TY - GEN AU - Avram,Florin AU - Avram,Florin TI - Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics SN - books978-3-03928-459-7 PY - 2021/// CY - Basel, Switzerland PB - MDPI - Multidisciplinary Digital Publishing Institute KW - Research & information: general KW - bicssc KW - Mathematics & science KW - Lévy processes KW - non-random overshoots KW - skip-free random walks KW - fluctuation theory KW - scale functions KW - capital surplus process KW - dividend payment KW - optimal control KW - capital injection constraint KW - spectrally negative Lévy processes KW - reflected Lévy processes KW - first passage KW - drawdown process KW - spectrally negative process KW - dividends KW - de Finetti valuation objective KW - variational problem KW - stochastic control KW - optimal dividends KW - Parisian ruin KW - log-convexity KW - barrier strategies KW - adjustment coefficient KW - logarithmic asymptotics KW - quadratic programming problem KW - ruin probability KW - two-dimensional Brownian motion KW - spectrally negative Lévy process KW - general tax structure KW - first crossing time KW - joint Laplace transform KW - potential measure KW - Laplace transform KW - first hitting time KW - diffusion-type process KW - running maximum and minimum processes KW - boundary-value problem KW - normal reflection KW - Sparre Andersen model KW - heavy tails KW - completely monotone distributions KW - error bounds KW - hyperexponential distribution KW - reflected Brownian motion KW - linear diffusions KW - drawdown KW - Segerdahl process KW - affine coefficients KW - spectrally negative Markov process KW - hypergeometric functions KW - capital injections KW - bankruptcy KW - reflection and absorption KW - Pollaczek–Khinchine formula KW - scale function KW - Padé approximations KW - Laguerre series KW - Tricomi–Weeks Laplace inversion N1 - Open Access N2 - 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) UR - https://mdpi.com/books/pdfview/book/3954 UR - https://directory.doabooks.org/handle/20.500.12854/76508 ER -