This paper is concerned with the particular hidden model: $X_{i+1}=b(X_i)+\sigma(X_i)\xi_{i+1}, Z_i=X_i+\varepsilon_i$, where $(\xi_i)$ and $(\varepsilon_i)$ are independent sequences of i.i.d. noise. Moreover, the sequences $(X_i)$ and $(\varepsilon_i)$ are independent and the distribution of $\varepsilon$ is known. Our aim is to estimate the functions $b$ and $\sigma^2$ when only observations $Z_1, \dots, Z_n$ are available. We propose to estimate $bf$ and $(b^2+\sigma^2)f$ and study the integrated mean square error of projection estimators of these functions on automatically selected projection spaces. By ratio strategy, estimators of $b$ and $\sigma^2$ are then deduced. The mean square risk of the resulting estimators are studied and their rates are discussed. Lastly, simulation experiments are provided: constants in the penalty functions defining the estimators are calibrated and the quality of the estimators is checked on several examples.