Numerical methods for stochastic Volterra integral equations with weakly singular kernels
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Abstract In this paper we first establish the existence, uniqueness and Hölder continuity of the solution to stochastic Volterra integral equations (SVIEs) with weakly singular kernels, with singularities $\alpha \in (0, 1)$ for the drift term and $\beta \in (0, 1/2)$ for the stochastic term. Subsequently, we propose a $\theta $-Euler–Maruyama scheme and a Milstein scheme to solve the equations numerically and obtain strong rates of convergence for both schemes in $L^{p}$ norm for any $p\geqslant 1$. For the $\theta $-Euler–Maruyama scheme the rate is $\min \big\{1-\alpha ,\frac{1}{2}-\beta \big\}~ $ and for the Milstein scheme is $\min \{1-\alpha ,1-2\beta \}$. These results on the rates of convergence are significantly different from those it is similar schemes for the SVIEs with regular kernels. The source of the difficulty is the lack of Itô formula for the equations. To get around this difficulty we use the Taylor formula subsequently carrying out a sophisticated analysis of the equation.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| 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