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.

翻译：由于随机过程的弱收敛未能考虑随时间增长的信息（由底层滤子表示），弱拓扑中一个微小的随机模型误差可能导致多期决策问题中的巨大损失。为解决此类不连续性，Aldous引入了扩展弱收敛，它能完整刻画随机过程的所有本质特性（包括滤子）；然而该方法一直被认为难以找到高效的数值实现。本文基于粗糙路径理论中的高秩路径发展方法，引入了一种称为高秩路径特征函数距离（HRPCFD）的新度量用于扩展弱收敛，该方法同时定义了测度值过程的特征函数。我们证明了该HRPCFD具有诸多优良解析性质，使我们能够设计从数据中训练HRPCFD的高效算法，并通过将HRPCFD作为判别器构建HRPCF-GAN以用于条件时间序列生成。在假设检验和生成建模方面的数值实验表明，与多种先进方法相比，我们的方法具有优越性能，凸显了其在合成时间序列生成的广泛应用中以及解决经典金融经济问题（如最优停止或效用最大化问题）方面的潜力。