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Levan Labadze

Publications and source records attributed to Levan Labadze.

3 recordsLinked to original sources

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

~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.

math.ST

Infinite-sample consistent estimations of parameters of the Wiener process with drift

We consider the Wiener process with drift $$ dX_t=μdt +σd W_t $$ with initial value problem $X_0=x_0$, where $x_0 \in R$, $ μ\in R$ and $σ> 0$ are parameters. By use values $(z_k)_{k \in N}$ of corresponding trajectories at a fixed positive moment $t$, the infinite-sample consistent estimates of each unknown parameter of the Wiener process with drift are constructed under assumption that all another parameters are known. Further, we propose a certain approach for estimation of unknown parameters $x_0,μ,σ$ of the Wiener process with drift by use the values $(z^{(1)}_k)_{k \in N}$ and $(z^{(2)}_k)_{k \in N}$ being the results of observations on the $2k$-th and $2k+1$-th trajectories of the Wiener process with drift at moments $t_1$ and $t_2$ , respectively.

math.ST

Estimation of the parameters of the Ornstein-Uhlenbeck's stochastic process

It is considered Ornstein-Uhlenbeck process $ x_t = x_0 e^{-θt} + μ(1-e^{-θt}) + σ\int_0^t e^{-θ(t-s)} dW_s$, where $x_0 \in R$, $θ>0$, $ μ\in R$ and $σ> 0$ are parameters. By use values $(z_k)_{k \in N}$ of corresponding trajectories at a fixed positive moment $t$, a consistent estimate of each unknown parameter of the Ornstein-Uhlenbeck's stochastic process is constructed under assumption that all another parameters are known.

math.ST