Optimal stopping of stochastic transport minimizing submartingale costs
Bibliographic record
Abstract
Given a stochastic state process <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X Subscript t Baseline right-parenthesis Subscript t"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(X_t)_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a real-valued submartingale cost process <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper S Subscript t Baseline right-parenthesis Subscript t"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(S_t)_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we characterize optimal stopping times <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau"> <mml:semantics> <mml:mi> τ </mml:mi> <mml:annotation encoding="application/x-tex">\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that minimize the expectation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript tau"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi> τ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">S_\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> while realizing given initial and target distributions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi> μ </mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu"> <mml:semantics> <mml:mi> ν </mml:mi> <mml:annotation encoding="application/x-tex">\nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , i.e., <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 0 tilde mu"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo> ∼ </mml:mo> <mml:mi> μ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X_0\sim \mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript tau Baseline tilde nu"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mi> τ </mml:mi> </mml:msub> <mml:mo> ∼ </mml:mo> <mml:mi> ν </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X_\tau \sim \nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A dual optimization problem is considered and shown to be attained under suitable conditions. The optimal solution of the dual problem then provides a contact set, which characterizes the location where optimal stopping can occur. The optimal stopping time is uniquely determined as the first hitting time of this contact set provided we assume a natural structural assumption on the pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X Subscript t Baseline comma upper S Subscript t Baseline right-parenthesis Subscript t"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(X_t, S_t)_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which generalizes the <italic>twist condition</italic> on the cost in optimal transport theory. This paper extends the Brownian motion settings studied in Ghoussoub, Kim, and Palmer [Calc. Var. Partial Differential Equations 58 (2019), Paper No. 113, 31] and Ghoussoub, Kim, and Palmer [A solution to the Monge transport problem for Brownian martingales, 2019] and deals with more general costs.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 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".