Signature moments to characterize laws of stochastic processes
This work addresses the challenge of comparing and testing the distributions of stochastic processes, which is important for fields like finance and machine learning, but it appears incremental as it builds on existing signature methods.
The authors tackled the problem of characterizing the laws of stochastic processes by introducing robust signature moments, which led to a maximum mean discrepancy metric that can be efficiently computed using signature kernels. As an application, they developed a non-parametric two-sample hypothesis test for these laws.
The sequence of moments of a vector-valued random variable can characterize its law. We study the analogous problem for path-valued random variables, that is stochastic processes, by using so-called robust signature moments. This allows us to derive a metric of maximum mean discrepancy type for laws of stochastic processes and study the topology it induces on the space of laws of stochastic processes. This metric can be kernelized using the signature kernel which allows to efficiently compute it. As an application, we provide a non-parametric two-sample hypothesis test for laws of stochastic processes.