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

TL;DR

This paper introduces 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.

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.

Figures (7)

• Figure 1: The high-level illustration of the high rank path development. Here the prediction process $\hat{X}_{t}:=\mathbb{P}(X | \mathcal{F}_t)$ for all $t \in [0, T]$, $\boldsymbol{\Phi}_{\hat{X}_t}(M_1)$ is the PCF of the prediction process and $\mathcal{U}_{M_1, M_2}(\hat{X})$ is the high rank development of the path $t \mapsto \boldsymbol{\Phi}_{\hat{X}_{t}}(M_1)$ under the linear map $M_2$.
• Figure 2: Flowchart of HRPCF-GAN for learning condition distribution $\mathbb{P}(X_{\text{future}}| X_{\text{past}})$.
• Figure 3: Sample plots of the conditional distribution $\mathbb{P}(X | \mathcal{F}_t)$ on fBM conditioned on the same past path, using both true and GAN models (arranged from top to bottom). Each column represents different $t$. The thick red /green line indicates the conditional mean of the future path estimated by model simulated samples/true models. The shaded red area presents the region of $\pm \text{std}$ of model simulated samples, whereas the shaded area shown corresponds to the region of $\pm \text{ theoretical std}$.
• Figure 4: $\mathbb{X}^n$ (left) converges to $\mathbb{X}$ (right) weakly, but the corresponding price of American options on $\mathbb{X}^n$ cannot converge to the counterpart on $\mathbb{X}$, see Example \ref{['example: OSP']} below. Therefore the usage of slightly erroneous models in weak topology may cause significant loss in decision making problems. This example is taken from Backhoff2020Adapted and bonnier2020adapted.
• Figure 5: Distributions of EPCFD (left) and EHRPCFD (right) under $H_0$ and $H_1$. The distribution consists of $100$ runs under both hypotheses. For EPCFD, fix $K_1 = 8$ and $n = 5$. For High Rank PCFD fix $K_1 = 1$, $n=3$, $K_2=10$, $m=13$.

Theorems & Definitions (31)

• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5: Characteristicity of laws
• Remark 2.6
• Definition 3.1
• Definition 3.2
• Theorem 3.3: Characteristicity of synonym
• Definition 3.4