Optimal State Equation for the Control With Two Distinct Dynamic Systems
Bibliographic record
Abstract
We consider a class of stochastic control problems which have been widely used in optimal foraging theory and financial modeling. The optimal state process has two distinct dynamics, characterized by two pairs of drift and diffusion coefficients, depending on whether it takes values bigger or smaller than a threshold value. Adopting a perturbation type approach, we find an expression for potential measure of the optimal state process. We then obtain an expression for the transition density of the optimal state process by inverting the associated Laplace transform. Properties including the stationary distribution of the optimal state process are discussed. Finally, an expression of the value function and the optimal control are given for such stochastic control problems. As an application, we transform the continuous-time two-armed bandit problem in finance into our control framework to derive the optimal investment strategy which maximizes the probability of reaching a specified threshold.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.005 | 0.003 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".