Adaptive estimation of the transition density of a particular hidden Markov chain

We study the following model of hidden Markov chain: $Y_i=X_i+\epsilon_i$, $i=1,...,n+1$ with $(X_i)$ a real-valued positive recurrent and stationary Markov chain and $(\epsilon_i)_{1\leq i\leq n+1}$ a noise independent of the sequence $(X_i)$ having a known distribution. We present an adaptive estimator of the transition density based on the quotient of a deconvolution estimator of the density of $X_i$ and an estimator of the density of $(X_i,X_{i+1})$. These estimators are obtained by contrast minimization and model selection. We evaluate the $L2$ risk and its rate of convergence for ordinary smooth and supersmooth noise with regard to ordinary smooth and supersmooth chains. Some examples are also detailed.