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Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics
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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).
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Keywords
- adjustment coefficient
- affine coefficients
- bankruptcy
- barrier strategies
- boundary-value problem
- capital injection constraint
- capital injections
- capital surplus process
- completely monotone distributions
- de Finetti valuation objective
- diffusion-type process
- dividend payment
- Dividends
- drawdown
- drawdown process
- error bounds
- first crossing time
- first hitting time
- first passage
- fluctuation theory
- general tax structure
- heavy tails
- hyperexponential distribution
- hypergeometric functions
- joint Laplace transform
- Laguerre series
- Laplace transform
- Lévy processes
- linear diffusions
- log-convexity
- logarithmic asymptotics
- Mathematics & science
- non-random overshoots
- normal reflection
- Optimal Control
- optimal dividends
- Padé approximations
- Parisian ruin
- Pollaczek–Khinchine formula
- potential measure
- quadratic programming problem
- Reference, information & interdisciplinary subjects
- reflected Brownian motion
- reflected Lévy processes
- reflection and absorption
- Research & information: general
- ruin probability
- running maximum and minimum processes
- scale function
- scale functions
- Segerdahl process
- skip-free random walks
- Sparre Andersen model
- spectrally negative Lévy process
- spectrally negative Lévy processes
- spectrally negative Markov process
- spectrally negative process
- stochastic control
- Tricomi–Weeks Laplace inversion
- two-dimensional Brownian motion
- variational problem
Links
DOI: 10.3390/books978-3-03928-459-7Editions