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.

翻译：随机签名已被提出作为成熟路径签名的一种灵活且易于实现的替代方案。本文采用随机签名方法，基于储层计算思想提出了一种金融时间序列数据的生成模型。具体而言，我们提出了一种基于离散时间随机签名的新型Wasserstein类距离度量。该概率度量空间上的距离度量能够捕捉（条件）分布之间的差异。其有效性通过我们在连续函数空间上提出的随机签名通用逼近定理得到验证，该定理以底层路径作为输入。随后，我们将该度量作为损失函数应用于基于储层神经随机微分方程的非对抗生成器模型，用于合成时间序列数据。最后，我们将模型结果与现有文献中的基准方法进行了对比分析。