Signature Wasserstein GAN (SigCWGAN)

The core innovation of SigCWGAN is the use of path signatures—a mathematical tool from rough path theory—to convert irregularly sampled time series into a fixed-length feature vector. This transform captures the sequential order and lead-lag relationships between data points, encoding the entire trajectory's geometric structure. Unlike raw returns or handcrafted features, signatures provide a universal, non-linear feature map that is invariant to time reparameterization, making them ideal for distinguishing real market paths from synthetic ones.

SigCWGAN replaces the standard GAN loss with the Wasserstein distance (Earth Mover's distance), computed in the signature space rather than raw data space. This metric measures the minimal cost of transforming one distribution into another, providing a meaningful and continuous signal even when the generator and real data distributions have disjoint supports. The result is dramatically improved training stability, elimination of mode collapse, and a loss curve that genuinely correlates with sample quality.

By conditioning generation on the global path structure encoded in signatures, SigCWGAN excels at producing long synthetic trajectories (thousands of time steps) that maintain statistical fidelity. Traditional GANs often suffer from accumulating errors that cause generated paths to diverge from realistic behavior over long horizons. SigCWGAN's signature-based discriminator evaluates the entire path holistically, ensuring that stylized facts like volatility clustering and mean reversion persist across extended sequences.

The architecture naturally extends to conditional generation by concatenating conditioning variables (e.g., market regime labels, macroeconomic indicators) to the signature features before passing them to the discriminator. This allows controlled synthesis of market scenarios such as:

• High-volatility stress periods for tail-risk testing
• Specific market microstructures like auction phases
• Regime-specific trajectories (bull, bear, sideways) The generator learns to produce paths consistent with the specified conditions while maintaining realistic path-level statistics.

In practice, signatures are truncated to a finite order (typically 3–5), balancing expressiveness with computational tractability. The truncated signature retains the most salient geometric information—drift, volatility, and cross-asset interactions—while discarding higher-order noise. This dimensionality reduction is crucial for scaling to high-frequency data with multiple assets. The truncation level acts as a hyperparameter controlling the trade-off between model capacity and computational cost, with higher orders capturing more intricate path dependencies.

SigCWGAN typically employs an Autoregressive Feedforward Neural Network (AR-FNN) as its generator, which produces time series one step at a time conditioned on past values and a noise vector. This architecture respects the temporal causality of financial data while the signature-based discriminator provides global feedback on the entire generated path. The combination of local autoregressive generation with global signature-based critique creates a powerful inductive bias that yields both short-term realism and long-term consistency.

A mathematical transform that maps a continuous path into an infinite series of iterated integrals, capturing the sequential structure and geometric roughness of a time series. It acts as a feature map that linearizes complex functions of the path, making it a powerful tool for encoding the order-dependent nature of financial data. Key properties include:

• Uniqueness: The signature uniquely determines the path up to tree-like equivalence.
• Invariance: It is invariant to time reparameterization, focusing purely on the shape of the trajectory.
• Truncation: In practice, the infinite series is truncated at a finite level, providing a fixed-dimensional feature vector for any path.

Also known as the Earth Mover's Distance, this metric measures the minimum cost required to transform one probability distribution into another. Unlike the Jensen-Shannon divergence used in standard GANs, the Wasserstein distance provides meaningful gradients even when the real and generated distributions have disjoint supports. This property dramatically improves training stability and correlates with the visual quality of generated samples, making it the preferred loss function for generating continuous financial time series.

An extension of the base SigCWGAN architecture that conditions the generation process on auxiliary information such as past price history, macroeconomic indicators, or a specific market regime. The generator receives a conditioning vector concatenated with the latent noise, while the discriminator evaluates the joint distribution of the condition and the path signature. This allows for controlled generation of synthetic trajectories that follow a specific historical context, enabling scenario-based stress testing.

The Autoregressive Feedforward Neural Network is the generator architecture commonly paired with the SigCWGAN framework. It generates a time series one step at a time, conditioning each subsequent point on the previously generated sequence. This autoregressive structure explicitly enforces the causal ordering of financial data, ensuring that future synthetic prices cannot influence past ones, a critical constraint for realistic market simulation.

The core objective function of the SigCWGAN discriminator. Instead of directly comparing raw data distributions, the discriminator computes the difference between the expected path signatures of real and generated time series. By matching these expected signatures, the generator is forced to replicate the sequential characteristics of the data. This metric is particularly sensitive to the lead-lag relationships and higher-order temporal dependencies that define market microstructure.

A compressed representation of the path signature that removes redundant information. The log-signature is a smaller, more computationally efficient feature vector that still uniquely determines the path. It is computed as the logarithm of the formal power series of the signature and is often used in practice within the SigCWGAN discriminator to reduce the dimensionality of the feature space while preserving the geometric information of the original financial trajectory.