We introduce a variational inference framework for Lévy-driven stochastic differential equations (SDEs) in which the approximate posterior is expressed as a tilted Lévy process, with the prior jump measure reweighted by a learned exponential factor and the Brownian component acquiring the standard score correction. Although our experiments focus on stable processes, the change-of-measure construction applies to the broader class of Lévy processes, opening a route to scalable posterior inference in models with general non-Gaussian jump structure. The quadratic neural parametrisation is the computational device that makes this tractable, yielding closed-form normalising constants and an exact rejection sampler for the tilted jump law.
Experimental Validation
We validate our framework on two settings that both exhibit pronounced heavy-tailed dynamics: (i) synthetic data with ground-truth parameters and (ii) challenging real-world forecasting task.
Multivariate Financial Forecasting
We evaluate on all ten stocks jointly (a 10-dimensional SDE) across 302 rolling evaluation windows, using the energy score as the primary multivariate scoring rule alongside per-dimension CRPS. NJ-SDE diverged in every multivariate run and is excluded. The table below reports the results.
| Model | CRPS (avg) | Energy Score (avg) |
|---|---|---|
| Our Model | value | value |
| Neural MJD | value | value |
Our model achieves state-of-the-art average performance, matching Neural MJD on CRPS (our value vs their value) and energy score (our value vs their value), with the distinction emerging in tail performance.
Conclusion
We introduced a variational inference framework for Lévy-driven SDEs in which the approximate posterior is expressed as a tilted Lévy process, with the prior jump measure reweighted by a learned exponential factor and the Brownian component acquiring the standard score correction. Although our experiments focus on stable processes, the change-of-measure construction applies to the broader class of Lévy processes, opening a route to scalable posterior inference in models with general non-Gaussian jump structure. The quadratic neural parametrisation is the computational device that makes this tractable, yielding closed-form normalising constants and an exact rejection sampler for the tilted jump law.
Optimisation Details
Both TS and Gaussian SDE models use the same neural drift parametrisation (a one-hidden-layer MLP of width 64). We found that increasing the drift width consistently degraded predictive performance, suggesting that a small drift network acts as an effective regulariser by preventing the drift from absorbing variability that should instead be attributed to the latent noise process.
Frequently Asked Questions
Does the tilting framework apply beyond stable Lévy processes?
Yes. The change-of-measure construction applies to the broader class of Lévy processes, not just stable processes. The experiments focus on stable processes for validation, but the theory extends to general non-Gaussian jump structure.
What computational advantage does the quadratic neural parametrisation provide?
The quadratic form yields closed-form normalising constants and enables an exact rejection sampler for the tilted jump law, making the approach tractable and scalable in practice.
Why did NJ-SDE diverge in multivariate financial forecasting?
NJ-SDE diverged in every multivariate run and was therefore excluded from the reported results. The divergence occurred across all 302 rolling evaluation windows.
How were the neural drift networks sized across competing models?
Both the TS and Gaussian SDE models used the same architecture: a one-hidden-layer MLP with width 64. The paper notes that increasing drift width degraded performance, acting as a regulariser.
What scoring metrics were used for multivariate forecasting?
The primary multivariate scoring rule was the energy score, supplemented by per-dimension CRPS. These metrics allowed comparison of full distributional forecasts.
Is the framework limited to financial time series?
No. The paper validates the framework on both synthetic data with ground-truth parameters and challenging real-world forecasting tasks, demonstrating broader applicability to any heavy-tailed dynamics setting.