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.

我们利用随机化签名引入金融时间序列数据的生成模型，该模型基于离散时间随机化签名建立了一种新的 Wasserstein 类型距离，作为概率测度空间中的距离度量，并将其作为非对抗生成模型的损失函数。我们通过将我们的模型与现有文献中的基准进行比较来验证其性能。

生成时间序列建模的通用随机签名

Universal randomised signatures for generative time series modelling

In this paper we introduce a Wasserstein-type distance on the set of Gaussian
mixture models. This distance is defined by restricting the set of possible
coupling measures in the optimal transport problem to Gaussian mixture models.
We derive a very simple discrete formulation for this distance, which makes it
suitable for high dimensional problems. We also study the corresponding
multi-marginal and barycenter formulations. We show some properties of this
Wasserstein-type distance, and we illustrate its practical use with some
examples in image processing.

此研究介绍了一种基于 Wasserstein 距离的方法，用于高维数问题中的 Gaussian 混合模型的优化问题，并讨论了它的性质和在图像处理中的应用。