Neural Stochastic Differential
Equations: continuous-time
dynamics, learned from data.
This division turns Stokhos Labs' broader AI product engineering into a focused risk analytics platform. The model replaces discrete autoregressive updates with an SDE whose drift and diffusion are parameterized by neural networks, trained end-to-end on U.S. Treasury yield data.
Trained before the crisis.
Tested during it.
Training setup
Daily U.S. Treasury yields (1M, 3M, 6M, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y, 20Y, 30Y) from the Federal Reserve H.15 Selected Interest Rates release, covering all available FRED history. Two exogenous macro variables (Effective Fed Funds Rate and 5-Year Breakeven Inflation) provide monetary policy and inflation context.
The 11 raw yields are compressed into 3 Nelson-Siegel factors (Level, Slope, Curvature), re-estimated daily. The model forecasts these factors, not raw yields. A 20-day sliding window provides look-back context.
Both fθ and gφ are single-hidden-layer MLPs (128 neurons, ReLU), trained jointly via backpropagation through the EM solver using torchsde. Adam optimizer, learning rate 0.001, 150 epochs.
Out-of-sample on February–April 2020. No parameter updates after January 1, 2020. Strict embargo, zero data leakage.
Why it generalizes to crises
The diffusion gφ is conditioned on Y(t), the current state of the yield curve. Even without seeing a crisis during training, the model recognizes that unusual yield levels imply elevated volatility and widens its uncertainty bounds. It generalizes to new regimes by design, not by chance.
State-dependent diffusion means uncertainty automatically widens as yields move outside the training distribution, with no threshold rules or regime flags hardcoded by the researcher.
Validated during the hardest
market stress test in a generation.
Reduction in mean squared forecast error during the March 2020 financial panic, the most extreme out-of-sample stress test available, versus the ARX-GARCH industry baseline. Out-of-sample. Zero retraining.
| Model | Architecture | Time Domain | Noise | Panic-Period MSE |
|---|---|---|---|---|
| ARX-GARCH | Linear autoregressive + GARCH volatility | Discrete | Gaussian | Baseline |
| Neural SDE Stokhos | fθ drift + gφ diffusion + EM solver | Continuous | State-dependent | −64.3% (MSE 0.692) |
| Neural SDE + Memory GRU | GRU drift + GRU diffusion + EM solver + hidden state | Continuous | Path-dependent | −57.4% (MSE 0.827) |
Evaluation on out-of-sample U.S. Treasury yield curve, Feb–Apr 2020. Trained on FRED daily data 2015–Jan 2020. No recalibration after Jan 1, 2020. ARX-GARCH baseline MSE: 1.943.
The GRU variant achieves lower day-to-day error during recovery (MAE 0.263 vs. 0.314) but reacts slower at shock onset. When the March 2020 panic severed every historical precedent, the memory state became a liability. The standard Neural SDE (carrying no pre-crisis history) proved more agile at the moment of maximum stress.
Peer-reviewed academic research.
Bachelor's Thesis, Colorado College, May 2026
Stress Testing the Yield Curve: The Failure of Discrete-Time Econometrics vs. Continuous-Time Neural Networks During Market Panics
All quantitative claims (architecture, training procedure, error metrics) are drawn directly from the thesis. Zero parameter updates after January 1, 2020 ensured the COVID-19 crisis remained a completely out-of-sample test.
See the numbers live.
Forecast vs. actual yield curves, cumulative error, and regime classification, pre-computed from the research.