Exit Problems for Levy and Markov Processes with One-Sided Jumps and Related Topics

Exit problems for one-dimensional Lévy processes are easier when jumps only occur in one direction. In the last few years this intuition became more precise: we know now that a wide variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two q-harmonic functions (or scale functions or positive martingales) W and Z. The proofs typically require not much more than the strong Markov property which hold in principle for the wider class of spectrally-negative strong Markov processes. This has been established already in particular cases such as random walks Markov additive processes Lévy processes with omega-state-dependent killing and certain Lévy processes with state dependent drift and seems to be true for general strong Markov processes subject to technical conditions. However computing the functions W and Z is still an open problem outside the Lévy and diffusion classes even for the simplest risk models with state-dependent parameters (say Ornstein-Uhlenbeck or Feller branching diffusion with phase-type jumps).Motivated by these considerations this Special Issue aims to review and push further the state-of-the-art progress on the following topics: W Z formulas for exit problems of the Lévy and diffusion classes (including drawdown problems)W Z formulas for quasi-stationary distributionsAsymptotic resultsExtensions to random walks Markov additive processes omega models processes with Parisian reflection or absorbtion processes with state-dependent drift etc.Optimal stopping dividends real options etc.Numeric computation of the scale functions