Latent Neural Stochastic Volterra Equations via Markovian Memory Factors

Stochastic Volterra equations (SVEs) offer a flexible framework for modelling stochastic systems with memory, extending classical stochastic differential equations (SDEs) by weighting past drift and diffusion contributions through a Volterra kernel. This flexibility comes at a cost: SVEs are generally non-Markovian, which complicates latent-variable inference. In this paper, we propose \emph{latent neural SVEs}, extending latent neural SDEs to systems with memory by incorporating finite-dimensional Markovian representations based on mixtures of Ornstein--Uhlenbeck (OU) processes with distinct mean-reversion rates. The resulting augmented state remains Markovian, allowing the variational inference machinery of latent neural SDEs to be applied directly, while the OU mixture induces a flexible approximation of Volterra memory kernels. The model is trained by variational inference from noisy observations, requiring access to neither the driving noise nor the underlying kernel. Experiments on fractional and multi-timescale OU dynamics and rough Heston variance dynamics show that explicitly parameterising memory improves the reproduction of temporal dependence and its extrapolation, particularly at small and moderate model capacities.