Approximation of the solution of the backward stochastic differential equation. Small noise, large sample and high frequency cases

We present a review of some recently obtained results on estimation of the solution of a backward stochastic differential equation (BSDE) in the Markovian case. We suppose that the forward equation depends on some finite-dimensional unknown parameter. We consider the problem of estimating this parameter and then use the proposed estimator to estimate the solution of the BSDE. This last estimator is constructed with the help of the solution of the corresponding partial differential equation. We are interested in three observation models admitting a consistent estimation of the unknown parameter: small noise, large samples and unknown volatility. In the first two cases we have a continuous time observation, and the unknown parameter is in the drift coefficient. In the third case the volatility of the forward equation depends on the unknown parameter, and we have discrete time observations. The presented estimators of the solution of the BSDE in the three casesmentioned are asymptotically efficient.