Fix GBM demo replication statistics and speed up the Gaussian transform

[larissensium]

Fixed a table in gbm_demo to correctly show results averaged over replications.

[sou-cheng-choi]

@larissensium

I have

• merged in latest changes from develop branch
• corrected some typos
• made the paths from QuantLib independent with different seed values in quantlib_util.py
• added unit tests

Please review and verify the changes.

[JiangruiKang]

The switch to Burley scrambling is the right way to obtain independent randomized-QMC replications, but one reproducibility defect remains at dimensions above the Jaeckel table. The current tests are green because they use n_steps=4; please cover the 32/33-dimensional boundary before merging.

[JiangruiKang]

[P2] Use a fixed nonzero underlying Sobol seed

QuantLib treats this second argument differently from the replication-specific scramble seed. Jaeckel direction integers are tabulated only through dimension 32; above that, SobolRsg uses this value to generate additional direction integers, and QuantLib defines 0 as a clock-selected random seed. Consequently, two calls with the same public seed can produce different paths when n_steps >= 33 (see lballabio/QuantLib#2732), even though the comment says the underlying seed is fixed. Please use a fixed nonzero value such as 42, vary only scrambleSeed, and add a regression test at n_steps=33 showing that equal scramble seeds reproduce exactly while different scramble seeds remain distinct.

[sou-cheng-choi]

@JiangruiKang

Thank you for your invaluable suggestion. I have implemented the changes with a unit test test_reproducible_above_direction_integer_table in test_demo_gbm.py. Please re-review when you have a moment.

@larissensium , @alegresor and @JiangruiKang :

Summary of changes:

1. Numerical Stability and Reproducibility: Fixed the QuantLib Sobol generator seeding (SOBOL_DIRECTION_SEED = 1) to ensure reproducible point sets, resolved the QuantLib replication leak using np.stack(ql_paths), and centralized the covariance calculation function for symmetry and testability.
2. Methodological Accuracy: Updated QuantLib to default to "exact" GBM evolution, replacing the biased Euler-Maruyama scheme. Implemented statistically rigorous variance calculations by correctly setting ddof=0 for replications and reserving ddof=1 for cross-replication standard error. Increased number of replications from 3 to 8 to reduce noise in results.
3. Speed: Use ndtri to speed up GBM.
4. Sampling and Analysis Expansion: Added support for the Halton low-discrepancy sequence in QuantLib, expanding sampler options. Introduced a new construction-ablation feature to isolate and analyze path-construction effects (e.g., PCA vs. BrownianBridge in QMCPy).
5. Code Robustness and Testing: Increased test coverage, growing test_demo_gbm.py. Refactored utility functions and corrected docstrings to reflect the exact-scheme change.
6. Updated software versions: Quantlib 1.43, QMCPy 2.4, etc.

[alegresor]

Nice change! Haven't heard of this ndtri before, but it is suppose to be much faster than norm.ppf

[sou-cheng-choi]

@alegresor

Thanks! ndtri is much faster indeed.