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EXIT PROBLEMS FOR LéVY AND MARKOV PROCESSES WITH ONE-SIDED J IBD

FLORIN AVRAM
MDPI AG
06 / 2021
9783039284580
Inglés

Sinopsis

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)áWáandáZ. 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 functionsáW andáZáis 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,áZáformulas for exit problems of the Lévy and diffusion classes (including drawdown problems)W,áZáformulas 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