# Structured Neural SDEs for Functional Calibration

arXiv: https://arxiv.org/abs/2609.34831 · HTML: https://mlsanity.com/paper/2609.34831/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 25 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 7 of 7)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Liquid Structural State-Space Models](https://mlsanity.com/paper/2209.12951/) (2022)
- [Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise](https://mlsanity.com/paper/1906.02355/) (2019)
- [Robust pricing and hedging via neural SDEs](https://mlsanity.com/paper/2007.04154/) (2020)
- [Arbitrage-free neural-SDE market models](https://mlsanity.com/paper/2105.11053/) (2021)
- [Structured Linear CDEs: Maximally Expressive and Parallel-in-Time Sequence Models](https://mlsanity.com/paper/2505.17761/) (2025)

## Abstract (arXiv)

Neural Stochastic Differential Equations (Neural SDEs) provide flexible continuous-time generative models, but generic neural drift and diffusion networks are costly to simulate on long horizons and can give unstable gradients when the training signal is a path functional rather than a pointwise observation. We introduce SLiSDE, a family of Neural SDE models built from structured linear stochastic layers. Parallel-in-time simulation is obtained at the layer level, while expressivity is recovered by gated in-flow stacking: previous-layer paths modulate the next layer's latent flow through learned gates. For functional calibration tasks in which rare paths dominate the loss, we add an optional Girsanov tilt that acts as a learned importance sampler with an exact likelihood-ratio correction. We prove well-posedness, a discretisation error bound, validity of the change of measure, and a universality result: the terminal laws of the gated stack are dense in the space of square-integrable laws. Experiments on functional calibration benchmarks show that the structured model outperforms fully neural SDE baselines while retaining parallel-time simulation and stable importance weights.