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

Dear Colleagues,

It has long been well-known that exit problems for one-dimensional Lévy processes are easier when there are jumps in one direction only. In the last few years, this intuition became more precise: We know now that a great 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 and 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, 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, 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 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 distributions
• Asymptotic results
• Extensions 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

Prof. Dr. Florin Avram
Guest Editor

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