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

TL;DR

This paper introduces a novel approximation algorithm for filtering and prediction in diffusion processes using conditional GANs combined with signatures from rough path theory, offering an efficient alternative to solving complex SPDEs.

Contribution

It extends GANs with signatures and neural differential equations to approximate the conditional law in diffusion filtering, providing a rigorous mathematical framework and demonstrating numerical efficiency.

Findings

The proposed method accurately approximates the conditional law in diffusion processes.

Numerical results confirm the efficiency and robustness of the algorithm.

The approach offers a practical alternative to traditional SPDE-based filtering methods.

Abstract

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.