Volatility estimation of hidden Markov processes and adaptive filtration (Q6559474)

In this paper, the author deals with a non-homogeneous partially observed linear system described by the equations \N\begin{align*}\N\mathrm{d}Y_t&=a\left(\vartheta ,t\right)Y_t\mathrm{d}t+b\left(\vartheta ,t\right)\mathrm{d}V_t,\quad Y_0=y_0,\quad 0\leq t\leq T,\\\N\mathrm{d}X_t&=f\left(\vartheta ,t\right)Y_t\mathrm{d}t+\varepsilon \sigma \left(t\right)\mathrm{d}W_t,\quad X_0=0,\quad 0\leq t\leq T, \N\end{align*}\Nwhere \(a\left(\cdot \right)\), \(b\left( \cdot \right)\), \( f\left(\cdot \right)\) and \(\sigma \left(\cdot \right)\) are known smooth functions, \(\varepsilon \in (0,1]\) is a small parameter, and where the Wiener processes \(V_t,0\leq t\leq T\) and \(W_t,0\leq t\leq T\) are independent. The process \(X^T=\left(X_t,0\leq t\leq T\right)\) is observed and the process \(Y^T=\left(Y_t,0\leq t\leq T\right)\) is hidden. \N\NThe problem is estimation of the finite-dimensional parameter \(\vartheta\in\Theta \subset \mathbb{R}^d\) and construction of an adaptive filter. The conditional expectation \(m\left(\vartheta ,t\right)=\mathbf{E}_\vartheta \left(Y_t|X_s,0\leq s\leq t\right)\) satisfies the equations of Kalman-Bucy filtering (see [\textit{R. E. Kalman} and \textit{R. S. Busy}, ``Mew results in linear filtering and prediction theory'', Trans. ASME, 83D, 95--100 (1961); \textit{R. S. Liptser} and \textit{A. N. Shiryaev}, Statistics of random processes. 2: Applications. Transl. from the Russian by A. B. Aries. 2nd rev. and exp. ed. Berlin: Springer (2001; Zbl 1008.62073)]). If the value of \(\vartheta \) is unknown, then to approximate \(m\left(\vartheta ,t\right)\) we can use some estimator \(\bar\vartheta _\varepsilon \) and to use something like \(m\left(\bar\vartheta _\varepsilon,t\right) \). The using of a series of MLE estimates \(\hat\vartheta _{t,\varepsilon }\) constructed from observations \(X^t=\left(X_s,0\leq s\leq t\right)\) for each \(t \in (0,T]\) can provide a good approximation of \(m\left(\vartheta,t\right)\), while the numerical realization of such algorithm can be a difficult problem. The author proposes a construction of another estimate \(\vartheta _{t,\varepsilon }^\star,0<t\leq T \), called One-step MLE-process, which depends on the observations \(X^t \), can be easily calculated and which is asymptotically as \(\varepsilon \rightarrow 0\) equivalent to the MLE estimate \(\hat\vartheta _{t,\varepsilon },0<t\leq T \). This estimator can be used for approximation \(m\left(\vartheta ,t\right)\). The error of approximation \(m\left(\vartheta,t\right)-m\left(\vartheta _{t,\varepsilon }^\star ,t\right) \) is described, the optimality of such approximation is discussed. This estimator is used for nonparametric estimation of the integral of the square of volatility of unobservable component.