Neural Fractional Stochastic Differential Equations

Fractional differential equations (FDEs) excel at modelling systems with long-range memory, while stochastic differential equations (SDEs) capture inherent randomness. Existing neural differential equation models address either memory (Neural FDEs) or stochasticity (Neural SDEs), but a unified framework combining both with a learnable scalar memory exponent and an efficient adjoint remains absent. We introduce the Neural Fractional Stochastic Differential Equation (Neural FSDE), which learns continuous-time dynamics with both memory and randomness from data via a Caputo FSDE with neural drift, neural diffusion, and a learnable fractional order. To enable scalable training, we derive a discrete adjoint method that computes gradients with respect to the initial state, the drift and diffusion network parameters, and the fractional order, with memory cost independent of the autograd graph depth. We validate the framework on two real-world domains. On the stochastic-fractional COVID-19 model of Bonyah et al., the NeuralFSDE provides the first end-to-end real-data calibration under a rolling-origin forecasting protocol, recovering a fractional order less than 1. On S&P~500 index option pricing, we recast rough Bergomi, rough Heston, Neural SDE, and NeuralFSDE as Caputo-FSDE special cases under a single calibration pipeline; the NeuralFSDE achieves the lowest implied-volatility error across schedules and instruments, with a matched-architecture comparison against the memoryless Neural SDE, isolating the contribution of fractional memory. Code is available at https://anonymous.4open.science/r/NeuralFSDE.