High Rank Path Development: an approach of learning the filtration of stochastic processes
This addresses a foundational challenge in stochastic modeling for applications like synthetic time series generation and financial problems, offering a novel and efficient solution.
The paper tackles the problem of extended weak convergence for stochastic processes, which is crucial for multi-period decision making but lacked efficient numerical implementations, by introducing the High Rank PCF Distance (HRPCFD) metric and showing it outperforms state-of-the-art methods in hypothesis testing and generative modeling.
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.