  In the present paper the problem of approximating the solution of BSDE is
considered in the case where the solution of forward equation is observed in
the presence of small Gaussian noise. We suppose that the volatility of the
forward equation depends on an unknown parameter. This approximation is made in
several steps. First we obtain a preliminary estimator of the unknown
parameter, then using Kalman-Bucy filtration equations and Fisher-score device
we construct an one-step MLE-process of this parameter. The solution of BSDE is
approximated by means of the solution of PDE and the One-step MLE-process. The
error of approximation is described in different metrics.
