**3.3. Sequential estimation procedure for** *ϑ* ∈ Θ<sup>3</sup>

We shall use the notation, introduced in the previous paragraph for the parameter *λ* = e*v*<sup>0</sup> and its estimator *λ<sup>t</sup>* as well as for the functions *Z*(*t*), *Z*˜(*t*), Ψ(*t*) and Ψ˜ (*t*).

Chose the non-random functions *ν*3(*n*,*ε*), *n* ≥ 1, *ε* > 0, satisfying the following conditions as *ε* → 0 or *n* → ∞ :

$$\nu\_{\mathfrak{J}}(\mathfrak{n}, \mathfrak{e}) = o(\varepsilon^{-1} \mathfrak{c}\_{\mathfrak{n}}), \quad \frac{\log^{1/2} \nu\_{\mathfrak{J}}(\mathfrak{n}, \mathfrak{e})}{\mathfrak{e}^{\mathfrak{D}\_{\mathfrak{0}} \nu\_{\mathfrak{J}}(\mathfrak{n}, \mathfrak{e})}} \varepsilon^{-1} \mathfrak{c}\_{\mathfrak{n}} = o(1). \tag{16}$$

– the stopping time

*N* ∑ *n*=1 *β*2

*<sup>β</sup>*3(*n*,*ε*) = ||*G*˜ <sup>−</sup><sup>1</sup>

<sup>−</sup>1*cn*)−1/2Ψ−1/2

In this case we write the deviation of *ϑ*3(*n*,*ε*) in the form

*ϑ*3(*n*,*ε*) − *ϑ* = (*ε*

<sup>3</sup> (*n*,*ε*)

*<sup>E</sup>ϑ*|| ˜

where *S*3(*N*) =

*G*˜

and we have

*following properties:* 1◦. *for any ε* > 0

where

where

<sup>3</sup>(*n*,*ε*)=(*ε*

˜

*ζ*3(*n*,*ε*) = Ψ−1/2

Define the sequential estimation plan of *ϑ* by

*ϑ* = (*a*, *b*)� *in* (1) *be such that ϑ* ∈ Θ3. *Then:*

2◦. *the following inequalities are valid:*

*<sup>σ</sup>*3(*ε*) = inf{*<sup>n</sup>* <sup>≥</sup> 1 : *<sup>S</sup>*3(*N*) <sup>&</sup>gt; (*ρ*3*δ*−<sup>1</sup>

<sup>3</sup>(*n*,*ε*), *δ*<sup>3</sup> ∈ (0, 1) is some fixed chosen number,

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

*τmin*(*n*,*ε*) 

*ν*3(*n*,*ε*)

sup *ϑ*∈Θ<sup>3</sup>

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

*εT*3(*ε*) ≤ lim

*ε*→0

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

*<sup>d</sup>*Δ˜ *<sup>Y</sup>*(*t*) = *aZ*∗(*t*)*dt* <sup>+</sup> <sup>Δ</sup>˜ *<sup>ξ</sup>*(*t*)*dt* <sup>+</sup> *dV*(*t*) <sup>−</sup> *dV*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>), *<sup>t</sup>* <sup>≥</sup> 2,

*<sup>Z</sup>*∗(*t*) = <sup>Δ</sup>˜ *<sup>Y</sup>*(*t*) <sup>−</sup> <sup>Δ</sup>˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>) for *<sup>t</sup>* <sup>≥</sup> 2, 0 for *t* < 2.

0 < lim *ε*→0

*II. For every ϑ* ∈ Θ<sup>3</sup> *the estimator ϑ*3(*n*,*ε*) *is strongly consistent:*

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

<sup>3</sup> (*n*,*ε*)||, *<sup>ρ</sup>*<sup>3</sup> <sup>=</sup> <sup>6</sup>(<sup>3</sup> <sup>+</sup> *<sup>R</sup>*2) ∑

On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 33

<sup>3</sup> (*n*,*ε*) ˜

<sup>3</sup> (*n*,*ε*)*G*3(*n*,*ε*), Ψ3(*n*,*ε*) = diag{*ε*

*<sup>ζ</sup>*3(*n*,*ε*)||<sup>2</sup> <sup>≤</sup> <sup>6</sup>(<sup>3</sup> <sup>+</sup> *<sup>R</sup>*2), *<sup>n</sup>* <sup>≥</sup> 1, *<sup>ε</sup>* <sup>&</sup>gt; 0.

**Proposition 3.3.** *Assume that the condition* (7) *on the sequence* (*cn*) *holds and let the parameter*

*I. For every ϑ* ∈ Θ<sup>3</sup> *the sequential plan* (*T*3(*ε*), *ϑ*3(*ε*)) *defined in* (20) *is closed and possesses the*

*<sup>E</sup>ϑ*||*ϑ*3(*ε*) <sup>−</sup> *<sup>ϑ</sup>*||<sup>2</sup> <sup>≤</sup> *<sup>δ</sup>*3*ε*;

<sup>3</sup> )1/2}, (19)

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

*n*≥1

*ζ*3(*n*,*ε*), *n* ≥ 1,

<sup>Ψ</sup>˜ (*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)(Δ˜ *<sup>ξ</sup>*(*t*)*dt* <sup>+</sup> *dV*(*t*) <sup>−</sup> *dV*(*<sup>t</sup>* <sup>−</sup> <sup>1</sup>))

*T*3(*ε*) = *τmax*(*σ*3(*ε*),*ε*), *ϑ*3(*ε*) = *ϑ*3(*σ*3(*ε*),*ε*). (20)

*εT*3(*ε*) < ∞ *P<sup>ϑ</sup>* − *a.s.;*

1 *cn* ,

Example: *ν*3(*n*,*ε*) = log2 *ε*−1*cn*.

We introduce several quantities:

– the parameter *α*<sup>3</sup> = *v*<sup>0</sup> and its estimator

$$\alpha\_3(n,\varepsilon) = \ln|\lambda\_{\nu\_3(n,\varepsilon)}|\cdot|$$

where *λ<sup>t</sup>* is defined in (11);

– the sequences of stopping times

$$\pi\_{31}(n,\varepsilon) = \inf\{T > 0 : \int\_{\nu\_3(n,\varepsilon)}^T \tilde{Z}^2(t-3)dt = \varepsilon^{-1}c\_n\},\tag{17}$$

$$\pi\_{32}(n,\varepsilon) = \inf \{ T > 0 : \int\_{\upsilon\_3(n,\varepsilon)}^T (\tilde{\Delta}Y(t-\mathfrak{Z}))^2 dt = \mathfrak{e}^{2a\_3(n,\varepsilon)\varepsilon^{-1}c\_n} \},\tag{18}$$

*τmin*(*n*,*ε*) = min{*τ*31(*n*,*ε*), *τ*32(*n*,*ε*)}, *τmax*(*n*,*ε*) = max{*τ*31(*n*,*ε*), *τ*32(*n*,*ε*)},

– the matrices

$$\begin{aligned} G\_3(S, T) &= \int\_S^T \Psi(t) \Psi(t) dt, \\ \Phi\_3(S, T) &= \int\_S^T \Psi(t) d\tilde{\Delta} Y(t), \\ G\_3(n, \varepsilon) &= G\_3(\nu\_3(n, \varepsilon), \tau\_{\min}(n, \varepsilon)), \\ \Phi\_3(n, \varepsilon) &= \Phi\_3(\nu\_3(n, \varepsilon), \tau\_{\min}(n, \varepsilon)); \end{aligned}$$

– the estimators

$$\vartheta\_3(n,\varepsilon) = G\_3^{-1}(n,\varepsilon)\Phi\_3(n,\varepsilon),\ n \ge 1,\ \varepsilon > 0;$$

– the stopping time

10 Will-be-set-by-IN-TECH

lim*n*∨*<sup>ε</sup> <sup>ϑ</sup>*2(*n*,*ε*) = *<sup>ϑ</sup> <sup>P</sup><sup>ϑ</sup>* <sup>−</sup> *a.s.*

We shall use the notation, introduced in the previous paragraph for the parameter *λ* = e*v*<sup>0</sup>

Chose the non-random functions *ν*3(*n*,*ε*), *n* ≥ 1, *ε* > 0, satisfying the following conditions as

<sup>−</sup>1*cn*), log1/2 *<sup>ν</sup>*3(*n*,*ε*)

*α*3(*n*,*ε*) = ln |*λν*3(*n*,*ε*)|,

 *T*

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

*ν*3(*n*,*ε*)

*τmin*(*n*,*ε*) = min{*τ*31(*n*,*ε*), *τ*32(*n*,*ε*)}, *τmax*(*n*,*ε*) = max{*τ*31(*n*,*ε*), *τ*32(*n*,*ε*)},

*G*3(*n*,*ε*) = *G*3(*ν*3(*n*,*ε*), *τmin*(*n*,*ε*)), Φ3(*n*,*ε*) = Φ3(*ν*3(*n*,*ε*), *τmin*(*n*,*ε*));

Ψ˜ (*t*)Ψ(*t*)*dt*,

Ψ˜ (*t*)*d*Δ˜ *Y*(*t*),

<sup>3</sup> (*n*,*ε*)Φ3(*n*,*ε*), *n* ≥ 1, *ε* > 0;

 *T*

*ν*3(*n*,*ε*)

 *T S*

 *T S*

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

[*T*2(*ε*) <sup>−</sup> <sup>1</sup>

2*v*<sup>1</sup> ln *ε*

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

<sup>−</sup>1*cn* = *o*(1). (16)

<sup>−</sup>1*cn*}, (17)

(Δ˜ *<sup>Y</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>3</sup>))2*dt* <sup>=</sup> <sup>e</sup>2*α*3(*n*,*ε*)*ε*−<sup>1</sup>*cn* }, (18)

<sup>−</sup>1] <sup>≤</sup> lim *ε*→0

and its estimator *λ<sup>t</sup>* as well as for the functions *Z*(*t*), *Z*˜(*t*), Ψ(*t*) and Ψ˜ (*t*).

2◦. *the inequalities below are valid:*

[*T*2(*ε*) <sup>−</sup> <sup>1</sup>

2*v*<sup>1</sup> ln *ε*

*II. For every ϑ* ∈ Θ<sup>2</sup> *the estimator ϑ*2(*n*,*ε*) *is strongly consistent:*

**3.3. Sequential estimation procedure for** *ϑ* ∈ Θ<sup>3</sup>

*ν*3(*n*,*ε*) = *o*(*ε*

*τ*31(*n*,*ε*) = inf{*T* > 0 :

*G*3(*S*, *T*) =

Φ3(*S*, *T*) =

*ϑ*3(*n*,*ε*) = *G*−<sup>1</sup>

*τ*32(*n*,*ε*) = inf{*T* > 0 :

0 < lim *ε*→0

*ε* → 0 or *n* → ∞ :

Example: *ν*3(*n*,*ε*) = log2 *ε*−1*cn*. We introduce several quantities:

where *λ<sup>t</sup>* is defined in (11); – the sequences of stopping times

– the matrices

– the estimators

– the parameter *α*<sup>3</sup> = *v*<sup>0</sup> and its estimator

$$\sigma\_3(\varepsilon) = \inf \{ n \ge 1 \, : \, \mathcal{S}\_3(N) > (\rho\_3 \delta\_3^{-1})^{1/2} \} \,\,\tag{19}$$

where *S*3(*N*) = *N* ∑ *n*=1 *β*2 <sup>3</sup>(*n*,*ε*), *δ*<sup>3</sup> ∈ (0, 1) is some fixed chosen number,

$$\beta\_{\mathfrak{Z}}(\mathfrak{n},\mathfrak{e}) = ||\tilde{\mathcal{G}}\_{\mathfrak{Z}}^{-1}(\mathfrak{n},\mathfrak{e})||\_{\prime} \quad \rho\_{\mathfrak{Z}} = 6(3+R^{2})\sum\_{n\geq 1} \frac{1}{\mathfrak{c}\_{n}} \prime$$

$$\tilde{\mathcal{G}}\_{\mathfrak{Z}}(\mathfrak{n},\mathfrak{e}) = (\varepsilon^{-1}\mathfrak{c}\_{\mathfrak{n}})^{-1/2} \Psi\_{\mathfrak{Z}}^{-1/2}(\mathfrak{n},\mathfrak{e}) \mathbf{G}\_{\mathfrak{Z}}(\mathfrak{n},\mathfrak{e}), \quad \Psi\_{\mathfrak{Z}}(\mathfrak{n},\mathfrak{e}) = \text{diag}\{\varepsilon^{-1}\mathfrak{c}\_{\mathfrak{n}\prime}\mathfrak{c}^{2\omega\_{3}(\mathfrak{n},\mathfrak{e})\varepsilon^{-1}\mathfrak{c}\_{\mathfrak{n}}}\}.$$

In this case we write the deviation of *ϑ*3(*n*,*ε*) in the form

$$
\vartheta\_{\mathfrak{Z}}(n,\varepsilon) - \vartheta = (\varepsilon^{-1}\varepsilon\_n)^{-1/2}\tilde{G}\_{\mathfrak{Z}}^{-1}(n,\varepsilon)\tilde{\zeta}\_{\mathfrak{Z}}(n,\varepsilon), \ n \ge 1,
$$

where

$$\tilde{\zeta}\_3(n,\varepsilon) = \Psi\_3^{-1/2}(n,\varepsilon) \int\_{\nu\_3(n,\varepsilon)}^{\tau\_{\min}(n,\varepsilon)} \tilde{\Psi}(t-\mathfrak{I})(\tilde{\Delta}\_{\varepsilon}^{\mathfrak{r}}(t)dt + dV(t) - dV(t-1)),$$

and we have

$$E\_{\theta} ||\tilde{\zeta}\_{\mathcal{B}}(n, \varepsilon)||^{2} \le 6(\mathfrak{Z} + \mathbb{R}^{2}), \ n \ge 1, \ \varepsilon > 0.$$

Define the sequential estimation plan of *ϑ* by

$$T\_{\mathfrak{J}}(\varepsilon) = \tau\_{\max}(\sigma\_{\mathfrak{J}}(\varepsilon), \varepsilon), \ \vartheta\_{\mathfrak{J}}(\varepsilon) = \vartheta\_{\mathfrak{J}}(\sigma\_{\mathfrak{J}}(\varepsilon), \varepsilon). \tag{20}$$

**Proposition 3.3.** *Assume that the condition* (7) *on the sequence* (*cn*) *holds and let the parameter ϑ* = (*a*, *b*)� *in* (1) *be such that ϑ* ∈ Θ3. *Then:*

*I. For every ϑ* ∈ Θ<sup>3</sup> *the sequential plan* (*T*3(*ε*), *ϑ*3(*ε*)) *defined in* (20) *is closed and possesses the following properties:*

1◦. *for any ε* > 0

$$\sup\_{\vartheta \in \overline{\Theta}\_3} E\_{\vartheta} ||\vartheta\_3(\varepsilon) - \vartheta||^2 \le \delta\_3 \varepsilon\_{\varepsilon}^2$$

2◦. *the following inequalities are valid:*

$$0 < \varliminf\_{\varepsilon \to 0} \varepsilon T\_{\mathfrak{I}}(\varepsilon) \le \overline{\lim}\_{\varepsilon \to 0} \varepsilon T\_{\mathfrak{I}}(\varepsilon) < \infty \mid P\_{\theta} - a.s.;$$

*II. For every ϑ* ∈ Θ<sup>3</sup> *the estimator ϑ*3(*n*,*ε*) *is strongly consistent:*

$$\lim\_{n \vee \varepsilon} \vartheta\_3(n, \varepsilon) = \vartheta\_1 P\_{\vartheta} - a.s.$$
