Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics
Avram, Florin
Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics - Basel, Switzerland MDPI - Multidisciplinary Digital Publishing Institute 2021 - 1 electronic resource (218 p.)
Open Access
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).
Creative Commons
English
9783039284580 9783039284597
Информационные технологии
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
004
Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics - Basel, Switzerland MDPI - Multidisciplinary Digital Publishing Institute 2021 - 1 electronic resource (218 p.)
Open Access
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).
Creative Commons
English
9783039284580 9783039284597
Информационные технологии
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
004