On explicit solutions to stochastic differential equations
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Bibliographic record
Abstract
This note is concerned with the study of explicit solutions to stochastic differential equations. Previously, Doss and Sussman showed that the unique strong solution to the scalar Itô equation X can be represented as a function ρ of a Brownian motion and an auxiliary stochastic process Yt determined, for every path of by ordinary differential equation (ODE). ρ itself is determined by a second differential equation. Now, it will be shown that X can be solved explicitly as with f(.) being a continuous real valued function, provided solves a differential equation related to the one defining ρ as well as a simple reaction-diffusion equation strongly. In particular, for a given dispersion coefficient σ(.), there will be a class drift coefficients b(.) are provided. The corresponding explicit solution xt for any given dispersion σ is also supplied
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 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) | 0.001 | 0.002 |
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