JumpLie: Discovering and Exploiting Jump-Diffusion Symmetries

Jump-diffusion (JD) processes provide a natural description of stochastic systems that combine continuous evolution with abrupt discrete events, a setting increasingly relevant to machine learning, while symmetry offers a powerful means of exposing structure and constructing inductive biases for learning. We introduce *JumpLie*, an end-to-end, data-driven framework for discovering and exploiting Lie symmetries of JD dynamics, building on the point-symmetry theory of A. Nass (2017). Our approach first learns the underlying JD dynamics from trajectory data and then discovers neural symmetry generators by enforcing the corresponding continuous and finite-jump invariance conditions. This formulation highlights a fundamental distinction from conventional diffusion-SDE symmetry: invariance of the continuous stochastic dynamics alone does not guarantee invariance under the discrete jump mechanism. We study recovery of JD symmetry structure from data and demonstrate its utility across three settings: improving out-of-distribution generalization, discovering and exploiting structure in jump-based stochastic generative models through Generator Matching, and symmetry-aware volatility estimation for financial time series. Together, these results establish JD symmetry as a learnable structural prior for systems with intertwined continuous and discrete stochastic dynamics. To the best of our knowledge, this is the first machine-learning work to introduce the data-driven discovery and downstream use of JD Lie symmetry, opening a new direction for symmetry learning beyond purely continuous dynamics.