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

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We consider the Wiener process with drift dXₜ=μdt +σd Wₜ with initial value problem X₀=x₀, where x₀ ∈ R, μ∈ R and σ> 0 are parameters. By use values (zₖ)ₖ ∈ 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 x₀,μ,σ of the Wiener process with drift by use the values (z⁽¹⁾ₖ)ₖ ∈ ₙ and (z⁽²⁾ₖ)ₖ ∈ ₙ being the results of observations on the 2k-th and 2k+1-th trajectories of the Wiener process with drift at moments t₁ and t₂, respectively.