Optimal prediction of the last r-excursion time of Brownian motion models
Bibliographic record
Abstract
This paper investigates the optimal prediction of the last $r$-excursion time for a Brownian motion model. The last $r$-excursion time, denoted by $l_{r}$, refers to the right endpoint of the last negative excursion lasting longer than a constant $r>0$. It reduces to the standard last passage time when $r\downarrow 0$. For a Brownian motion with drift $\mu >0$ and volatility $\sigma >0$, our goal is to identify an optimal stopping time that minimizes the ($L_{1}$) distance from the last $r$-excursion time $l_{r}$. We find that the optimal stopping barrier exhibits two distinct structures: a constant barrier (characterized as a solution of a non-linear equation) or a moving barrier (characterized by the unique solution to an integral equation) depending on the ratio $R=\frac{\mu \sqrt{r}}{\sigma }$ which integrates a firm's financial profitability, volatility, and risk tolerance to financial distress. To obtain the optimal stopping time, we examine the smooth fit condition, Lipschitz continuity of the barrier, and probability regularity of the boundary points. As an application in risk management, we develop a decision rule that informs the timing of business expansion and contraction.
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How this classification was reachedexpand
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".