Convergence and stability analysis of neural stochastic differential equations with Poisson jumps

Standard Neural SDEs are limited to continuous paths, failing to capture abrupt shocks, while traditional jump–diffusion models suffer from parametric inflexibility and the curse of dimensionality, restricting their applicability to complex high-dimensional systems. This paper proposes a Neural Stochastic Differential Equation with Jumps (Neural SDEJs) model driven by both Brownian motion and Poisson jumps to capture such discontinuous dynamics. Our main contributions are as follows: (i) We construct a novel jump–diffusion network architecture based on compensated Poisson martingales to model complex, heavy-tailed time series. (ii) We establish a rigorous theoretical foundation for the model, proving that its numerical solution converges in probability to the true solution. Crucially, by utilizing approximation theory in Barron spaces, we demonstrate that our method avoids the curse of dimensionality, offering significant theoretical advantages over traditional polynomial-based approaches. (iii) We theoretically analyze the stability of the learned dynamics and demonstrate the model’s robustness against input distribution shifts. Numerical experiments confirm our convergence bounds and show strong predictive performance on discontinuous data.