Pathwise convergence rate for numerical solutions of stochastic differential equations
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Bibliographic record
Abstract
Devoted to numerical solutions of stochastic differential equations (SDEs), this work constructs a sequences of re-embedded numerical solutions having the same distribution as those of the original SDE in a new probability space. It is shown that the re-embedded numerical solutions converge strongly to the solution of the SDE. Moreover, the rate of convergence is ascertained. The main theorem is obtained by deriving a number of technical lemmas, that are interesting in their own right. Different from the well-known results in numerical solutions of SDEs, in lieu of the usual Brownian motion increments in the algorithm, an easily implementable sequence of independent and identically distributed (i.i.d.) random variables is used. Being easier to implement compared to the construction of Brownian increments, such an i.i.d. sequence is preferable in the actual computation. As far as the convergence and uniform mean square error estimates are concerned, the use of the i.i.d. sequence does not introduce essential difficulties compared with that of the Brownian increments. Nevertheless, the analysis becomes much more difficult in the study of rates of convergence because one has to deal with the difference of the Brownian increments and the i.i.d. sequence in the almost sure sense. This paper presents a new approach to establishing rates of convergence.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| 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 it