Variational Inference for Lévy Process-Driven SDEs via Neural Tilting

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 provide a natural mathematical framework for capturing jumps and heavy tails, Bayesian inference for Lévy-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 Lévy-driven SDEs. Our approach constructs a flexible variational family by exponentially reweighting the Lévy 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.

Point process simulation of generalised hyperbolic Lévy processes

Generalised hyperbolic (GH) processes are a class of stochastic processes that are used to model the dynamics of a wide range of complex systems that exhibit heavy-tailed behavior, including systems in finance, economics, biology, and physics. In this paper, we present novel simulation methods based on subordination with a generalised inverse Gaussian (GIG) process and using a generalised shot-noise representation that involves random thinning of infinite series of decreasing jump sizes. Compared with our previous work on GIG processes, we provide tighter bounds for the construction of rejection sampling ratios, leading to improved acceptance probabilities in simulation. Furthermore, we derive methods for the adaptive determination of the number of points required in the associated random series using concentration inequalities. Residual small jumps are then approximated using an appropriately scaled Brownian motion term with drift. Finally the rejection sampling steps are made significantly more computationally efficient through the use of squeezing functions based on lower and upper bounds on the L\'evy density. Experimental results are presented illustrating the strong performance under various parameter settings and comparing the marginal distribution of the GH paths with exact simulations of GH random variates. The new simulation methodology is made available to researchers through the publication of a Python code repository.