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High Rank Path Development: an approach to learning the filtration of stochastic processes
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Introduces High Rank PCF Distance (HRPCFD), a metric for extended weak convergence built by applying path development twice (to conditional expectations), and uses it as the discriminator in HRPCF-GAN for conditional time series generation.
Problem
Weak convergence of stochastic processes ignores the filtration (information growth over time), so models close in weak topology can give very different values in multi-period decisions like optimal stopping; extended weak convergence fixes this but has lacked efficient numerical implementations.
Approach
Define a high rank path characteristic function for measure-valued prediction processes by applying the unitary (path development) feature twice: first to each conditional expectation of the path development, then to the resulting matrix-valued path. The resulting HRPCFD metrises extended weak convergence and is estimated via a sequence-to-sequence regression of conditional developments, with a GA
Key results
- HRPCF-GAN gets Auto-Correlation 0.082 and Cross-Correlation 0.013 on fBM, about 21.9% and 72.3% lower than the second-best model.
- HRPCF-GAN reports an average 34% (fBM) and 14% (Stock) improvement over PCFGAN across metrics.
- In the hypothesis test, High Rank PCFD achieves power 1 for H=0.4, 0.425, 0.575, 0.6 and outperforms signature and classical MMDs near H=0.5.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use HRPCF-GAN instead of RCGAN | conditional generation on fBM · fBM | Auto-C.: .105±.001 -> .082±.002 | Helped |
| Use HRPCF-GAN instead of PCFGAN | conditional generation on fBM · fBM | Cross-C.: .047±.001 -> .013±.001 | Helped |
| Use HRPCF-GAN instead of RCGAN | conditional generation on fBM · fBM | Discriminative Score: .207±.008 -> .151±.006 | Helped |
| Use HRPCF-GAN instead of PCFGAN | conditional generation on fBM · fBM | Sig-W1: .199±.004 -> .169±.009 | Helped |
| Use HRPCF-GAN instead of RCGAN | conditional generation on fBM · fBM | Conditional Expectation: 1.822±.023 -> 1.693±.021 | Helped |
| Use HRPCF-GAN instead of PCFGAN | conditional generation on Stock · Stock | Auto-C.: .198±.003 -> .189±.010 | Helped |
| Use HRPCF-GAN instead of PCFGAN | conditional generation on Stock · Stock | Cross-C.: .055±.004 -> .053±.005 | Helped |
| Use HRPCF-GAN instead of TimeGAN | conditional generation on Stock · Stock | Discriminative Score: .020±.021 -> .016±.005 | Helped |
| Use HRPCF-GAN instead of PCFGAN | conditional generation on Stock · Stock | Sig-W1: .005±.001 -> .004±.002 | Helped |
| Use HRPCF-GAN instead of PCFGAN | conditional generation on Stock · Stock | Conditional Expectation: .060±.001 -> .056±.002 | Helped |
| Use HRPCF-GAN instead of PCFGAN | conditional generation on Stock · Stock | Amer. Put: .202±.020 -> .179±.006 | Helped |
| Replace PCFD with High Rank PCFD | hypothesis testing of 3-dimensional Brownian motion versus fractional Brownian motion · H=0.45 | test power: 0.99± 0.02 -> 0.97± 0.04 | Hurt |
Surprising findings
- High Rank PCFD test power at H=0.475 and 0.525 is only about 0.30, showing that near-Brownian alternatives remain hard to detect.
Limitations
- Choice of generator/network architecture is acknowledged as crucial and not fully studied.
- Estimation of conditional expectation (regression module) has room for improvement in accuracy and training stability.
- Results are on a single synthetic process and five stocks only.
Experimental setup
Datasets: multivariate fractional Brownian motion (H=1/4) Yahoo Finance daily log returns of 5 US stocks (2010-2020)
Benchmarks: fBM H=1/4 conditional generation US stock daily log returns conditional generation hypothesis test power for Brownian vs fBM with H in [0.4,0.6]
Metrics: test power type-I error Auto-Correlation Cross-Correlation Discriminative Score Sig-W1 distance Conditional Expectation American put option price error
Baselines: RCGAN TimeGAN PCFGAN PCFD linear signature MMD RBF signature MMD High Rank signature MMD classical linear MMD classical RBF MMD
Builds on: PCF-GAN path characteristic function (PCF) extended weak convergence RCGAN TimeGAN
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