  Randomised signature has been proposed as a flexible and easily implementable
alternative to the well-established path signature. In this article, we employ
randomised signature to introduce a generative model for financial time series
data in the spirit of reservoir computing. Specifically, we propose a novel
Wasserstein-type distance based on discrete-time randomised signatures. This
metric on the space of probability measures captures the distance between
(conditional) distributions. Its use is justified by our novel universal
approximation results for randomised signatures on the space of continuous
functions taking the underlying path as an input. We then use our metric as the
loss function in a non-adversarial generator model for synthetic time series
data based on a reservoir neural stochastic differential equation. We compare
the results of our model to benchmarks from the existing literature.
