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

<p><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>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 </span><em style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>q</em><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>-harmonic functions (or scale functions or positive martingales) </span><em style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>W</em><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)> and </span><em style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>Z</em><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>. 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 </span><em style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>W</em><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)> and </span><em style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>Z</em><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)> 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).</span></p><p><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>Motivated by these considerations this Special Issue aims to review and push further the state-of-the-art progress on the following topics:</span></p><ul><li><em style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>W</em><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)> </span><em style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>Z</em><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)> formulas for exit problems of the Lévy and diffusion classes (including drawdown problems)</span></li><li><em style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>W</em><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)> </span><em style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>Z</em><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)> formulas for quasi-stationary distributions</span></li><li><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>Asymptotic results</span></li><li><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>Extensions to random walks Markov additive processes omega models processes with Parisian reflection or absorbtion processes with state-dependent drift etc.</span></li><li><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>Optimal stopping dividends real options etc.</span></li><li><span style=color: rgba(34 34 34 1); background-color: rgba(255 255 255 1)>Numeric computation of the scale functions</span></li></ul><p><br></p><p><br></p>