Structured Neural SDEs for Functional Calibration

2026-09-28

AI Summary

General neural stochastic differential equations (SDEs) face challenges of high computational costs in long-horizon simulations and unstable gradients when training on path functionals. This work proposes the SLiSDE framework, which leverages structured linear stochastic layers to enable time-parallel simulation and employs stacked gated inflows to restore model expressivity. Furthermore, it introduces importance sampling based on Girsanov measure transformations to optimize rare-event calibration. Rigorous theoretical error bounds and universality proofs are provided. Experimental results demonstrate that the proposed method significantly outperforms fully neural baselines on financial calibration benchmarks, effectively balancing parallel efficiency, weight stability, and the capacity to sample rare paths.

Abstract

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.

Problem

• Neural SDEs
• Functional Calibration
• Long-horizon Simulation
• Gradient Instability
• Rare Paths

Innovation

• Neural SDEs
• Parallel-in-time simulation
• Gated in-flow stacking
• Girsanov tilt
• Functional calibration