sup>X</sup>*(*s*)*ds* for *<sup>t</sup>* <sup>∈</sup> [0, 1) and *<sup>X</sup>*˜(*t*) = 0 for *<sup>t</sup>* <sup>∈</sup> [−1, 0). Based on

*x*˜0(*t* − *s*)*dW*(*s*)

), *γ*<sup>0</sup> < *v*0, *ϑ* ∈ Θ13,

), *γ*<sup>1</sup> < *v*1, *ϑ* ∈ Θ21,

), *γ*<sup>1</sup> < *v*1, *ϑ* ∈ Θ22,

), *γ* < 0, *ϑ* ∈ Θ3,

), *γ* < 0, *ϑ* ∈ Θ42,

), *γ* < 0, *ϑ* ∈ Θ41,

*t*−1

the representation above for the function *x*0(·), the subsequent properties of *x*0(*t*) the function

*<sup>t</sup>*−<sup>1</sup> *<sup>x</sup>*0(*s*)*ds* can be easily shown to fulfill *<sup>x</sup>*˜0(*t*) = 0, *<sup>t</sup>* <sup>∈</sup> [−1, 0] and as *<sup>t</sup>* <sup>→</sup> <sup>∞</sup>

*<sup>v</sup>*1(*a*−*v*1−1)*ev*1*<sup>t</sup>* <sup>+</sup> *<sup>o</sup>*(*eγ*1*<sup>t</sup>*

), *γ* < 0, *ϑ* ∈ Θ11,

*<sup>v</sup>*<sup>0</sup> ]*ev*0*<sup>t</sup>* <sup>+</sup> *<sup>o</sup>*(*eγ*0*<sup>t</sup>*

), *<sup>γ</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>v</sup>*0, *<sup>ϑ</sup>* <sup>∈</sup> <sup>Θ</sup>12, <sup>2</sup>

lim *ε*→0

lim *ε*→0

[*T*∗(*ε*) + <sup>1</sup>

lim *ε*→0

<sup>13</sup>(*ε*) *is defined in (30);*

By the definition of Δ˜ *Y*(*t*) we have:

*X*˜(*t*) = *x*˜0(*t*)*X*0(0) + *b*

*<sup>t</sup>*−<sup>1</sup> *<sup>X</sup>*0(*s*)*ds* <sup>+</sup> � *<sup>t</sup>*

*φ*˜0(*t*)*ev*0*<sup>t</sup>* + *o*(*eγ*0*<sup>t</sup>*

*– for ϑ* ∈ Θ<sup>12</sup> ∪ Θ<sup>42</sup>

*– for ϑ* ∈ Θ<sup>13</sup>

*the function* Ψ��

*– for ϑ* ∈ Θ<sup>2</sup>

**4. Proofs**

where

for *<sup>t</sup>* <sup>≥</sup> 1, *<sup>X</sup>*˜(*t*) = � <sup>0</sup>

*x*˜0(*t*) =

⎧

*o*(*eγ<sup>t</sup>*

<sup>1</sup>−*e*−*v*<sup>0</sup>

<sup>1</sup>−*e*−*v*<sup>0</sup>

<sup>1</sup>−*e*−*v*<sup>0</sup>

<sup>1</sup>−*e*−*v*<sup>0</sup> *<sup>v</sup>*0(*v*0−*a*+1)*ev*0*<sup>t</sup>* <sup>−</sup> <sup>1</sup>

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1 <sup>1</sup>−*<sup>a</sup>* <sup>+</sup> *<sup>o</sup>*(*eγ<sup>t</sup>*

*<sup>x</sup>*˜0(*t*) = � *<sup>t</sup>*

$$\tilde{\phi}\_{\dot{l}}(t) = \tilde{A}\_{\dot{l}} \cos \xi\_{\dot{l}} t + \tilde{B}\_{\dot{l}} \sin \xi\_{\dot{l}} t$$

and *A*˜ *i*, *B*˜ *<sup>i</sup>*, *ξ<sup>i</sup>* are some constants (see [10, 12]).

The processes *X*˜(*t*) and Δ˜ *V*(*t*) are mutually independent and the process *X*˜(*t*) has the representation similar to (3). Then, after some algebra similar to those in [10, 12] we get for the processes *<sup>X</sup>*˜(*t*), *<sup>Y</sup>*˜(*t*) = *<sup>X</sup>*˜(*t*) <sup>−</sup> *<sup>λ</sup>X*˜(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>), *<sup>λ</sup>* <sup>=</sup> <sup>e</sup>*v*<sup>0</sup> , <sup>Δ</sup>˜ *<sup>Y</sup>*(*t*) and

$$Z(t) = \begin{cases} \tilde{\Delta}Y(t) - \lambda \tilde{\Delta}Y(t-1) & \text{for} \quad t \ge 2, \\ 0 & \text{for} \quad t < 2 \end{cases}$$

in the case Θ<sup>13</sup> the following limits:

$$\lim\_{t \to \infty} t^{-1} \mathbf{e}^{-v\_0 t} \tilde{\Delta} Y(t) = \lim\_{t \to \infty} t^{-1} \mathbf{e}^{-v\_0 t} \tilde{X}(t) = \tilde{\mathbf{C}}\_X \quad P\_\theta-\text{a.s.} \tag{23}$$

$$\lim\_{t \to \infty} \mathbf{e}^{-v\_0 t} \tilde{Y}(t) = \mathbf{C}\_{Y'} \qquad \lim\_{t \to \infty} \mathbf{e}^{-v\_0 t} Z(t) = \tilde{\mathbf{C}}\_Z \quad P\_\theta-\text{a.s.}$$

and, as follows, for *u* ≥ 0

$$\lim\_{T \to \infty} \mid T^{-2} \mathbf{e}^{-2v\_0T} \int\_1^T \tilde{\Lambda} Y(t - u) \tilde{\Lambda} Y(t) dt - \frac{\tilde{C}\_X^2}{2v\_0} \left[ 1 - \frac{u}{T} \right] \mathbf{e}^{-uv\_0} \mid = 0 \quad P\_\theta-\text{a.s.}, \tag{24}$$

$$\lim\_{T \to \infty} \mid T^{-1} \mathbf{e}^{-2v\_0T} \int\_1^T \tilde{\Lambda} Y(t - u) Z(t) dt - \frac{\tilde{C}\_X \tilde{C}\_Z}{2v\_0} \left[ 1 - \frac{u}{T} \right] \mathbf{e}^{-uv\_0} \mid = 0 \quad P\_\theta-\text{a.s.}$$

where *C*˜*x*, *CY* and *C*˜ *<sup>Z</sup>* are some nonzero constants, which can be found from [10, 12]. From (24) we obtain the limits:

$$\lim\_{T \to \infty} \frac{1}{T^2 \mathbf{e}^{2v\_0T}} \mathbf{G}\_1(T, s) = \mathbf{G}\_{13}(s), \quad \lim\_{T \to \infty} T^{-1} \mathbf{e}^{-4v\_0T} |\mathbf{G}\_1(T, s)| = \mathbf{G}\_{13} \mathbf{e}^{-(3+11s)v\_0} \quad P\_\theta-a.s.,$$

$$\mathbf{G}\_{13}(s) = \frac{\tilde{\mathbf{C}}\_X^2}{2v\_0} \begin{pmatrix} \mathbf{e}^{-(2+5s)v\_0} & \mathbf{e}^{-(1+5s)v\_0} \\ \mathbf{e}^{-2(1+3s)v\_0} & \mathbf{e}^{-(1+6s)v\_0} \end{pmatrix}, \qquad \mathbf{G}\_{13} = \frac{s \tilde{\mathbf{C}}\_X^3 \tilde{\mathbf{C}}\_Z}{4v\_0^2}$$
 $\mathbf{e}^{-1}$ 

and, as follows, we can find

$$\lim\_{T \to \infty} T^{-1} \mathbf{e}^{2v\_0 T} G\_1^{-1}(T, \mathbf{s}) = \tilde{G}\_{13}(\mathbf{s}) \begin{array}{c} P\_\theta - a.s. \\\\ \mathbf{e} \end{array}$$

$$\tilde{G}\_{13}(\mathbf{s}) = \frac{2v\_0 \mathbf{e}^{(3+11s)v\_0}}{s \tilde{C}\_X \tilde{C}\_Z} \begin{pmatrix} \mathbf{e}^{-(1+6s)v\_0} & -\mathbf{e}^{-(1+5s)v\_0} \\\\ -\mathbf{e}^{-2(1+3s)v\_0} & \mathbf{e}^{-(2+5s)v\_0} \end{pmatrix}$$

is a non-random matrix function.

From (23) and by the definition of the stopping times *τ*1(*n*,*ε*) we have

$$\lim\_{n \vee \varepsilon} \frac{\tau\_1^2(n, \varepsilon) \mathbf{e}^{2\pi\_1(n, \varepsilon) v\_0}}{\varepsilon^{-1} \mathfrak{c}\_n} = \mathfrak{g}\_{13}^\* \quad P\_\theta-\text{a.s.}\tag{25}$$

#### 16 Will-be-set-by-IN-TECH 38 Stochastic Modeling and Control On Guaranteed Parameter Estimation of Stochastic Delay Differential Equations by Noisy Observations <sup>17</sup>

$$\text{where } g\_{13}^{\*} = 2v\_0 \tilde{\mathbb{C}}\_X^{-2} \left( \mathbf{e}^{-2v\_0(2+5h\_1)} + \mathbf{e}^{-4v\_0(1+3h\_1)} \right)^{-1} \text{ and, as follows,}$$

$$\lim\_{n \nearrow \varepsilon} \left[ \tau\_1(n, \varepsilon) + \frac{1}{v\_0} \ln \tau\_1(n, \varepsilon) - \frac{1}{2v\_0} \ln \varepsilon^{-1} c\_{\imath t} \right] = \frac{1}{2v\_0} \ln g\_{13}^\* \quad P\_\theta-\text{a.s.} \tag{26}$$

$$\lim\_{n \vee \varepsilon} \frac{\tau\_1(n, \varepsilon)}{\ln \varepsilon^{-1} c\_n} = \frac{1}{2v\_0} \quad P\_\theta-\text{a.s.}\tag{27}$$

lim *ε*→0

ln<sup>3</sup> *ε*−1*cn*

deviation

where

*<sup>ϑ</sup>*1(*n*,*ε*) <sup>−</sup> *<sup>ϑ</sup>* <sup>=</sup> <sup>1</sup>

[*T*1(*ε*) + <sup>1</sup>

*G*˜ <sup>−</sup><sup>1</sup> <sup>1</sup> (*n*,*ε*)·

*T*

0

*ζ*1(*n*,*ε*)

<sup>1</sup> (*n*,*ε*)e2*τ*1(*n*,*ε*)*v*<sup>0</sup>

*<sup>ζ</sup>*1(*T*,*s*) =

the square integrable martingales *ζ*1(*T*,*s*) :

lim*n*∨*ε*

assertion II of Proposition 3.1. Hence Proposition 3.1 is valid.

**Proof of Proposition 3.2.**

– for *ϑ* ∈ Θ<sup>21</sup>

– for *ϑ* ∈ Θ<sup>22</sup>

*v*<sup>1</sup> − *v*<sup>0</sup> *v*<sup>1</sup> + *v*<sup>0</sup>

*CZ*

*τ*−<sup>1</sup>

*<sup>t</sup>* <sup>→</sup> <sup>∞</sup> relations for the processes <sup>Δ</sup>˜ *<sup>Y</sup>*(*t*), *<sup>Z</sup>*(*t*) and *<sup>Z</sup>*˜(*t*) :

*v*<sup>0</sup> + *v*<sup>1</sup>

<sup>|</sup>Δ˜ *<sup>Y</sup>*(*t*) <sup>−</sup> *CYe*

0

*<sup>λ</sup><sup>t</sup>* <sup>−</sup> *<sup>λ</sup>* <sup>=</sup> <sup>2</sup>*v*0e*<sup>v</sup>*<sup>0</sup>

*<sup>λ</sup><sup>t</sup>* <sup>−</sup> *<sup>λ</sup>* <sup>=</sup> <sup>2</sup>*v*0e*<sup>v</sup>*0*C*−<sup>1</sup>

; *CZ*(*t*), *UZ*(*t*) = <sup>∞</sup>

periodic (with the period Δ > 1) functions.

Δ˜ *Y*(*t*) = *CYe*

*Z*(*t*) = *CZe*

*CZC*−<sup>1</sup> *<sup>Y</sup> e*

*Z*˜(*t*) = *C*˜


<sup>|</sup>*Z*˜(*t*) <sup>−</sup> *<sup>C</sup>*˜

*<sup>Y</sup> UZ*(*t*)*e*

*<sup>Z</sup>*(*t*)*e v*1*t*

*v*0

*τ*2

ln *T*1(*ε*) − Ψ��

<sup>13</sup>(*ε*)] ≤

On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 39

 ·

<sup>Ψ</sup>*s*(*<sup>t</sup>* <sup>−</sup> <sup>2</sup> <sup>−</sup> <sup>5</sup>*s*)(Δ˜ *<sup>ξ</sup>*(*t*)*dt* <sup>+</sup> *dV*(*t*) <sup>−</sup> *dV*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>)).

*ζ*1(*T*, *h*1)

*<sup>T</sup>*−1e2*<sup>v</sup>*0*<sup>T</sup>* <sup>=</sup> <sup>0</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.

) *P<sup>ϑ</sup>* − a.s.,

) *P<sup>ϑ</sup>* − a.s.

) *P<sup>ϑ</sup>* − a.s.,

) *P<sup>ϑ</sup>* − a.s.,

*<sup>Z</sup>*(*t*) = *CZ*(*t*) − 2*v*0*UZ*(*t*) are the

*<sup>Z</sup>* =

For the proof of the assertion II of Proposition 3.1 we will use the representation (9) for the

*ζ*1(*n*,*ε*) = *ζ*1(*τ*1(*n*,*ε*) − *k*1(*n*)*h*1, *h*1),

According to (25), (27) and (28) first three factors in the right-hand side of this equality have *P<sup>ϑ</sup>* − a.s. positive finite limits. The last factor vanishes in *P<sup>ϑ</sup>* − a.s. sense by the properties of

> = lim *T*→∞

Then the estimators *ϑ*1(*n*,*ε*) are strongly consistent as *ε* → 0 or *n* → ∞ and we obtain the

Similar to the proof of Proposition 3.1 and [7]–[16] we can get the following asymptotic as

*<sup>v</sup>*1*<sup>t</sup>* + *o*(*e*

*<sup>v</sup>*1*<sup>t</sup>* + *o*(*e*

<sup>−</sup>(*v*0−*v*1)*<sup>t</sup>* + *o*(*e*

*<sup>v</sup>*1*<sup>t</sup>* + *o*(*e*

*γt*

*γt*

*v*1*t*


<sup>−</sup>(*v*0−*v*1)*<sup>t</sup>* + *o*(*e*



*γt*

) *P<sup>ϑ</sup>* − a.s.,

) *P<sup>ϑ</sup>* − a.s.;

−(*v*0−*v*1+*γ*)*t*

) *P<sup>ϑ</sup>* − a.s.,

) *P<sup>ϑ</sup>* − a.s.,

−(*v*0−*v*1+*γ*)*t*

*<sup>v</sup>*0*<sup>t</sup>* + *CY*1*e*

*Ze*

*<sup>v</sup>*0*<sup>t</sup>* <sup>−</sup> *CY*1(*t*)*<sup>e</sup>*

*v*1*t*

where *CY* and *CY*<sup>1</sup> are some non-zero constants, 0 <sup>&</sup>lt; *<sup>γ</sup>* <sup>&</sup>lt; *<sup>v</sup>*1, *CZ* <sup>=</sup> *CY*1(<sup>1</sup> <sup>−</sup> <sup>e</sup>*v*0−*v*<sup>1</sup> ), *<sup>C</sup>*˜

*CZ*(*<sup>t</sup>* <sup>−</sup> *<sup>u</sup>*)e−(*v*0+*v*1)*udu* and *<sup>C</sup>*˜

<sup>1</sup> (*n*,*ε*)e2*<sup>τ</sup>*1(*n*,*ε*)*v*<sup>0</sup> *ε*−1*cn*

1 2*v*<sup>0</sup>

ln *ε*−1*cn τ*1(*n*,*ε*)

ln *g*∗

<sup>3</sup>

<sup>13</sup> *P<sup>ϑ</sup>* − a.s.

· <sup>1</sup> *τ*−<sup>1</sup>

<sup>1</sup> (*n*,*ε*)e2*τ*1(*n*,*ε*)*v*<sup>0</sup>

*ζ*1(*n*,*ε*),

$$\lim\_{n \ne \varepsilon} \left[ \frac{1}{\ln^3 \varepsilon^{-1} \mathcal{L}\_n} \tilde{\mathcal{G}}\_1^{-1}(n, \varepsilon) - [(2v\_0)^3 g\_{13}^\*]^{-1} \mathbf{e}^{-2v\_0 k\_1(n) \hbar\_1} \tilde{\mathcal{G}}\_{13}(\hbar\_1) \right] = 0 \quad P\_\vartheta-\text{a.s.} \tag{28}$$

From (8) and (28) it follows the *P<sup>ϑ</sup>* − a.s. finiteness of the stopping time *σ*1(*ε*) for every *ε* > 0. The proof of the assertion I.1 of Proposition 3.1 for the case Θ<sup>13</sup> is similar e.g. to the proof of corresponding assertion in [14, 16]:

$$\begin{split} &E\_{\theta}||\vartheta\_{1}(\varepsilon)-\theta||^{2} = E\_{\theta}\frac{1}{S^{2}(\sigma\_{1}(\varepsilon))}||\sum\_{n=1}^{\sigma\_{1}(\varepsilon)}\beta\_{1}^{2}(n,\varepsilon)(\theta\_{1}(n,\varepsilon)-\theta)||^{2} \leq \\ & \leq \varepsilon\frac{\delta\_{1}}{\rho\_{1}}E\_{\theta}\sum\_{n=1}^{\sigma\_{1}(\varepsilon)}\frac{1}{c\_{n}}\cdot\beta\_{1}^{2}(n,\varepsilon)\cdot||\tilde{G}\_{1}^{-1}(n,\varepsilon)||^{2}\cdot||\tilde{\zeta}\_{1}(n,\varepsilon)||^{2} \leq \\ & \leq \frac{\varepsilon\delta\_{1}}{\rho\_{1}}\sum\_{n\geq1}\frac{1}{c\_{n}}E\_{\theta}||\tilde{\zeta}\_{1}(n,\varepsilon)||^{2} \leq \varepsilon\delta\_{1}\frac{15(3+R^{2})}{\rho\_{1}}\sum\_{n\geq1}\frac{1}{c\_{n}} = \varepsilon\delta\_{1}. \end{split}$$

Now we prove the assertion I.2 for *ϑ* ∈ Θ13. Denote the number

$$\tilde{g}\_{13} = [(2v\_0)^3 g\_{13}^\*]^2 \rho\_1^{-1} \delta\_1 ||\tilde{G}\_{13}(h\_1)||^{-2}$$

and the times

$$\begin{aligned} \overline{\sigma}\_{13}^{'}(\varepsilon) &= \inf \{ n \ge 1 : \sum\_{n=1}^{N} \ln^6 \varepsilon^{-1} c\_n > \tilde{\mathfrak{g}}\_{13} \mathbf{e}^{4v\_0h\_1} \}, \\\overline{\sigma}\_{13}^{''}(\varepsilon) &= \inf \{ n \ge 1 : \sum\_{n=1}^{N} \ln^6 \varepsilon^{-1} c\_n > \tilde{\mathfrak{g}}\_{13} \mathbf{e}^{20v\_0h\_1} \}. \end{aligned}$$

From (8) and (28) it follows, that for *ε* small enough

$$
\overline{\sigma}\_{13}^{\prime}(\varepsilon) \le \sigma\_1(\varepsilon) \le \overline{\sigma}\_{13}^{\prime\prime}(\varepsilon) \quad P\_\theta-\text{a.s.}\tag{29}
$$

Denote

$$\Psi\_{13}^{\prime}(\varepsilon) = \frac{1}{2v\_0} \ln(\varepsilon^{-1} c\_{\overline{\sigma}\_{13}^{\prime}(\varepsilon)}), \quad \Psi\_{13}^{\prime\prime}(\varepsilon) = \frac{1}{2v\_0} \ln(\varepsilon^{-1} c\_{\overline{\sigma}\_{13}^{\prime\prime}(\varepsilon)}). \tag{30}$$

Then, from (8), (26) and (29) we obtain finally the assertion I.2 of Proposition 3.1:

$$\lim\_{\varepsilon \to 0} \left[ T\_1(\varepsilon) + \frac{1}{v\_0} \ln T\_1(\varepsilon) - \Psi\_{13}'(\varepsilon) \right] \ge \frac{1}{2v\_0} \ln g\_{13}^\* \quad P\_\theta-\text{a.s.}$$

38 Stochastic Modeling and Control On Guaranteed Parameter Estimation of Stochastic Delay Differential Equations by Noisy Observations <sup>17</sup> On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 39

$$\overline{\lim\_{\varepsilon \to 0}} \left[ T\_1(\varepsilon) + \frac{1}{v\_0} \ln T\_1(\varepsilon) - \Psi\_{13}^{\prime\prime}(\varepsilon) \right] \le \frac{1}{2v\_0} \ln g\_{13}^\* \quad P\_\theta-\text{a.s.}$$

For the proof of the assertion II of Proposition 3.1 we will use the representation (9) for the deviation

$$
\vartheta\_1(\boldsymbol{n}, \boldsymbol{\varepsilon}) - \vartheta = \frac{1}{\ln^3 \varepsilon^{-1} \mathfrak{c}\_{\boldsymbol{n}}} \tilde{\mathsf{G}}\_1^{-1}(\boldsymbol{n}, \boldsymbol{\varepsilon}) \cdot \frac{\mathsf{r}\_1^2(\boldsymbol{n}, \boldsymbol{\varepsilon}) \mathsf{e}^{2\mathsf{r}\_1(\boldsymbol{n}, \boldsymbol{\varepsilon}) \mathsf{r}\_0}}{\varepsilon^{-1} \mathfrak{c}\_{\boldsymbol{n}}} \left( \cdot \frac{\ln \varepsilon^{-1} \mathfrak{c}\_{\boldsymbol{n}}}{\boldsymbol{\tau}\_1(\boldsymbol{n}, \boldsymbol{\varepsilon})} \right)^3 \cdot \frac{1}{\boldsymbol{\tau}\_1^{-1}(\boldsymbol{n}, \boldsymbol{\varepsilon}) \mathsf{e}^{2\mathsf{r}\_1(\boldsymbol{n}, \boldsymbol{\varepsilon}) \mathsf{r}\_0}} \zeta\_1(\boldsymbol{n}, \boldsymbol{\varepsilon}),
$$

where

16 Will-be-set-by-IN-TECH

2*v*<sup>0</sup> ln *ε*

13]

From (8) and (28) it follows the *P<sup>ϑ</sup>* − a.s. finiteness of the stopping time *σ*1(*ε*) for every *ε* > 0. The proof of the assertion I.1 of Proposition 3.1 for the case Θ<sup>13</sup> is similar e.g. to the proof of

> *σ*1(*ε*) ∑ *n*=1 *β*2

<sup>=</sup> <sup>1</sup> 2*v*<sup>0</sup>

−<sup>1</sup>

and, as follows,

2*v*<sup>0</sup>

ln *g*∗

<sup>1</sup>(*n*,*ε*)(*ϑ*1(*n*,*ε*) <sup>−</sup> *<sup>ϑ</sup>*)||<sup>2</sup> <sup>≤</sup>

*<sup>ζ</sup>*1(*n*,*ε*)||<sup>2</sup> <sup>≤</sup>

1 *cn*

= *εδ*1.

<sup>13</sup>(*ε*) *P<sup>ϑ</sup>* − a.s. (29)

<sup>13</sup> *P<sup>ϑ</sup>* − a.s.,

13(*ε*)). (30)

<sup>13</sup> *P<sup>ϑ</sup>* − a.s., (26)

<sup>13</sup>(*h*1)] = 0 *P<sup>ϑ</sup>* − a.s. (28)

*P<sup>ϑ</sup>* − a.s., (27)

<sup>−</sup>1*cn*]= <sup>1</sup>

<sup>−</sup>1e−2*v*0*k*1(*n*)*h*1*G*˜

<sup>1</sup> (*n*,*ε*)||<sup>2</sup> · || ˜

15(3 + *R*2) *<sup>ρ</sup>*<sup>1</sup> ∑ *n*≥1

13(*h*1)||−<sup>2</sup>

<sup>−</sup>1*cn* <sup>&</sup>gt; *<sup>g</sup>*˜13e<sup>4</sup>*v*0*h*<sup>1</sup> },

<sup>−</sup>1*cn* <sup>&</sup>gt; *<sup>g</sup>*˜13e<sup>20</sup>*v*0*h*<sup>1</sup> }.

e−2*v*0(2+5*h*1) + e−4*v*0(1+3*h*1)

ln *<sup>τ</sup>*1(*n*,*ε*) <sup>−</sup> <sup>1</sup>

*τ*1(*n*,*ε*) ln *ε*−1*cn*

> 1 *<sup>S</sup>*2(*σ*1(*ε*))||

> > <sup>1</sup>(*n*,*ε*) · ||*G*˜ <sup>−</sup><sup>1</sup>

*<sup>ζ</sup>*1(*n*,*ε*)||<sup>2</sup> <sup>≤</sup> *εδ*<sup>1</sup>

13] <sup>2</sup>*ρ*−<sup>1</sup> <sup>1</sup> *<sup>δ</sup>*1||*G*˜

> *N* ∑ *n*=1

*N* ∑ *n*=1

13(*ε*)), Ψ��

<sup>13</sup>(*ε*)] ≥

<sup>13</sup>(*ε*) = <sup>1</sup>

1 2*v*<sup>0</sup>

2*v*<sup>0</sup> ln(*ε* <sup>−</sup>1*cσ*��

ln *g*∗

<sup>13</sup>(*ε*) ≤ *σ*1(*ε*) ≤ *σ*��

Then, from (8), (26) and (29) we obtain finally the assertion I.2 of Proposition 3.1:

ln *T*1(*ε*) − Ψ�

ln<sup>6</sup> *ε*

ln<sup>6</sup> *ε*

*v*0

*G*˜ <sup>−</sup><sup>1</sup>

*<sup>E</sup>ϑ*||*ϑ*1(*ε*) <sup>−</sup> *<sup>ϑ</sup>*||<sup>2</sup> <sup>=</sup> *<sup>E</sup><sup>ϑ</sup>*

lim*n*∨*ε*

<sup>1</sup> (*n*,*ε*) <sup>−</sup> [(2*v*0)3*g*<sup>∗</sup>

1 *cn* · *β*2

1 *cn <sup>E</sup>ϑ*|| ˜

Now we prove the assertion I.2 for *ϑ* ∈ Θ13. Denote the number

*g*˜13 = [(2*v*0)3*g*<sup>∗</sup>

<sup>13</sup>(*ε*) = inf{*n* ≥ 1 :

<sup>13</sup>(*ε*) = inf{*n* ≥ 1 :

*σ*�

2*v*<sup>0</sup> ln(*ε* <sup>−</sup>1*cσ*�

[*T*1(*ε*) + <sup>1</sup>

*v*0

where *g*∗

<sup>13</sup> <sup>=</sup> <sup>2</sup>*v*0*C*˜−<sup>2</sup> *X* 

> lim*n*∨*<sup>ε</sup>* [ <sup>1</sup> ln3 *ε*−1*cn*

corresponding assertion in [14, 16]:

≤ *ε δ*1 *ρ*1 *Eϑ σ*1(*ε*) ∑ *n*=1

> *σ* �

*σ* ��

Ψ�

lim *ε*→0

From (8) and (28) it follows, that for *ε* small enough

<sup>13</sup>(*ε*) = <sup>1</sup>

≤ *εδ*1 *<sup>ρ</sup>*<sup>1</sup> ∑ *n*≥1

and the times

Denote

lim*n*∨*<sup>ε</sup>* [*τ*1(*n*,*ε*) + <sup>1</sup>

$$
\begin{aligned}
\zeta\_1(n,\varepsilon) &= \zeta\_1(\tau\_1(n,\varepsilon) - k\_1(n)h\_1, h\_1), \\
\zeta\_1(T, s) &= \int\_0^T \Psi\_s(t - 2 - 5s)(\tilde{\Delta}\xi(t)dt + dV(t) - dV(t - 1)).
\end{aligned}
$$

According to (25), (27) and (28) first three factors in the right-hand side of this equality have *P<sup>ϑ</sup>* − a.s. positive finite limits. The last factor vanishes in *P<sup>ϑ</sup>* − a.s. sense by the properties of the square integrable martingales *ζ*1(*T*,*s*) :

$$\lim\_{n \ne \varepsilon} \frac{\zeta\_1(n, \varepsilon)}{\tau\_1^{-1}(n, \varepsilon) \mathbf{e}^{2\tau\_1(n, \varepsilon)\upsilon\_0}} = \lim\_{T \to \infty} \frac{\zeta\_1(T, h\_1)}{T^{-1} \mathbf{e}^{2\upsilon\_0 T}} = 0 \quad P\_\theta-\text{a.s.}$$

Then the estimators *ϑ*1(*n*,*ε*) are strongly consistent as *ε* → 0 or *n* → ∞ and we obtain the assertion II of Proposition 3.1.

Hence Proposition 3.1 is valid.

#### **Proof of Proposition 3.2.**

Similar to the proof of Proposition 3.1 and [7]–[16] we can get the following asymptotic as *<sup>t</sup>* <sup>→</sup> <sup>∞</sup> relations for the processes <sup>Δ</sup>˜ *<sup>Y</sup>*(*t*), *<sup>Z</sup>*(*t*) and *<sup>Z</sup>*˜(*t*) :

$$\begin{aligned} -\text{for } \theta \in \Theta\_{21} \\ \tilde{\Delta}Y(t) &= \mathbb{C}\_{Y}e^{v\_{0}t} + \mathbb{C}\_{Y1}e^{v\_{1}t} + o(e^{\gamma t}) \quad P\_{\theta}-\text{a.s.}, \\ Z(t) &= \mathbb{C}\_{Z}e^{v\_{1}t} + o(e^{\gamma t}) \quad P\_{\theta}-\text{a.s.}, \\ \lambda\_{t} - \lambda &= \frac{2v\_{0}e^{v\_{0}}}{v\_{0}+v\_{1}} \mathbb{C}\_{Z}\mathbb{C}\_{Y}^{-1}e^{-(v\_{0}-v\_{1})t} + o(e^{-(v\_{0}-v\_{1}+\gamma)t}) \quad P\_{\theta}-\text{a.s.}, \\ \tilde{Z}(t) &= \tilde{\mathsf{C}}e^{v\_{1}t} + o(e^{\gamma t}) \quad P\_{\theta}-\text{a.s.}; \end{aligned}$$

– for *ϑ* ∈ Θ<sup>22</sup>

$$\begin{aligned} \vert \tilde{\Delta}Y(t) - \mathbb{C}\_Y e^{v\_0 t} - \mathbb{C}\_{Y1}(t) e^{v\_1 t} \vert &= o(e^{\gamma t}) \quad P\_\theta-\text{a.s.}\\ \vert Z(t) - \mathbb{C}\_Z(t) e^{v\_1 t} \vert &= o(e^{\gamma t}) \quad P\_\theta-\text{a.s.}\\ \lambda\_t - \lambda &= 2v\_0 e^{v\_0} \mathbb{C}\_Y^{-1} \mathcal{U}\_Z(t) e^{-(v\_0 - v\_1)t} + o(e^{-(v\_0 - v\_1 + \gamma)t}) \quad P\_\theta-\text{a.s.}\\ \vert \tilde{Z}(t) - \tilde{\mathcal{C}}\_Z(t) e^{v\_1 t} \vert &= o(e^{\gamma t}) \quad P\_\theta-\text{a.s.} \end{aligned}$$

where *CY* and *CY*<sup>1</sup> are some non-zero constants, 0 <sup>&</sup>lt; *<sup>γ</sup>* <sup>&</sup>lt; *<sup>v</sup>*1, *CZ* <sup>=</sup> *CY*1(<sup>1</sup> <sup>−</sup> <sup>e</sup>*v*0−*v*<sup>1</sup> ), *<sup>C</sup>*˜ *<sup>Z</sup>* = *CZ v*<sup>1</sup> − *v*<sup>0</sup> *v*<sup>1</sup> + *v*<sup>0</sup> ; *CZ*(*t*), *UZ*(*t*) = <sup>∞</sup> 0 *CZ*(*<sup>t</sup>* <sup>−</sup> *<sup>u</sup>*)e−(*v*0+*v*1)*udu* and *<sup>C</sup>*˜ *<sup>Z</sup>*(*t*) = *CZ*(*t*) − 2*v*0*UZ*(*t*) are the

periodic (with the period Δ > 1) functions.

#### 18 Will-be-set-by-IN-TECH 40 Stochastic Modeling and Control On Guaranteed Parameter Estimation of Stochastic Delay Differential Equations by Noisy Observations <sup>19</sup>

Denote

$$\mathcal{U}\_{\tilde{Z}}(T) = \int\_0^\infty \tilde{\mathcal{C}}\_Z(T - \mathfrak{u}) \mathbf{e}^{-(v\_0 + v\_1)\mathfrak{u}} d\mathfrak{u}$$

$$\mathcal{U}\_{\tilde{Z}\mathcal{Z}}(S, T) = \int\_0^\infty \tilde{\mathcal{C}}\_Z(T - \mathfrak{u}) \mathcal{C}\_Z(S - \mathfrak{u}) \mathbf{e}^{-2v\_1\mathfrak{u}} d\mathfrak{u}, \quad \tilde{\mathcal{U}}\_Z(T) = \mathcal{U}\_{\tilde{Z}\mathcal{Z}}(T, T).$$

It should be noted that the functions *CZ*(*t*), *UZ*(*t*), *C*˜ *<sup>Z</sup>*(*t*) and *UZ*˜(*T*) have at most two roots on each interval from [0, ∞) of the length Δ. At the same time the function *UZZ*˜ (*S*, *T*) - at most four roots.

With *Pϑ*-probability one we have:

– for *ϑ* ∈ Θ<sup>2</sup>

$$\lim\_{T-S \to \infty} \mathbf{e}^{-2v\_0T} \int\_S^T (\tilde{\Delta}Y(t-3))^2 dt = \frac{C\_Y^2}{2v\_0} \mathbf{e}^{-6v\_0} \tag{31}$$

– for *ϑ* ∈ Θ<sup>21</sup>

$$\lim\_{T-S \to \infty} \mathbf{e}^{-2v\_1T} \int\_S^T \tilde{Z}^2(t-3)dt = \frac{\tilde{C}\_Z^2}{2v\_1} \mathbf{e}^{-6v\_1} \tag{32}$$

$$\lim\_{T \to S \to \infty} \tilde{G}\_2^{-1}(\mathbb{S}, T) = \tilde{G}\_{21 \nu} \tag{33}$$

Denote

and thus

deviation *α*2(*n*,*ε*) − *α* :

= *ν*2(*n*,*ε*)

(ln *ε*

*δv*<sup>1</sup>

= *ν*2(*n*,*ε*)

*α*<sup>1</sup> ≤ lim *n*∨*ε*

where *<sup>α</sup><sup>i</sup>* <sup>=</sup> <sup>1</sup>

⎛

⎜⎜⎜⎜⎝

*v*<sup>1</sup> ln e−2*v*0*ν*2(*n*,*ε*)

[*v*<sup>0</sup> ln *Cν<sup>i</sup>* − *v*<sup>1</sup> ln

Then for *ϑ* ∈ Θ<sup>2</sup>

*<sup>C</sup>ν*<sup>1</sup> <sup>=</sup> <sup>e</sup>−6*v*<sup>1</sup> min � *<sup>C</sup>*˜2

*Z* 2*v*<sup>1</sup> , inf *T*>0

≤ lim

1 2*v*<sup>1</sup>

<sup>≤</sup> lim*n*∨*<sup>ε</sup>* [*ν*2(*n*,*ε*) <sup>−</sup> *<sup>δ</sup>*

*ν*2(*n*,*ε*)(*α*2(*n*,*ε*) − *α*) = *ν*2(*n*,*ε*)

*C*−<sup>1</sup> *<sup>ν</sup>*<sup>2</sup> ≤ lim *n*∨*ε*

*Cν*<sup>1</sup> ≤ lim *T*−*S*→∞

*<sup>T</sup>*−*S*→<sup>∞</sup> <sup>e</sup>−2*v*1*<sup>T</sup>*

ln *C*−<sup>1</sup>

e2*v*1*ν*2(*n*,*ε*) (*ε*−1*cn*)*<sup>δ</sup>* <sup>≤</sup> lim*n*∨*<sup>ε</sup>*

> *<sup>ν</sup>*<sup>2</sup> ≤ lim *n*∨*ε*

> > 2*v*<sup>1</sup> ln *ε*

2*v*0*ν*2(*n*,*ε*) + ln e−2*v*0*ν*2(*n*,*ε*)

*ν*2(*n*,*ε*) � 0

<sup>−</sup>1*cn*) · (*<sup>α</sup>* <sup>−</sup> *<sup>α</sup>*2(*n*,*ε*))<sup>≤</sup> lim*n*∨*<sup>ε</sup>* (ln *<sup>ε</sup>*

*C*2 *Y* 2*v*<sup>0</sup>

2*v*<sup>2</sup>

and using the limit relations (31), (36) and (37) we obtain

2*v*1*ν*2(*n*,*ε*) + ln e−2*v*1*ν*2(*n*,*ε*)

<sup>1</sup>*ν*2(*n*,*ε*) + *<sup>v</sup>*<sup>1</sup> ln e−2*v*1*ν*2(*n*,*ε*)

e−6*v*<sup>0</sup> ], *i* = 1, 2.

and from the definition (13) of *ν*2(*n*,*ε*) and (32), (34) we have

*U*˜ *<sup>Z</sup>*(*T*)

� *T*

*S*

�

e−2*v*1*<sup>T</sup>* � *T*

*S*

e2*<sup>v</sup>*1*ν*2(*n*,*ε*) (*ε*−1*cn*)*<sup>δ</sup>* <sup>≤</sup> *<sup>C</sup>*−<sup>1</sup>

> 2*v*<sup>1</sup> ln *ε*

1 2*v*<sup>1</sup>

ln *C*−<sup>1</sup>

(Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>))2*dt*

*<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt*

(Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>))2*dt*

*<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt*

*ν*2(*n*,*ε*) � 0

[*ν*2(*n*,*ε*) <sup>−</sup> *<sup>δ</sup>*

<sup>−</sup>1*cn*] <sup>≤</sup>

By the definition (12) of *α*2(*n*,*ε*) we find the following normalized representation for the

⎛

ln

ln

*ν*2(*n*,*ε*) � 0

> *ν*2(*n*,*ε*) � 0

> *ν*2(*n*,*ε*) � 0

(Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>))2*dt* <sup>−</sup> *<sup>v</sup>*<sup>0</sup> ln e−2*v*1*ν*2(*n*,*ε*)

*ν*2(*n*,*ε*) � 0

⎜⎜⎜⎜⎝

, *<sup>C</sup>ν*<sup>2</sup> <sup>=</sup> <sup>e</sup>−6*v*<sup>1</sup> max � *<sup>C</sup>*˜2

On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 41

*<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt* <sup>≤</sup>

*Z* 2*v*<sup>1</sup>

*<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt* <sup>≤</sup> *<sup>C</sup>ν*<sup>2</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s. (36)

*<sup>ν</sup>*<sup>1</sup> *P<sup>ϑ</sup>* − a.s.

*<sup>ν</sup>*<sup>1</sup> *P<sup>ϑ</sup>* − a.s. (37)

⎞

⎟⎟⎟⎟⎠ =

⎞

⎟⎟⎟⎟⎠ =

*<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt*

<sup>−</sup> *<sup>v</sup>*<sup>0</sup> *v*1

> <sup>−</sup> *<sup>v</sup>*<sup>0</sup> *v*1

*ν*2(*n*,*ε*) � 0

*<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt*

<sup>−</sup>1*cn*) · (*<sup>α</sup>* <sup>−</sup> *<sup>α</sup>*2(*n*,*ε*)) <sup>≤</sup> *<sup>α</sup>*<sup>2</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.,

<sup>−</sup>1*cn*] <sup>≤</sup>

, sup *T*>0 *U*˜ *<sup>Z</sup>*(*T*)

� .

where

$$
\tilde{\mathbf{G}}\_{21} = \begin{pmatrix}
\frac{2v\_1(v\_1+v\_0)^2}{\mathbb{C}\_Z\mathbb{C}\_Z(v\_1-v\_0)^2}\mathbf{e}^{3v\_1} & -\frac{4v\_0v\_1(v\_1+v\_0)}{\mathbb{C}\_Z\mathbb{C}\_Y(v\_1-v\_0)^2}\mathbf{e}^{3v\_0} \\
\end{pmatrix}
$$

– for *ϑ* ∈ Θ<sup>22</sup>

$$\lim\_{T \to S \to \infty} \left| \mathbf{e}^{-2v\_1T} \int\_S^T \tilde{Z}^2(t-3)dt - \mathbf{e}^{-6v\_1} \tilde{U}\_Z(T-3) \right| = 0,\tag{34}$$

$$\lim\_{T \to S \to \infty} \left| \tilde{G}\_2^{-1}(S, T) - \tilde{G}\_{22}(T) \right| = 0,\tag{35}$$

where

$$
\tilde{\mathbf{G}}\_{22}(T) = \left[\frac{1}{2v\_0} \mathcal{U}\_{22}(T, T - 3) - \mathcal{U}\_{\tilde{Z}}(T - 3)\mathcal{U}\_{\tilde{Z}}(T)\right]^{-1} \cdot \begin{pmatrix} \frac{\mathbf{e}^{3v\_1}}{2v\_0} & -\frac{\mathbf{e}^{3v\_0}}{\mathbf{C}\_Y}\mathcal{U}\_{\tilde{Z}}(T) \\\ -\frac{\mathbf{e}^{v\_0 + 3v\_1}}{2v\_0} & \frac{\mathbf{e}^{4v\_0}}{\mathbf{C}\_Y}\mathcal{U}\_{\tilde{Z}}(T) \end{pmatrix} \cdot \boldsymbol{1}
$$

The matrix *G*˜ <sup>21</sup> is constant and non-zero and *G*˜ <sup>22</sup>(*T*) is the periodic matrix function with the period Δ > 1 (see [3], [10, 12, 14]) and may have infinite norm for four points on each interval of periodicity only.

The next step of the proof is the investigation of the asymptotic behaviour of the stopping times *ν*2(*n*,*ε*), *τ*2(*n*,*ε*) and the estimators *α*2(*n*,*ε*).

Denote

18 Will-be-set-by-IN-TECH

on each interval from [0, ∞) of the length Δ. At the same time the function *UZZ*˜ (*S*, *T*) - at most

(Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>))2*dt* <sup>=</sup> *<sup>C</sup>*<sup>2</sup>

*<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt* <sup>=</sup> *<sup>C</sup>*˜2

*<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt* <sup>−</sup> <sup>e</sup>−6*v*1*U*˜ *<sup>Z</sup>* (*<sup>T</sup>* <sup>−</sup> <sup>3</sup>)

<sup>22</sup>(*T*) � �

> �−<sup>1</sup> · ⎛ ⎝

e3*<sup>v</sup>*<sup>1</sup>

−e*v*0+3*v*<sup>1</sup> 2*v*<sup>0</sup>

<sup>2</sup> (*S*, *<sup>T</sup>*) = *<sup>G</sup>*˜

*CZC*˜*Z*(*v*1−*v*0)<sup>2</sup> <sup>e</sup>3*v*<sup>1</sup> <sup>−</sup> <sup>4</sup>*v*0*v*1(*v*1+*v*0)

*CZC*˜*Z*(*v*1−*v*0)<sup>2</sup> <sup>e</sup>*v*0+3*v*<sup>1</sup> <sup>4</sup>*v*0*v*1(*v*1+*v*0)

<sup>2</sup> (*S*, *<sup>T</sup>*) <sup>−</sup> *<sup>G</sup>*˜

period Δ > 1 (see [3], [10, 12, 14]) and may have infinite norm for four points on each interval

The next step of the proof is the investigation of the asymptotic behaviour of the stopping

*Y* 2*v*<sup>0</sup>

*Z* 2*v*<sup>1</sup>

*CZCY*(*v*1−*v*0)<sup>2</sup> <sup>e</sup>3*v*<sup>0</sup>

*CZCY*(*v*1−*v*0)<sup>2</sup> e4*<sup>v</sup>*<sup>0</sup>

*<sup>Z</sup>*(*<sup>T</sup>* <sup>−</sup> *<sup>u</sup>*)e−(*v*0+*v*1)*udu*,

*<sup>Z</sup>*(*<sup>T</sup>* <sup>−</sup> *<sup>u</sup>*)*CZ*(*<sup>S</sup>* <sup>−</sup> *<sup>u</sup>*)e−2*v*1*udu*, *<sup>U</sup>*˜ *<sup>Z</sup>*(*T*) = *UZ*˜ *<sup>Z</sup>*˜(*T*, *<sup>T</sup>*).

*<sup>Z</sup>*(*t*) and *UZ*˜(*T*) have at most two roots

e−6*v*<sup>0</sup> , (31)

e−6*v*<sup>1</sup> , (32)

= 0, (34)

*CY UZ*˜(*T*)

⎞ ⎠ .

21, (33)

� <sup>=</sup> 0, (35)

e4*<sup>v</sup>*<sup>0</sup> *CY UZ*˜(*T*)

⎞ ⎠ ,

� � � � � �

<sup>2</sup>*v*<sup>0</sup> <sup>−</sup>e3*<sup>v</sup>*<sup>0</sup>

<sup>22</sup>(*T*) is the periodic matrix function with the

*UZ*˜(*T*) =

�∞

0 *C*˜

It should be noted that the functions *CZ*(*t*), *UZ*(*t*), *C*˜

lim *<sup>T</sup>*−*S*→<sup>∞</sup> <sup>e</sup>−2*v*0*<sup>T</sup>*

> lim *<sup>T</sup>*−*S*→<sup>∞</sup> <sup>e</sup>−2*v*1*<sup>T</sup>*

⎛ ⎝

> � � � � � �

e−2*v*1*<sup>T</sup>* � *T*

> lim *T*−*S*→∞

<sup>21</sup> is constant and non-zero and *G*˜

times *ν*2(*n*,*ε*), *τ*2(*n*,*ε*) and the estimators *α*2(*n*,*ε*).

*UZZ*˜ (*S*, *T*) =

With *Pϑ*-probability one we have:

*G*˜ <sup>21</sup> =

lim *T*−*S*→∞

� 1 2*v*<sup>0</sup> �∞

0 *C*˜

> � *T*

*S*

� *T*

*S*

lim *<sup>T</sup>*−*S*→<sup>∞</sup> *<sup>G</sup>*˜ <sup>−</sup><sup>1</sup>

2*v*1(*v*1+*v*0)<sup>2</sup>

<sup>−</sup> <sup>2</sup>*v*1(*v*1+*v*0)<sup>2</sup>

*S*

� � � *G*˜ <sup>−</sup><sup>1</sup>

*UZZ*˜ (*T*, *T* − 3) − *UZ*(*T* − 3)*UZ*˜(*T*)

Denote

four roots.

– for *ϑ* ∈ Θ<sup>2</sup>

– for *ϑ* ∈ Θ<sup>21</sup>

where

where

*G*˜ <sup>22</sup>(*T*) =

The matrix *G*˜

of periodicity only.

– for *ϑ* ∈ Θ<sup>22</sup>

$$\mathbb{C}\_{\upsilon 1} = \mathbf{e}^{-6v\_1} \min \left\{ \frac{\tilde{\mathbb{C}}\_Z^2}{2v\_1}, \inf\_{T>0} \tilde{\mathcal{U}}\_Z(T) \right\}, \quad \mathbb{C}\_{\upsilon 2} = \mathbf{e}^{-6v\_1} \max \left\{ \frac{\tilde{\mathbb{C}}\_Z^2}{2v\_1}, \sup\_{T>0} \tilde{\mathcal{U}}\_Z(T) \right\}.$$

Then for *ϑ* ∈ Θ<sup>2</sup>

$$\mathcal{C}\_{\nu1} \le \lim\_{T \to -S \to \infty} \operatorname{e}^{-2v\_1 T} \int\_S \tilde{Z}^2 (t - 3) dt \le \delta$$

$$\le \overline{\lim\_{T \to S \to \infty}} \operatorname{e}^{-2v\_1 T} \int\_S \tilde{Z}^2 (t - 3) dt \le \mathcal{C}\_{\nu2} \quad P\_\theta-\text{a.s.} \tag{36}$$

and from the definition (13) of *ν*2(*n*,*ε*) and (32), (34) we have

$$\mathbb{C}\_{\nu 2}^{-1} \le \varliminf\_{n \vee \varepsilon} \frac{\mathbf{e}^{2\upsilon\_1 \nu\_2(n,\varepsilon)}}{(\varepsilon^{-1} \mathfrak{c}\_{\mathcal{U}})^{\delta}} \le \varlimsup\_{n \vee \varepsilon} \frac{\mathbf{e}^{2\upsilon\_1 \nu\_2(n,\varepsilon)}}{(\varepsilon^{-1} \mathfrak{c}\_{\mathcal{U}})^{\delta}} \le \mathsf{C}\_{\nu 1}^{-1} \quad P\_{\theta}-\text{a.s.}$$

and thus

$$\frac{1}{2v\_1} \ln \mathbb{C}\_{\nu 2}^{-1} \le \varliminf\_{\overline{n} \vee \varepsilon} \left[ \nu\_2(n, \varepsilon) - \frac{\delta}{2v\_1} \ln \varepsilon^{-1} c\_{\eta} \right] \le$$

$$\le \varliminf\_{\overline{n} \vee \varepsilon} \left[ \nu\_2(n, \varepsilon) - \frac{\delta}{2v\_1} \ln \varepsilon^{-1} c\_{\eta} \right] \le \frac{1}{2v\_1} \ln \mathbb{C}\_{\nu 1}^{-1} \quad P\_{\theta}-\text{a.s.} \tag{37}$$

By the definition (12) of *α*2(*n*,*ε*) we find the following normalized representation for the deviation *α*2(*n*,*ε*) − *α* :

*ν*2(*n*,*ε*)(*α*2(*n*,*ε*) − *α*) = *ν*2(*n*,*ε*) ⎛ ⎜⎜⎜⎜⎝ ln *ν*2(*n*,*ε*) � 0 (Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>))2*dt* ln *ν*2(*n*,*ε*) � 0 *<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt* <sup>−</sup> *<sup>v</sup>*<sup>0</sup> *v*1 ⎞ ⎟⎟⎟⎟⎠ = = *ν*2(*n*,*ε*) ⎛ ⎜⎜⎜⎜⎝ 2*v*0*ν*2(*n*,*ε*) + ln e−2*v*0*ν*2(*n*,*ε*) *ν*2(*n*,*ε*) � 0 (Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>))2*dt* 2*v*1*ν*2(*n*,*ε*) + ln e−2*v*1*ν*2(*n*,*ε*) *ν*2(*n*,*ε*) � 0 *<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt* <sup>−</sup> *<sup>v</sup>*<sup>0</sup> *v*1 ⎞ ⎟⎟⎟⎟⎠ = = *ν*2(*n*,*ε*) *v*<sup>1</sup> ln e−2*v*0*ν*2(*n*,*ε*) *ν*2(*n*,*ε*) � 0 (Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>))2*dt* <sup>−</sup> *<sup>v</sup>*<sup>0</sup> ln e−2*v*1*ν*2(*n*,*ε*) *ν*2(*n*,*ε*) � 0 *<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt* 2*v*<sup>2</sup> <sup>1</sup>*ν*2(*n*,*ε*) + *<sup>v</sup>*<sup>1</sup> ln e−2*v*1*ν*2(*n*,*ε*) *ν*2(*n*,*ε*) � 0 *<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt*

and using the limit relations (31), (36) and (37) we obtain

$$a\_1 \le \varliminf\_{\mathcal{U} \vee \mathcal{E}} (\ln \varepsilon^{-1} c\_{\mathcal{U}}) \cdot (a - a\_2(n, \varepsilon)) \le \varliminf\_{\mathcal{U} \vee \mathcal{E}} (\ln \varepsilon^{-1} c\_{\mathcal{U}}) \cdot (a - a\_2(n, \varepsilon)) \le a\_2 \quad P\_\theta-\text{a.s.},$$

$$\text{where } a\_i = \frac{1}{\delta v\_1} [v\_0 \ln \mathbb{C}\_{vi} - v\_1 \ln \frac{\mathbb{C}\_Y^2}{2v\_0} \mathbf{e}^{-\delta v\_0}], \quad i = 1, 2.$$

#### 20 Will-be-set-by-IN-TECH 42 Stochastic Modeling and Control On Guaranteed Parameter Estimation of Stochastic Delay Differential Equations by Noisy Observations <sup>21</sup>

Thus for *ϑ* ∈ Θ<sup>2</sup>

$$\mathbf{e}^{a\_1} \le \lim\_{n \vee \varepsilon} (\varepsilon^{-1} c\_n)^{(\mathfrak{a} - a\_2(n, \varepsilon))} \le \lim\_{n \vee \varepsilon} (\varepsilon^{-1} c\_n)^{(\mathfrak{a} - a\_2(n, \varepsilon))} \le \mathbf{e}^{a\_2} \quad P\_\theta-\text{a.s.}\tag{38}$$

Then, using (33), (35), (38), (39) and (41) we can find, similar to [12, 14], the lower and upper

�|*G*˜ <sup>−</sup><sup>1</sup>

On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 43

<sup>2</sup> )1/2}, *<sup>σ</sup>*<sup>22</sup> <sup>=</sup> inf{*<sup>n</sup>* <sup>≥</sup> 1 : *<sup>N</sup>* <sup>&</sup>gt; *<sup>g</sup>*˜

2*v*<sup>1</sup> ln *ε* <sup>−</sup>1] <sup>≤</sup>

[*T*2(*ε*) <sup>−</sup> <sup>1</sup>

*y*˜0(*t* − *s*)*dW*(*s*), *y*˜0(*s*) = *x*˜0(*s*) − *λx*˜0(*s* − 1),

−*v*0(*t*−*s*)

<sup>2</sup> (*n*,*ε*)|| ≤ *g*˜22, (42)

−1 <sup>21</sup> (*ρ*2*δ*−<sup>1</sup>

<sup>2</sup> )1/2}

ln *s*2*σ*<sup>22</sup> *P<sup>ϑ</sup>* − a.s.

*σ*2(*ε*) ≤ *σ*<sup>22</sup> *P<sup>ϑ</sup>* − a.s., (43)

1 2*v*<sup>1</sup>

*ds*, *Z*2(*t*) = *ZV*(*t*) + *Z*3(*t*),

−*v*0(*t*−*s*)

<sup>1</sup> (0),

<sup>1</sup> (0) − *EϑZ*1(0)*Z*3(−1).

*ds*,

*ds*, *<sup>Z</sup>*˜ <sup>1</sup>(*t*) = *<sup>Z</sup>*1(*t*) <sup>−</sup> <sup>2</sup>*v*0*Z*2(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>),

*Z*˜ <sup>1</sup>(*s*)*e*

*t*

−∞

<sup>0</sup> (*<sup>λ</sup>* <sup>−</sup> <sup>1</sup>)] + *<sup>E</sup>ϑZ*<sup>2</sup>

<sup>0</sup> (*<sup>λ</sup>* <sup>−</sup> <sup>1</sup>)] + *<sup>E</sup>ϑZ*˜ <sup>2</sup>

<sup>2</sup> (*n*,*ε*)|| ≤ lim*n*∨*<sup>ε</sup>*

and from (40) and (43) we obtain the second property of the assertion I in Proposition 3.2:

The assertions I.1 and II of Proposition 3.2 can be proved similar to the proof of the

Similar to the proof of Propositions 3.1, 3.2 and [7]–[16] we get for *ϑ* ∈ Θ<sup>3</sup> the needed asymptotic as *<sup>t</sup>* <sup>→</sup> <sup>∞</sup> relations for the processes <sup>Δ</sup>˜ *<sup>Y</sup>*(*t*), *<sup>Z</sup>*(*t*) and *<sup>Z</sup>*˜(*t*). To this end we introduce

bounds for the limits with *Pϑ*-probability one:

where *g*˜21 and *g*˜22 are positive finite numbers.

*σ*<sup>21</sup> = inf{*n* ≥ 1 : *N* > *g*˜

*ε*→0

Hence Proposition 3.2 is proven.

*ZV*(*t*) =

ln *s*1*σ*<sup>21</sup> ≤ lim

**Proof of Proposition 3.3.**

the following notation:

where

1 2*v*<sup>1</sup> *g*˜21 ≤ lim *n*∨*ε*

> *σ*<sup>21</sup> ≤ lim *ε*→0

> > −1 <sup>22</sup> (*ρ*2*δ*−<sup>1</sup>

2*v*<sup>1</sup> ln *ε*

[*T*2(*ε*) <sup>−</sup> <sup>1</sup>

corresponding statement of Proposition 3.1.

*<sup>Z</sup>*1(*t*) =

*t*

−∞

*<sup>Z</sup>*3(*t*) =

*t*

−∞

*t*

*Z*1(*s*)*e*

−∞

*C*˜

*CZZ*˜ <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>λ</sup>*<sup>2</sup> <sup>+</sup> <sup>2</sup>[*<sup>λ</sup>* <sup>−</sup> *<sup>v</sup>*−<sup>1</sup>

[Δ˜ *<sup>V</sup>*(*s*) <sup>−</sup> *<sup>λ</sup>*Δ˜ *<sup>V</sup>*(*<sup>s</sup>* <sup>−</sup> <sup>1</sup>)]*<sup>e</sup>*

−*v*0(*t*−*s*)

*<sup>Z</sup>*˜ <sup>2</sup>(*t*) = *ZV*(*t*) + *<sup>Z</sup>*˜ <sup>3</sup>(*t*), *<sup>Z</sup>*˜ <sup>3</sup>(*t*) =

*<sup>Z</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>λ</sup>*<sup>2</sup> <sup>+</sup> <sup>4</sup>[*<sup>λ</sup>* <sup>−</sup> *<sup>v</sup>*−<sup>1</sup>


Thus, by the definition (14) of the stopping time *σ*2(*ε*) and from (42) we have

*σ*2(*ε*) ≤ lim *ε*→0

> <sup>−</sup>1]<sup>≤</sup> lim *ε*→0

Let *s*<sup>1</sup> and *s*<sup>2</sup> be the positive roots of the following equations

$$\mathbf{C}\_{\mathsf{V}\mathsf{Z}} \cdot \mathbf{s} + \frac{\mathbf{C}\_{Y}^{2}}{2v\_{0}} \mathbf{e}^{-6v\_{0}} \cdot \mathbf{e}^{\mathfrak{a}\_{2}} \cdot \mathbf{s}^{\mathfrak{a}} = 1 \quad \text{and} \quad \mathbf{C}\_{\mathsf{V}1} \cdot \mathbf{s} + \frac{\mathbf{C}\_{Y}^{2}}{2v\_{0}} \mathbf{e}^{-6v\_{0}} \cdot \mathbf{e}^{\mathfrak{a}\_{1}} \cdot \mathbf{s}^{\mathfrak{a}} = 1$$

respectively. It is clear that 0 < *s*<sup>1</sup> ≤ *s*<sup>2</sup> < ∞.

By the definition of stopping times *τ*2(*n*,*ε*) we have

$$\begin{split} \lim\_{n \to \varepsilon} \left[ \frac{1}{\varepsilon^{-1} c\_{\eta}} \int\_{v\_{2}(n,\varepsilon)}^{\tau\_{2}(n,\varepsilon)} \frac{1}{(\varepsilon^{-1} c\_{\eta})^{a\_{2}(n,\varepsilon)}} \int\_{v\_{2}(n,\varepsilon)}^{\tau\_{2}(n,\varepsilon)} \int (\tilde{\Delta} Y(t-3))^{2} dt \right] &= \\ = \lim\_{n \to \varepsilon} \left[ \frac{1}{\mathfrak{e}^{2v\_{1}\tau\_{2}(n,\varepsilon)}} \int\_{0}^{\tau\_{2}(n,\varepsilon)} \tilde{Z}^{2}(t-3) dt \cdot \frac{\mathfrak{e}^{2v\_{1}\tau\_{2}(n,\varepsilon)}}{\varepsilon^{-1} c\_{\eta}} + \\ + \frac{1}{\mathfrak{e}^{2v\_{0}\tau\_{2}(n,\varepsilon)}} \int\_{0}^{\tau\_{2}(n,\varepsilon)} (\tilde{\Delta} Y(t-3))^{2} dt \cdot (\varepsilon^{-1} c\_{\eta})^{(a-a\_{2}(n,\varepsilon))} \cdot \left(\frac{\mathfrak{e}^{2v\_{1}\tau\_{2}(n,\varepsilon)}}{\varepsilon^{-1} c\_{\eta}}\right)^{a} \right] = 1. \end{split}$$

Then, using (38), for *ϑ* ∈ Θ<sup>2</sup> we have

$$s\_1 \le \lim\_{n \to \varepsilon} \frac{\mathbf{e}^{2v\_1 \tau\_2(n, \varepsilon)}}{\varepsilon^{-1} \mathfrak{c}\_{\mathcal{U}}} \le \overline{\lim\_{n \vee \varepsilon}} \frac{\mathbf{e}^{2v\_1 \tau\_2(n, \varepsilon)}}{\varepsilon^{-1} \mathfrak{c}\_{\mathcal{U}}} \le s\_2 \quad P\_\theta-\text{a.s.}\tag{39}$$

and thus

$$\frac{1}{2v\_1}\ln s\_1 \le \lim\_{n \ne \varepsilon} \left[\tau\_2(n, \varepsilon) - \frac{1}{2v\_1}\ln \varepsilon^{-1} c\_n\right] \le$$

$$\le \overline{\lim}\_{n \ne \varepsilon} \left[\tau\_2(n, \varepsilon) - \frac{1}{2v\_1}\ln \varepsilon^{-1} c\_n\right] \le \frac{1}{2v\_1}\ln s\_2 \quad P\_\theta-\text{a.s.}\tag{40}$$

From (37) and (40) it follows, in particular, that

$$\lim\_{n \nearrow \varepsilon} \left[ \tau\_2(n, \varepsilon) - \nu\_2(n, \varepsilon) \right] = \infty \quad P\_\theta-\text{a.s.}\tag{41}$$

By the definition of *G*˜ <sup>2</sup>(*n*,*ε*), the following limit relation can be proved

$$\begin{split} & \lim\_{\overline{n} \to \overline{\varepsilon}} \left[ ||\tilde{\mathcal{G}}\_{2}^{-1}(n, \varepsilon)||^{2} - (1 + \mathbf{e}^{2 \upsilon\_{1}}) \{ \left( \frac{\mathbf{e}^{2 \upsilon\_{1} \underline{\varepsilon} \left( n, \varepsilon \right)}}{\varepsilon^{-1} \mathfrak{c}\_{\scriptscriptstyle \rm I}} \right)^{-2} (< \tilde{\mathcal{G}}\_{2}^{-1}(0, \tau\_{2}(n, \varepsilon) - k\_{2}(n)h\_{2}) >\_{11} \rangle^{2} + \\ & \left( \frac{\mathbf{e}^{2 \upsilon\_{1} \underline{\varepsilon} \left( n, \varepsilon \right)}}{\varepsilon^{-1} \mathfrak{c}\_{\scriptscriptstyle \rm I}} \right)^{-(1+a)} (\varepsilon^{-1} \mathfrak{c}\_{\scriptscriptstyle \rm I})^{a\_{2} \left( n, \varepsilon \right) - a} (< \tilde{\mathcal{G}}\_{2}^{-1}(0, \tau\_{2}(n, \varepsilon) - k\_{2}(n)h\_{2}) >\_{12} \rangle^{2} \} ] = 0 \quad P\_{\theta} - \text{a.s.}, \end{split}$$

where < *G* >*ij* is the *ij*-th element of the matrix *G*.

Then, using (33), (35), (38), (39) and (41) we can find, similar to [12, 14], the lower and upper bounds for the limits with *Pϑ*-probability one:

$$\left|\mathfrak{F}\_{21} \leq \varliminf\_{n \in \mathbb{Z}} ||\tilde{G}\_2^{-1}(n, \varepsilon)|| \leq \varliminf\_{n \vee \varepsilon} ||\tilde{G}\_2^{-1}(n, \varepsilon)|| \leq \mathfrak{g}\_{22'} \tag{42}$$

where *g*˜21 and *g*˜22 are positive finite numbers.

Thus, by the definition (14) of the stopping time *σ*2(*ε*) and from (42) we have

$$
\sigma\_{21} \le \varprojlim\_{\varepsilon \to 0} \sigma\_2(\varepsilon) \le \varprojlim\_{\varepsilon \to 0} \sigma\_2(\varepsilon) \le \sigma\_{22} \quad P\_\theta-\text{a.s.}\tag{43}
$$

where

20 Will-be-set-by-IN-TECH

*<sup>α</sup>* <sup>=</sup> 1 and *<sup>C</sup>ν*<sup>1</sup> · *<sup>s</sup>* <sup>+</sup>

1 (*ε*−1*cn*)*α*2(*n*,*ε*)

*<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt* ·

<sup>−</sup>1*cn*)(*α*−*α*2(*n*,*ε*)) ·

e2*<sup>v</sup>*1*τ*2(*n*,*ε*) *ε*−1*cn*

> 2*v*<sup>1</sup> ln *ε*

> > 1 2*v*<sup>1</sup>

(< *G*˜ <sup>−</sup><sup>1</sup>

[*τ*2(*n*,*ε*) <sup>−</sup> <sup>1</sup>

<sup>−</sup>1*cn*] <sup>≤</sup>

<sup>2</sup>(*n*,*ε*), the following limit relation can be proved

�−<sup>2</sup>

e2*<sup>v</sup>*1*τ*2(*n*,*ε*) *ε*−1*cn*

<sup>−</sup>1*cn*)(*α*−*α*2(*n*,*ε*)) <sup>≤</sup> <sup>e</sup>*α*<sup>2</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s. (38)

<sup>e</sup>−6*v*<sup>0</sup> · <sup>e</sup>*α*<sup>1</sup> · *<sup>s</sup>*

(Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>))2*dt*

+

e2*<sup>v</sup>*1*τ*2(*n*,*ε*) *ε*−1*cn*

*<sup>α</sup>* = 1

⎤ ⎥ <sup>⎦</sup> <sup>=</sup>

�*α*�

≤ *s*<sup>2</sup> *P<sup>ϑ</sup>* − a.s. (39)

ln *s*<sup>2</sup> *P<sup>ϑ</sup>* − a.s. (40)

<sup>2</sup> (0, *<sup>τ</sup>*2(*n*,*ε*) <sup>−</sup> *<sup>k</sup>*2(*n*)*h*2) <sup>&</sup>gt;11)2<sup>+</sup>

<sup>2</sup> (0, *<sup>τ</sup>*2(*n*,*ε*) <sup>−</sup> *<sup>k</sup>*2(*n*)*h*2) <sup>&</sup>gt;12)2}] = <sup>0</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.,

= 1.

*C*2 *Y* 2*v*<sup>0</sup>

*τ*2(*n*,*ε*) �

*ν*2(*n*,*ε*)

e2*v*1*τ*2(*n*,*ε*) *ε*−1*cn*

�

<sup>−</sup>1*cn*] <sup>≤</sup>

lim*n*∨*<sup>ε</sup>* [*τ*2(*n*,*ε*) <sup>−</sup> *<sup>ν</sup>*2(*n*,*ε*)] = <sup>∞</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s. (41)

<sup>−</sup>1*cn*)(*α*−*α*2(*n*,*ε*)) <sup>≤</sup> lim*n*∨*<sup>ε</sup>* (*<sup>ε</sup>*

*<sup>Z</sup>*˜ <sup>2</sup>(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*dt* <sup>+</sup>

1 e2*<sup>v</sup>*1*τ*2(*n*,*ε*)

(Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>))2*dt* · (*<sup>ε</sup>*

e2*v*1*τ*2(*n*,*ε*) *ε*−1*cn*

ln *s*<sup>1</sup> ≤ lim *n*∨*ε*

�

<sup>−</sup>1*cn*)*α*2(*n*,*ε*)−*α*(< *G*˜ <sup>−</sup><sup>1</sup>

2*v*<sup>1</sup> ln *ε*

*τ*2(*n*,*ε*) �

0

<sup>≤</sup> lim*n*∨*<sup>ε</sup>*

Let *s*<sup>1</sup> and *s*<sup>2</sup> be the positive roots of the following equations

<sup>e</sup>−6*v*<sup>0</sup> · <sup>e</sup>*α*<sup>2</sup> · *<sup>s</sup>*

Thus for *ϑ* ∈ Θ<sup>2</sup>

<sup>e</sup>*α*<sup>1</sup> <sup>≤</sup> lim*n*∨*<sup>ε</sup>* (*<sup>ε</sup>*

*C*2 *Y* 2*v*<sup>0</sup>

respectively. It is clear that 0 < *s*<sup>1</sup> ≤ *s*<sup>2</sup> < ∞.

1 *ε*−1*cn*

By the definition of stopping times *τ*2(*n*,*ε*) we have

*τ*2(*n*,*ε*) �

*ν*2(*n*,*ε*)

⎡ ⎢ ⎣

<sup>=</sup> lim*n*∨*<sup>ε</sup>*

*τ*2(*n*,*ε*) �

0

*s*<sup>1</sup> ≤ lim *n*∨*ε*

From (37) and (40) it follows, in particular, that

<sup>2</sup> (*n*,*ε*)||<sup>2</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> <sup>e</sup>2*v*<sup>0</sup> ){

(*ε*

where < *G* >*ij* is the *ij*-th element of the matrix *G*.

�−(1+*α*)

1 2*v*<sup>1</sup>

<sup>≤</sup> lim*n*∨*<sup>ε</sup>* [*τ*2(*n*,*ε*) <sup>−</sup> <sup>1</sup>

*Cν*<sup>2</sup> · *s* +

⎡ ⎢ ⎣

1 e2*v*0*τ*2(*n*,*ε*)

Then, using (38), for *ϑ* ∈ Θ<sup>2</sup> we have

lim*n*∨*ε*

+

By the definition of *G*˜

[||*G*˜ <sup>−</sup><sup>1</sup>

e2*<sup>v</sup>*1*τ*2(*n*,*ε*) *ε*−1*cn*

lim *n*∨*ε*

+ �

and thus

$$
\sigma\_{21} = \inf\{ n \ge 1 \colon N > \tilde{\varrho}\_{22}^{-1} (\rho\_2 \delta\_2^{-1})^{1/2} \}, \qquad \sigma\_{22} = \inf\{ n \ge 1 \colon N > \tilde{\varrho}\_{21}^{-1} (\rho\_2 \delta\_2^{-1})^{1/2} \}
$$

and from (40) and (43) we obtain the second property of the assertion I in Proposition 3.2:

$$\frac{1}{2v\_1} \ln s\_1 \sigma\_{21} \le \varliminf\_{\varepsilon \to 0} \left[ T\_2(\varepsilon) - \frac{1}{2v\_1} \ln \varepsilon^{-1} \right] \le \varlimsup\_{\varepsilon \to 0} \left[ T\_2(\varepsilon) - \frac{1}{2v\_1} \ln \varepsilon^{-1} \right] \le \frac{1}{2v\_1} \ln s\_2 \sigma\_{22} \quad P\_\theta-\text{a.s.}$$

The assertions I.1 and II of Proposition 3.2 can be proved similar to the proof of the corresponding statement of Proposition 3.1.

Hence Proposition 3.2 is proven.

#### **Proof of Proposition 3.3.**

Similar to the proof of Propositions 3.1, 3.2 and [7]–[16] we get for *ϑ* ∈ Θ<sup>3</sup> the needed asymptotic as *<sup>t</sup>* <sup>→</sup> <sup>∞</sup> relations for the processes <sup>Δ</sup>˜ *<sup>Y</sup>*(*t*), *<sup>Z</sup>*(*t*) and *<sup>Z</sup>*˜(*t*). To this end we introduce the following notation:

$$Z\_1(t) = \int\_{-\infty}^t \ddot{y}\_0(t-s)dW(s), \ \dot{y}\_0(s) = \ddot{x}\_0(s) - \lambda \ddot{x}\_0(s-1),$$

$$Z\_V(t) = \int\_{-\infty}^t [\ddot{\Delta}V(s) - \lambda \ddot{\Delta}V(s-1)]e^{-v\_0(t-s)}ds, \ \ Z\_2(t) = Z\_V(t) + Z\_3(t),$$

$$Z\_3(t) = \int\_{-\infty}^t Z\_1(s)e^{-v\_0(t-s)}ds, \ \ \dot{Z}\_1(t) = Z\_1(t) - 2v\_0Z\_2(t-1),$$

$$Z\_2(t) = Z\_V(t) + Z\_3(t), \ \ Z\_3(t) = \int\_{-\infty}^t Z\_1(s)e^{-v\_0(t-s)}ds,$$

$$\dot{\zeta}\_Z = 1 + \lambda^2 + 4[\lambda - v\_0^{-1}(\lambda - 1)] + E\_\theta \dot{Z}\_1^2(0),$$

$$C\_{\tilde{Z}Z} = 1 + \lambda^2 + 2[\lambda - v\_0^{-1}(\lambda - 1)] + E\_\theta Z\_1^2(0) - E\_\theta Z\_1(0)Z\_3(-1).$$

It should be noted that in the considered case <sup>Θ</sup><sup>3</sup> all the introduced processes *<sup>Z</sup>*1(·),..., *<sup>Z</sup>*˜ <sup>3</sup>(·) are stationary Gaussian processes, continuous in probability, having a spectral density and, as follows, ergodic, see [21].

According to the definition of the set Θ<sup>3</sup> as *t* → ∞ we have:

$$
\tilde{\Delta}Y(t) = \mathbb{C}\_Y e^{v\_0 t} + o(e^{\gamma t}) \ \ P\_\theta-\text{a.s.},
$$

$$
$$

where *CY* and *γ* < *v*<sup>0</sup> are some constants.

Using this properties and the representation for the deviation

$$
\lambda\_t - \lambda = \frac{\int\_0^t Z(s)\tilde{\Delta}Y(s-1)ds}{\int\_0^t (\tilde{\Delta}Y(s-1))^2 ds}
$$

of the estimator *λ<sup>t</sup>* defined in (11), it is easy to obtain with *Pϑ*-probability one the following limit relations:

$$\lim\_{T \to \infty} \frac{1}{e^{2v\_0T}} \int\_0^T \tilde{\Delta}Y(t-u)\tilde{\Delta}Y(t-s)dt = \frac{C\_Y^2}{2v\_0}e^{-v\_0(u+s)}, \ u, s \ge 0,\tag{44}$$

$$\lim\_{T \to \infty} \left| \frac{1}{\mathcal{E}^{\eta\_0 T}} \int\_0^T Z(t)\tilde{\Delta}Y(t-\mu)dt - \mathcal{C}\_Y e^{-\upsilon\_0 \mu} Z\_2(T) \right| = 0, \ \mu \ge 0,\tag{45}$$

$$\lim\_{t \to \infty} |e^{v\_0 t}(\lambda\_t - \lambda) - 2v\_0 e^{v\_0} C\_Y^{-1} Z\_2(t)| = 0,\tag{46}$$

Denote

and, as follows,

*Q* =

lim *<sup>T</sup>*→<sup>∞</sup> *<sup>T</sup>* · *<sup>G</sup>*−<sup>1</sup>

*Z*2(*ν*3(*n*,*ε*))*ε*−1*cn*

lim

*ν*3(*n*,*ε*) and from (49) we find

lim*n*∨*<sup>ε</sup>* ln e2*<sup>α</sup>*3(*n*,*ε*)*ε*−<sup>1</sup>*cn* e2*α*3*ε*−<sup>1</sup>*cn*

= 2*C*−<sup>1</sup> <sup>3</sup> lim*n*∨*<sup>ε</sup>*

enough it follows

where

and then

From (44), (45), (47), (48) with *Pϑ*-probability one holds

 1 1 −*λ* 0

lim*n*∨*ε*

<sup>e</sup>*v*0*ν*3(*n*,*ε*) <sup>=</sup> <sup>2</sup>*C*−<sup>1</sup>

lim*n*∨*ε*

lim*n*∨*<sup>ε</sup>* [*τ*32(*n*,*ε*) <sup>−</sup> *<sup>ε</sup>*

Then, from (50)–(52) with *Pϑ*-probability one we obtain

lim*n*∨*<sup>ε</sup> <sup>G</sup>*˜ <sup>−</sup><sup>1</sup>

*<sup>T</sup>*→<sup>∞</sup> *<sup>ϕ</sup>*−1/2(*T*) · *<sup>G</sup>*3(0, *<sup>T</sup>*) · *<sup>Q</sup>* · *<sup>ϕ</sup>*−1/2(*T*) = diag{*CZZ*˜ ,

*ZZ*˜ ·

Further, by the definition (17) of stopping times *τ*31(*n*,*ε*), first condition in (16) on the function

= *C*˜−<sup>1</sup>

For the investigation of asymptotic properties of stopping times *τ*32(*n*,*ε*) with *Pϑ*-probability

one we show, using the second condition in (16) on the function *ν*3(*n*,*ε*) and (46), that

<sup>3</sup> lim*n*∨*<sup>ε</sup>*

<sup>−</sup>1*cn*] = <sup>1</sup>

2*v*<sup>0</sup>

*<sup>Z</sup>* <sup>∨</sup> <sup>1</sup>}*C*−<sup>1</sup> *ZZ*˜ ·

where *a* ∨ *b* = max(*a*, *b*) and, by the definition (19) of the stopping time *σ*3(*ε*), for *ε* small

*σ*3(*ε*) = *σ*<sup>3</sup> *P<sup>ϑ</sup>* − a.s.,

e2*α*3(*n*,*ε*)*ε*−<sup>1</sup>*cn* e2*v*0*ε*−<sup>1</sup>*cn*

Thus, by the definition (18) of stopping times *τ*32(*n*,*ε*) and from (44) we find

<sup>3</sup> (*n*,*ε*) = {*C*˜

*<sup>σ</sup>*<sup>3</sup> <sup>=</sup> inf{*<sup>n</sup>* <sup>≥</sup> 1 : *<sup>N</sup>* <sup>&</sup>gt; *<sup>g</sup>*−<sup>1</sup>

<sup>3</sup> (0, *<sup>T</sup>*) = *<sup>C</sup>*−<sup>1</sup>

*τ*31(*n*,*ε*) *ε*−1*cn*

<sup>=</sup> lim*n*∨*<sup>ε</sup>* <sup>2</sup>(*α*3(*n*,*ε*) <sup>−</sup> *<sup>α</sup>*)*<sup>ε</sup>*

, *<sup>ϕ</sup>*(*T*) = diag{*T*, e2*v*0*T*}.

On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 45

 1 0 −*λ* 0

*Z*2(*ν*3(*n*,*ε*)) log1/2 *ν*3(*n*,*ε*)

= 1 *P<sup>ϑ</sup>* − a.s.

ln <sup>2</sup>*v*0e6*<sup>v</sup>*<sup>0</sup> *C*2 *Y*

<sup>3</sup> (*ρ*3*δ*−<sup>1</sup>

 1 0 −*λ* 0

<sup>3</sup> )1/2}

 ,

*C*2 *Y* 2*v*<sup>0</sup> }

*<sup>Z</sup> P<sup>ϑ</sup>* − a.s. (51)

log1/2 *ν*3(*n*,*ε*) <sup>e</sup>*v*0*ν*3(*n*,*ε*) · *<sup>ε</sup>*

<sup>−</sup>1*cn*<sup>=</sup> lim*n*∨*<sup>ε</sup>* <sup>2</sup>*λ*−1(*λν*3(*n*,*ε*) <sup>−</sup> *<sup>λ</sup>*)*<sup>ε</sup>*

·

*P<sup>ϑ</sup>* − a.s. (50)

<sup>−</sup>1*cn* =

<sup>−</sup>1*cn* = 0

*P<sup>ϑ</sup>* − a.s. (52)

$$\lim\_{t \to \infty} |\tilde{Z}(t) - (\tilde{\Delta}V(t) - \lambda \tilde{\Delta}V(t-1)) - \tilde{Z}\_1(t)| = 0\_\lambda$$

$$\lim\_{T \to \infty} \left| \frac{1}{e^{\upsilon\_0 T}} \int\_0^T \tilde{Z}(t) \tilde{\Delta} Y(t) dt - \mathbb{C}\_Y \tilde{Z}\_2(T) \right| = 0,\tag{47}$$

$$\lim\_{T \to \infty} \frac{1}{T} \int\_{0}^{T} \tilde{Z}(t)Z(t)dt = \mathbb{C}\_{\mathbb{Z}Z'} \tag{48}$$

$$\lim\_{T \to \infty} \frac{1}{T} \int\_0^T \tilde{Z}^2(t)dt = \tilde{C}\_Z. \tag{49}$$

For the investigation of the asymptotic properties of the components of sequential plan we will use Propositions 2 and 3 from [14]. According to these propositions the processes *Zi*(·), *<sup>Z</sup>*˜*i*(·), *<sup>i</sup>* <sup>=</sup> 1, 3 and *ZV*(·) defined above are *<sup>O</sup>*((log *<sup>t</sup>*) 1 <sup>2</sup> ) as *t* → ∞ *P<sup>ϑ</sup>* − a.s.

Denote

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

It should be noted that in the considered case <sup>Θ</sup><sup>3</sup> all the introduced processes *<sup>Z</sup>*1(·),..., *<sup>Z</sup>*˜ <sup>3</sup>(·) are stationary Gaussian processes, continuous in probability, having a spectral density and, as

*<sup>v</sup>*0*<sup>t</sup>* + *o*(*e*

<sup>|</sup>*Z*(*t*) <sup>−</sup> [Δ˜ *<sup>V</sup>*(*t*) <sup>−</sup> *<sup>λ</sup>*Δ˜ *<sup>V</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>)] <sup>−</sup> *<sup>Z</sup>*1(*t*)<sup>|</sup> <sup>=</sup> *<sup>o</sup>*(1) *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.,

 *t* 0

> *t* 0

<sup>Δ</sup>˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>u</sup>*)Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)*dt* <sup>=</sup> *<sup>C</sup>*<sup>2</sup>

*<sup>Z</sup>*(*t*)Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>u</sup>*)*dt* <sup>−</sup> *CYe*

lim *<sup>t</sup>*→<sup>∞</sup> <sup>|</sup>*<sup>e</sup> v*0*t*

0

lim *T*→∞

1 *T T*

0

1 *T T*

0

lim *T*→∞

For the investigation of the asymptotic properties of the components of sequential plan we will use Propositions 2 and 3 from [14]. According to these propositions the processes

of the estimator *λ<sup>t</sup>* defined in (11), it is easy to obtain with *Pϑ*-probability one the following

*γt*

*<sup>Z</sup>*(*s*)Δ˜ *<sup>Y</sup>*(*<sup>s</sup>* <sup>−</sup> <sup>1</sup>)*ds*

(Δ˜ *<sup>Y</sup>*(*<sup>s</sup>* <sup>−</sup> <sup>1</sup>))2*ds*

(*λ<sup>t</sup>* − *λ*) − 2*v*0*e*

*<sup>t</sup>*→<sup>∞</sup> <sup>|</sup>*Z*˜(*t*) <sup>−</sup> (Δ˜ *<sup>V</sup>*(*t*) <sup>−</sup> *<sup>λ</sup>*Δ˜ *<sup>V</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>)) <sup>−</sup> *<sup>Z</sup>*˜ <sup>1</sup>(*t*)<sup>|</sup> <sup>=</sup> 0,

*<sup>Z</sup>*˜(*t*)Δ˜ *<sup>Y</sup>*(*t*)*dt* <sup>−</sup> *CYZ*˜ <sup>2</sup>(*T*)

*Y* 2*v*<sup>0</sup> *e* −*v*0(*u*+*s*)

<sup>−</sup>*v*0*uZ*2(*T*)

 

*<sup>v</sup>*0*C*−<sup>1</sup>

 

*Z*˜ <sup>2</sup>(*t*)*dt* = *C*˜

1

, *u*,*s* ≥ 0, (44)

<sup>=</sup> 0, *<sup>u</sup>* <sup>≥</sup> 0, (45)

*<sup>Y</sup> Z*2(*t*)| = 0, (46)

<sup>=</sup> 0, (47)

*<sup>Z</sup>*. (49)

*<sup>Z</sup>*˜(*t*)*Z*(*t*)*dt* <sup>=</sup> *CZZ*˜ , (48)

<sup>2</sup> ) as *t* → ∞ *P<sup>ϑ</sup>* − a.s.

) *P<sup>ϑ</sup>* − a.s.,

follows, ergodic, see [21].

limit relations:

According to the definition of the set Θ<sup>3</sup> as *t* → ∞ we have:

Using this properties and the representation for the deviation

where *CY* and *γ* < *v*<sup>0</sup> are some constants.

lim *T*→∞

> lim *T*→∞ 1 *ev*0*<sup>T</sup> T*

1 *e*2*v*0*<sup>T</sup>*  *T*

0

0

lim

lim *T*→∞ 1 *ev*0*<sup>T</sup> T*

*Zi*(·), *<sup>Z</sup>*˜*i*(·), *<sup>i</sup>* <sup>=</sup> 1, 3 and *ZV*(·) defined above are *<sup>O</sup>*((log *<sup>t</sup>*)

Δ˜ *Y*(*t*) = *CYe*

*λ<sup>t</sup>* − *λ* =

$$Q = \begin{pmatrix} 1 & 1 \\ -\lambda & 0 \end{pmatrix}, \qquad \varphi(T) = \text{diag}\{T, \mathbf{e}^{2v\_0 T}\}.$$

From (44), (45), (47), (48) with *Pϑ*-probability one holds

$$\lim\_{T \to \infty} \varrho^{-1/2}(T) \cdot \mathcal{G}\_{\mathfrak{d}}(0, T) \cdot \mathcal{Q} \cdot \varrho^{-1/2}(T) = \text{diag}\{\mathbb{C}\_{\mathbb{Z}Z'} \frac{\mathbb{C}\_Y^2}{2v\_0}\}$$

and, as follows,

$$\lim\_{T \to \infty} \ T \cdot G\_3^{-1}(0, T) = \mathbb{C}\_{ZZ}^{-1} \cdot \begin{pmatrix} 1 & 0 \\ -\lambda & 0 \end{pmatrix} \quad P\_\theta-\text{a.s.}\tag{50}$$

Further, by the definition (17) of stopping times *τ*31(*n*,*ε*), first condition in (16) on the function *ν*3(*n*,*ε*) and from (49) we find

$$\lim\_{n \vee \varepsilon} \frac{\tau\_{31}(n, \varepsilon)}{\varepsilon^{-1} \mathfrak{c}\_n} = \tilde{\mathfrak{C}}\_{\mathbb{Z}}^{-1} \quad P\_{\theta}-\text{a.s.}\tag{51}$$

For the investigation of asymptotic properties of stopping times *τ*32(*n*,*ε*) with *Pϑ*-probability one we show, using the second condition in (16) on the function *ν*3(*n*,*ε*) and (46), that

$$\lim\_{n\to\varepsilon}\ln\frac{\mathfrak{e}^{2\mathfrak{a}\_{3}(\eta,\varepsilon)\varepsilon^{-1}\mathfrak{c}\_{\mathfrak{t}}}}{\mathfrak{e}^{2\mathfrak{a}\_{3}\varepsilon^{-1}\mathfrak{c}\_{\mathfrak{t}}}}=\lim\_{n\to\varepsilon}2(\mathfrak{a}\_{3}(\mathfrak{n},\varepsilon)-\mathfrak{a})\varepsilon^{-1}\mathfrak{c}\_{\mathfrak{t}}=\lim\_{n\to\varepsilon}2\lambda^{-1}(\lambda\_{\upsilon\_{3}(\eta,\varepsilon)}-\lambda)\varepsilon^{-1}\mathfrak{c}\_{\mathfrak{t}}=\\=2\mathfrak{C}\_{3}^{-1}\lim\_{n\to\varepsilon}\frac{Z\_{2}(\upsilon\_{3}(\eta,\varepsilon))}{\log^{1/2}\upsilon\_{3}(\eta,\varepsilon)}\cdot\frac{\log^{1/2}\upsilon\_{3}(\eta,\varepsilon)}{\mathsf{e}^{\mathfrak{p}\_{0}\upsilon\_{3}(\eta,\varepsilon)}}\cdot\varepsilon^{-1}\mathfrak{c}\_{\mathfrak{t}}=0$$

and then

$$\lim\_{n \vee \varepsilon} \frac{\mathbf{e}^{2\iota\_3(n\varepsilon)\varepsilon^{-1}\varepsilon\_n}}{\mathbf{e}^{2\upsilon\_0\varepsilon^{-1}\varepsilon\_n}} = 1 \quad P\_\theta-\text{a.s.}$$

Thus, by the definition (18) of stopping times *τ*32(*n*,*ε*) and from (44) we find

$$\lim\_{n \nearrow \varepsilon} \left[ \tau\_{32}(n, \varepsilon) - \varepsilon^{-1} c\_{\mathfrak{n}} \right] = \frac{1}{2v\_0} \ln \frac{2v\_0 \mathfrak{e}^{\delta v\_0}}{\mathcal{C}\_Y^2} \quad P\_\theta-\text{a.s.} \tag{52}$$

Then, from (50)–(52) with *Pϑ*-probability one we obtain

$$\lim\_{n \vee \varepsilon} \tilde{G}\_3^{-1}(n, \varepsilon) = \{\tilde{C}\_Z \vee 1\} C\_{ZZ}^{-1} \cdot \begin{pmatrix} 1 & 0 \\ -\lambda & 0 \end{pmatrix} \cdot \delta$$

where *a* ∨ *b* = max(*a*, *b*) and, by the definition (19) of the stopping time *σ*3(*ε*), for *ε* small enough it follows

$$
\sigma\_3(\varepsilon) = \overline{\sigma}\_3 \quad P\_\theta-\text{a.s.}
$$

where

$$\overline{\sigma}\_3 = \inf \{ n \ge 1 \, : \, N > \mathcal{g}\_3^{-1} (\rho\_3 \delta\_3^{-1})^{1/2} \} $$

#### 24 Will-be-set-by-IN-TECH 46 Stochastic Modeling and Control On Guaranteed Parameter Estimation of Stochastic Delay Differential Equations by Noisy Observations <sup>25</sup>

and *<sup>g</sup>*<sup>3</sup> <sup>=</sup> {*C*˜ *<sup>Z</sup>* <sup>∨</sup> <sup>1</sup>}2*C*−<sup>2</sup> *ZZ*˜ (<sup>1</sup> <sup>+</sup> *<sup>λ</sup>*2).

Thus we obtain the *Pϑ*-finiteness of the stopping time *T*3(*ε*) and the assertion II.2 of Proposition 3.3:

$$\lim\_{\varepsilon \to 0} \varepsilon T\_{\mathfrak{I}}(\varepsilon) = \{ \tilde{C}\_{Z}^{-1} \vee 1 \} c\_{\overline{\sigma}\_{3}} \quad P\_{\theta}-\text{a.s.}$$

The assertions I.1 and II of Proposition 3.3 can be proved similar to the proofs of Propositions 3.1 and 3.2.

Hence Proposition 3.3 is proven.

#### **Proof of Proposition 3.4.**

This case is a scalar analogue of the case Θ<sup>11</sup> ∪ Θ12.

By the definition,

$$Z^\*(t) = \tilde{X}(t) - \tilde{X}(t-1) + \tilde{\Delta}V(t) - \tilde{\Delta}V(t-1).$$

According to the asymptotic properties of the process (*X*˜(*t*)), for *u* = 0, 2 we have:

– for *ϑ* ∈ Θ<sup>41</sup> :

exist the positive constant limits

$$\lim\_{T \to \infty} \frac{1}{T} \int\_{0}^{T} Z^\*(t) Z^\*(t - u) dt = \mathbb{C}\_{41}^\*(u) \text{ } P\_\theta-\text{a.s.};\tag{53}$$

lim*n*∨*ε*

lim*n*∨*<sup>ε</sup>* [*τ*4(*n*,*ε*) <sup>−</sup> <sup>1</sup>

*<sup>σ</sup>*<sup>41</sup> <sup>=</sup> inf{*<sup>n</sup>* <sup>≥</sup> 1 : *<sup>N</sup>* <sup>&</sup>gt; (*ρ*4*δ*−<sup>1</sup>

41(2)(*C*<sup>∗</sup>

with *Pϑ*-probability one for *ε* small enough:

3.4, which follows from (55), (57) and (58):

<sup>41</sup>(0))−<sup>1</sup> <sup>≤</sup> lim

1 2*v*<sup>0</sup>

<sup>≤</sup> lim*n*∨*<sup>ε</sup>* [*T*4(*ε*) <sup>−</sup> <sup>1</sup>

*n*∨*ε*

ln *<sup>c</sup>σ*<sup>41</sup> <sup>2</sup>*v*<sup>0</sup> *C*∗ <sup>42</sup>(0)

> 2*v*<sup>0</sup> ln *ε* <sup>−</sup>1] <sup>≤</sup>

and assertion 3 of Theorem 3.1 follow from Propositions 3.1-3.4 directly.

(Δ˜ *<sup>Y</sup>*(*t*))2*dt* <sup>=</sup> <sup>∞</sup> and <sup>∞</sup>

*cσ*<sup>41</sup> (*C*<sup>∗</sup>

Hence Proposition 3.4 is valid.

noted, that the integrals

[3], [7]–[16]).

∞

0

Denote

where *g*<sup>41</sup> = [*C*<sup>∗</sup>

– for *ϑ* ∈ Θ<sup>41</sup> :

– for *ϑ* ∈ Θ<sup>42</sup> :

e2*v*0*τ*4(*n*,*ε*) *ε*−1*cn*

> 2*v*<sup>0</sup> ln *ε*

As follows, the stopping times *τ*4(*n*,*ε*) are *P<sup>ϑ</sup>* − a.s. finite for all *ϑ* ∈ Θ4.

<sup>4</sup> )1/2*<sup>g</sup>*

−1

<sup>41</sup>(0))−<sup>1</sup> <sup>∨</sup> <sup>2</sup>*v*0e−2*v*<sup>0</sup> ], *<sup>g</sup>*<sup>42</sup> = [*C*<sup>∗</sup>

*a* ∧ *b* = min(*a*, *b*) and by the definition of the stopping time *σ*4(*ε*) (21) as well as from (53)–(56) follows the *P<sup>ϑ</sup>* − a.s. finiteness of *σ*4(*ε*) and the following inequalities, which hold

Then we obtain the finiteness of the stopping time *T*4(*ε*) and the assertion I.2 of Proposition

*<sup>ε</sup>T*4(*ε*) <sup>≤</sup> lim*n*∨*<sup>ε</sup> <sup>ε</sup>T*4(*ε*) <sup>≤</sup> *<sup>c</sup>σ*<sup>42</sup> (*C*<sup>∗</sup>

≤ lim *n*∨*ε*

> 1 2*v*<sup>0</sup>

**Proof of Theorem 3.1.** The closeness of the sequential estimation plan SEP�(*ε*) (assertion 1)

Now we prove the assertion 2. To this end we show first, that all the stopping times *τi*(*n*,*ε*), *i* = 1, 2, 4 and *τ*3*i*(*n*,*ε*), *i* = 1, 2 are *P<sup>ϑ</sup>* − a.s.-finite for every *ϑ* ∈ Θ. It should be

0

in all the cases Θ1,..., Θ<sup>4</sup> and, as follows, all the stopping times *τi*(*n*,*ε*), *i* = 1, 2, 4 and *τ*32(*n*,*ε*) are *Pϑ*-a.s.-finite for every *ε* > 0 and all *n* ≥ 1. The properties (59) can be established by using the asymptotic properties of the process (*X*(*t*),*Y*(*t*)) (see proofs of Propositions 3.1–3.4 and

The stopping times *τ*31(*n*,*ε*) are finite in the region Θ<sup>2</sup> ∪ Θ<sup>3</sup> according to Propositions 3.2, 3.3. As follows, it remind only to verify the finiteness of the stopping time *τ*31(*n*,*ε*) for *ϑ* ∈ Θ<sup>1</sup> ∪ Θ4.

[*T*4(*ε*) <sup>−</sup> <sup>1</sup>

ln *<sup>c</sup>σ*<sup>42</sup> <sup>2</sup>*v*<sup>0</sup> *C*∗ <sup>42</sup>(0)

2*v*<sup>0</sup> ln *ε* <sup>−</sup>1] <sup>≤</sup>

<sup>2</sup>

<sup>2</sup>

= 2*v*<sup>0</sup> *C*∗ 42 <sup>2</sup>

On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 47

*v*0

ln <sup>2</sup>*v*<sup>0</sup> *C*∗ 42

<sup>41</sup> }, *<sup>σ</sup>*<sup>42</sup> <sup>=</sup> inf{*<sup>n</sup>* <sup>≥</sup> 1 : *<sup>N</sup>* <sup>&</sup>gt; (*ρ*4*δ*−<sup>1</sup>

41(2)(*C*<sup>∗</sup>

*σ*<sup>41</sup> ≤ *σ*4(*ε*) ≤ *σ*<sup>42</sup> *P<sup>ϑ</sup>* − a.s. (58)

<sup>41</sup>(0))−<sup>1</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.;

*P<sup>ϑ</sup>* − a.s.

(*Z*∗(*t*))2*dt* <sup>=</sup> <sup>∞</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s. (59)

<sup>−</sup>1*cn*] = <sup>1</sup>

*P<sup>ϑ</sup>* − a.s., (56)

*P<sup>ϑ</sup>* − a.s. (57)

<sup>4</sup> )1/2*<sup>g</sup>*

<sup>41</sup>(0))−<sup>1</sup> <sup>∧</sup> <sup>2</sup>*v*0e−2*v*<sup>0</sup> ], where

−1 <sup>42</sup> },

– for *ϑ* ∈ Θ<sup>42</sup> :

$$\mathbf{e}^{-\upsilon\_0 t} Z^\*(t) = \mathbf{C}\_{42}^\* + o(\mathbf{e}^{-(\upsilon\_0 - \gamma)t}) \text{ as } t \to \infty \quad P\_\vartheta-\text{a.s.}$$

$$\lim\_{T \to \infty} \frac{1}{\mathbf{e}^{2\upsilon\_0 T}} \int\_0^T Z^\*(t) Z^\*(t - u) dt = \frac{(\mathbf{C}\_{42}^\*)^2 \mathbf{e}^{-\upsilon\_0 u}}{2\upsilon\_0} \quad P\_\vartheta-\text{a.s.} \tag{54}$$

where

$$\mathbf{C}\_{42}^{\*} = \frac{1 - \mathbf{e}^{v\_0}}{v\_0(v\_0 - a + 1)}.$$

Assertions I.1 and II of Proposition 3.4 can be proved similar to Proposition 3.1.

Now we prove the closeness of the plan (22) and assertion I.2 of Proposition 3.4. To this end we shall investigate the asymptotic properties of the stopping times *τ*4(*n*,*ε*) and *σ*4(*ε*).

From the definition of *τ*4(*n*,*ε*) and (53), (54) we have

– for *ϑ* ∈ Θ<sup>41</sup> :

$$\lim\_{n \vee \varepsilon} \frac{\tau\_4(n, \varepsilon)}{\varepsilon^{-1} \varepsilon\_n} = (\mathbb{C}\_{41}^\*(0))^{-1} \ P\_\vartheta - \text{a.s.} \tag{55}$$

– for *ϑ* ∈ Θ<sup>42</sup> :

46 Stochastic Modeling and Control On Guaranteed Parameter Estimation of Stochastic Delay Differential Equations by Noisy Observations <sup>25</sup> On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 47

$$\lim\_{n \vee \varepsilon} \frac{\mathbf{e}^{2v\_0 \tau\_4(n,\varepsilon)}}{\varepsilon^{-1} \mathfrak{c}\_n} = \left(\frac{2v\_0}{\mathbb{C}\_{42}^\*}\right)^2 \quad P\_\theta-\text{a.s.}\tag{56}$$

$$\lim\_{n \ne \varepsilon} \left[ \pi\_4(n, \varepsilon) - \frac{1}{2v\_0} \ln \varepsilon^{-1} c\_n \right] = \frac{1}{v\_0} \ln \frac{2v\_0}{C\_{42}^\*} \quad P\_\theta-\text{a.s.}\tag{57}$$

As follows, the stopping times *τ*4(*n*,*ε*) are *P<sup>ϑ</sup>* − a.s. finite for all *ϑ* ∈ Θ4.

Denote

24 Will-be-set-by-IN-TECH

Thus we obtain the *Pϑ*-finiteness of the stopping time *T*3(*ε*) and the assertion II.2 of Proposition

The assertions I.1 and II of Proposition 3.3 can be proved similar to the proofs of Propositions

*<sup>Z</sup>*∗(*t*) = *<sup>X</sup>*˜(*t*) <sup>−</sup> *<sup>X</sup>*˜(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>) + <sup>Δ</sup>˜ *<sup>V</sup>*(*t*) <sup>−</sup> <sup>Δ</sup>˜ *<sup>V</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>).

*Z*∗(*t*)*Z*∗(*t* − *u*)*dt* = *C*<sup>∗</sup>

<sup>42</sup> <sup>+</sup> *<sup>o</sup>*(e−(*v*0−*γ*)*<sup>t</sup>*

*<sup>Z</sup>*∗(*t*)*Z*∗(*<sup>t</sup>* <sup>−</sup> *<sup>u</sup>*)*dt* <sup>=</sup> (*C*<sup>∗</sup>

<sup>42</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>e</sup>*v*<sup>0</sup>

Now we prove the closeness of the plan (22) and assertion I.2 of Proposition 3.4. To this end we shall investigate the asymptotic properties of the stopping times *τ*4(*n*,*ε*) and *σ*4(*ε*).

= (*C*∗

*v*0(*v*<sup>0</sup> − *a* + 1)

According to the asymptotic properties of the process (*X*˜(*t*)), for *u* = 0, 2 we have:

*<sup>Z</sup>* ∨ 1}*cσ*<sup>3</sup> *P<sup>ϑ</sup>* − a.s.

<sup>41</sup>(*u*) *P<sup>ϑ</sup>* − a.s.; (53)

*P<sup>ϑ</sup>* − a.s., (54)

)) as *t* → ∞ *P<sup>ϑ</sup>* − a.s.,

<sup>41</sup>(0))−<sup>1</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.; (55)

42)2e−*v*0*<sup>u</sup>* 2*v*<sup>0</sup>

.

*<sup>ε</sup>T*3(*ε*) = {*C*˜−<sup>1</sup>

and *<sup>g</sup>*<sup>3</sup> <sup>=</sup> {*C*˜

3.1 and 3.2.

3.3:

*<sup>Z</sup>* <sup>∨</sup> <sup>1</sup>}2*C*−<sup>2</sup>

Hence Proposition 3.3 is proven.

exist the positive constant limits

**Proof of Proposition 3.4.**

By the definition,

– for *ϑ* ∈ Θ<sup>41</sup> :

– for *ϑ* ∈ Θ<sup>42</sup> :

where

– for *ϑ* ∈ Θ<sup>41</sup> :

– for *ϑ* ∈ Θ<sup>42</sup> :

*ZZ*˜ (<sup>1</sup> <sup>+</sup> *<sup>λ</sup>*2).

This case is a scalar analogue of the case Θ<sup>11</sup> ∪ Θ12.

lim *T*→∞

e−*v*0*<sup>t</sup>*

lim *T*→∞ 1 *T T*

0

*Z*∗(*t*) = *C*∗

 *T*

0

*C*∗

*τ*4(*n*,*ε*) *ε*−1*cn*

Assertions I.1 and II of Proposition 3.4 can be proved similar to Proposition 3.1.

1 e2*v*0*<sup>T</sup>*

From the definition of *τ*4(*n*,*ε*) and (53), (54) we have

lim*n*∨*ε*

lim *ε*→0

$$
\sigma\_{41} = \inf \{ n \ge 1 \colon N > (\rho\_4 \delta\_4^{-1})^{1/2} \mathfrak{g}\_{41}^{-1} \}, \qquad \sigma\_{42} = \inf \{ n \ge 1 \colon N > (\rho\_4 \delta\_4^{-1})^{1/2} \mathfrak{g}\_{42}^{-1} \},
$$

where *g*<sup>41</sup> = [*C*<sup>∗</sup> 41(2)(*C*<sup>∗</sup> <sup>41</sup>(0))−<sup>1</sup> <sup>∨</sup> <sup>2</sup>*v*0e−2*v*<sup>0</sup> ], *<sup>g</sup>*<sup>42</sup> = [*C*<sup>∗</sup> 41(2)(*C*<sup>∗</sup> <sup>41</sup>(0))−<sup>1</sup> <sup>∧</sup> <sup>2</sup>*v*0e−2*v*<sup>0</sup> ], where *a* ∧ *b* = min(*a*, *b*) and by the definition of the stopping time *σ*4(*ε*) (21) as well as from (53)–(56) follows the *P<sup>ϑ</sup>* − a.s. finiteness of *σ*4(*ε*) and the following inequalities, which hold with *Pϑ*-probability one for *ε* small enough:

$$
\sigma\_{41} \le \sigma\_4(\varepsilon) \le \sigma\_{42} \quad P\_\theta-\text{a.s.}\tag{58}
$$

Then we obtain the finiteness of the stopping time *T*4(*ε*) and the assertion I.2 of Proposition 3.4, which follows from (55), (57) and (58):

 $\text{--for } \theta \in \Theta\_{41}:$ 
$$c\_{\sigma\_{41}}(\mathbb{C}\_{41}^\*(0))^{-1} \le \varliminf\_{\overline{n \vee \varepsilon}} \varepsilon T\_4(\varepsilon) \le \overline{\lim}\_{n \vee \varepsilon} \varepsilon T\_4(\varepsilon) \le c\_{\sigma\_{42}}(\mathbb{C}\_{41}^\*(0))^{-1} \text{ } P\_{\theta}-\text{a.s.} $$

– for *ϑ* ∈ Θ<sup>42</sup> :

$$\frac{1}{2v\_0} \ln c\_{\sigma\_{41}} \left( \frac{2v\_0}{C\_{42}^\*(0)} \right)^2 \le \varliminf\_{n \vee \varepsilon} \left[ T\_4(\varepsilon) - \frac{1}{2v\_0} \ln \varepsilon^{-1} \right] \le$$

$$\le \varliminf\_{n \vee \varepsilon} \left[ T\_4(\varepsilon) - \frac{1}{2v\_0} \ln \varepsilon^{-1} \right] \le \frac{1}{2v\_0} \ln c\_{\sigma\_{42}} \left( \frac{2v\_0}{C\_{42}^\*(0)} \right)^2 \quad P\_\theta-\text{a.s.}$$

Hence Proposition 3.4 is valid.

**Proof of Theorem 3.1.** The closeness of the sequential estimation plan SEP�(*ε*) (assertion 1) and assertion 3 of Theorem 3.1 follow from Propositions 3.1-3.4 directly.

Now we prove the assertion 2. To this end we show first, that all the stopping times *τi*(*n*,*ε*), *i* = 1, 2, 4 and *τ*3*i*(*n*,*ε*), *i* = 1, 2 are *P<sup>ϑ</sup>* − a.s.-finite for every *ϑ* ∈ Θ. It should be noted, that the integrals

$$\int\_0^\infty (\tilde{\Delta}Y(t))^2 dt = \infty \quad \text{and} \quad \int\_0^\infty (Z^\*(t))^2 dt = \infty \quad P\_\theta-\text{a.s.}\tag{59}$$

in all the cases Θ1,..., Θ<sup>4</sup> and, as follows, all the stopping times *τi*(*n*,*ε*), *i* = 1, 2, 4 and *τ*32(*n*,*ε*) are *Pϑ*-a.s.-finite for every *ε* > 0 and all *n* ≥ 1. The properties (59) can be established by using the asymptotic properties of the process (*X*(*t*),*Y*(*t*)) (see proofs of Propositions 3.1–3.4 and [3], [7]–[16]).

The stopping times *τ*31(*n*,*ε*) are finite in the region Θ<sup>2</sup> ∪ Θ<sup>3</sup> according to Propositions 3.2, 3.3. As follows, it remind only to verify the finiteness of the stopping time *τ*31(*n*,*ε*) for *ϑ* ∈ Θ<sup>1</sup> ∪ Θ4. According to the definition (17) of these stopping times it is enough to show the divergence of the following integral

Due to the obtained finiteness properties of all the stopping times in these sums all the

This chapter presents a sequential approach to the guaranteed parameter estimation problem of a linear stochastic continuous-time system. We consider a concrete stochastic delay

At the same time for the construction of the sequential estimation plans we used mainly the structure and the asymptotic behaviour of the solution of the system. Analogously, the presented method can be used for the guaranteed accuracy parameter estimation problem of the linear ordinary and delay stochastic differential equations of an arbitrary order with and

The obtained estimation procedure can be easily generalized, similar to [9, 11, 13, 16], to estimate the unknown parameters with preassigned accuracy in the sense of the *Lq*-norm (*q* ≥ 2). The estimators with such properties may be used in various adaptive procedures

[1] Arato, M. [1982]. *Linear Stochastic Systems with Constant Coefficients. A Statistical Approach*,

[2] Galtchouk, L. & Konev, V. [2001]. On sequential estimation of parameters in semimartingale regression models with continuous time parameter, *The Annals of*

[3] Gushchin, A. A. & Küchler, U. [1999]. Asymptotic inference for a linear stochastic

[4] Kallianpur, G. [1987]. *Stochastic Filtering Theory*, Springer Verlag, New York, Heidelberg,

[5] Kolmanovskii, V. & A. Myshkis. [1992]. *Applied Theory of Functional Differential Equations*,

differential equation with time delay, *Bernoulli* Vol. (5), 6: 1059–1098.

differential equation driven by an additive Wiener process with noisy observations.

1 *cn* sup *ϑ*∈Θ

On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 49

1 *cn* = *ε* 4 ∑ *j*=1

*<sup>E</sup>ϑ*|| ˜

*<sup>ζ</sup>j*(*n*,*ε*)||<sup>2</sup> <sup>≤</sup>

*δ<sup>j</sup>* = *ε*.

4 ∑ *j*=1 *ρ*−<sup>1</sup> *<sup>j</sup> <sup>δ</sup><sup>j</sup>* ∑ *n*≥1

4 ∑ *j*=1 *ρ*−<sup>1</sup> *<sup>j</sup> <sup>δ</sup>jμ<sup>j</sup>* ∑ *n*≥1

mathematical expectations are well-defined and we can estimate finally

*<sup>E</sup>ϑ*||*ϑ*(*ε*) <sup>−</sup> *<sup>ϑ</sup>*||<sup>2</sup> <sup>≤</sup> *<sup>ε</sup>*

<sup>≤</sup> <sup>15</sup>(<sup>3</sup> <sup>+</sup> *<sup>R</sup>*2)*<sup>ε</sup>*

sup *ϑ*∈Θ

Hence Theorem 3.1 is proven.

without noises in observations.

(control, prediction, filtration).

*Humboldt University Berlin, Germany*

*Statistics* Vol. (29), 5: 1508–1536.

Springer Verlag, Berlin, Heidelberg, New York.

**Author details**

Vyacheslav A. Vasiliev *Tomsk State University, Russia*

**6. References**

Berlin.

Kluwer Acad. Pabl.

Uwe Küchler

**5. Conclusion**

$$\int\_0^\infty \tilde{Z}^2(t)dt = \infty \quad P\_\theta-\text{a.s.}$$

where *<sup>Z</sup>*˜(*t*) = <sup>Δ</sup>˜ *<sup>Y</sup>*(*t*) <sup>−</sup> *<sup>λ</sup>t*Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>).

This property follows from the following facts:

– for *ϑ* ∈ Θ<sup>11</sup>

$$\lim\_{t \to \infty} \lambda\_t = \tilde{\lambda}\_{\prime} \quad P\_{\theta}-\text{a.s.}$$

where *λ*˜ is some constant and the process *Z*˜(*t*) can be approximated, similar to the case Θ<sup>3</sup> (see the proof of Proposition 3.3) by a Gaussian stationary process;

$$- \text{ for } \theta \in \Theta\_{12}$$

$$\lim\_{t \to \infty} |\lambda\_t - C\_1(t)| = 0 \quad P\_\theta-\text{a.s.}$$

and then

$$\lim\_{t \to \infty} \left| \mathbf{e}^{-v\_0 t} \tilde{Z}(t) - \mathbf{C}\_2(t) \right| = 0 \quad P\_\theta-\text{a.s.}$$

where *C*1(*t*) and *C*2(*t*) are some periodic bounded functions;

– for Θ<sup>13</sup>

$$\lim\_{t \to \infty} t(\lambda\_t - \lambda) = \mathbb{C}\_{\mathbb{S}} \quad P\_{\theta}-\text{a.s.}$$

and

$$\lim\_{t \to \infty} \mathbf{e}^{-v\_0 t} \tilde{Z}(t) = \mathbf{C}\_4 \quad P\_\theta-\text{a.s.}$$

where *C*<sup>3</sup> and *C*<sup>4</sup> are some non-zero constants;

– for Θ<sup>41</sup>

$$\lim\_{t \to \infty} \left| \tilde{Z}(t) - \frac{1-\lambda}{1-a} \left( X\_0(0) + b \int\_{-1}^0 X\_0(s) ds \right) - \lambda \right| $$

$$-\frac{1}{1-a} (\mathcal{W}(t) - \lambda \mathcal{W}(t-1)) - (\tilde{\Delta}V(t) - \lambda \tilde{\Delta}V(t-1)) \right| = 0 \quad P\_\theta-\text{a.s.} $$

– for Θ<sup>42</sup>

$$\lim\_{t \to \infty} \text{ e }^{-\upsilon\_0 t} \tilde{Z}(t) = \mathbb{C}\_5 \quad P\_\theta-\text{a.s.}$$

where *C*<sup>5</sup> is some non-zero constant.

Denote *μ*<sup>1</sup> = *μ*<sup>2</sup> = 1, *μ*<sup>3</sup> = *μ*<sup>4</sup> = 2/5.

Now we can verify the second property of the estimator *ϑ*(*ε*). By the definition of stopping times *σj*(*ε*), *j* = 1, 4, we get

$$\begin{split} \sup\_{\theta \in \overline{\Theta}} E\_{\theta} ||\theta(\varepsilon) - \theta||^{2} &\leq \varepsilon \sup\_{\theta \in \overline{\Theta}} E\_{\theta} \rho\_{j^{\*}}^{-1} \delta\_{j^{\*}} \sum\_{n=1}^{\sigma\_{j^{\*}}(\varepsilon)} \frac{1}{c\_{n}} \beta\_{j^{\*}}^{2} (n, \varepsilon) \cdot ||\tilde{\mathcal{G}}\_{j^{\*}}^{-1} (n, \varepsilon)||^{2} \cdot ||\tilde{\zeta}\_{j^{\*}} (n, \varepsilon)||^{2} \leq \varepsilon \\ &\leq \varepsilon \sup\_{\theta \in \overline{\Theta}} E\_{\theta} \rho\_{j^{\*}}^{-1} \delta\_{j^{\*}} \sum\_{n\geq 1} \frac{1}{c\_{n}} ||\tilde{\zeta}\_{j^{\*}} (n, \varepsilon)||^{2} .\end{split}$$

Due to the obtained finiteness properties of all the stopping times in these sums all the mathematical expectations are well-defined and we can estimate finally

$$\sup\_{\theta \in \overline{\Theta}} E\_{\theta} ||\theta(\varepsilon) - \theta||^2 \le \varepsilon \sum\_{j=1}^4 \rho\_j^{-1} \delta\_j \sum\_{n \ge 1} \frac{1}{c\_n} \sup\_{\theta \in \overline{\Theta}} E\_{\theta} ||\tilde{\zeta}\_j(n, \varepsilon)||^2 \le \varepsilon$$

$$\le 15(3 + R^2)\varepsilon \sum\_{j=1}^4 \rho\_j^{-1} \delta\_j \mu\_j \sum\_{n \ge 1} \frac{1}{c\_n} = \varepsilon \sum\_{j=1}^4 \delta\_j = \varepsilon.$$

Hence Theorem 3.1 is proven.

## **5. Conclusion**

26 Will-be-set-by-IN-TECH

According to the definition (17) of these stopping times it is enough to show the divergence of

*<sup>t</sup>*→<sup>∞</sup> *<sup>λ</sup><sup>t</sup>* <sup>=</sup> *<sup>λ</sup>*˜ , *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s., where *λ*˜ is some constant and the process *Z*˜(*t*) can be approximated, similar to the case Θ<sup>3</sup>

*<sup>t</sup>*→<sup>∞</sup> <sup>|</sup>*λ<sup>t</sup>* <sup>−</sup> *<sup>C</sup>*1(*t*)<sup>|</sup> <sup>=</sup> <sup>0</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.,

*<sup>t</sup>*→<sup>∞</sup> *<sup>t</sup>*(*λ<sup>t</sup>* <sup>−</sup> *<sup>λ</sup>*) = *<sup>C</sup>*<sup>3</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.

⎛

(*W*(*t*) <sup>−</sup> *<sup>λ</sup>W*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>)) <sup>−</sup> (Δ˜ *<sup>V</sup>*(*t*) <sup>−</sup> *<sup>λ</sup>*Δ˜ *<sup>V</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>))

⎝*X*0(0) + *b*

*<sup>Z</sup>*˜(*t*) <sup>−</sup> *<sup>C</sup>*2(*t*)<sup>|</sup> <sup>=</sup> <sup>0</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.,

*<sup>Z</sup>*˜(*t*) = *<sup>C</sup>*<sup>4</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.,

*<sup>Z</sup>*˜(*t*) = *<sup>C</sup>*<sup>5</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.,

Now we can verify the second property of the estimator *ϑ*(*ε*). By the definition of stopping

1 *cn β*2

> 1 *cn* || ˜

*σj*<sup>∗</sup> (*ε*) ∑ *n*=1

*<sup>j</sup> <sup>δ</sup><sup>j</sup>* ∑ *n*≥1 � 0

*X*0(*s*)*ds*

⎞ ⎠ −

� � �

� <sup>=</sup> <sup>0</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.;

*<sup>j</sup>*<sup>∗</sup> (*n*,*ε*)||<sup>2</sup> · || ˜

*<sup>ζ</sup><sup>j</sup>* (*n*,*ε*)||<sup>2</sup> <sup>≤</sup>

−1

*<sup>j</sup>*<sup>∗</sup> (*n*,*ε*) · ||*G*˜ <sup>−</sup><sup>1</sup>

*<sup>ζ</sup><sup>j</sup>* (*n*,*ε*)||2.

*<sup>Z</sup>*˜ <sup>2</sup>(*t*)*dt* <sup>=</sup> <sup>∞</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> a.s.,

� ∞ 0

lim

(see the proof of Proposition 3.3) by a Gaussian stationary process;

lim

lim *<sup>t</sup>*→<sup>∞</sup> <sup>|</sup>e−*v*0*<sup>t</sup>*

where *C*<sup>3</sup> and *C*<sup>4</sup> are some non-zero constants;

<sup>−</sup> <sup>1</sup> 1 − *a*

times *σj*(*ε*), *j* = 1, 4, we get

sup *ϑ*∈Θ

where *C*<sup>5</sup> is some non-zero constant. Denote *μ*<sup>1</sup> = *μ*<sup>2</sup> = 1, *μ*<sup>3</sup> = *μ*<sup>4</sup> = 2/5.

*<sup>E</sup>ϑ*||*ϑ*(*ε*) <sup>−</sup> *<sup>ϑ</sup>*||<sup>2</sup> <sup>≤</sup> *<sup>ε</sup>* sup

lim *t*→∞ � � � � � �

where *C*1(*t*) and *C*2(*t*) are some periodic bounded functions;

lim

lim *<sup>t</sup>*→<sup>∞</sup> <sup>e</sup>−*v*0*<sup>t</sup>*

*<sup>Z</sup>*˜(*t*) <sup>−</sup> <sup>1</sup> <sup>−</sup> *<sup>λ</sup>* 1 − *a*

> lim *<sup>t</sup>*→<sup>∞</sup> <sup>e</sup>−*v*0*<sup>t</sup>*

*ϑ*∈Θ

≤ *ε* sup *ϑ*∈Θ

*<sup>E</sup>ϑρ*−<sup>1</sup> *<sup>j</sup> δ<sup>j</sup>*

*<sup>E</sup>ϑρ*−<sup>1</sup>

the following integral

– for *ϑ* ∈ Θ<sup>11</sup>

– for *ϑ* ∈ Θ<sup>12</sup>

and then

– for Θ<sup>13</sup>

– for Θ<sup>41</sup>

– for Θ<sup>42</sup>

and

where *<sup>Z</sup>*˜(*t*) = <sup>Δ</sup>˜ *<sup>Y</sup>*(*t*) <sup>−</sup> *<sup>λ</sup>t*Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>).

This property follows from the following facts:

This chapter presents a sequential approach to the guaranteed parameter estimation problem of a linear stochastic continuous-time system. We consider a concrete stochastic delay differential equation driven by an additive Wiener process with noisy observations.

At the same time for the construction of the sequential estimation plans we used mainly the structure and the asymptotic behaviour of the solution of the system. Analogously, the presented method can be used for the guaranteed accuracy parameter estimation problem of the linear ordinary and delay stochastic differential equations of an arbitrary order with and without noises in observations.

The obtained estimation procedure can be easily generalized, similar to [9, 11, 13, 16], to estimate the unknown parameters with preassigned accuracy in the sense of the *Lq*-norm (*q* ≥ 2). The estimators with such properties may be used in various adaptive procedures (control, prediction, filtration).

## **Author details**

Uwe Küchler *Humboldt University Berlin, Germany*

Vyacheslav A. Vasiliev *Tomsk State University, Russia*

## **6. References**


**Coherent Upper Conditional Previsions Defined by**

**Chapter 3**

Coherent conditional previsions and probabilities are tools to model and quantify uncertainties; they have been investigated in de Finetti [3], [4], Dubins [10] Regazzini [13], [14] and Williams [20]. Separately coherent upper and lower conditional previsions have been introduced in Walley [18], [19] and models of upper and lower conditional previsions

In the subjective probabilistic approach coherent probability is defined on an arbitrary class of sets and any coherent probability can be extended to a larger domain. So in this framework no measurability condition is required for random variables. In the sequel, bounded random variables are bounded real-valued functions (these functions are called *gambles* in Walley [19] or *random quantities* in de Finetti [3]). When a measurability condition for a random variable is required, for example to define the Choquet integral, it is explicitly mentioned through the

Separately coherent upper conditional previsions are functionals on a linear space of bounded random variables satisfying the axioms of separate coherence. They cannot always be defined as an extension of conditional expectation of measurable random variables defined by the Radon-Nikodym derivative, according to the axiomatic definition. It occurs because one of the defining properties of the Radon-Nikodym derivative, that is to be measurable with respect to the *σ*-field of the conditioning events, contradicts a necessary condition for coherence (see

So the necessity to find a new mathematical tool in order to define coherent upper conditional

In Doria [8], [9] a new model of coherent upper conditional prevision is proposed in a metric space. It is defined by the Choquet integral with respect to the *s*-dimensional Hausdorff

> ©2012 Doria, 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 Doria, 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.

have been analysed in Vicig et al. [17] and Miranda and Zaffalon [12].

**Hausdorff Outer Measures to Forecast in Chaotic**

**Dynamical Systems**

http://dx.doi.org/10.5772/50326

Doria [9, Theorem 1], Seidenfeld [16]).

Additional information is available at the end of the chapter

Serena Doria

**1. Introduction**

paper.

previsions arises.

