On approximation of BSDE and multi-step MLE-processes

We consider the problem of approximation of the solution of the backward stochastic differential equations in Markovian case. We suppose that the forward equation depends on some unknown finite-dimensional parameter. This approximation is based on the solution of the partial differential equations and multi-step estimator-processes of the unknown parameter. As the model of observations of the forward equation we take a diffusion process with small volatility. First we establish a lower bound on the errors of all approximations and then we propose an approximation which is asymptotically efficient in the sense of this bound. The obtained results are illustrated on the example of the Black and Scholes model.