  We consider the filtering and prediction problem for a diffusion process. The
signal and observation are modeled by stochastic differential equations (SDEs)
driven by correlated Wiener processes. In classical estimation theory,
measure-valued stochastic partial differential equations (SPDEs) are derived
for the filtering and prediction measures. These equations can be hard to solve
numerically. We provide an approximation algorithm using conditional generative
adversarial networks (GANs) in combination with signatures, an object from
rough path theory. The signature of a sufficiently smooth path determines the
path completely. As a result, in some cases, GANs based on signatures have been
shown to efficiently approximate the law of a stochastic process. For our
algorithm we extend this method to sample from the conditional law, given
noisy, partial observation. Our generator is constructed using neural
differential equations (NDEs), relying on their universal approximator
property. We show well-posedness in providing a rigorous mathematical
framework. Numerical results show the efficiency of our algorithm.

Machine Learning;Machine Learning;Optimization and Control