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

• Other Authors: Avram, Florin (Editor)
• Format: Electronic Book Chapter
• Language: English
• Published: Basel, Switzerland MDPI - Multidisciplinary Digital Publishing Institute 2021
• Subjects: Research & information: general; Mathematics & science; Lévy processes; non-random overshoots; skip-free random walks; fluctuation theory; scale functions; capital surplus process; dividend payment; optimal control; capital injection constraint; spectrally negative Lévy processes; reflected Lévy processes; first passage; drawdown process; spectrally negative process; dividends; de Finetti valuation objective; variational problem; stochastic control; optimal dividends; Parisian ruin; log-convexity; barrier strategies; adjustment coefficient; logarithmic asymptotics; quadratic programming problem; ruin probability; two-dimensional Brownian motion; spectrally negative Lévy process; general tax structure; first crossing time; joint Laplace transform; potential measure; Laplace transform; first hitting time; diffusion-type process; running maximum and minimum processes; boundary-value problem; normal reflection; Sparre Andersen model; heavy tails; completely monotone distributions; error bounds; hyperexponential distribution; reflected Brownian motion; linear diffusions; drawdown; Segerdahl process; affine coefficients; spectrally negative Markov process; hypergeometric functions; capital injections; bankruptcy; reflection and absorption; Pollaczek-Khinchine formula; scale function; Padé approximations; Laguerre series; Tricomi-Weeks Laplace inversion
• Online Access: DOAB: download the publication; DOAB: description of the publication

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).