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# Adaptive Kalman Filter for Systems with Unknown Initial Values

Yury A Kutoyants

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Yury A Kutoyants. Adaptive Kalman Filter for Systems with Unknown Initial Values. 2025. hal-05419906

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https://univ-lemans.hal.science/hal-05419906v1

Preprint submitted on 16 Dec 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Adaptive Kalman Filter for Systems with Unknown Initial Values

YURY A. KUTOYANTS

Le Mans University, Le Mans, France

Tomsk State University, Tomsk, Russia

## Abstract

The models of partially observed linear stochastic differential equations with unknown initial values of the non-observed component are considered in two situations. In the first problem, the initial value is deterministic, and in the second problem, it is assumed to be a Gaussian random variable. The main problem is the computation of adaptive Kalman filters and the discussion of their asymptotic optimality. The realization of this program for both models is done in several steps. First, a preliminary estimator of the unknown parameter is constructed by observations on some learning interval. Then, this estimator is used for the calculation of recurrent one-step MLE estimators, which are subsequently substituted in the equations of Kalman filtration.

Key words: Hidden Markov process, Kalman-Bucy filter, unknown initial value, One-step MLE-process, on-line approximation, adaptive filter.

MSC 2000 Classification: 62M05, 60G35, 62M15.

## 1 Introduction

Consider the partially observed continuous time system

$$dX_t = f(\vartheta, t) Y_t dt + \sigma(t) dW_t, \quad X_0 = 0, \quad 0 \le t \le T, \quad (1)$$

$$dY_t = a(\vartheta, t) Y_t dt + b(\vartheta, t) dV_t, \quad y_0, \quad 0 \le t \le T, \quad (2)$$

where $W_t, 0 \le t \le T$ and $V_t, 0 \le t \le T$ are independent Wiener processes, all functions $f(\cdot), \sigma(\cdot), a(\cdot), b(\cdot)$ are known and $\vartheta$ is some unknown finite dimensional parameter. The process $X^T = (X_t, 0 \le t \le T)$ is observed and the process $Y^T = (Y_t, 0 \le t \le T)$ is “hidden”. The main problem is to find a good approximation of the unobserved component $Y_t$ by observations $X^t = (X_s, 0 \le s \le t)$. If $\vartheta$ and $y_0$ are known then the solution of the problem is well-known and was given by Kalman and Bucy [15]: the conditional expectation $m(\vartheta, t) =$
$E_\vartheta (Y_t|X^t)$ is solution of the system of equations

$$dm(\vartheta, t) = a(\vartheta, t) m(\vartheta, t) dt + \frac{f(\vartheta, t) \gamma(\vartheta, t)}{\sigma(t)^2} [dX_t - f(\vartheta, t) m(\vartheta, t) dt], \quad (3)$$

$$\frac{d\gamma(\vartheta, t)}{dt} = 2a(\vartheta, t)\gamma(\vartheta, t) - \frac{f(\vartheta, t)^2 \gamma(\vartheta, t)^2}{\sigma(t)^2} + b(\vartheta, t)^2 \quad (4)$$

subject to the initial conditions $m(\vartheta, 0) = y_0$ and $\gamma(\vartheta, 0) = 0$. The construction of the adaptive filters (with $\vartheta$ unknown and $y_0$ known) for both models is given in two steps. First, the unknown parameter is estimated, and then this estimator is substituted into the corresponding equations, such as (3) and (4). Due to its practical importance, there exists a vast literature on adaptive filtering for discrete-time models; see [6], [10], [33], and references therein. Continuous-time systems are less well studied.

General results concerning parameter estimation and adaptive filtering can be found, for example, in the works [4], [6], [10], [31], and [33].

The problems related with the construction of the filtering equations in the situations where the initial values of the hidden system are unknown or wrong occupy a special place in the filtering theory. The different statements and results can be found in the works [18], [35] (see as well references therein).

The properties of estimators for continuous time models like (1), (2) are studied in different asymptotics. The simplest case corresponds to small noise in both equations, i.e., where $\sigma(t) \to \varepsilon\sigma(t)$, $b(\vartheta, t) \to \varepsilon b(t)$, and $\varepsilon \to 0$. The properties of parameter estimators were described in many works, see, for example, [17], [20], [21], [23], [29], [34], and references therein. The parameter estimation in the case of a homogeneous partially observed linear system was studied in the asymptotic of large samples in the works [1], [2], [3], [5], [7], [13], [19], [20], [22], [25], [31]. The third model of observations: small noise in the observation equations only was studied in the works [8], [9], [16], [24], [26], [32]. Remark that for these three models the question of asymptotically efficient adaptive filtration was discussed in the works [29], [27], and [26] respectively.

In this work we consider three problems of adaptive filtration in the cases where the initial value of $Y_0 = y_0$ is supposed to be unknown. Note that in the mentioned above works [20], [21], [29] it was always supposed that $y_0$ is known. The chosen models are relatively simple but nevertheless the construction of adaptive filters is non trivial. All filters are constructed in four steps. The first step is to obtain a preliminary consistent estimator of the unknown parameter, the second step - this estimator is used for the construction of the recurrent One-Step MLE-process and the third step is the substitution of this One-step estimator-process in the Kalman-Bucy equations. The solution of these equations gives us the adaptive Kalman filter. The last step is to describe the error of approximation of the conditional expectation. The similar problems of asymptotically efficient estimation of $m(\vartheta, t)$ were considered in [29] (small noise in both equations), in [26] (small noise in observations only), in [27] (large samples statement) and in [28] (hidden autoregressive
process). The main advantage of the proposed procedure is its relative computational simplicity because after the small learning interval the estimator of the unknown parameter and the adaptive filter have recurrent structure.

# 2 Model of observations

Consider the partially observed linear system of equations

$$dX_t = f(t) Y_t dt + \varepsilon \sigma(t) dW_t, \quad X_0 = 0, \quad 0 \le t \le T, \qquad (5)$$

$$dY_t = a(t) Y_t dt + \varepsilon b(t) dV_t, \quad Y_0. \qquad (6)$$

where $W_t, V_t, 0 \le t \le T$ are independent Wiener processes. The functions $f(t), a(t), \sigma(t), b(t); t \in [0, T]$ and the parameter $\varepsilon \in (0, 1]$ are supposed to be known. The initial value $\vartheta \in \Theta$ is the unknown parameter. The observations are $X^T = (X_t, 0 \le t \le T)$ and the process $Y^T = (Y_t, 0 \le t \le T)$ is “hidden”. Here $\varepsilon \in (0, 1]$ is some known small parameter and the asymptotic corresponds to the limit $\varepsilon \to 0$. Therefore we consider a linear system with small noise in both equations.

## 2.1 Deterministic initial value

### 2.1.1 Unknown initial value

Suppose that the partially observed system is

$$dX_t = f(t) Y_t dt + \varepsilon \sigma(t) dW_t, \quad X_0 = 0, \quad 0 \le t \le T,$$

$$dY_t = a(t) Y_t dt + \varepsilon b(t) dV_t, \quad Y_0 = \vartheta,$$

where the functions $f(t), \sigma(t), a(t), b(t)$ are known and satisfy the

Conditions $\mathcal{A}_*$. The functions $f(t), \sigma(t), a(t), b(t), t \in [0, T]$ are bounded and the functions $f(\cdot), \sigma(\cdot)$ are separated from 0.

The unknown parameter is the initial value $Y_0 = \vartheta \in \Theta = (\alpha, \beta)$ and we have to estimate $\vartheta$ by observations $X^T$. Moreover this estimator will be used later for the construction of the adaptive Kalman filter. Our goal is to obtain a recurrent adaptive filter. We need as well a recurrent presentation of the estimator, because at each moment $t \in [0, T]$ we need an estimator constructed by the observations $X_s, 0 \le s \le t$. Then this estimator will be used in the construction of the filter.

Recall that the conditional expectation $m(\vartheta, t) = E_{\vartheta}(Y_t|\mathfrak{F}_t^X)$ (here $\mathfrak{F}_t^X$ is the $\sigma$-algebra generated by the observations $X^t = (X_s, 0 \le s \le t)$) satisfies the equations of Kalman-Bucy filtration (3),(4)

$$dm(\vartheta, t) = a(t) m(\vartheta, t) dt + \frac{f(t) \gamma(t)}{\varepsilon^2 \sigma(t)^2} [dX_t - f(t) m(\vartheta, t) dt], \quad m(\vartheta, 0) = \vartheta,$$
$$ \frac{\partial \gamma(t)}{\partial t} = 2a(t) \gamma(t) - \frac{f(t)^2 \gamma(t)^2}{\varepsilon^2 \sigma(t)^2} + \varepsilon^2 b(t)^2, \quad \gamma(0) = 0, \quad 0 \le t \le T. $$

These equations can be slightly simplified by introducing the function $\gamma_*(t) = \varepsilon^{-2}\gamma(t)$. Then we can write

$$ dm(\vartheta, t) = a(t) m(\vartheta, t) dt + \frac{f(t) \gamma_*(t)}{\sigma(t)^2} [dX_t - f(t) m(\vartheta, t) dt], \quad m(\vartheta, 0) = \vartheta, \quad (7) $$

$$ \frac{\partial \gamma_*(t)}{\partial t} = 2a(t) \gamma_*(t) - \frac{f(t)^2 \gamma_*(t)^2}{\sigma(t)^2} + b(t)^2, \quad \gamma_*(0) = 0, \quad 0 \le t \le T. \quad (8) $$

The solution of the equation (7) is

$$ m(\vartheta, t) = \vartheta \Phi(0, t) + \int_{0}^{t} \Phi(s, t) f(s) \gamma_{*}(s) \sigma(s)^{-2} dX_{s} = \vartheta \Phi(0, t) + H(t, X^{t}) \quad (9) $$

where

$$ \Phi(s,t) = \exp \left( \int_s^t A(r) dr \right), \quad A(t) = a(t) - f(t)^2 \gamma_*(t) \sigma(t)^{-2}, \\ H(t, X^t) = \int_0^t \Phi(s,t) f(s) \gamma_*(s) \sigma(s)^{-2} dX_s. $$

Therefore the good approximation of $m(\vartheta, t)$ needs a good estimator of the parameter $\vartheta$.

One way to estimate $\vartheta$ is use the MLE $\hat{\vartheta}_{t,\varepsilon}$. The measures $(\mathbf{P}_{\vartheta}^{(t,\varepsilon)}, \vartheta \in \Theta)$ induced by observations $X^t$ in the measurable space of continuous functions $(\mathcal{C}[0,t], \mathfrak{B}_t)$ under condition $\mathcal{A}_*$ are equivalent and the likelihood ratio (LR) function is [30]

$$ L(\vartheta, X^t) = \exp \left( \int_0^t \frac{f(s) m(\vartheta, s)}{\varepsilon^2 \sigma(s)^2} dX_s - \int_0^t \frac{f(s)^2 m(\vartheta, s)^2}{2\varepsilon^2 \sigma(s)^2} ds \right), \quad \vartheta \in \Theta. \quad (10) $$

Hence the Fisher information

$$ I^t(\vartheta) = \varepsilon^2 \mathbf{E}_{\vartheta} \left[ \frac{\partial \ln L(\vartheta, X^t)}{\partial \vartheta} \right]^2 = \int_0^t \frac{f(s)^2 \Phi(0, s)^2}{\sigma(s)^2} ds $$

does not depend on $\vartheta$ and the MLE has an explicit expression

$$ \hat{\vartheta}_{t,\varepsilon} = \left( \int_{0}^{t} \frac{f(s)^2 \Phi(0, s)^2}{\sigma(s)^2} ds \right)^{-1} \int_{0}^{t} \frac{f(s) \Phi(0, s)}{\sigma(s)^2} [dX_s - f(s) H(s, X^s) ds]. \quad (11) $$

We write $I^t(\vartheta)$ to mention the estimated parameter and $\vartheta_0$ for the true value of $\vartheta$. It is easy to see that

$$ \varepsilon^{-1} (\hat{\vartheta}_{t,\varepsilon} - \vartheta_0) = \left( \int_0^t \frac{f(s)^2 \Phi(0, s)^2}{\sigma(s)^2} ds \right)^{-1} \int_0^t \frac{f(s) \Phi(0, s)}{\sigma(s)} d\bar{W}_s \sim \mathcal{N} (0, I^t(\vartheta_0)^{-1}). $$
Here we used the representation of observations

$$dX_s = f(s)m(\vartheta_0, s)ds + \varepsilon\sigma(s)d\bar{W}_s, \quad X_0 = 0, \quad 0 \le s \le t$$

where $\bar{W}_s, 0 \le s \le t$ is an innovation Wiener process (see Theorem 7.6 in [30]).

The estimator $\hat{\vartheta}_{t,\varepsilon}$ can be written in the recurrent form as follows:

$$\hat{\vartheta}_{t,\varepsilon} = \frac{S(t)}{I^t}$$

where

$$dS(t) = \frac{f(t)\Phi(0,t)}{\sigma(t)^2} [dX_t - f(t)H(t,X^t)dt], \quad S(0) = 0,$$

$$dI^t = \frac{f(t)^2 \Phi(0,t)^2}{\sigma(t)^2} dt, \quad I^0 = 0,$$

$$dH(t, X^t) = A(t)H(t, X^t)dt + f(t)\gamma_*(t)\sigma(t)^{-2}dX_t.$$

It is possible to write the stochastic differential of the process $\hat{\vartheta}_{t,\varepsilon}, 0 \le t \le T$. Let us multiply both sides of (11) by $I^t(\vartheta)$, denote $g(t) = f(t)\Phi(0,t)\sigma(t)^{-1}$ and take the differential, then we obtain

$$\hat{\vartheta}_{t,\varepsilon} \frac{f(t)^2 \Phi(0,t)^2}{\sigma(t)^2} dt + I^t(\vartheta) d\hat{\vartheta}_{t,\varepsilon} = \frac{f(t) \Phi(0,t)}{\sigma(t)^2} [dX_t - f(t) H(t, X^t) dt]$$

and

$$d\hat{\vartheta}_{t,\varepsilon} = -\frac{g(t)^2}{I^t(\vartheta)} \hat{\vartheta}_{t,\varepsilon} dt + \frac{g(t)}{I^t(\vartheta)\sigma(t)} [dX_t - f(t) H(t, X^t) dt]. \quad (12)$$

In these two representations there is a question of the initial value because $\hat{\vartheta}_{0,\varepsilon} = 0/0$. To avoid this problem we can fix some (small) $\tau$ and calculate the preliminary estimator

$$\hat{\vartheta}_{\tau,\varepsilon} = \frac{1}{I^\tau} \int_0^\tau \frac{g(s)}{\sigma(s)} [dX_s - f(s) H(s, X^s) ds] \quad (13)$$

and then use the equation (12) to compute $\hat{\vartheta}_{t,\varepsilon}, \tau \le t \le T$.

Let us verify that (12)-(13) gives the same estimator, i.e., that $\hat{\vartheta}_{\tau,\varepsilon} \sim N(\vartheta_0, \varepsilon^2 I^t (\vartheta_0)^{-1})$. It is sufficient to calculate $E_{\vartheta_0}(\hat{\vartheta}_{\tau,\varepsilon} - \vartheta_0)^2 = \varepsilon^2 I^t (\vartheta_0)^{-1}$. We have

$$\hat{\vartheta}_{t,\varepsilon} = \hat{\vartheta}_{\tau,\varepsilon} \tilde{\Phi}(\tau, t) + \int_{\tau}^{t} \tilde{\Phi}(s, t) \frac{g(s)}{I^s \sigma(s)} [dX_s - f(s) H(s, X^s) ds],$$
where

$$ \tilde{\Phi}(s,t) = \exp \left( - \int_s^t \frac{g(r)^2}{I^r} dr \right). $$

Therefore the substitution of the observations gives the expression

$$ \hat{\vartheta}_{t,\varepsilon} = \hat{\vartheta}_{\tau,\varepsilon} \tilde{\Phi}(\tau,t) + \vartheta_0 \int_{\tau}^{t} \tilde{\Phi}(s,t) \frac{g(s)^2}{I^s} ds + \varepsilon \int_{\tau}^{t} \tilde{\Phi}(s,t) \frac{g(s)}{I^s} d\bar{W}_s. $$

Remark that

$$ \int_s^t \frac{g(r)^2}{I^r} dr = \int_s^t \frac{1}{I^r} d\left(\int_0^r g(q)^2 dq\right) = \int_s^t \frac{1}{I^r} dI^r = \ln I^t - \ln I^s $$

and

$$ \tilde{\Phi}(s,t) = \frac{I^s}{I^t}. $$

Therefore

$$ \int_{\tau}^{t} \tilde{\Phi}(s,t) \frac{g(s)^2}{I^s} ds = \frac{1}{I^t} \int_{\tau}^{t} g(s)^2 ds = \frac{I^t - I^\tau}{I^t} $$

and we obtain the relation

$$ \hat{\vartheta}_{t,\varepsilon} = \hat{\vartheta}_{\tau,\varepsilon} \frac{I^\tau}{I^t} + \vartheta_0 \frac{I^t - I^\tau}{I^t} + \frac{\varepsilon}{I^t} \int_{\tau}^{t} g(s) d\bar{W}_s. $$

Hence

$$ \varepsilon^{-1} (\hat{\vartheta}_{t,\varepsilon} - \vartheta_0) = \varepsilon^{-1} (\hat{\vartheta}_{\tau,\varepsilon} - \vartheta_0) \frac{I^\tau}{I^t} + \frac{1}{I^t} \int_{\tau}^{t} g(s) d\bar{W}_s $$

and

$$ \begin{aligned} \varepsilon^{-2} \mathbf{E}_{\vartheta_0} (\hat{\vartheta}_{t,\varepsilon} - \vartheta_0)^2 &= \varepsilon^{-2} \mathbf{E}_{\vartheta_0} (\hat{\vartheta}_{\tau,\varepsilon} - \vartheta_0)^2 \left(\frac{I^\tau}{I^t}\right)^2 + \frac{1}{(I^t)^2} \int_{\tau}^{t} g(s)^2 ds \\ &= \frac{I^\tau}{(I^t)^2} + \frac{I^t - I^\tau}{(I^t)^2} = \frac{1}{I^t}. \end{aligned} $$

$$ \text{Therefore } \varepsilon^{-1} (\hat{\vartheta}_{t,\varepsilon} - \vartheta_0) \sim \mathcal{N} (0, I^t (\vartheta_0)^{-1}). $$

We have the recurrent equations for the estimator of the parameter and the adaptive Kalman filter is defined in two steps. First we introduce the random process

$$ \mathrm{d}n_t^* = \left[ a(t) - \frac{f(t)^2 \gamma_*(t)}{\sigma(t)^2} \right] n_t^* \mathrm{d}t + \frac{f(t) \gamma_*(t)}{\sigma(t)^2} \mathrm{d}X_t, \quad n_0^* = 0. \quad (14) $$
The solution of this equation is (see (9))

$$n_t^* = \int_0^t \Phi(s,t) f(s) \gamma_*(s) \sigma(s)^{-2} dX_s.$$

The adaptive Kalman filter defined by the equation

$$m_t^* = \hat{\vartheta}_{t,\varepsilon} \Phi(0,t) + n_t^* \quad (15)$$

The error of estimation is

$$\varepsilon^{-1} (m_t^* - m(\vartheta_0, t)) = \varepsilon^{-1} (\hat{\vartheta}_{t,\varepsilon} - \vartheta_0) \Phi(0,t) = \frac{\Phi(0,t)}{\Gamma^t} \int_0^t \frac{f(s) \gamma_*(s)}{\sigma(s)} d\bar{W}_s$$

The lower bound in this problem is

$$\lim_{\nu \to 0} \lim_{\varepsilon \to 0} \sup_{|\vartheta - \vartheta_0| \le \nu} \varepsilon^{-2} \mathbf{E}_{\vartheta} [\bar{m}_{t,\varepsilon} - m(\vartheta, t)]^2 \ge \frac{\Phi(0,t)^2}{\Gamma^t(\vartheta_0)}.$$

To prove this bound we follow, e.g., [27]. We have an elementary inequality

$$\sup_{|\vartheta - \vartheta_0| \le \nu} \mathbf{E}_{\vartheta} [\bar{m}_{t,\varepsilon} - m(\vartheta, t)]^2 \ge \int_{\vartheta_0 - \nu}^{\vartheta_0 + \nu} \mathbf{E}_{\vartheta} [\bar{m}_{t,\varepsilon} - m(\vartheta, t)]^2 p_{\nu}(\vartheta) d\vartheta$$

where we introduced the continuous positive density $p_{\nu}(\vartheta)$, $(\vartheta_0 - \nu, \vartheta_0 + \nu)$. Denote $\tilde{m}(t)$ the Bayes estimator. Then by the definition of Bayes estimator

$$\int_{\vartheta_0 - \nu}^{\vartheta_0 + \nu} \mathbf{E}_{\vartheta} [\bar{m}_{t,\varepsilon} - m(\vartheta, t)]^2 p_{\nu}(\vartheta) d\vartheta \ge \int_{\vartheta_0 - \nu}^{\vartheta_0 + \nu} \mathbf{E}_{\vartheta} [\tilde{m}(t) - m(\vartheta, t)]^2 p_{\nu}(\vartheta) d\vartheta.$$

Recall that

$$\tilde{m}(t) = \mathbf{E}(m(\vartheta, t) | \mathfrak{F}_t^X) = \int_{\alpha}^{\beta} m(\theta, t) p_{\nu}(\theta | X^t) d\theta = \frac{\int_{\alpha}^{\beta} m(\theta, t) p_{\nu}(\theta) L(\theta, X^t) d\theta}{\int_{\alpha}^{\beta} p_{\nu}(\theta) L(\theta, X^t) d\theta} \\ = H(t, X^t) + \Phi(0, t) \frac{\int_{\alpha}^{\beta} \theta p_{\nu}(\theta) L(\theta, X^t) d\theta}{\int_{\alpha}^{\beta} p_{\nu}(\theta) L(\theta, X^t) d\theta}.$$

If we change the variable $\theta = \vartheta + \varepsilon u$ and denote $\theta_u = \vartheta + \varepsilon u$, $Z_{\varepsilon}(\vartheta, u, t) = L(\theta_u, X^t) / L(\vartheta, X^t)$, $U_{\varepsilon} = [\varepsilon^{-1}(\alpha - \vartheta), \varepsilon^{-1}(\beta - \vartheta)]$ then

$$\frac{\tilde{m}(t) - m(\vartheta, t)}{\varepsilon \Phi(0, t)} = \frac{\int_{U_{\varepsilon}} up_{\nu}(\theta_u) Z_{\varepsilon}(\vartheta, u, t) du}{\int_{U_{\varepsilon}} p_{\nu}(\theta_u) Z_{\varepsilon}(\vartheta, u, t) du} = \frac{\int_{U_{\varepsilon}} uZ_{\varepsilon}(\vartheta, u, t) du}{\int_{U_{\varepsilon}} Z_{\varepsilon}(\vartheta, u, t) du} (1 + o(1))$$
$$ \implies \frac{\int_{\mathbb{R}} uZ(\vartheta, u, t) \, du}{\int_{\mathbb{R}} Z(\vartheta, u, t) \, du} = I^t(\vartheta_0)^{-1} \int_0^t \frac{f(s) \Phi(0, s)}{\sigma(s)} dW(s) $$

Here we used the continuity of $p_ν(·)$ and denoted $W(s), 0 ≤ s ≤ t$ a Wiener process. It can be shown that the moments of BE converge too (see [11]).

Therefore

$$ \lim_{\nu \to 0} \lim_{\varepsilon \to 0} \sup_{|\vartheta - \vartheta_0| \le \nu} \varepsilon^{-2} \mathbf{E}_{\vartheta} [\bar{m}_{t, \varepsilon} - m(\vartheta, t)]^2 \ge \frac{\Phi(0, t)^2}{I^t(\vartheta_0)}. $$

As

$$ \varepsilon^{-2} \mathbf{E}_{\vartheta} [m_t^* - m(\vartheta, t)]^2 = \frac{\Phi(0, t)^2}{I^t(\vartheta)} $$

the approximation $m_t^*$ is asymptotically efficient.

### 2.1.2 Unknown initial value and parameter

Suppose that the partially observed system is

$$ dX_t = f(t) Y_t dt + \varepsilon \sigma(t) dW_t, \quad X_0 = 0, \quad 0 \le t \le T, \quad (16) $$

$$ dY_t = a(\theta_2, t) Y_t dt + \varepsilon b(t) dV_t, \quad Y_0 = \theta_1. \quad (17) $$

The unknown parameter is $\vartheta = (\theta_1, \theta_2) \in \Theta = (\alpha_1, \beta_1) \times (\alpha_2, \beta_2)$, $\alpha_1 > 0$. The functions $f(·), \sigma(·), a(·), b(·)$ satisfy the condition $\mathcal{A}_*$. Our goal is to find an approximation of the conditional expectation $m(\vartheta, t) = \mathbf{E}_{\vartheta} (Y_t | \mathfrak{F}_t^X)$ and describe its properties in the asymptotic $\varepsilon \to 0$.

Remark that the LR formula for $L(\vartheta, X^t)$ is (10), where the function $m(\vartheta, s), 0 \le s \le t$ is solution of the equations like (7), (8), where $a(t) = a(\theta_1, t)$. Therefore the calculation of the MLE $\hat{\vartheta}_{\varepsilon,t}$ defined by the equation

$$ L(\hat{\vartheta}_{\varepsilon,t}, X^t) = \sup_{\vartheta \in \Theta} L(\vartheta, X^t) $$

is a very difficult computational problem. It requires the solutions of filtration equations for all or many $\vartheta \in \Theta$ and for all $t \in [0, T]$. That is why we propose below the construction of the estimator-process which needs such solutions just for one value of $\vartheta$. Moreover this estimator has recurrent expression which reduces one more the computation of the adaptive filter.

Introduce conditions $\mathcal{B}$.
B1. The function $a(\vartheta, \cdot), \vartheta \in \bar{\Theta}$ has two continuous derivatives on $\vartheta$.

B2. The functions $f(\cdot), \sigma(\cdot), a(\cdot), b(\cdot)$ have continuous derivatives in $t \in [0, T]$.

The adaptive recurrent K-B filter is constructed in several steps. First we estimate the parameter $\theta_2$ by the observations on the small interval $[0, \tau_\epsilon]$, where $\tau_\epsilon \to 0$. Then we use this estimator in the construction of the parameter $\theta_1$ by observations on the interval $[0, \tau]$. The obtained estimator of $\vartheta$ is used to introduce the One-step MLE-process. The last step is to substitute this One-step estimator-process in the equations of Kalman-Bucy filtration.

The limit system ($\epsilon = 0$) is, of course,

$$ 
\begin{aligned}
\frac{dx_t(\vartheta_0)}{dt} &= f(t)y_t(\vartheta_0), & x_0(\vartheta_0) &= 0, & 0 \le t \le T, \\
\frac{dy_t(\vartheta_0)}{dt} &= a(\theta_{2,0}, t)y_t(\vartheta_0), & y_0(\vartheta_0) &= \theta_{1,0}.
\end{aligned}
$$

Here $\vartheta_0 = (\theta_{1,0}, \theta_{2,0})$ is the true value of $\vartheta$.

Note that the limit function of the process $Y_t$ is

$$ y_t(\vartheta_0) = \theta_{1,0} \phi(\theta_{2,0}, 0, t), \quad \phi(\theta_{2,0}, s, t) = \exp \left( \int_s^t a(\theta_{2,0}, v) \, dv \right) $$

and

$$ Y_t = \theta_{1,0} \phi(\theta_{2,0}, 0, t) + \epsilon \int_0^t \phi(\theta_{2,0}, s, t) b(s) \, dV_s. $$

The observations have the following representation as $\tau_\epsilon \to 0$

$$ 
\begin{aligned}
X_{\tau_{\epsilon}} &= \theta_{1,0} \int_{0}^{\tau_{\epsilon}} f(t) \phi(\theta_{2,0}, 0, t) \, dt + \epsilon \int_{0}^{\tau_{\epsilon}} \sigma(s) \, dW_s \\
&\quad + \epsilon \int_{0}^{\tau_{\epsilon}} f(t) \int_{0}^{t} \phi(\theta_{2,0}, s, t) b(s) \, dV_s \, dt \\
&= \theta_{1,0} f(0) \tau_{\epsilon} + \epsilon \sigma(0) W_{\tau_{\epsilon}} + \epsilon \tau_{\epsilon}^{3/2} f(0) b(0) \int_{0}^{1} r \, dv_{r,\epsilon} + O(\epsilon \tau_{\epsilon}^{3/2}),
\end{aligned}
$$

where we used the Taylor formula and introduced $v_{r,\epsilon}, 0 \le r \le 1$ as a Wiener process independent of $W_{\tau_\epsilon}$. Here and below if the object is random then convergences in $O(\cdot)$ and $o(\cdot)$ are understood in probability, for example, $\tau_\epsilon^{-1}o(\tau_\epsilon) \to 0$ in probability.

Hence the method of moments estimator (MME) can be

$$ \theta_{1,\tau_{\epsilon}}^* = \frac{X_{\tau_{\epsilon}}}{f(0)\tau_{\epsilon}} = \theta_{1,0} + \frac{\epsilon}{\sqrt{\tau_{\epsilon}}} \sigma(0) \xi_{1,\epsilon} + \tau_{\epsilon} O(1) + \epsilon \tau_{\epsilon}^{1/2} b(0) \int_{0}^{1} r \, dv_{r,\epsilon} + \epsilon \tau_{\epsilon} O(1). $$
The solution of the equation $\varepsilon/\sqrt{\tau_{\varepsilon}} = \tau_{\varepsilon}$ is the rate $\tau_{\varepsilon} = \varepsilon^{2/3}$. Therefore, we obtain asymptotic normality

$$ \varepsilon^{-2/3} (\theta_{1,\tau_{\varepsilon}}^* - \theta_{1,0}) \implies \mathcal{N}(q, d^2) $$

with some parameters $q, d^2$. The particular values of these parameters are not interesting. Moreover, we have the convergence of all polynomial moments too.

Let us fix some (small) $\tau > 0$ and define the random process

$$ x_{t,\varepsilon}(\theta_2) = \theta_{1,\tau_{\varepsilon}}^* \int_0^t f(s) \phi(\theta_2, 0, s) \, ds, \quad 0 \le t \le \tau, $$

and define the minimum distance estimator (MDE) $\check{\theta}_{2,\tau}$ by the equation

$$ \int_0^\tau [X_t - x_{t,\varepsilon}(\check{\theta}_{2,\tau})]^2 dt = \inf_{\alpha_2 < \theta_2 < \beta_2} \int_0^\tau [X_t - x_{t,\varepsilon}(\theta_2)]^2 dt. $$

Introduce the identifiability condition

B3. For a given $\tau$, any $\nu > 0$ and any $\theta_{2,0} \in [\alpha_2, \beta_2]$

$$ g(\vartheta_0, \nu) = \inf_{|\theta_2 - \theta_{2,0}| \ge \nu} \int_0^\tau [F(\theta_2, t) - F(\theta_{2,0}, t)]^2 dt > 0. $$

Here

$$ F(\theta, t) = \int_0^t f(s) \phi(\theta, 0, s) \, ds. $$

We have

$$ \begin{aligned} X_t - x_{t,\varepsilon}(\check{\theta}_{2,\tau}) &= x_t(\vartheta_0) - x_{t,\varepsilon}(\check{\theta}_{2,\tau}) + \varepsilon \int_0^t \sigma(s) \, dW_s \\ &\quad + \varepsilon \int_0^t f(s) \int_0^s \phi(\theta_{2,0}, r, s) b(r) \, dV_r \, ds \\ &= (\theta_{1,0} - \theta_{1,\tau_{\varepsilon}}^*) F(\theta_{2,0}, t) + \theta_{1,\tau_{\varepsilon}}^* [F(\theta_{2,0}, t) - F(\check{\theta}_{2,\tau}, t)] + \varepsilon \eta_t, \end{aligned} $$

where $\eta_t, 0 \le t \le \tau$ is a Gaussian process with bounded variance defined by the last equality.

Below $\|h_t\|_{\tau}$ is $\mathcal{L}_2[0, \tau]$-norm. To prove the consistency of the estimator $\check{\theta}_{2,\tau}$ we write

$$ \begin{aligned} \mathbf{P}_{\vartheta_0} (|\check{\theta}_{2,\tau} - \theta_{2,0}| \ge \nu) &= \mathbf{P}_{\vartheta_0} \left( \inf_{|\theta_2 - \theta_{2,0}| \le \nu} \|X_t - x_{t,\varepsilon}(\theta_2)\|_{\tau} > \inf_{|\theta_2 - \theta_{2,0}| \ge \nu} \|X_t - x_{t,\varepsilon}(\theta_2)\|_{\tau} \right) \\ &\le \mathbf{P}_{\vartheta_0} \left( \inf_{|\theta_2 - \theta_{2,0}| \le \nu} (\|X_t - x_t(\vartheta_0)\|_{\tau} + \|x_t(\vartheta_0) - x_{t,\varepsilon}(\theta_{2,0})\|_{\tau} + \|x_{t,\varepsilon}(\theta_2) - x_{t,\varepsilon}(\theta_{2,0})\|_{\tau}) \right) \end{aligned} $$
$$ > \inf_{|\theta_2 - \theta_{2,0}| \ge \nu} \|x_{t,\epsilon}(\theta_2) - x_{t,\epsilon}(\theta_{2,0})\|_{\tau} - \|X_t - x_t(\vartheta_0)\|_{\tau} - \|x_t(\vartheta_0) - x_{t,\epsilon}(\theta_{2,0})\|_{\tau} \\ \le \mathbf{P}_{\vartheta_0} \left( 2 \|X_t - x_t(\vartheta_0)\|_{\tau} + 2 \|x_t(\vartheta_0) - x_{t,\epsilon}(\theta_{2,0})\|_{\tau} > \alpha_1 g(\theta_{2,0}, \nu) \right). $$

By Tchebyshev inequality we have

$$ \mathbf{P}_{\vartheta_0} (|\check{\theta}_{2,\tau} - \theta_{2,0}| \ge \nu) \le \frac{8}{\alpha_1^2 g(\vartheta_0, \nu)^2} (\mathbf{E}_{\vartheta_0} \|X_t - x_t(\vartheta_0)\|_{\tau}^2 + \mathbf{E}_{\vartheta_0} \|x_t(\vartheta_0) - x_{t,\epsilon}(\theta_{2,0})\|_{\tau}^2) \\ \le \frac{C\epsilon^2 + C\epsilon^{4/3}}{\alpha_1^2 g(\vartheta_0, \nu)^2} \to 0. $$

Therefore the estimator $\bar{\vartheta}_{\tau,\epsilon} = (\theta_{1,\tau\epsilon}, \theta_{2,\tau})$ is consistent. It can be shown that the difference $\epsilon^{-2/3}(\bar{\vartheta}_{\tau,\epsilon} - \vartheta_0)$ is asymptotically normal and the moments $\epsilon^{-2p/3}\mathbf{E}_{\vartheta_0} \|\bar{\vartheta}_{\tau,\epsilon} - \vartheta_0\|^p$ are bounded for any $p > 0$. This estimator will be used for the construction of One-step MLE-process. We need the equations of Kalman-Bucy filter for this model of observations:

$$ dm(\vartheta, t) = A(\theta_2, t) m(\vartheta, t) dt + \frac{f(t) \gamma_*(\theta_2, t)}{\sigma(t)^2} dX_t, \quad m(\vartheta, 0) = \theta_1, \quad (18) $$

$$ \frac{d\gamma_*(\theta_2, t)}{dt} = 2a(\theta_2, t)\gamma_*(\theta_2, t) - \frac{f(t)^2\gamma_*(\theta_2, t)^2}{\sigma(t)^2} + b(t)^2, \quad \gamma_*(\theta_2, 0) = 0, \quad (19) $$

where

$$ A(\theta_2, t) = a(\theta_2, t) - \frac{f(t)^2 \gamma_*(\theta_2, t)}{\sigma(t)^2}. $$

The derivatives $\dot{m}_1(\vartheta, t) = \partial m(\vartheta, t)/\partial \theta_1$ and $\dot{m}_2(\vartheta, t) = \partial m(\vartheta, t)/\partial \theta_2$ satisfy the equations

$$ \frac{d\dot{m}_1(\vartheta, t)}{dt} = A(\theta_2, t) \dot{m}_1(\vartheta, t), \quad \dot{m}_1(\vartheta, t) = 1, \\ \frac{d\dot{m}_2(\vartheta, t)}{dt} = A(\theta_2, t) \dot{m}_2(\vartheta, t) dt + \dot{A}_2(\theta_2, t) m(\vartheta, t) dt + \frac{f(t) \dot{\gamma}_*(\theta_2, t)}{\sigma(t)^2} dX_t, \quad \dot{m}_2(\vartheta, 0) = 0. $$

Therefore

$$ \dot{m}_1(\vartheta, t) = \Phi(\theta_2, 0, t), \quad \Phi(\theta_2, s, t) = \exp\left(\int_s^t A(\theta_2, r) dr\right), \\ \dot{m}_2(\vartheta, t) = \int_0^t \Phi(\theta_2, s, t) \dot{A}_2(\theta_2, s) m(\vartheta, s) ds + \int_0^t \Phi(\theta_2, s, t) \frac{f(s) \dot{\gamma}_*(\theta_2, s)}{\sigma(s)^2} dX_s. $$

The limit ($\epsilon = 0$) values are: for $\dot{m}_1(\vartheta, t) = \Phi(\theta_2, 0, t) \equiv \dot{y}_1(\vartheta, t)$ and for $\dot{m}_2(\vartheta, t)$ we obtain the expression

$$ \dot{y}_2(\vartheta, \vartheta_0, t) = \int_0^t \Phi(\theta_2, s, t) \dot{A}_2(\theta_2, s) y_s(\vartheta, \vartheta_0) ds $$
$$+ \int_{0}^{t} \Phi(\theta_{2}, s, t) \frac{f(s)^{2} \dot{\gamma}_{*}(\theta_{2}, s)}{\sigma(s)^{2}} y_{s}(\vartheta_{0}) \, ds$$

where

$$y_t(\vartheta_0) = \theta_{1,0}\phi(\theta_{2,0}, 0, t), \quad \phi(\theta_{2,0}, 0, t) = \exp\left(\int_0^t a(\theta_{2,0}, s) \, ds\right),$$

$$y_t(\vartheta, \vartheta_0) = \theta_1 \Phi(\theta_2, 0, t) + \theta_{1,0} \int_0^t \Phi(\theta_2, s, t) \frac{f(s)^2 \dot{\gamma}_*(\theta_2, s)}{\sigma(s)^2} \phi(\theta_{2,0}, 0, s) \, ds.$$

The function $\dot{y}_2(\vartheta, \vartheta_0, t)$ at the point $\vartheta = \vartheta_0$ takes the value $\dot{y}_2(\vartheta_0, t)$ where

$$\begin{aligned} \dot{y}_2(\vartheta_0, t) &= \int_0^t \Phi(\theta_{2,0}, s, t) \dot{a}(\theta_{2,0}, s) y_s(\vartheta_0) \, ds \\ &= \theta_{1,0} \int_0^t \Phi(\theta_{2,0}, s, t) \phi(\theta_{2,0}, 0, s) \dot{a}(\theta_{2,0}, s) \, ds. \end{aligned}$$

The Fisher information matrix is

$$\mathbf{I}_{\tau}^{t}(\vartheta) = \begin{pmatrix} \int_{\tau}^{t} \frac{f(s)^2 \dot{y}_1(\vartheta, s)^2}{\sigma(s)^2} \mathrm{d}s & \int_{\tau}^{t} \frac{f(s)^2 \dot{y}_1(\vartheta, s) \dot{y}_2(\vartheta, s)}{\sigma(s)^2} \mathrm{d}s \\ \int_{\tau}^{t} \frac{f(s)^2 \dot{y}_1(\vartheta, s) \dot{y}_2(\vartheta, s)}{\sigma(s)^2} \mathrm{d}s & \int_{\tau}^{t} \frac{f(s)^2 \dot{y}_2(\vartheta, s)^2}{\sigma(s)^2} \mathrm{d}s \end{pmatrix}.$$

Introduce the condition

B_3. For a given $\tau$ and any $t \in (\tau, T]$ and any $\vartheta \in \Theta$ the Fisher information is non degenerate

$$\inf_{\|e\|=1, e \in \mathbb{R}^2} e^{\top} \mathbf{I}_{\tau}^{t}(\vartheta) e > 0.$$

The One-step MLE-process is

$$\vartheta_{t, \varepsilon}^{*} = \bar{\vartheta}_{\tau, \varepsilon} + \mathbf{I}_{\tau}^{t} (\bar{\vartheta}_{\tau, \varepsilon})^{-1} \int_{\tau}^{t} \frac{f(s) \dot{m}(\bar{\vartheta}_{\tau, \varepsilon}, s)}{\sigma(s)^2} [dX_s - f(s) m(\bar{\vartheta}_{\tau, \varepsilon}, s) ds]. \quad (20)$$

Here the vector $\dot{m}(\vartheta, s) = (\dot{m}_1(\vartheta, t), \dot{m}_2(\vartheta, t))^{\top}$. Remark that the vector $\dot{m}(\vartheta, s)$ can be replaced by the vector $\dot{y}(\vartheta, s) = (\dot{y}_1(\vartheta, s), \dot{y}_2(\vartheta, s))^{\top}$ and the asymptotic properties of the estimator will be the same.

Define the Gaussian process $\eta_t(\vartheta_0) = (\eta_{1,t}(\vartheta_0), \eta_{2,t}(\vartheta_0))^{\top}$

$$\eta_t(\vartheta_0) = \mathbf{I}_{\tau}^t(\vartheta)^{-1} \int_{\tau}^{t} \frac{f(s) \dot{y}_s(\vartheta_0)}{\sigma(s)} dw(s), \quad \tau \le t \le T,$$

where $\dot{y}_t(\vartheta_0) = (\dot{y}_1(\vartheta_0, t), \dot{y}_2(\vartheta_0, t))^{\top}$ and $w(s), 0 \le s \le T$ is some Wiener process.
**Proposition 1.** Suppose that the conditions $\mathcal{A}_*$ and $\mathcal{B}_1 - \mathcal{B}_3$ are satisfied. Then the One-step MLE-process $\vartheta_{t,\varepsilon}^*, \tau < t \le T$ is consistent, asymptotically normal

$$ \eta_{t,\varepsilon}(\vartheta_0) = (\eta_{1,t,\varepsilon}(\vartheta_0), \eta_{2,t,\varepsilon}(\vartheta_0))^\top \equiv \varepsilon^{-1} (\vartheta_{t,\varepsilon}^* - \vartheta_0) \implies \mathcal{N}(0, \mathbf{I}_\tau^t (\vartheta_0)^{-1}) \quad (21) $$

and the process $\eta_{t,\varepsilon}(\vartheta_0), \tau_* \le t \le T$ for any $\tau_* \in (\tau, T]$ converges in distribution in $(C(0,T), \mathfrak{B})$ to the Gaussian process

$$ \eta_{t,\varepsilon}(\vartheta_0) \implies \eta_t(\vartheta_0), \quad \tau_* \le t \le T. \quad (22) $$

*Proof.* Consider the difference

$$ \begin{aligned} \varepsilon^{-1} (\vartheta_{t,\varepsilon}^* - \vartheta_0) = \varepsilon^{-1} (\bar{\vartheta}_{\tau,\varepsilon} - \vartheta_0) + \mathbf{I}_{\tau}^t (\bar{\vartheta}_{\tau,\varepsilon})^{-1} \int_{\tau}^{t} \frac{f(s) \dot{m}(\bar{\vartheta}_{\tau,\varepsilon}, s)}{\sigma(s)} d\bar{W}_s \\ + \mathbf{I}_{\tau}^t (\bar{\vartheta}_{\tau,\varepsilon})^{-1} \int_{\tau}^{t} \frac{f(s)^2 \dot{m}(\bar{\vartheta}_{\tau,\varepsilon}, s)}{\varepsilon\sigma(s)^2} [m(\vartheta_0, s) - m(\bar{\vartheta}_{\tau,\varepsilon}, s)] ds. \end{aligned} \quad (23) $$

The consistency of the estimator $\bar{\vartheta}_{\tau,\varepsilon}$ and the central limit theorem for stochastic integrals allow us to write

$$ \pi_{t,\varepsilon} \equiv \mathbf{I}_{\tau}^t (\bar{\vartheta}_{\tau,\varepsilon})^{-1} \int_{\tau}^{t} \frac{f(s) \dot{m}(\bar{\vartheta}_{\tau,\varepsilon}, s)}{\sigma(s)} d\bar{W}_s \implies \eta_t(\vartheta_0) \sim \mathcal{N}(0, \mathbf{I}_{\tau}^t (\vartheta_0)^{-1}). $$

If we consider the vector $\pi_{t_1,\varepsilon}, \dots, \pi_{t_k,\varepsilon}$, where $\tau \le t_1 \le t_2 \le \dots \le t_k \le T$ then we obtain the convergence

$$ (\pi_{t_1,\varepsilon}, \dots, \pi_{t_k,\varepsilon}) \implies (\eta_{t_1}(\vartheta_0), \dots, \eta_{t_k}(\vartheta_0)). $$

For the ordinary integral we have the relations

$$ \begin{aligned} & \mathbf{I}_{\tau}^t (\bar{\vartheta}_{\tau,\varepsilon})^{-1} \int_{\tau}^{t} \frac{f(s)^2 \dot{m}(\bar{\vartheta}_{\tau,\varepsilon}, s)}{\varepsilon\sigma(s)^2} [m(\vartheta_0, s) - m(\bar{\vartheta}_{\tau,\varepsilon}, s)] ds \\ &= \mathbf{I}_{\tau}^t (\vartheta_0)^{-1} \int_{\tau}^{t} \frac{f(s)^2 \dot{m}(\vartheta_0, s)}{\varepsilon\sigma(s)^2} [m(\vartheta_0, s) - m(\bar{\vartheta}_{\tau,\varepsilon}, s)] ds (1 + O(\varepsilon^{2/3})) \\ &= \mathbf{I}_{\tau}^t (\vartheta_0)^{-1} \int_{\tau}^{t} \frac{f(s)^2 \dot{m}(\vartheta_0, s)}{\sigma(s)^2} \dot{m}(\vartheta_0, s)^\top ds \frac{(\vartheta_0 - \bar{\vartheta}_{\tau,\varepsilon})}{\varepsilon} (1 + O(\varepsilon^{2/3})) \\ &= \frac{(\vartheta_0 - \bar{\vartheta}_{\tau,\varepsilon})}{\varepsilon} + \frac{(\vartheta_0 - \bar{\vartheta}_{\tau,\varepsilon})}{\varepsilon} O(\varepsilon^{2/3}) = \frac{(\vartheta_0 - \bar{\vartheta}_{\tau,\varepsilon})}{\varepsilon} + O(\varepsilon^{1/3}) \end{aligned} $$

because

$$ \int_{\tau}^{t} \frac{f(s)^2 \dot{m}(\vartheta_0, s) \dot{m}(\vartheta_0, s)^{\top}}{\sigma(s)^2} ds \longrightarrow \mathbf{I}_{\tau}^{t}(\vartheta_0). $$
The substitution of these limits in (23) proves (21).

To prove the weak convergence (22) we have to verify the estimate

$$ \mathbf{E}_{\vartheta_0} \| \eta_{t_2, \varepsilon} (\vartheta_0) - \eta_{t_1, \varepsilon} (\vartheta_0) \|^4 \leq C |t_2 - t_1|^2. $$

The calculations are direct but cumbersome. It uses the non deneracy of the Fisher information, boundedness of the derivatives and the elementary properties of stochastic and ordinary integrals. □

Remark that the equations of the adaptive filter (24),(25) are given in recurrent form. It is possible to write the One-step MLE-process in the recurrent form too. To simplify its calculation we replace in $(\bar{\vartheta}_{\tau, \varepsilon}, s)$ by $\dot{\bar{\vartheta}}(\bar{\vartheta}_{\tau, \varepsilon}, s)$ in (20). Then we multiply both sides of obtained equation by $\mathbf{I}_{\tau}^t(\bar{\vartheta}_{\tau, \varepsilon})$, denote the matrix

$$ \mathbf{L}(\vartheta, t) = \frac{f(t)^2 \dot{\vartheta}(\vartheta, t) \dot{\vartheta}(\vartheta, t)^{\top}}{\sigma(t)^2} $$

and take the differential of the equation

$$ \mathbf{I}_{\tau}^{t}(\bar{\vartheta}_{\tau, \varepsilon}) \vartheta_{t, \varepsilon}^{*} = \mathbf{I}_{\tau}^{t}(\bar{\vartheta}_{\tau, \varepsilon}) \bar{\vartheta}_{\tau, \varepsilon} + \int_{\tau}^{t} \frac{f(s) \dot{\vartheta}(\bar{\vartheta}_{\tau, \varepsilon}, s)}{\sigma(s)^2} [dX_s - f(s) m(\bar{\vartheta}_{\tau, \varepsilon}, s) ds]. $$

Then we obtain

$$ \mathbf{I}_{\tau}^{t}(\bar{\vartheta}_{\tau, \varepsilon}) d\vartheta_{t, \varepsilon}^{*} + \mathbf{L}(\bar{\vartheta}_{\tau, \varepsilon}, t) \vartheta_{t, \varepsilon}^{*} dt = \mathbf{L}(\bar{\vartheta}_{\tau, \varepsilon}, t) \bar{\vartheta}_{\tau, \varepsilon} \\ + \frac{f(s) \dot{\vartheta}(\bar{\vartheta}_{\tau, \varepsilon}, s)}{\sigma(s)^2} [dX_s - f(s) m(\bar{\vartheta}_{\tau, \varepsilon}, s) ds] $$

and

$$ d\vartheta_{t, \varepsilon}^{*} = \mathbf{I}_{\tau}^{t}(\bar{\vartheta}_{\tau, \varepsilon})^{-1} \mathbf{L}(\bar{\vartheta}_{\tau, \varepsilon}, t) [\bar{\vartheta}_{\tau, \varepsilon} - \vartheta_{t, \varepsilon}^{*}] dt \\ + \mathbf{I}_{\tau}^{t}(\bar{\vartheta}_{\tau, \varepsilon})^{-1} \frac{f(t) \dot{\vartheta}(\bar{\vartheta}_{\tau, \varepsilon}, t)}{\sigma(t)^2} [dX_t - f(t) m(\bar{\vartheta}_{\tau, \varepsilon}, t) ds]. $$

The Fisher information can be written in the recurrent form.

### 2.1.3 Adaptive filter

Suppose that we already have the estimators $\bar{\vartheta}_{\tau, \varepsilon}$ and $\vartheta_{t, \varepsilon}^* = (\theta_{1,t,\varepsilon}^*, \theta_{2,t,\varepsilon}^*)^\top$, $\tau < t \leq T$. Then the adaptive filter $m_{t,\varepsilon}^*, \tau_* < t \leq T$, where $\tau_* > \tau$ is

$$ dm_{t, \varepsilon}^* = a(\theta_{2,t, \varepsilon}^*, t) m_{t, \varepsilon}^* dt + \frac{f(t) \gamma_*^*(t)}{\sigma(t)^2} [dX_t - f(t) m_{t, \varepsilon}^* dt], \quad m_{\tau_*, \varepsilon}^*, \quad (24) $$
$$ \frac{d\gamma_{*}^{*}(t)}{dt} = 2a(\theta_{2,t,\varepsilon}^{*}, t)\gamma_{*}^{*}(t) - \frac{f(t)^2\gamma_{*}^{*}(t)^2}{\sigma(t)^2} + b(t)^2, \quad \gamma_{*}^{*}(\tau_{*}). \quad (25) $$

The initial value $\gamma_{*}^{*}(\tau_{*})$ is the solution of the equation

$$ \frac{d\gamma_{*}^{*}(s)}{ds} = 2a(\theta_{2,\tau,\varepsilon}^{*}, s) \gamma_{*}^{*}(s) - \frac{f(s)^2 \gamma_{*}^{*}(s)^2}{\sigma(s)^2} + b(s)^2, \quad \gamma_{*}^{*}(0) = 0, \quad 0 \le s \le \tau_{*} \quad (26) $$

at the point $s = \tau_{*}$.

To calculate $m_{\tau_{*},\varepsilon}^{*}$ we denote

$$ R(\theta_2, t) = \Phi(\theta_2, 0, t)^{-1} f(t) \gamma_*(\theta_2, t) \sigma(t)^{-2} = \Phi(\theta_2, 0, t)^{-1} B(\theta_2, t) $$

and remark that

$$ \begin{aligned} m(\vartheta, t) &= \Phi(\theta_2, 0, t) \left[ \theta_1 + \int_0^t R(\theta_2, s) \, dX_s \right] \\ &= \Phi(\theta_2, 0, t) \left[ \theta_1 + R(\theta_2, t) X_t - \int_0^t R'_s(\theta_2, s) X_s \, ds \right] \\ &= \theta_1 \Phi(\theta_2, 0, t) + \frac{f(t) \gamma_*(\theta_2, t)}{\sigma(t)^2} X_t - \int_0^t \Phi(\theta_2, s, t) [A'_s(\theta_2, s) + B'_s(\theta_2, s)] X_s \, ds. \end{aligned} $$

Here $R'_s(\vartheta, s)$ and $A'_s(\vartheta, s)$ are the derivatives of $R(\vartheta, s)$, $A(\vartheta, s)$ w.r.t. $s$.

The initial value is calculated as follows

$$ \begin{aligned} m_{\tau_{*}, \varepsilon}^{*} = \theta_{1, \tau_{*}, \varepsilon}^{*} \Phi(\theta_{2, \tau_{*}, \varepsilon}^{*}, 0, \tau_{*}) &+ \frac{f(\tau_{*}) \gamma_{*}(\theta_{2, \tau_{*}, \varepsilon}^{*}, \tau_{*})}{\sigma(\tau_{*})^{2}} X_{\tau_{*}} \\ &- \int_{0}^{\tau_{*}} \Phi(\theta_{2, \tau_{*}, \varepsilon}^{*}, s, \tau_{*}) [A'_{s}(\theta_{2, \tau_{*}, \varepsilon}^{*}, s) + B'_{s}(\theta_{2, \tau_{*}, \varepsilon}^{*}, s)] X_{s} \, ds. \end{aligned} \quad (27) $$

**Theorem 1.** Let the conditions $A_{*}$ and $B_{1} - B_{3}$ hold. Then the error of approximation

$$ \varepsilon^{-1} (m_{t, \varepsilon}^{*} - m(\vartheta_0, t)) \implies \xi_m (\vartheta_0, t), \quad \tau_{*} \le t \le T $$

where the Gaussian process $\xi_m(\vartheta_0, t)$, $\tau_{*} \le t \le T$ is defined below in (28).

*Proof.* Denote $\delta_m(t) = m_{t,\varepsilon}^*(t) - m(\vartheta_0, t)$, $\delta_{\vartheta}(t) = \vartheta_{t,\varepsilon}^* - \vartheta_0$, $\delta_{\theta_2}(t) = \theta_{2,t,\varepsilon}^* - \theta_{2,0,t,\varepsilon}$, $\delta_{\gamma}(t) = \gamma_{*}^*(t) - \gamma_*(\vartheta_{2,0}, t)$. We have already the estimate $\delta_{\vartheta}(t) = O(\varepsilon)$. The equations for $\delta_m(t)$ and $\delta_{\gamma}(t)$ we obtain as difference of equations for corresponding functions

$$ \begin{aligned} d\delta_m(t) ={}& A(\theta_{2,t,\varepsilon}^*, t) \delta_m(t) dt + [a(\theta_{2,t,\varepsilon}^*, t) - a(\theta_{2,0}, t)] m(\vartheta_0) dt \\ &+ \varepsilon \frac{f(t) \delta_{\gamma}(t)}{\sigma(t)} d\bar{W}_t, \quad \delta_m(\tau_*) \end{aligned} $$
$$ \frac{d\delta_{\gamma}(t)}{dt} = 2[a(\theta_{2,t,\varepsilon}^{*}, t) - a(\theta_{2,0}, t)]\gamma_{*}(\theta_{2,0}, t) + \left[2a(\theta_{2,0}, t) - \frac{f(t)^{2}[\gamma_{*}^{*}(t) + \gamma_{*}(\vartheta_{2,0}, t)]}{\sigma(t)^{2}}\right]\delta_{\gamma}(t), \quad \delta_{\gamma}(\tau_{*}). $$

We have $a(\theta_{2,t,\varepsilon}^*, t) - a(\theta_{2,0}, t) = \dot{a}(\theta_{2,0}, t)\delta_{\theta_2}(t)(1+O(\varepsilon))$. Using these relations, boundedness of the derivatives and Grönwell lemma it can be shown that $\delta_{\gamma}(t) = O(\varepsilon)$, $\delta_m(t) = O(\varepsilon)$.

Therefore

$$ d\delta_m(t) = A(\theta_{2,0}, t) \delta_m(t) dt + \dot{a}(\theta_{2,0}, t) m(\vartheta_0, t) \delta_{\vartheta}(t) dt + \varepsilon \frac{f(t) \delta_{\gamma}(t)}{\sigma(t)} d\bar{W}_t + O(\varepsilon^2), $$

$$ \frac{d\delta_{\gamma}(t)}{dt} = 2\dot{a}(\theta_{2,0}, t) \gamma_{*}(\theta_{2,0}, t) \delta_{\vartheta}(t) + 2A(\theta_{2,0}, t) \delta_{\gamma}(t) + O(\varepsilon^2). $$

The solutions of these linear equations are

$$ \delta_{\gamma}(t) = \delta_{\gamma}(\tau_{*}) \Phi(\theta_{2,0}, \tau_{*}, t)^2 + 2 \int_{\tau_{*}}^{t} \Phi(\theta_{2,0}, s, t)^2 \dot{a}(\theta_{2,0}, s) \gamma_{*}(\theta_{2,0}, s) \delta_{\vartheta}(s) ds + O(\varepsilon^2), $$

$$ \delta_m(t) = \delta_m(\tau_*) \Phi(\theta_{2,0}, \tau_*, t) + \int_{\tau_*}^{t} \Phi(\theta_{2,0}, s, t) \dot{a}(\theta_{2,0}, s) m(\vartheta_0, s) \delta_{\vartheta}(s) ds \\ + \varepsilon \int_{\tau_*}^{t} \Phi(\theta_{2,0}, s, t) f(s) \delta_{\gamma}(s) \sigma(s)^{-1} d\bar{W}_s + O(\varepsilon^2). $$

Solution of the equation related with (26) gives us the representation

$$ \begin{aligned} \varepsilon^{-1} \delta_{\gamma}(\tau_{*}) &= 2\varepsilon^{-1} \int_{0}^{\tau_{*}} \Phi(\theta_{2,0}, s, \tau_{*})^2 \dot{a}(\theta_{2,0}, s) \gamma_{*}(\theta_{2,0}, s) \delta_{\theta_2}(\tau_{*}) ds + O(\varepsilon) \\ &= 2 \int_{0}^{\tau_{*}} \Phi(\theta_{2,0}, s, \tau_{*})^2 \dot{a}(\theta_{2,0}, s) \gamma_{*}(\theta_{2,0}, s) \eta_{2,\tau_{*}}(\theta_{2}) ds \equiv \xi_{\gamma}(\vartheta_{0}). \end{aligned} $$

Hence

$$ \begin{aligned} \varepsilon^{-1} \delta_{\gamma}(t) &\implies \xi_{\gamma}(\vartheta_0) \Phi(\theta_{2,0}, \tau_{*}, t)^2 + 2 \int_{\tau_{*}}^{t} \Phi(\theta_{2,0}, s, t)^2 \dot{a}(\theta_{2,0}, s) \gamma_{*}(\theta_{2,0}, s) \eta_{2,s}(\vartheta_0) ds \\ &\equiv \xi_{\gamma}(\vartheta_0, t), \quad \tau_{*} \le t \le T. \end{aligned} $$

Let us denote $C(\theta_2, s) = A'_s(\theta_2, s) + B'_s(\theta_2, s)$ then we can write (see (27))

$$ \begin{aligned} \varepsilon^{-1} \delta_m(\tau_*) &\implies \Phi(\theta_{2,0}, 0, \tau_*) \eta_{1,\tau_*}(\vartheta_0) + \theta_{1,0} \dot{\Phi}(\theta_{2,0}, 0, \tau_*) \eta_{2,\tau_*}(\vartheta_0) + \frac{f(\tau_*)}{\sigma(\tau_*)^2} \xi_\gamma(\vartheta_0) x_{\tau_*}(\vartheta_0) \\ &\quad - \eta_{2,\tau_*}(\vartheta_0) \int_0^{\tau_*} \left[ \dot{\Phi}(\theta_{2,0}, 0, \tau_*) C(\theta_{2,0}, s) + \Phi(\theta_{2,0}, 0, \tau_*) \dot{C}(\theta_{2,0}, s) \right] x_s(\vartheta_0) ds \end{aligned} $$
$$ \equiv \xi_m(\vartheta_0). $$

Therefore

$$ \begin{aligned} \varepsilon^{-1}\delta_m(t) \implies & \xi_m(\vartheta_0)\Phi(\theta_{2,0}, \tau_*, t) + \int_{\tau_*}^t \Phi(\theta_{2,0}, s, t) \dot{a}(\theta_{2,0}, s) y_s(\vartheta_0) \eta_{2,s}(\vartheta_0) \, ds \\ & + \int_{\tau_*}^t \Phi(\theta_{2,0}, s, t) f(s) \sigma(s)^{-1} \xi_\gamma(\vartheta_0, s) \, dw(s) \equiv \xi_m(\vartheta_0, t), \quad \tau_* \le t \le T. \end{aligned} \quad (28) $$

□

## 2.2 Random initial value

Let us consider an example of partially observed system with the random initial value of non observed component:

$$ dX_t = fY_t dt + \varepsilon\sigma dW_t, \quad X_0 = 0, \quad 0 \le t \le T, \quad (29) $$

$$ dY_t = \vartheta Y_t dt + \varepsilon b dV_t, \quad Y_0 = y_0 \sim \mathcal{N}(0, d^2). \quad (30) $$

Here $f > 0, \sigma > 0, b > 0, d^2 > 0$ and $\vartheta \in \Theta = (\alpha, \beta), \alpha > 0$. The random variable $Y_0$ in independent on the Wiener processes $W_t, V_t; 0 \le t \le T$. The sample value of $Y_0$ we denote $y_0$. As usual, we have to estimate the parameter $\vartheta$ by observations $X^T = (X_t, 0 \le t \le T)$.

### 2.2.1 Parameter estimators

Remark that these processes can be written in explicit form

$$ X_t = \frac{f y_0}{\vartheta} [e^{\vartheta t} - 1] + \varepsilon f b \int_0^t \int_0^s e^{\vartheta(s-r)} dV_r \, ds + \varepsilon \sigma W_t, \quad 0 \le t \le T, $$

$$ Y_t = y_0 e^{\vartheta t} + \varepsilon b \int_0^t e^{\vartheta(t-s)} dV_s. $$

The limit system ($\vartheta_0$ is the true value)

$$ \frac{dx_t(\vartheta_0)}{dt} = f y_t(\vartheta_0), \quad x_0(\vartheta_0) = 0, \quad 0 \le t \le T, $$

$$ \frac{dy_t(\vartheta_0)}{dt} = \vartheta_0 y_t(\vartheta_0), \quad y_0(\vartheta_0) \sim \mathcal{N}(0, d^2) $$

gives us a couple of random processes $x_t(\vartheta_0), y_t(\vartheta_0), 0 \le t \le T$. As $y_t(\vartheta) = y_0 e^{\vartheta t}$, we have the equality

$$ x_t(\vartheta) = \frac{y_0 f}{\vartheta} (e^{\vartheta t} - 1), \quad 0 \le t \le T. $$
There are several ways to construct a preliminary estimator of $\vartheta_0$. Note that we can not define the MDE by the equality

$$ \check{\vartheta}_{\tau, \varepsilon} = \arg \inf_{\vartheta \in \Theta} \int_{0}^{\tau} [X_t - x_t(\vartheta)]^2 dt $$

because $y_0$ is unknown and it's conditional expectation is always $m(\vartheta, 0) = 0$.

Let us denote $x'_t(\vartheta)$ the derivative of $x_t(\vartheta)$ w.r.t. $t$ and remark that $(t_1 < t_2)$

$$ \vartheta_0 = \frac{\ln |x'_{t_2}(\vartheta_0)| - \ln |x'_{t_1}(\vartheta_0)|}{|t_2 - t_1|} $$

Therefore if we have a consistent estimator $X'_{t,\varepsilon}$ of the derivative, then the MME

$$ \vartheta_{t_2, \varepsilon}^* = \frac{\ln |X'_{t_2, \varepsilon}| - \ln |X'_{t_1, \varepsilon}|}{|t_2 - t_1|} \quad (31) $$

can be consistent too.

Note that for the estimator of the derivative $X'_{t,\varepsilon} = (X_{t+\delta_\varepsilon} - X_t)/\delta_\varepsilon$ we have

$$ \begin{aligned} X'_{t,\varepsilon} &= \frac{f y_0}{\delta_\varepsilon} \int_t^{t+\delta_\varepsilon} e^{\vartheta_0 s} ds + \frac{f b \varepsilon}{\delta_\varepsilon} \int_t^{t+\delta_\varepsilon} \int_0^s e^{\vartheta_0 (s-r)} dV_r ds + \frac{\varepsilon}{\delta_\varepsilon} \sigma [W_{t+\delta_\varepsilon} - W_t] \\ &= f y_0 e^{\vartheta_0 t} + \frac{f \vartheta_0 y_0}{2} e^{\vartheta_0 t} \delta_\varepsilon + f b \varepsilon \int_0^t e^{\vartheta_0 (t-r)} dV_r + \frac{\varepsilon}{\sqrt{\delta_\varepsilon}} \sigma \xi_{t,\varepsilon} + O\left(\varepsilon \sqrt{\delta_\varepsilon}\right) \end{aligned} $$

where $\xi_{t,\varepsilon} \sim \mathcal{N}(0, 1)$. The correct rate $\delta_\varepsilon = \varepsilon^{2/3}$ we obtain as solution of the equation $\delta_\varepsilon = \varepsilon/\sqrt{\delta_\varepsilon}$. Taylor expansion allows to write

$$ \begin{aligned} \ln |X'_{t,\varepsilon}| &= \ln \left| f y_0 e^{\vartheta_0 t} \left( 1 + \left[ \frac{\vartheta_0}{2} + \frac{\sigma \xi_{t,\varepsilon}}{f y_0} e^{-\vartheta_0 t} \right] \varepsilon^{2/3} (1 + o(1)) \right) \right| \\ &= \ln (f |y_0|) + \vartheta_0 t + \ln \left| 1 + \left[ \frac{\vartheta_0}{2} + \frac{\sigma \xi_{t,\varepsilon}}{f y_0} e^{-\vartheta_0 t} \right] \varepsilon^{2/3} (1 + o(1)) \right| \\ &= \ln (f |y_0|) + \vartheta_0 t + \left[ \frac{\vartheta_0}{2} + \frac{\sigma \xi_{t,\varepsilon}}{f y_0} e^{-\vartheta_0 t} \right] \varepsilon^{2/3} (1 + o(1)), \end{aligned} $$

where the last equality is valid asymptotically as $\varepsilon \to 0$.

Therefore we proved the following proposition.

**Proposition 2.** For the MME $\vartheta_{t_2,\varepsilon}^*$ the following relation holds

$$ \varepsilon^{-2/3} (\vartheta_{t_2,\varepsilon}^* - \vartheta_0) \implies \frac{\sigma}{f(t_2 - t_1)} \left[ \frac{\xi_{t_2}}{y_0} e^{-\vartheta_0 t_2} - \frac{\xi_{t_1}}{y_0} e^{-\vartheta_0 t_1} \right]. $$

Here $\xi_{t_1} \sim \mathcal{N}(0, 1)$, $\xi_{t_2} \sim \mathcal{N}(0, 1)$ and $y_0 \sim \mathcal{N}(0, d^2)$ are independent Gaussian random variables.
Remark that $\xi_{t_1}d/y_0$ and $\xi_{t_2}d/y_0$ are Cauchy variables and hence there is no convergence of moments of this estimator.

It is easy to see that $\varepsilon^{-2/3}y_0(\vartheta_{t_2,\varepsilon}^* - \vartheta_0)$ is asymptotically normal.

**Remark 1.** Introduce $t_{i,\varepsilon} = p_i/\ln(1/\varepsilon) \to 0$, $i = 1, 2$. Here $p_1 > 0, p_2 > p_1$ and $0 < t_{i,\varepsilon} < T$. Then the normalizing and limit change

$$ \ln \left( \frac{1}{\varepsilon} \right) \varepsilon^{-2/3} (\vartheta_{t_2, \varepsilon}^* - \vartheta_0) \implies \frac{\sigma}{f(p_2 - p_1)} \left[ \frac{\xi_{p_2}}{y_0} - \frac{\xi_{p_1}}{y_0} \right]. \quad (32) $$

Here $\xi_{p_1} \sim \mathcal{N}(0, 1)$, $\xi_{p_2} \sim \mathcal{N}(0, 1)$ are independent random variables.

The Kalman-Bucy filter for conditional expectation $m(\vartheta, t), 0 \le t \le T$ is

$$ dm(\vartheta, t) = \vartheta m(\vartheta, t) dt + \frac{f\gamma(\vartheta, t)}{\varepsilon^2\sigma^2} [dX_t - fm(\vartheta, t)dt], \quad m(\vartheta, 0) = 0, \quad (33) $$

$$ \frac{\partial \gamma(\vartheta, t)}{\partial t} = 2\vartheta \gamma(\vartheta, t) - \frac{f^2 \gamma(\vartheta, t)^2}{\varepsilon^2 \sigma^2} + \varepsilon^2 b^2, \quad \gamma(\vartheta, 0) = d^2. \quad (34) $$

The solution of the equation (33) can be written explicitly

$$ \begin{aligned} m(\vartheta, t) &= \int_0^t \Phi(\vartheta, s, t) \frac{f^2 \gamma(\vartheta, s)}{\varepsilon^2 \sigma^2} Y_s \, ds + \int_0^t \Phi(\vartheta, s, t) \frac{f \gamma(\vartheta, s)}{\varepsilon \sigma} \, dW_s \\ &= y_0 \int_0^t \Phi(\vartheta, s, t) \frac{f^2 \gamma(\vartheta, s)}{\varepsilon^2 \sigma^2} e^{\vartheta_0 s} \, ds + \int_0^t \Phi(\vartheta, s, t) \frac{f \gamma(\vartheta, s)}{\varepsilon \sigma} \, dW_s \\ &\quad + \int_0^t \Phi(\vartheta, s, t) \frac{f^2 \gamma(\vartheta, s)}{\varepsilon \sigma^2} \int_0^s e^{\vartheta_0(s-r)} dV_r \, ds \end{aligned} $$

where

$$ \Phi(\vartheta, s, t) = \exp \left( \int_s^t \left[ \vartheta - \frac{f^2 \gamma(\vartheta, r)}{\varepsilon^2 \sigma^2} \right] dr \right). $$

Let us see the asymptotic of the main term

$$ I(\tau, \varepsilon) \equiv y_0 \int_0^\tau \Phi(\vartheta, s, \tau) \frac{f^2 \gamma(\vartheta, s)}{\varepsilon^2 \sigma^2} e^{\vartheta_0 s} \, ds $$

for the small values of $\tau$. Integrating by parts we obtain

$$ \begin{aligned} I(\tau, \varepsilon) &= y_0 \Phi(\vartheta, 0, \tau) \int_0^\tau \exp \left( \int_0^s \frac{f^2 \gamma(\vartheta, r)}{\varepsilon^2 \sigma^2} dr \right) \frac{f^2 \gamma(\vartheta, s)}{\varepsilon^2 \sigma^2} e^{(\vartheta_0 - \vartheta)s} ds \\ &= y_0 \Phi(\vartheta, 0, \tau) \int_0^\tau e^{(\vartheta_0 - \vartheta)s} d \exp \left( \int_0^s \frac{f^2 \gamma(\vartheta, r)}{\varepsilon^2 \sigma^2} dr \right) \\ &= y_0 e^{\vartheta_0 \tau} - y_0 \Phi(\vartheta, 0, \tau) - y_0 (\vartheta_0 - \vartheta) \int_0^\tau \Phi(\vartheta, s, \tau) e^{\vartheta_0 s} ds \end{aligned} $$
$$ = y_0 e^{\vartheta_0 \tau} + O(\varepsilon^2). \qquad (35) $$

The function $\gamma_*(\vartheta, t) = \varepsilon^{-2}\gamma(\vartheta, t)$ is solution of the Riccati equation

$$ \frac{\partial \gamma_*(\vartheta, t)}{\partial t} = 2\vartheta \gamma_*(\vartheta_*, t) - \frac{f^2 \gamma_*(\vartheta, t)^2}{\sigma^2} + b^2, \qquad \gamma_*(\vartheta, 0) = \frac{d^2}{\varepsilon^2}. $$

Recall that Riccati equation with constant coefficients has the following explicite solution (see [1])

$$ \gamma_*(\vartheta, t) = e^{-2r(\vartheta)t} \left[ \frac{1}{\frac{d^2}{\varepsilon^2} - \frac{\sigma^2}{f^2}(\vartheta + r(\vartheta))} + \frac{f^2(1 - e^{-2r(\vartheta)t})}{2\sigma^2 r(\vartheta)} \right]^{-1} + \frac{\sigma^2}{f^2}(\vartheta + r(\vartheta)). $$

Here $r(\vartheta) = (\vartheta^2 + b^2 f^2 \sigma^{-2})^{1/2}$. The function $\gamma_*(\vartheta, t)$ for any $t \in (0, T]$ has the asymptotic expansion

$$ \gamma_*(\vartheta, t) = \frac{\sigma^2}{f^2} \left[ \vartheta + r(\vartheta) + \frac{2r(\vartheta)}{e^{2r(\vartheta)t} - 1} \right] (1 + O(\varepsilon^2)) = \gamma^*(\vartheta, t) (1 + O(\varepsilon^2)) $$

where the function $\gamma^*(\vartheta, t)$ is defined by the last equality. Moreover, for any $\tau \in (0, T]$ we have the uniform convergence too

$$ \sup_{\tau \le t \le T} |\gamma_*(\vartheta, t) - \gamma^*(\vartheta, t)| \le C\varepsilon^2. $$

Therefore for any $t \in (0, T)$ we have the approximation

$$ \gamma(\vartheta, t) = \varepsilon^2 \gamma^*(\vartheta, t) (1 + O(\varepsilon^2)). $$

Omitting the terms of order $O(\varepsilon^2)$ we obtain the representation

$$ \begin{aligned} m^*(\vartheta, t) = m(\vartheta, \tau) \Phi^*(\vartheta, \tau, t) &+ y_0 \int_{\tau}^{t} \Phi^*(\vartheta, s, t) \frac{f^2 \gamma^*(\vartheta, s)}{\sigma^2} e^{\vartheta_0 s} \, ds \\ &+ \varepsilon \int_{\tau}^{t} \Phi^*(\vartheta, s, t) \frac{f \gamma(\vartheta, s)}{\sigma} \, dW_s \\ &+ \varepsilon \int_{\tau}^{t} \Phi^*(\vartheta, s, t) \frac{f^2 \gamma^*(\vartheta, s)}{\sigma^2} \int_{0}^{s} e^{\vartheta_0 (s-r)} \, dV_r \, ds \\ = & y_0 e^{\vartheta_0 \tau} \Phi^*(\vartheta, \tau, t) + y_0 \int_{\tau}^{t} \Phi^*(\vartheta, s, t) \frac{f^2 \gamma^*(\vartheta, s)}{\sigma^2} e^{\vartheta_0 s} \, ds + o(1), \end{aligned} $$

where

$$ \Phi^*(\vartheta, s, t) = \exp \left( \int_s^t \left[ \vartheta - \frac{f^2 \gamma^*(\vartheta, r)}{\sigma^2} \right] \, dr \right). $$
The initial value $m(\vartheta, \tau)$ we obtain as solution of the equation (33).

For the derivative $\dot{m}_t(\vartheta) = \partial m_t(\vartheta) / \partial \vartheta$ we have the equation

$$d\dot{m}_t(\vartheta) = \left[\vartheta - \frac{f^2\gamma^*(\vartheta, t)}{\sigma^2}\right] \dot{m}_t(\vartheta) dt + \frac{f\dot{\gamma}^*(\vartheta, t)}{\sigma^2} dX_t + \left[1 - \frac{f^2\dot{\gamma}^*(\vartheta, t)}{\sigma^2}\right] m_t(\vartheta) dt$$

and obtain the expression

$$\begin{aligned} \dot{m}_t(\vartheta) &= y_0 \int_{\tau}^{t} \Phi^*(\vartheta, s, t) \left[ t - s + \frac{f^2}{\sigma^2} \left( \dot{\gamma}^*(\vartheta, s) - \int_{s}^{t} \dot{\gamma}^*(\vartheta, s) dr \right) \right] e^{\vartheta_0 s} ds + o(1) \\ &= y_0 H(\vartheta, \vartheta_0, t) + o(1) \rightarrow y_0 H(\vartheta, \vartheta_0, t) \end{aligned}$$

as $\varepsilon \to 0$.

Therefore the empirical Fisher information function converges in probability

$$\begin{aligned} I_{\tau}^{t}(\vartheta_{0}) &= \frac{f^2}{\sigma^2} \int_{\tau}^{t} \dot{m}_{s}(\vartheta_{0})^2 ds \rightarrow \frac{y_{0}^{2} f^2}{\sigma^2} \int_{\tau}^{t} H(\vartheta_{0}, \vartheta_{0}, s)^2 ds \\ &= y_{0}^{2} G(\vartheta_{0}, T). \end{aligned}$$

**Remark 2.** The family of measures $(\mathbf{P}_{\vartheta}, \vartheta \in \Theta)$ is locally asymptotically mixing normal (LAMN). The normalized likelihood ratio function constructed by the observations $X_{\tau}^T = (X_t, \tau \le t \le T)$

$$Z_{\varepsilon}(\vartheta_0, u) = \frac{L(\vartheta_0 + \varepsilon u, X_{\tau}^T)}{L(\vartheta_0, X_{\tau}^T)}, \quad u \in \mathbb{U}_{\varepsilon} = \left( \frac{\alpha - \vartheta_0}{\varepsilon}, \frac{\beta - \vartheta_0}{\varepsilon} \right)$$

admits the representation

$$Z_{\varepsilon}(\vartheta_0, u) = \exp \left( u y_0 \Delta_{\varepsilon}(\vartheta_0, X_{\tau}^T) - \frac{u^2}{2} y_0^2 G(\vartheta_0, T) + r_{\varepsilon} \right).$$

Here $r_{\varepsilon} \to 0$ and

$$\Delta_{\varepsilon}(\vartheta_0, X^T) \implies \Delta(\vartheta_0) \sim \mathcal{N}(0, G(\vartheta_0, T)).$$

We do not study the MLE but note that it has the limit Cauchy distribution

$$\frac{\hat{\vartheta}_{\varepsilon} - \vartheta_{0}}{\varepsilon} \implies \hat{u} = \frac{\Delta(\vartheta_{0})}{y_{0} G(\vartheta_{0}, T)}$$

The K-B filter on the time interval $[\tau, T] (\tau > 0)$ we re-write as follows

$$d m_{*}(\vartheta, t) = \left[ \vartheta - \frac{f^{2} \gamma^{*}(\vartheta, t)}{\sigma^{2}} \right] m_{*}(\vartheta, t) d t + \frac{f \gamma^{*}(\vartheta, t)}{\sigma^{2}} d X_{t}. \quad (36)$$
The initial value $m_*(\vartheta, \tau) = m(\vartheta, \tau)$ can be calculated by solving the system (33)-(34) on the interval $[0, \tau]$.

The equation for the derivative $\dot{m}_*(\vartheta, t)$ is

$$ \begin{aligned} \mathrm{d}\dot{m}_*(\vartheta, t) = & [1 - f^2\dot{\gamma}^*(\vartheta, t)\sigma^{-2}] m_*(\vartheta, t) \mathrm{d}t + [\vartheta - f^2\gamma^*(\vartheta, t)\sigma^{-2}] \dot{m}_*(\vartheta, t) \mathrm{d}t \\ & + f\dot{\gamma}^*(\vartheta, t)\sigma^{-2}\mathrm{d}X_t, \quad \dot{m}_*(\vartheta, \tau), \quad \tau < t \le T. \end{aligned} \quad (37) $$

Note that the calculation of the $m_*(\vartheta, t)$ and $\dot{m}_*(\vartheta, t)$ by these equations does not depend on $y_0$, which appears in the limits only.

If $\varepsilon \to 0$ then

$$ \begin{aligned} m_*(\vartheta, t) \to y_t(\vartheta, \vartheta_0) = y_0 e^{\vartheta_0 \tau} (\vartheta, \vartheta_0) \Phi^*(\vartheta, \tau, t) \\ + y_0 \int_{\tau}^{t} \Phi^*(\vartheta, s, t) f^2 \gamma^*(\vartheta, s) \sigma^{-2} e^{\vartheta_0 s} \mathrm{d}s = y_0 h_t(\vartheta, \vartheta_0), \end{aligned} $$

where the function $h_t(\vartheta, \vartheta_0)$ is defined by the last equality.

The expression for the limit of the derivative $\dot{m}(\vartheta, t) \to \dot{y}_t(\vartheta, \vartheta_0)$ is the following random function

$$ \begin{aligned} \dot{y}_t(\vartheta, \vartheta_0) &= \dot{y}_\tau(\vartheta, \vartheta_0) \Phi^*(\vartheta, 0, \tau) \\ &\quad + y_0 \int_{\tau}^{t} \Phi^*(\vartheta, s, t) \left[ h_s(\vartheta, \vartheta_0) + \frac{f^2 \dot{\gamma}^*(\vartheta, s)}{\sigma^2} [e^{\vartheta_0 s} - h_s(\vartheta, \vartheta_0)] \right] \mathrm{d}s \\ &= \dot{y}_0 e^{\vartheta_0 \tau} \tau \Phi^*(\vartheta, 0, \tau) + y_0 \bar{H}_t(\vartheta, \vartheta_0) \equiv y_0 \bar{G}_t(\vartheta, \vartheta_0), \end{aligned} $$

where $\bar{H}_t(\vartheta, \vartheta_0)$ and $\bar{G}_t(\vartheta, \vartheta_0)$ are defined by the last two equalities. If $\vartheta = \vartheta_0$, then $h_s(\vartheta_0, \vartheta_0) = e^{\vartheta_0 s}$ and

$$ \begin{aligned} \dot{y}_t(\vartheta_0, \vartheta_0) &= \dot{y}_\tau(\vartheta_0, \vartheta_0) \Phi^*(\vartheta_0, 0, \tau) + y_0 e^{\vartheta_0 t} \int_{\tau}^{t} \exp \left( -\frac{f^2}{\sigma^2} \int_{s}^{t} \gamma^*(\vartheta_0, r) \mathrm{d}r \right) \mathrm{d}s \\ &= y_0 e^{\vartheta_0 \tau} \tau \Phi^*(\vartheta_0, 0, \tau) + y_0 \bar{H}_t(\vartheta_0, \vartheta_0) = y_0 \bar{G}_t(\vartheta_0, \vartheta_0). \end{aligned} $$

These limits are proved for the integrals on the intervals $[\tau, T]$, but the value $\tau$ can be taken close to 0. It can be verified that the integral on the interval $[0, \tau]$ can be done less than any constant by choosing sufficiently small $\tau$. That is why in the expressions for the limits we use the integrals on the intervals $(0, T]$.

The Fisher information we write in two forms: for $\varepsilon > 0$ and $\varepsilon = 0$

$$ \begin{aligned} I_{\tau, \varepsilon}^t(\vartheta) &= \frac{f^2}{\sigma^2} \int_{\tau}^{t} \dot{m}_*(\vartheta, s)^2 \mathrm{d}s, \quad \tau < t \le T, \\ I_{\tau}^t(\vartheta_0) &= y_0^2 J_{\tau}^t(\vartheta_0), \end{aligned} $$
where $J_{\tau}^{t}(\vartheta_{0})$ is the corresponding deterministic function. We have $I_{\tau, \varepsilon}^{t}(\vartheta_{0}) \to I_{\tau}^{t}(\vartheta_{0})$ as $\varepsilon \to 0$.

The One-step MLE-process $\vartheta_{t, \varepsilon}^{*}, \tau < t \le T$ can be defined by the equality

$$ \vartheta_{t, \varepsilon}^{*} = \vartheta_{\tau_{*}, \varepsilon}^{*} + I_{\tau_{*}, \varepsilon}^{t} (\vartheta_{\tau, \varepsilon}^{*})^{-1} \frac{f}{\sigma^2} \int_{\tau}^{t} \dot{m}_{*}(\vartheta_{\tau_{*}, \varepsilon}^{*}, s) [dX_{s} - f m_{*}(\vartheta_{\tau_{*}, \varepsilon}^{*}, s) ds]. $$

Here the MME $\vartheta_{\tau_{*}, \varepsilon}^{*}$ is calculated for the values $t_2 = \frac{3\tau}{4}$ and $t_1 = \frac{\tau}{2}$. The random processes $m_{*}(\vartheta_{\tau_{*}, \varepsilon}^{*}, s), \tau \le t \le T$ and $\dot{m}_{*}(\vartheta_{\tau_{*}, \varepsilon}^{*}, s), \tau \le t \le T$ are obtained as solutions of (36) and (37) subject to the initial values calculated as solutions of the same equations but written without stochastic integrals as it was done in (27).

We have the representation

$$ \frac{\vartheta_{t, \varepsilon}^{*} - \vartheta_{0}}{\varepsilon} = \frac{\vartheta_{\tau_{*}, \varepsilon}^{*} - \vartheta_{0}}{\varepsilon} + I_{\tau_{*}, \varepsilon}^{t} (\vartheta_{\tau, \varepsilon}^{*})^{-1} \frac{f}{\sigma} \int_{\tau}^{t} \dot{m}_{*}(\vartheta_{\tau_{*}, \varepsilon}^{*}, s) d\bar{W}_{s} \\ + I_{\tau_{*}, \varepsilon}^{t} (\vartheta_{\tau, \varepsilon}^{*})^{-1} \frac{f^2}{\varepsilon\sigma^2} \int_{\tau}^{t} \dot{m}_{*}(\vartheta_{\tau_{*}, \varepsilon}^{*}, s) [m(\vartheta_{0}, s) - m_{*}(\vartheta_{\tau_{*}, \varepsilon}^{*}, s)] ds. $$

Following the main steps of the proof of the Proposition 1 it can be shown that this estimator for any $t \in (\tau, T]$ is asymptotically conditionally normal:

$$ \hat{\eta}_{t, \varepsilon}(\vartheta_0) = \frac{\vartheta_{t, \varepsilon}^{*} - \vartheta_0}{\varepsilon} \implies \hat{\eta}_t(\vartheta_0) = I_{\tau}^{t}(\vartheta_0)^{-1} \frac{f}{\sigma} \int_{\tau}^{t} \dot{y}(\vartheta_0, \vartheta_0, s) dw(s), \\ = y_0^{-1} J_{\tau}^{t}(\vartheta_0)^{-1} \frac{f}{\sigma} \int_{\tau}^{t} \bar{G}_s(\vartheta_0, \vartheta_0) dw(s) \equiv y_0^{-1} \zeta_{\vartheta}(\vartheta_0, t), $$

where $\zeta_m(\vartheta_0, t), \tau \le t \le T$ is a Gaussian process.

We have as well the following asymptotic normality

$$ \frac{y_0 (\vartheta_{t, \varepsilon}^* - \vartheta_0)}{\varepsilon} \implies \zeta_{\vartheta}(\vartheta_0, t) \sim \mathcal{N} (0, J_{\tau}^t (\vartheta_0)^{-1}), \quad \tau < t \le T. $$

### 2.2.2 Adaptive filter

The adaptive Kalman filter is

$$ dm_{t, \varepsilon}^{*} = [\vartheta_{t, \varepsilon}^{*} - f^2 \gamma^{*} (\vartheta_{t, \varepsilon}^{*}, t) \sigma^{-2}] m_{t, \varepsilon}^{*} dt + f \gamma^{*} (\vartheta_{t, \varepsilon}^{*}, t) \sigma^{-2} dX_{t}, \quad \tau \le t \le T. $$

The initial value $m_{\tau, \varepsilon}^{*} = m(\vartheta_{\tau, \varepsilon}^{*}, \tau)$ is calculated as it was explained above in (27).

**Proposition 3.** The error of approximation $m_{t, \varepsilon}^{*} - m(\vartheta_0, t)$ has the following asymptotics

$$ \varepsilon^{-1} (m_{t, \varepsilon}^{*} - m(\vartheta_0, t)) \implies \xi_{m}^{*} (\vartheta_0, t), \quad \tau \le t \le T, $$

where the Gaussian process $\xi_m^*(\vartheta_0, t)$ defined in (38) below does not depend on $y_0$.
Proof. This equation and the equation (36) with $\vartheta = \vartheta_0$ allow us to write the equation for the error of estimation $\delta_m(t) = m_{t,\varepsilon}^* - m(\vartheta_0, t)$

$$ \delta_m(t) = \delta_m(\tau) \Phi^*(\vartheta_0, 0, \tau) + \int_{\tau}^{t} \Phi(\vartheta_0, s, t) [A(\vartheta_{t,\varepsilon}^*, s) - A(\vartheta_0, s)] m_{s,\varepsilon}^* ds \\ + f\sigma^{-2} \int_{\tau}^{t} \Phi^*(\vartheta_0, s, t) [\gamma^*(\vartheta_{s,\varepsilon}^*, s) - \gamma^*(\vartheta_0, s)] dX_s. $$

Recall that the difference between $\gamma(\vartheta, t)$ and $\varepsilon^2\gamma^*(\vartheta, t)$ is of order $\varepsilon^2$. Therefore the difference $m^*(\vartheta_0, t) - m_*(\vartheta_0, t)$ is of the same order. We have

$$ \delta_m(t) = \dot{m}(\vartheta_0, \tau) \delta_\vartheta(\tau) \Phi^*(\vartheta_0, 0, \tau) + \int_{\tau}^{t} \Phi^*(\vartheta_0, s, t) \dot{A}(\vartheta_0, s) m_{s,\varepsilon}^* \delta_\vartheta(s) ds \\ + f\sigma^{-2} \int_{\tau}^{t} \Phi^*(\vartheta_0, s, t) \dot{\gamma}^*(\vartheta_0, s) \delta_\vartheta(s) dX_s + O(\varepsilon^2) $$

where $\delta_\vartheta(s) = \vartheta_{s,\varepsilon}^* - \vartheta_0$. Therefore

$$ \begin{aligned} \varepsilon^{-1}\delta_m(t) &= \dot{m}(\vartheta_0, \tau) \Phi^*(\vartheta_0, 0, \tau) \varepsilon^{-1}\delta_\vartheta(\tau) + \int_{\tau}^{t} \Phi^*(\vartheta_0, s, t) \dot{A}(\vartheta_0, s) m_{s,\varepsilon}^* \varepsilon^{-1}\delta_\vartheta(s) ds \\ &\quad + f\sigma^{-2} \int_{\tau}^{t} \Phi^*(\vartheta_0, s, t) \dot{\gamma}^*(\vartheta_0, s) \varepsilon^{-1}\delta_\vartheta(s) dX_s + O(\varepsilon) \\ \implies \dot{y}(\vartheta_0, \vartheta_0, \tau) \Phi^*(\vartheta_0, 0, \tau) \eta_\tau(\vartheta_0) + \int_{\tau}^{t} \Phi^*(\vartheta_0, s, t) \dot{A}(\vartheta_0, s) y_s(\vartheta_0) \eta_s(\vartheta_0) ds \\ &\quad + f^2\sigma^{-2} \int_{\tau}^{t} \Phi^*(\vartheta_0, s, t) \dot{\gamma}^*(\vartheta_0, s) y_s(\vartheta_0) \eta_s(\vartheta_0) ds \\ &= G_\tau(\vartheta_0, \vartheta_0) \Phi^*(\vartheta_0, 0, \tau) \zeta(\vartheta_0, \tau) \\ &\quad + \int_{\tau}^{t} \Phi^*(\vartheta_0, s, t) \dot{A}(\vartheta_0, s) H_s(\vartheta_0, \vartheta_0) \zeta(\vartheta_0, s) ds \\ &\quad + f^2\sigma^{-2} \int_{\tau}^{t} \Phi^*(\vartheta_0, s, t) \dot{\gamma}^*(\vartheta_0, s) H_s(\vartheta_0, \vartheta_0) \zeta(\vartheta_0, s) ds \\ &= G_\tau(\vartheta_0, \vartheta_0) \Phi^*(\vartheta_0, 0, \tau) \zeta(\vartheta_0, \tau) + \int_{\tau}^{t} \Phi^*(\vartheta_0, s, t) H_s(\vartheta_0, \vartheta_0) \zeta(\vartheta_0, s) ds \\ &= \xi_m^*(\vartheta_0, t). \end{aligned} \tag{38} $$

<figure><img src="image_3.png" /></figure>

### 3 Discussion

We considered three relatively simple models. The obtained results can be generalized for more general, say, multidimensional partially observed systems. The principal question in each model is: how to construct the preliminary estimator.
Remark, that in the third problem (random initial value) the family of measures of this statistical experiment with unknown parameter $\vartheta$ is locally asymptotically mixing normal (LAMN) (see Remark 2) The study of MLE can be an interesting problem but we will not do it here.

Of course there is a problem of definition of asymptotic efficiency of adaptive filter in the problem of estimation of $m(\vartheta_0, t)$. Remind that in the construction of the lower bound we obtain on the right hand side the expression like

$$ \mathbf{E}_{\vartheta} [\dot{m}(\vartheta, t) I_0^t (\vartheta)^{-1} \Delta_{\varepsilon} (\vartheta, X^t)]^2 $$

where the random variable $y_0$ vanishes in the limit. This probably can be proved but we do not occupy by this problem here. In any case the proposed here One-step MLE-process with fixed $\tau$ is not asymptotically efficient.

Hence the MLE $\hat{\vartheta}_{\varepsilon}$ can be asymptotically conditionally normal, but in the problem of estimation $m(\vartheta_0, t)$ the asymptotically efficient estimator can be asymptotically normal. Of course there is a problem of definition of asymptotic efficiency in the problem of estimation $m(\vartheta_0, t)$ (see [26]-[28] where the question of asymptotic optimality of the adaptive filters was discussed).

**Acknowledgments.** I am grateful to Reviwers for useful comments. This research was financially supported by the Russian Science Foundation research project No 24-11-00191.

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