On fluctuation theory for spectrally negative Lévy processes with Parisian reflection below, and applications
Bibliographic record
Abstract
As is well known, all functionals of a Markov process may be expressed in terms of the generator operator, modulo some analytic work. In the case of spectrally negative Markov processes, however, it is conjectured that everything can be expressed in a more direct way using the W W scale function which intervenes in the two-sided first passage problem, modulo performing various integrals. This conjecture arises from work on Levy processes [ 6, 7, 12, 16, 28–30, 50 ] where the W W scale function has explicit Laplace transform, and is therefore easily computable; furthermore it was found in the papers above that a second scale function Z Z introduced in [ 7 ] (this is an exponential transform ( 8 ) of W W ) greatly simplifies first passage laws, especially for reflected processes. Z Z is a harmonic function of the Lévy process (like W W ), corresponding to exterior boundary conditions w ( x ) = e θ x w(x)=e^{\theta x} ( 9 ) and is also a particular case of a “smooth Gerber–Shiu function” S w \mathcal {S}_w . The concept of the Gerber–Shiu function was introduced in [ 26 ]; we will use it however here in the more restricted sense of [ 15 ], who define this to be a “smooth” harmonic function of the process, which fits the exterior boundary condition w ( x ) w(x) and simultaneously solves the problems ( 17 ), ( 18 ). It has been conjectured that similar laws govern other classes of spectrally negativeprocesses, but it is quite difficult to find assumptions which allow proving this for general classes of Markov processes. However, we show below that in the particular case of spectrally negative Lévy processes with Parisian absorption and reflection from below [ 6, 16, 21 ], this conjecture holds true, once the appropriate W W and Z Z are identified (this observation seems new). This paper gathers a collection of first passage formulas for spectrally negative Parisian Lévy processes, expressed in terms of W W , Z Z
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".