On a consistent estimator of a useful signal in Ornstein-Uhlenbeck model in $\mathbb{C}[-l,l[$

Abstract

~It is considered a transmittion process of a useful signal in Ornstein-Uhlenbeck model in $\mathbb{C}[-l,l[$ defined by the stochastic differential equation $$ d\Psi(t,x,\omega)=\sum_{n=0}^{2m} A_n\frac{\partial^{n}}{\partial x^{n}}\Psi(t,x,\omega)dt +\sigma d W(t,\omega) $$ with initial condition $$\Psi(0,x,\omega)=\Psi_0(x) \in FD^{(0)}[-l,l[, $$ where $m \ge 1$, $(A_n)_{0 \le n \le 2m} \in \mathbb{R}^+\times \mathbb{R}^{2m-1}$,$~((t,x,\omega) \in [0,+\infty[\times [-l,l[ \times \Omega)$, $\sigma \in \mathbb{R}^+$, $\mathbb{C}[-l,l[$ is Banach space of all real-valued bounded continuous functions on $[-l,l[$, $FD^{(0)}[-l,l[ \subset \mathbb{C}[-l,l[ $ is class of all real-valued bounded continuous functions on $[-l,l[$ whose Fourier series converges to himself everywhere on $[-l,l[$, $(W(t,\omega))_{t \ge 0}$ is a Wiener process and $\Psi_0(x)$ is a useful signal. By use a sequence of transformed signals $(Z_k)_{k \in N}=(\Psi(t_0,x,\omega_k))_{k \in N}$ at moment $t_0>0$, consistent and infinite-sample consistent estimations of the useful signal $\Psi_0$ is constructed under assumption that parameters $(A_n)_{0 \le n \le 2m}$ and $\sigma$ are known. Animation and simulation of the Ornstein-Uhlenbeck process in $\mathbb{C}[-l,l[$ and an estimation of a useful signal are also presented.