Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics
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...
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| Формат: | Online |
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| Хэл сонгох: | англи |
| Хэвлэсэн: |
MDPI - Multidisciplinary Digital Publishing Institute
2022
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| Нөхцлүүд: | |
| Онлайн хандалт: | ONIX_20220111_9783039284580_244 |
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Шошго байхгүй, Энэхүү баримтыг шошголох эхний хүн болох!
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| _version_ | 1869515692084035584 |
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| collection | Directory of Open Access Books |
| description | 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). |
| format | Online |
| id | doab-20.500.12854ir-76508 |
| institution | Directory of Open Access Books |
| language | eng |
| publishDate | 2022 |
| publishDateRange | 2022 |
| publishDateSort | 2022 |
| publisher | MDPI - Multidisciplinary Digital Publishing Institute |
| publisherStr | MDPI - Multidisciplinary Digital Publishing Institute |
| record_format | ojs |
| spelling | doab-20.500.12854ir-765082024-03-28T03:32:11Z Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics Avram, Florin 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 thema EDItEUR::G Reference, Information and Interdisciplinary subjects::GP Research and information: general thema EDItEUR::P Mathematics and Science 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). 2022-01-11T13:33:50Z 2022-01-11T13:33:50Z 2021 book ONIX_20220111_9783039284580_244 9783039284580 9783039284597 https://directory.doabooks.org/handle/20.500.12854/76508 eng image/jpeg Attribution 4.0 International https://mdpi.com/books/pdfview/book/3954 https://mdpi.com/books/pdfview/book/3954 MDPI - Multidisciplinary Digital Publishing Institute 10.3390/books978-3-03928-459-7 10.3390/books978-3-03928-459-7 46cabcaa-dd94-4bfe-87b4-55023c1b36d0 9783039284580 9783039284597 218 Basel, Switzerland open access |
| spellingShingle | 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 thema EDItEUR::G Reference, Information and Interdisciplinary subjects::GP Research and information: general thema EDItEUR::P Mathematics and Science Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics |
| title | Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics |
| title_full | Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics |
| title_fullStr | Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics |
| title_full_unstemmed | Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics |
| title_short | Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics |
| title_sort | exit problems for levy and markov processes with one sided jumps and related topics |
| topic | 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 thema EDItEUR::G Reference, Information and Interdisciplinary subjects::GP Research and information: general thema EDItEUR::P Mathematics and Science |
| topic_facet | 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 thema EDItEUR::G Reference, Information and Interdisciplinary subjects::GP Research and information: general thema EDItEUR::P Mathematics and Science |
| url | ONIX_20220111_9783039284580_244 |