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 $$ dΨ(t,x,ω)=\sum_{n=0}^{2m} A_n\frac{\partial^{n}}{\partial x^{n}}Ψ(t,x,ω)dt +σd W(t,ω) $$ with initial condition $$Ψ(0,x,ω)=Ψ_0(x) \in FD^{(0)}[-l,l[, $$ where $m \ge 1$, $(A_n)_{0 \le n \le 2m} \in \mathbb{R}^+\times \mathbb{R}^{2m-1}$,$~((t,x,ω) \in [0,+\infty[\times [-l,l[ \times Ω)$, $σ\in \mathbb{R}^+$, $\mathbb{C}[-l,l[$ 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 $[-l,l[$, $(W(t,ω))_{t \ge 0}$ is a Wiener process and $Ψ_0(x)$ is a useful signal. By use a sequence of transformed signals $(Z_k)_{k \in N}=(Ψ(t_0,x,ω_k))_{k \in N}$ at moment $t_0>0$, consistent and infinite-sample consistent estimations of the useful signal $Ψ_0$ is constructed under assumption that parameters $(A_n)_{0 \le n \le 2m}$ and $σ$ 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.