Approximation and duality problems of refracted processes

Using excursion theory, we construct a Markov process with no positive jumps whose positive and negative motions are given by different standard processes. The resulting process is a generalization of Kyprianou-Loeffen's refracted L\évy processes. We discuss approximation problem for our generalized refracted Lévy processes by removing small jumps and taking the limit as the removal level tends to zero. We also discuss conditions for refracted processes to have dual processes.