Weak Convergence Methods for Approximation of the Evaluation of Path-Dependent Functionals
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Bibliographic record
Abstract
In many applications, one needs to evaluate a path-dependent objective functional $V$ associated with a continuous-time stochastic process $X$. Due to the nonlinearity and possible lack of Markovian property, more often than not, $V$ cannot be evaluated analytically, and only Monte Carlo simulation or numerical approximation is possible. In addition, such calculations often require the handling of stopping times, the usual dynamic programming approach may fall apart, and the continuity of the functional becomes the main issue. Denoting by $h$ the stepsize of the approximation sequence, this work develops a numerical scheme so that an approximating sequence of path-dependent functionals $V^h$ converges to $V$. By a natural division of labors, the main task is divided into two parts. Given a sequence $X^h$ that converges weakly to $X$, the first part provides sufficient conditions for the convergence of the sequence of path-dependent functionals $V^h$ to $V$. The second part constructs a sequence of approximations $X^h$ using Markov chain approximation methods and demonstrates the weak convergence of $X^h$ to $X$, when $X$ is the solution of a stochastic differential equation. As a demonstration, combining the results of the two parts above, approximation of option pricing for the discrete-monitoring-barrier option underlying stochastic volatility model is provided. Different from the existing literature, the weak convergence analysis is carried out by using the Skorohod topology together with the continuous mapping theorem. The advantage of this approach is that the functional under study may be a function of stopping times, projection of the underlying diffusion on a sequence of random times, and/or maximum/minimum of the underlying diffusion.
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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) | 0.000 | 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 it