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

Assume (*cn*)*n*≥<sup>1</sup> is an unboundedly increasing sequence of positive numbers satisfying the condition (7).

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

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

$$\lambda\_t = \frac{\int\_t^t \tilde{\Delta}Y(s)\tilde{\Delta}Y(s-1)ds}{\int\_t^t (\tilde{\Delta}Y(s-1))^2ds}, \quad t > 2, \ \lambda\_t = 0 \text{ otherwise};\tag{11}$$

where

where

– the stopping time

˜

and we have

*ϑ*2(*S*, *T*) = *G*−<sup>1</sup>

proposition.

*G*˜

*N* ∑ *n*=1 *β*2

*ζ*2(*n*,*ε*) = Ψ−1/2

*<sup>β</sup>*2(*n*,*ε*) = ||*G*˜ <sup>−</sup><sup>1</sup>

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

*τ*2(*n*,*ε*) −*k*2(*n*)*h*<sup>2</sup>

*ν*2(*n*,*ε*)

In this case we write the deviation of *ϑ*2(*n*,*ε*) in the form

<sup>2</sup> (*n*,*ε*)

*<sup>E</sup>ϑ*|| ˜

<sup>2</sup> (*S*, *<sup>T</sup>*)Φ2(*S*, *<sup>T</sup>*) = <sup>e</sup>−*v*1*TG*˜

<sup>2</sup>(*S*, *T*) = e−*v*1*T*Ψ−1/2

*ϑ* ∈ Θ22, similar to the case *ϑ* ∈ Θ12, the function

on every interval of periodicity of the length Δ.

*ϑ* = (*a*, *b*)� *in* (1) *be such that ϑ* ∈ Θ2. *Then:*

*possesses the following properties:*

Define the sequential estimation plan of *ϑ* by

*ϑ*2(*n*,*ε*) − *ϑ* = (*ε*

where *S*2(*N*) =

*G*2(*n*,*ε*) = *G*2(*n*, *k*2(*n*),*ε*), Φ2(*n*,*ε*) = Φ2(*n*, *k*2(*n*),*ε*);

<sup>2</sup>(*n*,*ε*), *ρ*<sup>2</sup> = *ρ*1, *δ*<sup>2</sup> ∈ (0, 1) is some fixed chosen number,

On Guaranteed Parameter Estimation of Stochastic Delay Diff erential Equations by Noisy Observations 31

<sup>2</sup> (*n*,*ε*) ˜

<sup>−</sup>1*cn*)−1/2Ψ−1/2

*ζ*2(*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*2(*ε*) = *τ*2(*σ*2(*ε*),*ε*), *ϑ*2(*ε*) = *ϑ*2(*σ*2(*ε*),*ε*). (15)

<sup>2</sup>(*S*, *<sup>T</sup>*)Φ˜ <sup>2</sup>(*S*, *<sup>T</sup>*), *<sup>T</sup>* <sup>&</sup>gt; *<sup>S</sup>* <sup>≥</sup> 0, where

<sup>2</sup> )1/2}, (14)

<sup>2</sup> (*n*,*ε*)*G*2(*n*,*ε*).

<sup>2</sup> (*T*)Φ2(*S*, *T*)

*<sup>σ</sup>*2(*ε*) = inf{*<sup>n</sup>* <sup>≥</sup> 1 : *<sup>S</sup>*2(*N*) <sup>&</sup>gt; (*ρ*2*δ*−<sup>1</sup>

<sup>2</sup>(*n*,*ε*)=(*ε*

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

*<sup>ζ</sup>*2(*n*,*ε*)||<sup>2</sup> <sup>≤</sup> <sup>15</sup>(<sup>3</sup> <sup>+</sup> R2), *<sup>n</sup>* <sup>≥</sup> 1, *<sup>ε</sup>* <sup>&</sup>gt; 0.

The construction of the sequential estimator *ϑ*2(*ε*) is based on the family of estimators

and <sup>Ψ</sup>2(*T*) = diag{e*v*1*T*, e*v*0*T*}. We have taken the discretization step *<sup>h</sup>* as above, because for

has some periodic (with the period Δ > 1) matrix function as a limit almost surely (see (35)). This limit matrix function may have an infinite norm only for four values of their argument *T*

We state the results concerning the estimation of the parameter *ϑ* ∈ Θ<sup>2</sup> in the following

**Proposition 3.2.** *Assume that the condition* (7) *on the sequence* (*cn*) *holds as well as the parameter*

*I. For any ε* > 0 *and every ϑ* ∈ Θ<sup>2</sup> *the sequential plan* (*T*2(*ε*), *ϑ*2(*ε*)) *defined by* (15) *is closed and*

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

1◦. sup *ϑ*∈Θ<sup>2</sup>

*f*2(*S*, *T*) = *G*˜ <sup>−</sup><sup>1</sup>

<sup>2</sup> (*T*)*G*2(*S*, *<sup>T</sup>*), <sup>Φ</sup>˜ <sup>2</sup>(*S*, *<sup>T</sup>*) = <sup>Ψ</sup>−1/2

<sup>2</sup> (*S*, *T*)

– the functions

$$\begin{aligned} Z(t) &= \begin{cases} \tilde{\Delta}Y(t) - \lambda \tilde{\Delta}Y(t-1) & \text{for} \quad t \ge 2, \\ 0 & \text{for} \quad t < 2; \end{cases} \\ \tilde{Z}(t) &= \begin{cases} \tilde{\Delta}Y(t) - \lambda\_t \tilde{\Delta}Y(t-1) & \text{for} \quad t \ge 2, \\ 0 & \text{for} \quad t < 2, \end{cases} \\ \Psi(t) &= \begin{cases} (\tilde{\Delta}Y(t), \tilde{\Delta}Y(t-1))' & \text{for} \quad t \ge 2, \\ (0,0)' & \text{for} \quad t < 2, \end{cases} \\ \tilde{\Psi}(t) &= \begin{cases} (Z(t), \tilde{\Delta}Y(t))' & \text{for} \quad t \ge 2, \\ (0,0)' & \text{for} \quad t < 2; \end{cases} \end{aligned}$$

– the parameter *α* = *v*0/*v*<sup>1</sup> and its estimator

$$w\_2(n, \varepsilon) = \frac{\ln \int\_{4}^{\nu\_2(n, \varepsilon)} (\tilde{\Delta}Y(t - 3))^2 dt}{\delta \ln \varepsilon^{-1} c\_n},\tag{12}$$

where

$$\nu\_2(\eta, \varepsilon) = \inf \{ T > 4 \, : \, \int\_4^T \tilde{Z}^2(t - 3)dt = (\varepsilon^{-1} \mathfrak{c}\_{\mathbb{M}})^\delta \}, \tag{13}$$

*δ* ∈ (0, 1) is a given number;

– the sequence of stopping times

$$\pi\_2(n,\varepsilon) = h\_2 \inf \{ k > h\_2^{-1} \nu\_2(n,\varepsilon) : \int\_{\substack{\nu\_2(n,\varepsilon) \\ \nu\_2(n,\varepsilon)}}^{h\_2} ||\Psi\_2^{-1/2}(n,\varepsilon)\Psi(t-\mathfrak{I})||^2 dt \ge 1 \},$$

where suppose *h*<sup>2</sup> = 1/5 and

$$\Psi\_2(n,\varepsilon) = \text{diag}\{\varepsilon^{-1}\varepsilon\_{n\prime}\left(\varepsilon^{-1}\varepsilon\_n\right)^{\alpha\_2(n,\varepsilon)}\};$$

– the matrices

$$\delta G\_2(S, T) = \int\_S^T \Psi(t - 3)\Psi'(t)dt, \quad \Phi\_2(S, T) = \int\_S^T \Psi(t - 3)d\tilde{\Delta}Y(t),$$

*G*2(*n*, *k*,*ε*) = *G*2(*ν*2(*n*,*ε*), *τ*2(*n*,*ε*) − *kh*2), Φ2(*n*, *k*,*ε*) = Φ2(*ν*2(*n*,*ε*), *τ*2(*n*,*ε*) − *kh*2);

– the times

$$k\_2(n) = \arg\min\_{k=\overline{\mathbb{T}}\overline{\mathbb{S}}} ||G\_2^{-1}(n,k,\varepsilon)||\_\prime \ n \ge 1;$$

– the estimators

$$
\vartheta\_2(n, \varepsilon) = G\_2^{-1}(n, \varepsilon) \Phi\_2(n, \varepsilon), \ n \ge 1,
$$

where

8 Will-be-set-by-IN-TECH

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

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

 (*Z*˜(*t*), <sup>Δ</sup>˜ *<sup>Y</sup>*(*t*))� for *<sup>t</sup>* <sup>≥</sup> 2, (0, 0)� for *t* < 2;

> *ν*2(*n*,*ε*) 4

> > *T* 4

> > > *kh*<sup>2</sup>

*ν*2(*n*,*ε*)

<sup>−</sup>1*cn*,(*ε*

(*t*)*dt*, Φ2(*S*, *T*) =

*G*2(*n*, *k*,*ε*) = *G*2(*ν*2(*n*,*ε*), *τ*2(*n*,*ε*) − *kh*2), Φ2(*n*, *k*,*ε*) = Φ2(*ν*2(*n*,*ε*), *τ*2(*n*,*ε*) − *kh*2);


(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* = (*<sup>ε</sup>*


<sup>−</sup>1*cn*)*α*2(*n*,*ε*)

<sup>2</sup> (*n*, *k*,*ε*)||, *n* ≥ 1;

<sup>2</sup> (*n*,*ε*)Φ2(*n*,*ε*), *n* ≥ 1,

 *T S*

*δ* ln *ε*−1*cn*

, *t* > 2, *λ<sup>t</sup>* = 0 otherwise; (11)

, (12)

<sup>−</sup>1*cn*)*δ*}, (13)

<sup>2</sup> (*n*,*ε*)Ψ˜ (*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)||2*dt* <sup>≥</sup> <sup>1</sup>},

<sup>Ψ</sup>˜ (*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)*d*Δ˜ *<sup>Y</sup>*(*t*),

};

*λ<sup>t</sup>* =

– the functions

where

*δ* ∈ (0, 1) is a given number; – the sequence of stopping times

where suppose *h*<sup>2</sup> = 1/5 and

*G*2(*S*, *T*) =

 *T S*

– the matrices

– the times

– the estimators

 *t* 2

*Z*(*t*) =

*Z*˜(*t*) =

Ψ(*t*) =

– the parameter *α* = *v*0/*v*<sup>1</sup> and its estimator

*<sup>τ</sup>*2(*n*,*ε*) = *<sup>h</sup>*<sup>2</sup> inf{*<sup>k</sup>* <sup>&</sup>gt; *<sup>h</sup>*−<sup>1</sup>

Ψ˜ (*t*) =

*α*2(*n*,*ε*) =

*ν*2(*n*,*ε*) = inf{*T* > 4 :

 *t* 2

<sup>Δ</sup>˜ *<sup>Y</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*

0 for *t* < 2;

0 for *t* < 2,

(0, 0)� for *t* < 2,

ln

<sup>2</sup> *ν*2(*n*,*ε*) :

Ψ2(*n*,*ε*) = diag{*ε*

<sup>Ψ</sup>˜ (*<sup>t</sup>* <sup>−</sup> <sup>3</sup>)Ψ�

*k*2(*n*) = arg min

*ϑ*2(*n*,*ε*) = *G*−<sup>1</sup>

*k*=1,5

$$G\_2(n, \varepsilon) = G\_2(n, k\_2(n), \varepsilon), \quad \Phi\_2(n, \varepsilon) = \Phi\_2(n, k\_2(n), \varepsilon);$$

– the stopping time

$$\sigma\_2(\varepsilon) = \inf \{ n \ge 1 \colon S\_2(N) > (\rho\_2 \delta\_2^{-1})^{1/2} \},\tag{14}$$

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

$$\beta\_2(n,\varepsilon) = ||\tilde{G}\_2^{-1}(n,\varepsilon)||\_\prime \quad \tilde{G}\_2(n,\varepsilon) = (\varepsilon^{-1}\varepsilon\_n)^{-1/2}\Psi\_2^{-1/2}(n,\varepsilon)G\_2(n,\varepsilon).$$

In this case we write the deviation of *ϑ*2(*n*,*ε*) in the form

$$
\vartheta\_2(n,\varepsilon) - \vartheta = (\varepsilon^{-1}\varepsilon\_n)^{-1/2}\tilde{G}\_2^{-1}(n,\varepsilon)\tilde{\zeta}\_2(n,\varepsilon), \ n \ge 1,
$$

where

$$\tilde{\zeta}\_2(n,\varepsilon) = \Psi\_2^{-1/2}(n,\varepsilon) \int\_{\nu\_2(n,\varepsilon)}^{\tau\_2(n,\varepsilon) - k\_2(n)h\_2} \Psi(t-\mathfrak{A})(\tilde{\Delta}\_{\mathfrak{F}}^{\mathfrak{z}}(t)dt + dV(t) - dV(t-1))dt$$

and we have

$$E\_{\theta} ||\tilde{\zeta}\_2(n, \varepsilon)||^2 \le 15(3 + \mathbb{R}^2), \ n \ge 1, \ \varepsilon > 0.$$

Define the sequential estimation plan of *ϑ* by

$$T\_2(\varepsilon) = \tau\_2(\sigma\_2(\varepsilon), \varepsilon), \ \vartheta\_2(\varepsilon) = \vartheta\_2(\sigma\_2(\varepsilon), \varepsilon). \tag{15}$$

The construction of the sequential estimator *ϑ*2(*ε*) is based on the family of estimators *ϑ*2(*S*, *T*) = *G*−<sup>1</sup> <sup>2</sup> (*S*, *<sup>T</sup>*)Φ2(*S*, *<sup>T</sup>*) = <sup>e</sup>−*v*1*TG*˜ <sup>2</sup>(*S*, *<sup>T</sup>*)Φ˜ <sup>2</sup>(*S*, *<sup>T</sup>*), *<sup>T</sup>* <sup>&</sup>gt; *<sup>S</sup>* <sup>≥</sup> 0, where

$$\tilde{\mathbf{G}}\_2(\mathbf{S}, T) = \mathbf{e}^{-\upsilon\_1 T} \mathbf{Y}\_2^{-1/2}(T) \mathbf{G}\_2(\mathbf{S}, T), \quad \tilde{\Phi}\_2(\mathbf{S}, T) = \mathbf{Y}\_2^{-1/2}(T) \Phi\_2(\mathbf{S}, T)$$

and <sup>Ψ</sup>2(*T*) = diag{e*v*1*T*, e*v*0*T*}. We have taken the discretization step *<sup>h</sup>* as above, because for *ϑ* ∈ Θ22, similar to the case *ϑ* ∈ Θ12, the function

$$f\_2(\mathcal{S}, T) = \tilde{G}\_2^{-1}(\mathcal{S}, T)$$

has some periodic (with the period Δ > 1) matrix function as a limit almost surely (see (35)). This limit matrix function may have an infinite norm only for four values of their argument *T* on every interval of periodicity of the length Δ.

We state the results concerning the estimation of the parameter *ϑ* ∈ Θ<sup>2</sup> in the following proposition.

**Proposition 3.2.** *Assume that the condition* (7) *on the sequence* (*cn*) *holds as well as the parameter ϑ* = (*a*, *b*)� *in* (1) *be such that ϑ* ∈ Θ2. *Then:*

*I. For any ε* > 0 *and every ϑ* ∈ Θ<sup>2</sup> *the sequential plan* (*T*2(*ε*), *ϑ*2(*ε*)) *defined by* (15) *is closed and possesses the following properties:*

$$1^\diamond . \sup\_{\vartheta \in \overline{\Theta}\_2} E\_\vartheta ||\vartheta\_2(\varepsilon) - \vartheta||^2 \le \delta\_2 \varepsilon;$$

2◦. *the inequalities below are valid:*

$$0 < \varliminf\_{\varepsilon \to 0} \left[ T\_2(\varepsilon) - \frac{1}{2v\_1} \ln \varepsilon^{-1} \right] \\
\le \varlimsup\_{\varepsilon \to 0} \left[ T\_2(\varepsilon) - \frac{1}{2v\_1} \ln \varepsilon^{-1} \right] \\
< \infty \text{ } P\_\vartheta - a.s.;$$

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

$$\lim\_{n \vee \varepsilon} \vartheta\_2(n, \varepsilon) = \vartheta\_1 P\_{\vartheta} - a.s.$$
