On the Joint Distributions of the Time to Ruin, the Surplus Prior to Ruin, and the Deficit at Ruin in the Classical Risk Model
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Bibliographic record
Abstract
Abstract The seminal paper by Gerber and Shiu (1998) unified and extended the study of the event of ruin and related quantities, including the time at which the event of ruin occurs, the deficit at the time of ruin, and the surplus immediately prior to ruin. The first two of these quantities are fundamentally important for risk management techniques that utilize the ideas of Value-at-Risk and Tail Value-at-Risk. As is well known, calculation of these and related quantities requires knowledge of the associated probability distributions. In this paper we derive an explicit expression for the joint (defective) distribution of the time to ruin, the surplus immediately prior to ruin, and the deficit at ruin in the classical compound Poisson risk model. As a by-product, we obtain expressions for the three bivariate distributions generated by the time to ruin, the surplus prior to ruin, and the deficit at ruin. Finally, we consider mixed Erlang claim sizes and show how the joint (defective) distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin can be calculated.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.008 | 0.009 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it