Representation for Martingales Living after a Random Time with Applications
Bibliographic record
Abstract
Our financial setting consists of a market model with two flows of information. The smallest flow 𝔽 is the “public” flow of information which is available to all agents, while the larger flow 𝔾 has additional information about the occurrence of a random time τ. This random time can model the default time in credit risk or death time in life insurance. Hence the filtration 𝔾 is the progressive enlargement of 𝔽 with τ. In this framework, when τ is a finite honest time, we describe explicitly how 𝔾-local martingales can be represented in terms of 𝔽-local martingales and parameters of τ. This representation complements the recent work of Choulli, Daveloose and Vanmaele to the case when martingales live “after τ.” Under some mild assumptions on the pair (𝔽, τ), we fully elaborate the application of these results to the explicit parametrization of all deflators under 𝔾. The results are illustrated in the case of a jump-diffusion model and a discrete-time market model.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.003 |
| 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".