Organizations: BNPP and Ecole Polytechnique. · BNPP and Monash University · BNPP and ENSAE Paris · LPSM, Université Paris Cité and Sorbonne University. · Ecole Polytechnique, CMAP.
We study the problem of generating synthetic time series that reproduce both marginal distributions and temporal dynamics, a central challenge in financial machine learning. Existing approaches typically fail to jointly model drift and stochastic volatility, as diffusion-based methods fix the volatility while martingale transport models ignore drift. We introduce the Schrödinger-Bass Bridge for Time Series (SBBTS), a unified framework that extends the Schrödinger-Bass formulation to multi-step time series. The method constructs a diffusion process that jointly calibrates drift and volatility and admits a tractable decomposition into conditional transport problems, enabling efficient learning. Numerical experiments on the Heston model demonstrate that SBBTS accurately recovers stochastic volatility and correlation parameters that prior SchrödingerBridge methods fail to capture. Applied to S&P 500 data, SBBTS-generated synthetic time series consistently improve downstream forecasting performance when used for data augmentation, yielding higher classification accuracy and Sharpe ratio compared to real-data-only training. These results show that SBBTS provides a practical and effective framework for realistic time series generation and data augmentation in financial applications.
Figure 2: Distribution of estimated Heston parameters using MLE. We show in blue, orange and green the density respectively from the data, SBTS, and SBBTS samples.
Metric
Zero-Shot
Real + Noise
Real
SBBTS
Classification metrics
Accuracy ( ↑ )
0.494
0.518(0.008)
0.521(0.006)
0.532(0.005)
Log Loss ( ↓ )
0.756
0.695(0.004)
0.693(0.003)
0.691(0.002)
ROC AUC ( ↑ )
0.486
0.494(0.012)
0.497(0.008)
0.521(0.007)
Financial metrics
Avg Daily Return (%) ( ↑ )
−0.020
0.086(0.035)
0.112(0.020)
0.143(0.015)
Table 1: test set performance of TabICL under different training configurations. Results are averaged over 5 seeds; standard deviations are reported in parentheses. ↓ indicates lower is better, ↑ indicates higher is better. Best result per row is bold .
Figure 3: Cumulative return (left) and cumulative excess return (right) on the test period, with S&P 500 index baseline.
Figure 4: Validation ( left ) and test set ( right ) performance as a function of the amount of synthetic data generated by SBBTS . Results are averaged over 5 seeds; error bars indicate one standard deviation.
Appendix figures & tables7 assets
Supplementary material from the paper’s appendix.
Appendix
K
T
T~
nepoch
Batch Size
lr
dmodel
nhead
Nπ
5
1
0.99
1000
128
10−3
128
16
50
Appendix
Table 2: Parameters used in the numerical experiments.
κ
θ
ξ
ρ
r
[0.5,4]
[0.5,1.5]
[0.1,0.9]
[−0.9,0.9]
[0.01,0.1]
Appendix
Table 3: Parameter ranges used for simulating the Heston process.
Figure 5: Autocorrelation functions of returns and squared returns for real and synthetic data across clusters.
Figure 6: Distribution of cluster factors in real and synthetic data.
Figure 7: Correlation matrices of returns for real and synthetic data.
Real
SBBTS
SBTS
VaR99% (%)
3.60
3.57
3.49
VaR95% (%)
2.11
2.19
2.17
ES99% (%)
4.65
4.44
4.36
ES95% (%)
3.15
3.18
3.07
Ann. Ret (%)
19.02
16.31
14.68
Ann. Std (%)
24.22
24.38
23.06
Appendix
Table 4: Comparison of tail-risk metrics (VaR, ES) and annualized performance statistics averaged across instruments.
Real
SBBTS
Validation
[−1.73,1.28]
[−0.62,2.30]
Test
[0.28,3.32]
[0.58,3.34]
Appendix
Table 5: 95% bootstrap confidence intervals for the Sharpe ratios obtained under real-only training and SBBTS -based data augmentation.
In recent years, financial institutions and firms have increasingly adopted synthetic data to address data scarcity and to generate counterfactual market scenarios. However, reproducing all the statistical properties of financial time series, commonly known as stylized facts, remains an open challenge for many existing general-purpose architectures. In this paper, we present a quality-aware generative framework that combines two classes of generative methods, demonstrating how their integration addresses existing limitations while enhancing the realism of synthetic data. Specifically, we first introduce CoMeTS-GAN (Correlated Multivariate Time Series GAN), a Conditional Generative Adversarial Network (C-GAN) designed to jointly generate mid-price and volume time-series for correlated stocks. We then show how our GAN architecture can be incorporated into state-of-the-art diffusion models to enhance the quality of generated correlation structures. Specifically, the GAN's Critic serves as a quality evaluation module that guides the diffusion process, enforcing learned correlation structures in the generated time-series. Our framework offers a lightweight and responsive solution for realistic stock market simulation, explicitly modeling inter-asset correlation structures. We experimentally validate our framework against leading generative architectures, showing that it more effectively captures the stylized facts of stock markets and models inter-asset correlations.
Giuseppe Masi, Andrea Coletta, Novella Bartolini
Sapienza University of Rome, Italy · Banca d’Italia, Italy
Schrödinger bridges for time series (SBTS) generate synthetic paths by projecting, in relative entropy, a Brownian reference onto the path laws that match the joint distribution of the data on the observation grid. The Brownian reference, however, fixes the quadratic variation of the generated paths, which is restrictive when stochastic volatility, correlated noise, or rank-deficient covariance structures must be reproduced. We introduce "Triangular-Reference Schrödinger Bridges for Time Series" (TR-SBTS), which keeps the entropy-projection backbone of SBTS but replaces the Brownian reference by a triangular, volatility-informed, intervalwise frozen reference on a state augmented with latent covariance descriptors. The construction remains a single entropy projection on the augmented state: the minimiser is the h-transform of the reference, and on each frozen interval the optimal drift has the logarithmic-gradient form b⋆(t,x)=A∇logH(t,x), intrinsic to the active covariance directions when the frozen covariance A is degenerate. We prove stability of the frozen approximation and consistency of the associated regularised kernel estimators, describe a reference-aware Nadaraya--Watson implementation of the conditional next-increment law, and evaluate the construction on numerical experiments.
Synthetic data has transformed language model training, yet its role in time series forecasting remains poorly understood. We present a large-scale empirical study: nine experiment groups, 4,218 runs systematically evaluating synthetic time series augmentation across five architectures, four synthetic signals and seven datasets. The effect is sharply architecture-conditional: channel-mixing models (TimesNet, iTransformer) benefit in the majority of trials, while channel-independent models (DLinear, PatchTST) are consistently degraded. In selected low-resource settings the gains are striking: TimesNet trained on only 10% of Weather data with synthetic augmentation surpasses the full-data baseline (4 of 16 sparsity-dataset combinations). Averaged across all architectures, augmentation hurts in 67% of trials. We further find that only the Seasonal-Trend generator reliably helps across the tested benchmarks, and that hard curriculum switching is actively harmful (+24% MSE degradation). These results provide concrete, actionable guidelines on how to use synthetic data: use synthetic augmentation with channel-mixing architectures, use gradual annealing schedules, and treat low-resource augmentation as architecture- and dataset-dependent. Code is available at \href{https://github.com/hugoiscracked/synthetic-ts/tree/main}
Hugo Cazaux, Eyjólfur Ingi Ásgeirsson, Hlynur Stefánsson
Department of Engineering Reykjavík University Iceland, Menntavegur 1, 102