First Hitting Problems for Markov Chains That Converge to a Geometric Brownian Motion
Bibliographic record
Abstract
We consider a discrete-time Markov chain with state space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>Δ</mml:mi><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mi>Δ</mml:mi><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi>N</mml:mi><mml:mo>}</mml:mo></mml:mrow></mml:math>. We compute explicitly the probability <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:math> that the chain, starting from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>j</mml:mi><mml:mi>Δ</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:math>, will hit N before 1, as well as the expected number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:math> of transitions needed to end the game. In the limit when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Δ</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:math> and the time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Δ</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:math> between the transitions decrease to zero appropriately, the Markov chain tends to a geometric Brownian motion. We show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mi>Δ</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:math> tend to the corresponding quantities for the geometric Brownian motion.
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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 | 0.004 | 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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 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".