First exit from an open set for a matrix-exponential Lévy process

We study the first exit from a general open set for a one-dimensional Lévy process, where the Lévy measure is proportional to a two-sided matrix-exponential distribution. Under appropriate conditions on the Lévy measure, we obtain an explicit solution for the joint distribution of the first-exit time and the position of the Lévy process upon first exit, in terms of the zeros and poles of the corresponding Laplace exponent. The present result complements several earlier works on the use of exit sets for Lévy processes with algebraically similar Laplace exponents, where exits from open intervals are the main focus.