Mathematical model for inference

The innovation estimator.^[1] for SDEs is defined in the framework of continuous-discrete state space models.^[2] These models arise as natural mathematical representation of the temporal evolution of continuous random phenomena and their measurements in a succession of time instants. In the simplest formulation, these continuous-discrete models ^[2] are expressed in term of a SDE of the form

${\displaystyle \qquad \qquad d\mathbf {x} (t)=\mathbf {f} (t,\mathbf {x} (t);\theta )dt+\sum _{i=1}^{m}\mathbf {g} _{i}\ (t,\mathbf {x} (t);\theta )\ d\mathbf {w} ^{i}(t)\qquad \qquad (1)}$

describing the time evolution of ${\displaystyle d}$ state variables ${\displaystyle \mathbf {x} }$ of the phenomenon for all time instant ${\displaystyle t\geq t_{0}}$, and an observation equation

${\displaystyle \qquad \qquad \mathbf {z} _{t_{k}}=\mathbf {Cx} (t_{k})+\mathbf {e} _{t_{k}}\qquad \qquad (2)}$

describing the time series of measurements ${\displaystyle \mathbf {z} _{t_{0}},...,\mathbf {z} _{t_{M-1}}}$of at least one of the variables ${\displaystyle \mathbf {x} }$ of the random phenomenon on ${\displaystyle M}$ time instants ${\displaystyle t_{0}\ ,...,\ t_{M-1}}$. In the model (1)-(2), ${\displaystyle \mathbf {f} }$ and ${\displaystyle \mathbf {g} _{i}}$ are differentiable functions, ${\displaystyle \mathbf {w} =(\mathbf {w} ^{1},...,\mathbf {w} ^{m})}$ is an ${\displaystyle m}$-dimensional standard Wiener process, ${\displaystyle \theta \in \mathbb {R} ^{p}}$ is a vector of ${\displaystyle p}$ parameters, ${\displaystyle \{\mathbf {e} _{t_{k}}:\mathbf {e} _{t_{k}}\sim \mathrm {N} (0,\Pi _{t_{k}})\}_{k=0,...,M-1}}$ is a sequence of ${\displaystyle r}$-dimensional i.i.d. Gaussian random vectors independent of ${\displaystyle \mathbf {w} }$, ${\displaystyle \Pi _{t_{k}}}$ an ${\displaystyle r\times r}$ positive definite matrix, and ${\displaystyle \mathbf {C} }$ an ${\displaystyle r\times d}$ matrix.

Statistical problem to solve

Once the dynamics of a phenomenon is described by a state equation as (1) and the way of measurement the state variables specified by an observation equation as (2), the inference problem to solve is the following:^[1][3] given ${\displaystyle M}$ partial and noisy observations ${\displaystyle \mathbf {z} _{t_{0}},...,\mathbf {z} _{t_{M-1}}}$ of the stochastic process ${\displaystyle \mathbf {x} }$ on the observation times ${\displaystyle t_{0},...,t_{M-1}}$, estimate the unobserved state variable of ${\displaystyle \mathbf {x} }$ and the unknown parameters ${\displaystyle \theta }$ in (1) that better fit to the given observations.

Differences with the maximum likelihood estimator

The maximum likelihood estimator of the parameters ${\displaystyle \theta }$ in the model (1)-(2) involves the evaluation of the - usually unknown - transition density function ${\displaystyle p_{\theta }(t_{k+1}-t_{k},\mathbf {x} (t_{k}),\mathbf {x} (t_{k+1}))}$ between the states ${\displaystyle \mathbf {x} (t_{k})}$ and ${\displaystyle \mathbf {x} (t_{k+1})}$ of the diffusion process ${\displaystyle \mathbf {x} }$ for all the observation times ${\displaystyle t_{k}}$ and ${\displaystyle t_{k+1}}$.^[7] Instead of this, the innovation estimator (5) is obtained by maximizing the likelihood of the discrete-time innovation process ${\displaystyle \{\nu _{t_{k}}\}_{k=1,...,M-1},}$ taking into account that ${\displaystyle \nu _{t_{1}},...,\nu _{t_{M-1}}}$ are Gaussian and independent random vectors. Remarkably, whereas the transition density function ${\displaystyle p_{\theta }(t_{k+1}-t_{k},\mathbf {x} (t_{k}),\mathbf {x} (t_{k+1}))}$ changes when the SDE for ${\displaystyle \mathbf {x} }$ does, the transition density function ${\displaystyle {\mathfrak {p}}_{\theta }(t_{k+1}-t_{k},\nu _{t_{k}},\nu _{t_{k+1}})}$ for the innovation process remains Gaussian independently of the SDEs for ${\displaystyle \mathbf {x} }$. Only in the case that the diffusion ${\displaystyle \mathbf {x} }$ is described by a linear SDE with additive noise, the density function ${\displaystyle p_{\theta }(t_{k+1}-t_{k},\mathbf {x} (t_{k}),\mathbf {x} (t_{k+1}))}$ is Gaussian and equal to ${\displaystyle {\mathfrak {p}}_{\theta }(t_{k+1}-t_{k},\nu _{t_{k}},\nu _{t_{k+1}}),}$ and so the maximum likelihood and the innovation estimator coincide.^[5] Otherwise,^[5] the innovation estimator is an approximation to the maximum likelihood estimator and, in this sense, the innovation estimator is a Quasi-Maximum Likelihood estimator. In addition, the innovation method is a particular instance of the Prediction Error method according to the definition given in.^[8] Therefore, the asymptotic results obtained in for that general class of estimators are valid for the innovation estimators.^[1][9][10] Intuitively, by following the typical control engineering viewpoint, it is expected that the innovation process - viewed as a measure of the prediction errors of the fitted model - be approximately a white noise process when the models fit the data,^[11][3] which can be used as a practical tool for designing of models and for optimal experimental design.^[6]

Properties

The innovation estimator (5) has a number of important attributes:

• Under conventional regularity conditions, the innovation estimator (5) is consistent and asymptoticaly normal distributed.^[1][10][12]

• For selecting models,^[11] the maximum log-likelihood ${\displaystyle -U_{M,h}({\widehat {\theta }}_{M},Z_{t_{M-1}})}$ of the innovation estimator (5) can be used to compute the Akaike or Bayesian information criterion.

• The ${\displaystyle 100(1-\alpha )\%}$ confidence limits ${\displaystyle {\widehat {\theta }}_{M}\pm \bigtriangleup }$ for the innovation estimator ${\displaystyle {\widehat {\theta }}_{M}}$ is estimated with^[6]

${\displaystyle \ \qquad \qquad \bigtriangleup =t_{1-\alpha ,M-\rho -1}{\sqrt {\frac {diag(Var({\widehat {\theta }}_{M}))}{M-p}}},}$

where ${\displaystyle t_{1-\alpha ,M-p-1}}$ is the t-student distribution with ${\displaystyle 100(1-\alpha )\%}$ significance level, and ${\displaystyle M-p-1}$ degrees of freedom . Here, ${\displaystyle {\text{V}}ar({\widehat {\theta }}_{M})=(I({\widehat {\theta }}_{M}))^{-1}}$ denotes the variance of the innovation estimator ${\displaystyle {\widehat {\theta }}_{M}}$, where

${\displaystyle \ \qquad \qquad I({\widehat {\theta }}_{M})=\sum _{k=1}^{M-1}I_{k}({\widehat {\theta }}_{M})}$

is the Fisher Information matrix the innovation estimator ${\displaystyle {\widehat {\theta }}_{M}}$ of ${\displaystyle \theta _{0}}$ and

${\displaystyle \qquad \qquad \lbrack I_{k}({\widehat {\theta }}_{M})]_{m,n}={\frac {\partial \mu ^{\intercal }}{\partial \theta _{m}}}\Sigma ^{-1}{\frac {\partial \mu }{\partial \theta _{n}}}+{\frac {1}{2}}trace(\Sigma ^{-1}{\frac {\partial \Sigma }{\partial \theta _{m}}}\Sigma ^{-1}{\frac {\partial \Sigma }{\partial \theta _{n}}})}$

is the entry ${\displaystyle (m,n)}$ of the matrix ${\displaystyle I_{k}({\widehat {\theta }}_{M})}$ with ${\displaystyle \mu =\mathbf {Cx} _{t_{k}/t_{k-1}}({\widehat {\theta }}_{M})}$ and ${\displaystyle \Sigma =\mathbf {\Sigma } _{t_{k}}({\widehat {\theta }}_{M})}$, for ${\displaystyle 1\leq m,n\leq p}$.

• The distribution of the fitting-innovation process ${\displaystyle \{\mathbf {\nu } _{t_{k}}:\mathbf {\nu } _{t_{k}}=\mathbf {z} _{t_{k}}-\mathbf {Cx} _{t_{k}/t_{k-1}}({\widehat {\theta }}_{M})\}_{k=1,\ldots M-1}}$ measures the goodness of fit of the model to the data.^{[1][3][11][6]}

• For smooth enough function ${\displaystyle \mathbf {h} }$, nonlinear observation equations of the form

${\displaystyle \qquad \qquad \mathbf {z} _{t_{k}}=\mathbf {h} (t_{k}{\text{, }}\mathbf {x} (t_{k}))+\mathbf {e} _{t_{k}},\qquad \qquad (6)}$

can be transformed to the simpler one (2), and the innovation estimator (5) can be applied.^[5]

Main conventional-type estimators

Conventional-type innovation estimators are those (9) derived from conventional-type continuous-discrete or discrete-discrete approximate filtering algorithms. With approximate continuous-discrete filters there are the innovation estimators based on Local Linearization (LL) filters,^[1][14][5] on the extended Kalman filter,^[15][16] and on the second order filters.^[3][16] Approximate innovation estimators based on discrete-discrete filters result from the discretization of the SDE (1) by means of a numerical scheme.^[17][18] Typically, the effectiveness of these innovation estimators is directly related to the stability of the involved filtering algorithms.

A shared drawback of these conventional-type filters is that, once the observations are given, the error between the approximate and the exact innovation process is fixed and completely settled by the time distance between observations.^[12] This might set a large bias of the approximate innovation estimators in some applications, bias that cannot be corrected by increasing the number of observations. However, the conventional-type innovation estimators are useful in many practical situations for which only medium or low accuracy for the parameter estimation is required.^[12]

Order-β innovation estimators

Let us consider the finer time discretization ${\displaystyle \left(\tau \right)_{h>0}=\{\tau _{n}:\tau _{n+1}-\tau _{n}\leq h{\text{ for }}n=0,1,\ldots ,N\}}$ of the time interval ${\displaystyle [t_{0},t_{M-1}]}$ satisfying the condition ${\displaystyle \left(\tau \right)_{h}\supset \{t\}_{M}}$. Further, let ${\displaystyle \mathbf {y} _{n}}$ be the approximate value of ${\displaystyle \mathbf {x} (\tau _{n})}$ obtained from a discretization of the equation (1) for all ${\displaystyle \left(\tau \right)_{h}}$, and

${\displaystyle \qquad \qquad \mathbf {y} =\{\mathbf {y} (t),t\in \lbrack t_{0},t_{M-1}]:\mathbf {y} (\tau _{n})=\mathbf {y} _{n},\quad }$ for all ${\displaystyle \quad \tau _{n}\in \left(\tau \right)_{h}\}\qquad \qquad (10)}$

a continuous-time approximation to ${\displaystyle \mathbf {x} }$.

A order-${\displaystyle \beta }$ LMV filter.^[13] is an approximate LMV filter for which ${\displaystyle \mathbf {y} }$ is an order-${\displaystyle \beta }$ weak approximation to ${\displaystyle \mathbf {x} }$ satisfying (10) and the weak convergence condition

${\displaystyle \qquad \qquad {\underset {t_{k}\leq t\leq t_{k+1}}{\sup }}\left\vert E\left(g(\mathbf {x} (t))|Z_{t_{k}}\right)-E\left(g(\mathbf {y} (t))|Z_{t_{k}}\right)\right\vert \leq L_{k}h^{\beta }}$

for all ${\displaystyle t_{k},t_{k+1}\in \{t\}_{M}}$ and any ${\displaystyle 2(\beta +1)}$ times continuously differentiable functions ${\displaystyle g:\mathbb {R} ^{d}\rightarrow \mathbb {R} }$ for which ${\displaystyle g}$ and all its partial derivatives up to order ${\displaystyle 2(\beta +1)}$ have polynomial growth, being ${\displaystyle L_{k}}$ a positive constant. This order-${\displaystyle \beta }$ LMV filter converges with rate ${\displaystyle \beta }$ to the exact LMV filter as ${\displaystyle h}$ goes to zero,^[13] where ${\displaystyle h}$ is the maximum stepsize of the time discretization ${\displaystyle (\tau )_{h}\supset \{t\}_{M}}$ on which the approximation ${\displaystyle \mathbf {y} }$ to ${\displaystyle \mathbf {x} }$ is defined.

A order-${\displaystyle \beta }$ innovation estimator is an approximate innovation estimator (9) for which the approximations to the discrete-time innovation (3) and innovation variance (4), respectively, resulting from an order-${\displaystyle \beta }$ LMV filter.^[12]

Approximations ${\displaystyle \mathbf {y} }$ of any kind converging to ${\displaystyle \mathbf {x} }$ in a weak sense (as, e.g., those in ^[19][13]) can be used to design an order-${\displaystyle \beta }$ LMV filter and, consequently, an order-${\displaystyle \beta }$ innovation estimator. These order-${\displaystyle \beta }$ innovation estimators are intended for the recurrent practical situation in which a diffusion process should be identified from a reduced number of observations distant in time or when high accuracy for the estimated parameters is required.

Properties

An order-${\displaystyle \beta }$ innovation estimator ${\displaystyle {\widehat {\mathbf {\theta } }}_{M}(h)}$ has a number of important properties:^[12][6]

• For each given data ${\displaystyle Z_{t_{M-1}}}$ of ${\displaystyle M}$ observations, ${\displaystyle {\widehat {\mathbf {\theta } }}_{M}(h)}$ converges to the exact innovation estimator ${\displaystyle {\widehat {\mathbf {\theta } }}_{M}}$ as the maximum stepsize ${\displaystyle h}$ of the time discretization ${\displaystyle \left(\tau \right)_{h}\supset \{t\}_{M}}$ goes to zero.

• For finite samples of ${\displaystyle M}$ observations, the expected value of ${\displaystyle {\widehat {\mathbf {\theta } }}_{M}(h)}$ converges to the expected value of the exact innovation estimator ${\displaystyle {\widehat {\mathbf {\theta } }}_{M}}$ as ${\displaystyle h}$ goes to zero.

• For an increasing number of observations, ${\displaystyle {\widehat {\mathbf {\theta } }}_{M}(h)}$ is asymptotically normal distributed and its bias decreases when ${\displaystyle h}$ goes to zero.

• Likewise to the convergence of the order-${\displaystyle \beta }$ LMV filter to the exact LMV filter, for the convergence and asymptotic properties of ${\displaystyle {\widehat {\mathbf {\theta } }}_{M}(h)}$ there are no constraints on the time distance ${\displaystyle t_{k+1}-t_{k}}$ between two consecutive observations ${\displaystyle \mathbf {z} _{t_{k}}}$ and ${\displaystyle \mathbf {z} _{t_{k+1}}}$, nor on the time discretization ${\displaystyle (\tau )_{h}\supset \{t\}_{M}.}$

• Approximations for the Akaike or Bayesian information criterion and confidence limits are directly obtained by replacing the exact estimator ${\displaystyle {\widehat {\mathbf {\theta } }}_{M}}$ by its approximation ${\displaystyle {\widehat {\mathbf {\theta } }}_{M}(h)}$. These approximations converge to the corresponding exact one when the maximum stepsize ${\displaystyle h}$ of the time discretization ${\displaystyle \left(\tau \right)_{h}\supset \{t\}_{M}}$ goes to zero.

• The distribution of the approximate fitting-innovation process ${\displaystyle \{{\widetilde {\mathbf {\nu } }}_{t_{k}}:{\widetilde {\mathbf {\nu } }}_{t_{k}}=\mathbf {z} _{t_{k}}-\mathbf {Cy} _{t_{k}/t_{k-1}}({\widehat {\theta }}_{M}(h))\}_{k=1,\ldots M-1}}$ measures the goodness of fit of the model to the data, which is also used as a practical tool for designing of models and for optimal experimental design.

• For smooth enough function ${\displaystyle \mathbf {h} }$, nonlinear observation equations of the form (6) can be transformed to the simpler one (2), and the order-${\displaystyle \beta }$ innovation estimator can be applied.

Fig. 1 Histograms of the differences ${\displaystyle ({\widehat {\alpha }}_{M}-{\widehat {\alpha }}_{h,M}^{D},{\widehat {\sigma }}_{M}-{\widehat {\sigma }}_{h,M}^{D})}$ and ${\displaystyle ({\widehat {\alpha }}_{M}-{\widehat {\alpha }}_{h,M},{\widehat {\sigma }}_{M}-{\widehat {\sigma }}_{h,M})}$ between the exact innovation estimator ${\displaystyle ({\widehat {\alpha }}_{M},{\widehat {\sigma }}_{M})}$ with the conventional ${\displaystyle ({\widehat {\alpha }}_{h,M}^{D},{\widehat {\sigma }}_{h,M}^{D})}$ and order-${\displaystyle 1}$ ${\displaystyle ({\widehat {\alpha }}_{h,M},{\widehat {\sigma }}_{h,M})}$ innovation estimators for the parameters ${\displaystyle (\alpha ,\sigma )}$ of the model (11)-(12) given ${\displaystyle 100}$ time series of ${\displaystyle M=10}$ noisy observations on the time interval ${\displaystyle [0.5,0.5+M-1]}$ with sampling period ${\displaystyle \Delta =1}$.

Figure 1 presents the histograms of the differences ${\displaystyle ({\widehat {\alpha }}_{M}-{\widehat {\alpha }}_{h,M}^{D},{\widehat {\sigma }}_{M}-{\widehat {\sigma }}_{h,M}^{D})}$ and ${\displaystyle ({\widehat {\alpha }}_{M}-{\widehat {\alpha }}_{h,M},{\widehat {\sigma }}_{M}-{\widehat {\sigma }}_{h,M})}$ between the exact innovation estimator ${\displaystyle ({\widehat {\alpha }}_{M},{\widehat {\sigma }}_{M})}$ with the conventional ${\displaystyle ({\widehat {\alpha }}_{h,M}^{D},{\widehat {\sigma }}_{h,M}^{D})}$ and order-${\displaystyle 1}$ ${\displaystyle ({\widehat {\alpha }}_{h,M},{\widehat {\sigma }}_{h,M})}$ innovation estimators for the parameters ${\displaystyle \alpha =-0.1}$ and ${\displaystyle \sigma =0.1}$ of the equation^[12]

${\displaystyle \qquad dx=txdt+\sigma {\sqrt {t}}xdw\quad (11)}$

obtained from 100 time series ${\displaystyle z_{t_{0}},..,z_{t_{M-1}}}$ of ${\displaystyle M}$ noisy observations

${\displaystyle \qquad z_{t_{k}}=x(t_{k})+e_{t_{k}},{\text{ for }}k=0,1,..,M-1,\quad (12)}$

of ${\displaystyle x}$ on the observation times ${\displaystyle \{t\}_{M=10}=\{t_{k}=0.5+k\Delta :k=0,\ldots M-1}$, ${\displaystyle \Delta =1\}}$, with ${\displaystyle x(0.5)=1}$ and ${\displaystyle \Pi _{k}=0.0001}$. The classical and the order-${\displaystyle 1}$ Local Linearization filters of the innovation estimators ${\displaystyle ({\widehat {\alpha }}_{h,M}^{D},{\widehat {\sigma }}_{h,M}^{D})}$ and ${\displaystyle ({\widehat {\alpha }}_{h,M},{\widehat {\sigma }}_{h,M})}$ are defined as in,^[12] respectively, on the uniform time discretizations ${\displaystyle \left(\tau \right)_{h=\Delta }\equiv \{t\}_{M}}$ and ${\displaystyle \left(\tau \right)_{h=\Delta /2,\Delta /8,\Delta /32}=\{\tau _{n}:\tau _{n}=0.5+nh}$, with ${\displaystyle n=0,1,\ldots ,(M-1)/h\}}$. The number of stochastic simulations of the order-${\displaystyle 1}$ Local Linearization filter is estimated via an adaptive sampling algorithm with moderate tolerance. The Figure 1 illustrates the convergence of the order-${\displaystyle 1}$ innovation estimator ${\displaystyle ({\widehat {\alpha }}_{h,M},{\widehat {\sigma }}_{h,M})}$ to the exact innovation estimators ${\displaystyle ({\widehat {\alpha }}_{M},{\widehat {\sigma }}_{M})}$ as ${\displaystyle h}$ decreases, which substantially improves the estimation provided by the conventional innovation estimator ${\displaystyle ({\widehat {\alpha }}_{\Delta ,M}^{D},{\widehat {\sigma }}_{\Delta ,M}^{D})}$.

Deterministic approximations

The order-${\displaystyle \beta }$ innovation estimators overcome the drawback of the conventional-type innovation estimators concerning the impossibility of reducing bias.^[12] However, the viable bias reduction of an order-${\displaystyle \beta }$ innovation estimators might eventually require that the associated order-${\displaystyle \beta }$ LMV filter performs a large number of stochastic simulations.^[13] In situations where only low or medium precision approximate estimators are needed, an alternative deterministic filter algorithm - called deterministic order-${\displaystyle \beta }$ LMV filter ^[13] - can be obtained by tracking the first two conditional moments ${\displaystyle \mu }$ and ${\displaystyle \Lambda }$ of the order-${\displaystyle \beta }$ weak approximation ${\displaystyle \mathbf {y} }$ at all the time instants ${\displaystyle \tau _{n}\in \left(\tau \right)_{h}}$ in between two consecutive observation times ${\displaystyle t_{k}}$ and ${\displaystyle t_{k+1}}$. That is, the value of the predictions ${\displaystyle \mathbf {y} _{t_{k+1}/t_{k}}}$ and ${\displaystyle \mathbf {P} _{t_{k+1}/t_{k}}}$ in the filtering algorithm are computed from the recursive formulas

${\displaystyle \qquad \qquad \mathbf {y} _{\tau _{n+1}/t_{k}}=\mu (\tau _{n},\mathbf {y} _{\tau _{n}/t_{k}};h_{n})\quad }$ and ${\displaystyle \quad \mathbf {P} _{\tau _{n+1}/t_{k}}=\Lambda (\tau _{n},\mathbf {P} _{\tau _{n}/t_{k}};h_{n}),\quad }$ with ${\displaystyle \tau _{n},\tau _{n+1}\in (\tau )_{h}\cap \lbrack t_{k,}t_{k+1}],}$

and with ${\displaystyle h_{n}=\tau _{n+1}-\tau _{n}}$. The approximate innovation estimators ${\displaystyle {\widehat {\mathbf {\theta } }}_{h,M}}$ defined with these deterministic order-${\displaystyle \beta }$ LMV filters not longer converge to the exact innovation estimator, but allow a significant bias reduction in the estimated parameters for a given finite sample with a lower computational cost.

Fig. 2 Histograms and confidence limits for the innovation estimators ${\displaystyle ({\widehat {\alpha }}_{h,M},{\widehat {\sigma }}_{h,M})}$ and ${\displaystyle ({\widehat {\alpha }}_{\cdot ,M},{\widehat {\sigma }}_{\cdot ,M})}$ of ${\displaystyle (\alpha ,\sigma )}$ computed with the deterministic order-1 LL filter on uniform ${\displaystyle \left(\tau \right)_{h,M}}$ and adaptive ${\displaystyle \left(\tau \right)_{\cdot ,M}}$ time discretizations, respectively, from ${\displaystyle 100}$ noisy realizations of the Van der Pol model (13)-(15) with sampling period ${\displaystyle \Delta =1}$ on the time interval ${\displaystyle [0,M-1]}$ and ${\displaystyle M=30}$. Observe the bias reduction of the estimated parameter as ${\displaystyle h}$ decreases.

Figure 2 presents the histograms and the confidence limits of the approximate innovation estimators ${\displaystyle ({\widehat {\alpha }}_{h,M},{\widehat {\sigma }}_{h,M})}$ and ${\displaystyle ({\widehat {\alpha }}_{\cdot ,M},{\widehat {\sigma }}_{\cdot ,M})}$ for the parameters ${\displaystyle \alpha =1}$ and ${\displaystyle \sigma =1}$ of the Van der Pol oscillator with random frequency^[12]

${\displaystyle \qquad dx_{1}=x_{2}dt\quad (13)}$

${\displaystyle \qquad dx_{2}=(-(x_{1}^{2}-1)x_{2}-\alpha x_{1})dt+\sigma x_{1}dw\quad (14)}$

obtained from 100 time series ${\displaystyle z_{t_{0}},..,z_{t_{M-1}}}$ of ${\displaystyle M}$ partial and noisy observations

${\displaystyle \qquad z_{t_{k}}=x_{1}(t_{k})+e_{t_{k}},{\text{ for }}k=0,1,..,M-1,\quad (15)}$

of ${\displaystyle x}$ on the observation times ${\displaystyle \{t\}_{M=30}=\{t_{k}=k\Delta :k=0,\ldots M-1}$, ${\displaystyle \Delta =1\}}$, with ${\displaystyle (x_{1}(0),x_{1}(0))=(1,1)}$ and ${\displaystyle \Pi _{k}=0.001}$. The deterministic order-${\displaystyle 1}$ Local Linearization filter of the innovation estimators ${\displaystyle ({\widehat {\alpha }}_{h,,M},{\widehat {\sigma }}_{h,M})}$ and ${\displaystyle ({\widehat {\alpha }}_{\cdot ,M},{\widehat {\sigma }}_{\cdot ,M})}$ is defined,^[12] for each estimator, on uniform time discretizations ${\displaystyle \left(\tau \right)_{h}=\{\tau _{n}:\tau _{n}=nh}$, with ${\displaystyle n=0,1,\ldots ,(M-1)/h\}}$ and on an adaptive time-stepping discretization ${\displaystyle \left(\tau \right)_{\cdot }}$ with moderate relative and absolute tolerances, respectively. Observe the bias reduction of the estimated parameter as ${\displaystyle h}$ decreases.