Universal approximation property of neural stochastic differential equations
This addresses the problem of approximating complex stochastic processes for researchers in machine learning and mathematical modeling, though it appears incremental as it builds on existing neural network and SDE theory.
The paper demonstrates that neural stochastic differential equations can approximate general stochastic differential equations arbitrarily well, with quantitative error estimates provided for cases with regular coefficients.
We identify various classes of neural networks that are able to approximate continuous functions locally uniformly subject to fixed global linear growth constraints. For such neural networks the associated neural stochastic differential equations can approximate general stochastic differential equations, both of Itô diffusion type, arbitrarily well. Moreover, quantitative error estimates are derived for stochastic differential equations with sufficiently regular coefficients.