Approximation and duality problems of refracted processes

For given two standard processes with no positive jumps, we construct, using the excursion theory, a Markov process whose positive and negative motions have the same law as the two processes. The resulting process is a generalization of Kyprianou–Loeffen’s refracted L'evy processes. We discuss approximation problem for our refracted processes coming from L'evy 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.