Asymptotic behavior of the Unit Root Bilinear model with fractional Gaussian noise. Euler's scheme with small perturbations

We consider the Unit Root Bilinear model with a sequence of innovations given by the fractional Gaussian noise (increases of the fractional Brownian motion). For such a model we prove a variant of the Donsker–Prohorov limit theorem and obtain convergence of the model in probability to solution of a proper stochastic differential equation with fBm. The proof is based on the result about convergence of the Euler's scheme with ‘small perturbations’ for SDE with fBm, which is also proved.