Forward and reverse representations for Markov chains

In this paper we carry over the concept of reverse probabilistic representations developed in Milstein, Schoenmakers, Spokoiny (2004) for diffusion processes, to discrete time Markov chains. We outline the construction of reverse chains in several situations and apply this to processes which are connected with jump-diffusion models and finite state Markov chains. By combining forward an reverse representations we then construct transition density estimators for chains which have root-N accuracy in any dimension and consider some applications.