Variational Inference for L\'evy Process-Driven SDEs via Neural Tilting

TL;DR

This paper introduces a neural exponential tilting method for variational inference in Le9vy process-driven SDEs, effectively modeling jumps and heavy tails in complex stochastic systems.

Contribution

It proposes a flexible neural tilting framework that preserves jump structures and enables scalable, accurate inference for Le9vy-driven SDEs, surpassing Gaussian-based methods.

Findings

Accurately captures jump dynamics in synthetic and real data.

Provides a scalable inference method with closed-form normalization.

Outperforms Gaussian-based variational approaches in heavy-tailed regimes.

Abstract

Modelling extreme events and heavy-tailed phenomena is central to building reliable predictive systems in domains such as finance, climate science, and safety-critical AI. While L\'evy processes provide a natural mathematical framework for capturing jumps and heavy tails, Bayesian inference for L\'evy-driven stochastic differential equations (SDEs) remains intractable with existing methods: Monte Carlo approaches are rigorous but lack scalability, whereas neural variational inference methods are efficient but rely on Gaussian assumptions that fail to capture discontinuities. We address this tension by introducing a neural exponential tilting framework for variational inference in L\'evy-driven SDEs. Our approach constructs a flexible variational family by exponentially reweighting the L\'evy measure using neural networks. This parametrization preserves the jump structure of the underlying process while remaining computationally tractable. To enable efficient inference, we develop a quadratic neural parametrization that yields closed-form normalization of the tilted measure, a conditional Gaussian representation for stable processes that facilitates simulation, and symmetry-aware Monte Carlo estimators for scalable optimization. Empirically, we demonstrate that the method accurately captures jump dynamics and yields reliable posterior inference in regimes where Gaussian-based variational approaches fail, on both synthetic and real-world datasets.