Learning the conditional law: signatures and conditional GANs in filtering and prediction of diffusion processes

We consider the filtering and prediction problem for a diffusion process. The signal and observation are modeled by stochastic differential equations (SDEs) driven by 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) and signatures, an object from rough path theory. The signature of a sufficiently smooth path determines the path completely. In some cases, GANs based on signatures have been shown to efficiently approximate the law of a stochastic process. In this paper we extend this method to approximate the prediction measure conditional to noisy observation. We use controlled differential equations (CDEs) as universal approximators to propose an estimator for the conditional and prediction law. We show well-posedness in providing a rigorous mathematical framework. Numerical results show the efficiency of our algorithm.