Since the weak convergence for stochastic processes does not account for the growth of information over time which is represented by the underlying filtration, a slightly erroneous stochastic model in weak topology may cause huge loss in multi-periods decision making problems. To address such discontinuities Aldous introduced the extended weak convergence, which can fully characterise all essential properties, including the filtration, of stochastic processes; however was considered to be hard to find efficient numerical implementations. In this paper, we introduce a novel metric called High Rank PCF Distance (HRPCFD) for extended weak convergence based on the high rank path development method from rough path theory, which also defines the characteristic function for measure-valued processes. We then show that such HRPCFD admits many favourable analytic properties which allows us to design an efficient algorithm for training HRPCFD from data and construct the HRPCF-GAN by using HRPCFD as the discriminator for conditional time series generation. Our numerical experiments on both hypothesis testing and generative modelling validate the out-performance of our approach compared with several state-of-the-art methods, highlighting its potential in broad applications of synthetic time series generation and in addressing classic financial and economic challenges, such as optimal stopping or utility maximisation problems.

本文介绍了一种基于粗路径理论中的高阶路径开发方法的新型度量标准，称为高阶PCF距离（HRPCFD），用于扩展弱收敛性的测量，该度量标准还定义了测度值过程的特征函数。然后展示了HRPCFD具有许多有利的分析性质，使得我们能够设计出一种有效的从数据训练HRPCFD和构建HRPCF-GAN的算法，其中HRPCFD被用作有条件时间序列生成的鉴别器。我们对假设检验和生成建模进行的数值实验验证了我们的方法相较于几种最先进的方法的优越性，突显出它在合成时间序列生成以及解决经典金融和经济挑战（如最优停止或效用最大化问题）方面的潜力。

高级路径发展：一种学习随机过程过滤的方法