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 Lévy processes offer the natural mathematical foundation for capturing jumps and heavy tails, Bayesian inference for Lévy-driven stochastic differential equations (SDEs) remains challenging. Monte Carlo approaches provide rigour and are well-established for low-dimensional problems, but scale poorly to high-dimensional settings. Neural variational inference frameworks achieve greater computational efficiency but typically rely on Gaussian assumptions that fail to capture discontinuities and heavy tails. We resolve this tension by introducing a neural exponential tilting framework for variational inference in Lévy-driven SDEs. Our approach derives the optimal variational family as an exponential reweighting of the Lévy measure, parametrised by neural networks. To ensure tractability, we develop (i) a quadratic neural parametrisation that enables closed-form computation of normalising constants, (ii) a conditionally Gaussian representation of stable processes for efficient forward simulation, and (iii) symmetry-exploiting Monte Carlo schemes for scalable loss approximation. The resulting tilted Lévy processes retain heavy-tailed behaviour while guaranteeing finite moments, thus combining mathematical expressiveness with computational feasibility. We demonstrate the effectiveness of our method on synthetic datasets, showing that it accurately captures jump dynamics and provides reliable posterior inference where existing approaches struggle.