  We derive consistency and asymptotic normality results for quasi-maximum
likelihood methods for drift parameters of ergodic stochastic processes
observed in discrete time in an underlying continuous-time setting. The special
feature of our analysis is that the stochastic integral part is unobserved and
non-parametric. Additionally, the drift may depend on the (unknown and
unobserved) stochastic integrand. Our results hold for ergodic semi-parametric
diffusions and backward SDEs. Simulation studies confirm that the methods
proposed yield good convergence results.
