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

We consider the Wiener process with drift $dX_t=\mu dt +\sigma d W_t$ with initial value problem $X_0=x_0$, where $x_0 \in R$, $ \mu \in R$ and $\sigma > 0$ 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 $x_0,\mu,\sigma$ 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.