A SIMPLIFIED TREATMENT OF BROWNIAN MOTION AND STOCHASTIC DIFFERENTIAL EQUATIONS ARISING IN FINANCIAL MATHEMATICS
Bibliographic record
Abstract
ABSTRACT Brownian motion is an important stochastic process used in modelling the random evolution of stock prices. In their 1973 seminal paper—which led to the awarding of the 1997 Nobel prize in Economic Sciences—Fischer Black and Myron Scholes assumed that the random stock price process is described (i.e., generated) by Brownian motion. Despite its relative simplicity, the description of Brownian motion in advanced textbooks sometimes lacks an intuitive basis. The present exposition attempts to provide a simplified construction of standard Brownian motion based on a gambling analogy. This is followed by a description and explicit solution of two stochastic differential equations (known as arithmetic and geometric Brownian motion processes) that are driven by the standard Brownian motion process. The paper also illustrates the use of the Maple computer algebra system to simulate the standard and geometric Brownian motion processes.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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 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".