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

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

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

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

$$\sum\_{n\geq 1} \frac{1}{c\_n} < \infty. \tag{7}$$

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

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

$$\Psi\_s(t) = \begin{cases} (\tilde{\Delta}Y(t), \tilde{\Delta}Y(t-s))' & \text{for} \quad t \ge 1+s, \\ (0,0)' & \text{for} \quad t < 1+s; \end{cases}$$

– the sequence of stopping times

– the functions

$$\tau\_1(n,\varepsilon) = h\_1 \inf\{k \ge 1 : \int\_0^{kh\_1} ||\Psi\_{h\_1}(t-2-5h\_1)||^2 dt \ge \varepsilon^{-1} c\_n\} \quad \text{for} \quad n \ge 1;$$

#### 6 Will-be-set-by-IN-TECH 28 Stochastic Modeling and Control On Guaranteed Parameter Estimation of Stochastic Delay Differential Equations by Noisy Observations <sup>7</sup>

– the matrices

$$\mathcal{G}\_1(T,s) = \int\_0^T \Psi\_s(t-2-5s)\Psi\_1'(t)dt,\quad \Phi\_1(T,s) = \int\_0^T \Psi\_s(t-2-5s)d\tilde{\Delta}Y(t),$$

$$\mathcal{G}\_1(n,k,\varepsilon) = \mathcal{G}\_1(\tau\_1(n,\varepsilon)-kh\_1,h\_1),\ \Phi\_1(n,k,\varepsilon) = \Phi\_1(\tau\_1(n,\varepsilon)-kh\_1,h\_1);$$
 $\varepsilon-\text{the times}$ 

for every *s* ≥ 0 have some periodic matrix functions as a limit on *T* almost surely. These limit matrix functions are finite and may be infinite on the norm only for four values of their argument *T* on every interval of periodicity of the length Δ > 1 (see the proof of Theorem 3.2

On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 29

of similar expressions we shall use, similar to [12, 14, 16], the unifying notation lim*n*∨*<sup>ε</sup> <sup>a</sup>*(*n*,*ε*) for

We state the results concerning the estimation of the parameter *ϑ* ∈ Θ<sup>1</sup> in the following

**Proposition 3.1.** *Assume that the condition* (7) *on the sequence* (*cn*) *holds and let the parameter*

*I. For any ε* > 0 *and every ϑ* ∈ Θ<sup>1</sup> *the sequential plan* (*T*1(*ε*), *ϑ*1(*ε*)) *defined by* (10) *is closed*

*<sup>E</sup>ϑ*||*ϑ*1(*ε*) <sup>−</sup> *<sup>ϑ</sup>*||<sup>2</sup> <sup>≤</sup> *<sup>δ</sup>*1*ε*;

[*T*1(*ε*) <sup>−</sup> <sup>1</sup>

[*T*1(*ε*) + <sup>1</sup>

*ε* · *T*1(*ε*) < ∞ *P<sup>ϑ</sup>* − *a.s.*,

2*v*<sup>0</sup> ln *ε*

*v*0

ln *T*1(*ε*) − Ψ��

*a*(*n*,*ε*) will be used. To avoid repetitions

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

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

in [10, 12]).

proposition.

*– for ϑ* ∈ Θ<sup>11</sup>

*– for ϑ* ∈ Θ<sup>12</sup>

*– for ϑ* ∈ Θ<sup>13</sup>

*the functions* Ψ�

condition (7).

0 < lim *ε*→0

*Then:*

In the sequel limits of the type lim*n*→<sup>∞</sup> *<sup>a</sup>*(*n*,*ε*) or lim*ε*→<sup>0</sup>

(*T*1(*ε*) < ∞ *P<sup>ϑ</sup>* − *a.s.*) *and possesses the following properties:*

0 < lim *ε*→0

[*T*1(*ε*) <sup>−</sup> <sup>1</sup>

2*v*<sup>0</sup> ln *ε*

ln *T*1(*ε*) − Ψ�

**3.2. Sequential estimation procedure for** *ϑ* ∈ Θ<sup>2</sup>

We introduce for every *ε* > 0 several quantities:

– the parameter *λ* = e*v*<sup>0</sup> and its estimator

*II. For every ϑ* ∈ Θ<sup>1</sup> *the estimator ϑ*1(*n*,*ε*) *is strongly consistent:*

1◦. sup *ϑ*∈Θ<sup>1</sup>

*ε* · *T*1(*ε*) ≤ lim

<sup>−</sup>1] <sup>≤</sup> lim *ε*→0

<sup>13</sup>(*ε*)], lim

<sup>13</sup>(*ε*) *are defined in (30).*

*ε*→0

*ε*→0

lim*n*∨*<sup>ε</sup> <sup>ϑ</sup>*1(*n*,*ε*) = *<sup>ϑ</sup> <sup>P</sup><sup>ϑ</sup>* <sup>−</sup> *<sup>a</sup>*.*s*.

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

both of those limits if their meaning is obvious.

*ϑ* = (*a*, *b*)� *in* (1) *be such that ϑ* ∈ Θ1.

2◦. *the inequalities below are valid:*

0 < lim *ε*→0

[*T*1(*ε*) + <sup>1</sup>

*v*0

<sup>13</sup>(*ε*) *and* Ψ��

$$\text{The first-order coupling between the two-dimensional } \mathcal{N} \text{-matrices is the only possible } \mathcal{N} \text{-matrices with } \mathcal{N} = \{0, 1, 2, \dots, N\} \text{ and } \mathcal{N} = \{0, 1, 2, \dots, N\}.$$

$$k\_1(n) = \arg\min\_{k=\overline{1,5}} ||G\_1^{-1}(n,k,\varepsilon)||\_\prime \ n \ge 1;$$

– the estimators

$$\Phi\_1(\mathfrak{n}, \varepsilon) = \mathbb{G}\_1^{-1}(\mathfrak{n}, \varepsilon) \Phi\_1(\mathfrak{n}, \varepsilon), \; n \ge 1, \quad \mathbb{G}\_1(\mathfrak{n}, \varepsilon) = \mathbb{G}\_1(\mathfrak{n}, k\_1(\mathfrak{n}), \varepsilon), \; \Phi\_1(\mathfrak{n}, \varepsilon) = \Phi\_1(\mathfrak{n}, k\_1(\mathfrak{n}), \varepsilon);$$

– the stopping time

$$\sigma\_1(\varepsilon) = \inf \{ N \ge 1 \colon S\_1(N) > (\rho\_1 \delta\_1^{-1})^{1/2} \},\tag{8}$$
  $\text{where } S\_1(N) = \sum\_{n=1}^N \beta\_1^2(n, \varepsilon),$  
$$\varepsilon$$

$$\beta\_1(n,\varepsilon) = ||\tilde{\mathcal{G}}\_1^{-1}(n,\varepsilon)||\_\prime \quad \tilde{\mathcal{G}}\_1(n,\varepsilon) = (\varepsilon^{-1}\varepsilon\_n)^{-1} \mathcal{G}\_1(n,k\_1(n),\varepsilon)$$

and *δ*<sup>1</sup> ∈ (0, 1) is some fixed chosen number,

$$\rho\_1 = 15(3 + \mathbb{R}^2) \sum\_{n \ge 1} \frac{1}{c\_n}.$$

The deviation of the 'first-step estimators' *ϑ*1(*n*,*ε*) has the form:

$$
\vartheta\_1(n, \varepsilon) - \vartheta = (\varepsilon^{-1} c\_{\imath \iota})^{-1/2} \tilde{G}\_1^{-1}(n, \varepsilon) \tilde{\zeta}\_1(n, \varepsilon), \ n \ge 1,\tag{9}
$$

$$
\tilde{\zeta}\_1(\mathfrak{u}, \mathfrak{e}) = (\mathfrak{e}^{-1} \mathfrak{e}\_n)^{-1/2} \int\_0^{\pi\_1(\mathfrak{u}, \mathfrak{e}) - k\_1(\mathfrak{u})h\_1} \Psi\_{h\_1}(t - 2 - 5h\_1) (\tilde{\Delta}\_\mathfrak{s}^\sharp(t)dt + dV(t) - dV(t - 1)) \dots
$$

By the definition of stopping times *<sup>τ</sup>*1(*n*,*ε*) <sup>−</sup> *<sup>k</sup>*1(*n*)*h*<sup>1</sup> we can control the noise ˜ *ζ*1(*n*,*ε*) :

> *<sup>E</sup>ϑ*|| ˜ *<sup>ζ</sup>*1(*n*,*ε*)||<sup>2</sup> <sup>≤</sup> <sup>15</sup>(<sup>3</sup> <sup>+</sup> <sup>R</sup>2), *<sup>n</sup>* <sup>≥</sup> 1, *<sup>ε</sup>* <sup>&</sup>gt; <sup>0</sup>

and by the definition of the stopping time *σ*1(*ε*) - the first factor *G*˜ <sup>−</sup><sup>1</sup> <sup>1</sup> (*n*,*ε*) in the representation of the deviation (9).

Define the sequential estimation plan of *ϑ* by

$$T\_1(\varepsilon) = \tau\_1(\sigma\_1(\varepsilon), \varepsilon), \ \theta\_1(\varepsilon) = \frac{1}{S(\sigma\_1(\varepsilon))} \sum\_{n=1}^{\sigma\_1(\varepsilon)} \beta\_1^2(n, \varepsilon) \theta\_1(n, \varepsilon). \tag{10}$$

We can see that the construction of the sequential estimator *ϑ*1(*ε*) is based on the family of estimators *ϑ*(*T*,*s*) = *G*−<sup>1</sup> <sup>1</sup> (*T*,*s*)Φ(*T*,*s*), *s* ≥ 0. We have taken the discretization step *h*<sup>1</sup> as above, because for *ϑ* ∈ Θ<sup>12</sup> the functions

$$f(T,s) = e^{2\upsilon\_0 T} \, G\_1^{-1}(T,s).$$

for every *s* ≥ 0 have some periodic matrix functions as a limit on *T* almost surely. These limit matrix functions are finite and may be infinite on the norm only for four values of their argument *T* on every interval of periodicity of the length Δ > 1 (see the proof of Theorem 3.2 in [10, 12]).

In the sequel limits of the type lim*n*→<sup>∞</sup> *<sup>a</sup>*(*n*,*ε*) or lim*ε*→<sup>0</sup> *a*(*n*,*ε*) will be used. To avoid repetitions of similar expressions we shall use, similar to [12, 14, 16], the unifying notation lim*n*∨*<sup>ε</sup> <sup>a</sup>*(*n*,*ε*) for both of those limits if their meaning is obvious.

We state the results concerning the estimation of the parameter *ϑ* ∈ Θ<sup>1</sup> in the following proposition.

**Proposition 3.1.** *Assume that the condition* (7) *on the sequence* (*cn*) *holds and let the parameter ϑ* = (*a*, *b*)� *in* (1) *be such that ϑ* ∈ Θ1.

*Then:*

6 Will-be-set-by-IN-TECH

<sup>1</sup>(*t*)*dt*, Φ1(*T*,*s*) =

*G*1(*n*, *k*,*ε*) = *G*1(*τ*1(*n*,*ε*) − *kh*1, *h*1), Φ1(*n*, *k*,*ε*) = Φ1(*τ*1(*n*,*ε*) − *kh*1, *h*1);


 *T* 0

<sup>1</sup> (*n*, *k*,*ε*)||, *n* ≥ 1;

<sup>1</sup> (*n*,*ε*)Φ1(*n*,*ε*), *n* ≥ 1, *G*1(*n*,*ε*) = *G*1(*n*, *k*1(*n*),*ε*), Φ1(*n*,*ε*) = Φ1(*n*, *k*1(*n*),*ε*);

<sup>1</sup>(*n*,*ε*)=(*ε*

*n*≥1

<sup>1</sup> (*n*,*ε*) ˜

1 *cn* . <sup>Ψ</sup>*s*(*<sup>t</sup>* <sup>−</sup> <sup>2</sup> <sup>−</sup> <sup>5</sup>*s*)*d*Δ˜ *<sup>Y</sup>*(*t*),

<sup>1</sup> )1/2}, (8)

*ζ*1(*n*,*ε*), *n* ≥ 1, (9)

*ζ*1(*n*,*ε*) :

<sup>1</sup> (*n*,*ε*) in the representation

<sup>1</sup>(*n*,*ε*)*ϑ*1(*n*,*ε*). (10)

<sup>−</sup>1*cn*)−1*G*1(*n*, *k*1(*n*),*ε*)

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

– the matrices

– the times

– the estimators

*ϑ*1(*n*,*ε*) = *G*−<sup>1</sup>

– the stopping time

*N* ∑ *n*=1 *β*2 <sup>1</sup>(*n*,*ε*),

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

and *δ*<sup>1</sup> ∈ (0, 1) is some fixed chosen number,

<sup>−</sup>1*cn*)−1/2

Define the sequential estimation plan of *ϑ* by

above, because for *ϑ* ∈ Θ<sup>12</sup> the functions

where *S*1(*N*) =

˜

*ζ*1(*n*,*ε*)=(*ε*

of the deviation (9).

estimators *ϑ*(*T*,*s*) = *G*−<sup>1</sup>

*G*1(*T*,*s*) =

 *T* 0

Ψ*s*(*t* − 2 − 5*s*)Ψ�

*k*1(*n*) = arg min

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

The deviation of the 'first-step estimators' *ϑ*1(*n*,*ε*) has the form:

*ϑ*1(*n*,*ε*) − *ϑ* = (*ε*

*<sup>E</sup>ϑ*|| ˜

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

0

and by the definition of the stopping time *σ*1(*ε*) - the first factor *G*˜ <sup>−</sup><sup>1</sup>

*<sup>T</sup>*1(*ε*) = *<sup>τ</sup>*1(*σ*1(*ε*),*ε*), *<sup>ϑ</sup>*1(*ε*) = <sup>1</sup>

By the definition of stopping times *<sup>τ</sup>*1(*n*,*ε*) <sup>−</sup> *<sup>k</sup>*1(*n*)*h*<sup>1</sup> we can control the noise ˜

*f*(*T*,*s*) = *e*

*k*=1,5

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

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

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

*<sup>ζ</sup>*1(*n*,*ε*)||<sup>2</sup> <sup>≤</sup> <sup>15</sup>(<sup>3</sup> <sup>+</sup> <sup>R</sup>2), *<sup>n</sup>* <sup>≥</sup> 1, *<sup>ε</sup>* <sup>&</sup>gt; <sup>0</sup>

*S*(*σ*1(*ε*))

We can see that the construction of the sequential estimator *ϑ*1(*ε*) is based on the family of

<sup>2</sup>*v*0*<sup>T</sup> G*−<sup>1</sup>

*σ*1(*ε*) ∑ *n*=1 *β*2

<sup>1</sup> (*T*,*s*)Φ(*T*,*s*), *s* ≥ 0. We have taken the discretization step *h*<sup>1</sup> as

<sup>1</sup> (*T*,*s*)

*I. For any ε* > 0 *and every ϑ* ∈ Θ<sup>1</sup> *the sequential plan* (*T*1(*ε*), *ϑ*1(*ε*)) *defined by* (10) *is closed* (*T*1(*ε*) < ∞ *P<sup>ϑ</sup>* − *a.s.*) *and possesses the following properties:*

$$1^\diamond \text{.} \sup\_{\vartheta \in \overline{\Theta}\_1} E\_{\vartheta} ||\vartheta\_1(\varepsilon) - \vartheta||^2 \le \delta\_1 \varepsilon;$$

2◦. *the inequalities below are valid:*

*– for ϑ* ∈ Θ<sup>11</sup> 0 < lim *ε*→0 *ε* · *T*1(*ε*) ≤ lim *ε*→0 *ε* · *T*1(*ε*) < ∞ *P<sup>ϑ</sup>* − *a.s.*, *– for ϑ* ∈ Θ<sup>12</sup> 0 < lim *ε*→0 [*T*1(*ε*) <sup>−</sup> <sup>1</sup> 2*v*<sup>0</sup> ln *ε* <sup>−</sup>1] <sup>≤</sup> lim *ε*→0 [*T*1(*ε*) <sup>−</sup> <sup>1</sup> 2*v*<sup>0</sup> ln *ε* <sup>−</sup>1] <sup>&</sup>lt; <sup>∞</sup> *<sup>P</sup><sup>ϑ</sup>* <sup>−</sup> *a.s.*, *– for ϑ* ∈ Θ<sup>13</sup> 0 < lim *ε*→0 [*T*1(*ε*) + <sup>1</sup> *v*0 ln *T*1(*ε*) − Ψ� <sup>13</sup>(*ε*)], lim *ε*→0 [*T*1(*ε*) + <sup>1</sup> *v*0 ln *T*1(*ε*) − Ψ�� <sup>13</sup>(*ε*)] < ∞ *P<sup>ϑ</sup>* − *a.s.,*

*the functions* Ψ� <sup>13</sup>(*ε*) *and* Ψ�� <sup>13</sup>(*ε*) *are defined in (30).*

*II. For every ϑ* ∈ Θ<sup>1</sup> *the estimator ϑ*1(*n*,*ε*) *is strongly consistent:*

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