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 $$d\Psi (t,x,\omega )=\sum _{n=0}^{2m}{A}_n\frac{{\partial }^n}{\partial x^n}\Psi (t,x,\omega )dt+\sigma dW(t,\omega )$$ with initial condition $$\Psi (0,x,\omega )={\Psi }_0(x)\in F{D}^{(0)}[-l,l[,$$ 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 $[-l,l[$, $F{D}^{(0)}[-l,l[\subset \mathbb{C}[-l,l[$ 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.

20 pages, 5 figues

References in corpus (2)

• Estimation of the parameters of the Ornstein-Uhlenbeck's stochastic process
• Infinite-sample consistent estimations of parameters of the Wiener process with drift