Completely asymmetric stable processes conditioned to avoid an interval

Abstract

In the recent article D\"oring et al. [4] the authors conditioned a stable process with two-sided jumps to avoid an interval. As usual the strategy was to find an invariant function for the process killed on entering the interval and to show that the corresponding h-transformed process is indeed the process conditioned to avoid an interval in a meaningful way. In the present article we consider the case of a completely asymmetric stable process. It turns out that the invariant function found in [4] does not exist or is not invariant but nonetheless, we will characterize the conditioned process as a Markov process.