High Rank Path Development: an approach of learning the filtration of stochastic processes

TL;DR

This paper introduces a novel metric called HRPCFD based on high rank path development from rough path theory, enabling better extended weak convergence analysis and improved time series generation.

Contribution

The paper proposes the HRPCFD metric for extended weak convergence, along with an efficient training algorithm and the HRPCF-GAN for conditional time series generation.

Findings

HRPCFD exhibits favorable analytic properties.

The HRPCF-GAN outperforms state-of-the-art methods.

Validated on hypothesis testing and generative modelling.

Abstract

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.